149

Программирование: теоремы и задачи

  • Upload
    devtype

  • View
    720

  • Download
    1

Embed Size (px)

Citation preview

Page 1: Программирование: теоремы и задачи

��

� � � �

� � � � � � � � � � � �

� � � � � � � � � � � � � �

� � � � � ! " # $ % $ !& ' ( % � " ) ! � � $ ! � $ ( $ ) � ! � � $ !

* $ ' + " �

� � � � # ! ) , ' # " $ * - . * /

0 1 1 2

Page 2: Программирование: теоремы и задачи

� � � � � �� � � � � � � �� ��� � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � ! � " � # �� $ % � � ! " � & � ' ( � � �

" � (� $ ) � � ) * + ) , & - - � $ � � '� � � .�/ 0 1 2 3 4 5 6 7 3 8 4 9 6 6 4 9

: ; < = > ? @ A B C D < E F > A > G < H @ H C @ = C > I I < C @ J > ; < K C > F L < G ; @ M E C N A ; @ ? E < OP @ L Q R < ; ? E J @ F > A > G H C < J @ A S E ? S ? C B R B ; < S I < O T B L Q U ; < = < V ; > N G < E Q @ ? ; @ J W; X I I B E @ A > I H @ ? E C @ B ; < S U @ C C B U E ; X Y < Z X ? E C X Y > L = @ C < E I @ J O[ L S N G < E B L B M < ; \ @ C I > E < U < ] ? E > C R B U L > ? ? ; < U @ J ] ? E N A B ; E @ J I L > A R < Y U N C W? @ J J X ? R < Y N G B Z ; X Y F > J B A B ; < M O @ ? @ Z < B I @ D B E Z X E Q < ? H @ L Q F @ J > ; @ ; > U C N D WU @ J X Y < \ > U N L Q E > E < J ; X Y F > ; S E < S Y J @ Z _ B @ Z C > F @ J > E B L Q ; X Y N G C B D A B ; < S Y ] JR U @ L > Y ? N = L N Z L ; ; X I < F N G B ; < B I I > E B I > E < U < < < ; \ @ C I > E < U < ] > E > U D B J< ; X Y a B L S Y ] ; B H C @ E < J @ C B G > _ < Y F > U @ ; @ A > E B L Q ? E J N b c O

� � � �

d e f g h i j k l h m i n k k i n o© � � � p q � & � � � � & - -

r s t u v w x u v y z { s | z } ~ � � { s t � v � � s � ~ t w v � ~ �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �   ¡ � � � � �

� � � ¢ £ � � � � � � � � � £ � � � � ¢ � � � � ¤ ¡ � � � � � � � � � � � � ¢ £ � � � � � � � � � �

¢ � � � � � � � � � � ¥ ¦ � � � � � � � � � � � � � � � � � � � � ¢ � � � � � � � � � � ¥ ¢ � � � � � � � � � ��   § © ª « ª ¥ ¬ � � ­ � � ®  

� � � � � � � � � � � � � � ¢ � ° £ � � � � ¥ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � £ � � � ¥� � � � � £ � ± � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ¥ � � � � � � � £ � �

� � � � � � � � � � � � � � � � � � � � � � � � � �  

� � � ² � � � £ � � ° £ � � � � � � � ¥ � � � � � ° � � ¢ � £ � � � � � � � � � � � � � � � � � � � � � � � � � ¥� � � � � � � � � � � � � � � � � � � � � � � � � �   ³ £ � � � � ­ � � � � � � � � � � � � � � � � � � �

� � � £ � � � � � � � � � � � � � � ¥ � � � µ � � � � � � �  

¶ � � � � ­ � � � ¥ � � � � � ° � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � ¢ � � � � � � � � � ¢ � � � � � � � � � � � � � � � � � � � � �   ³ £ � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � ¢ � � � ´ � � � � � � � � � � � � ¢ � � � � � £ � � ¢ � � � �

� � � � � � � � � � � � � � � � � � � � � � �  

¶ � � � � � � � ° � � � � � � � � � � � � � � � £ � � � � � � � � � � � � � � � � � � � ·

� � � � � � � � � � � � � � � � � � � � � � � ¹ ¥ � � � � � � � � � � � � � � � � � � � � � £ � � ¢ º � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ¢ � � � � £ � » � � � ­ � � � � � � � �

� � � � � � � �   ¼ � � � � � � � £ � � ° � � � � � � � � � � � � � � � � ¢ � � � � � � � � � � � � � � � �

� � � � � � µ � � � � � � � � � � � � � � � � � ¥ £ � � � ¢ � � � � £ � � � � £ � » � � � ­ � � � � � � � � � � � � �  

� � ° � � � � ¥ ½ � � � � � � � � � ¥ � � � µ � � � � � � � � � � � � � � � � � � � � � � � � � � � � µ � � � �

� � � � � � � � ¢ � � � � � � � � � � � � � � ¥ � � � ¥ � � � � � � � � � � � � � � � � � � � � � � � � � � � �

¢ � � � � � � � � ¢ £ � � �   ¾ � � � ¥ � � � � � � � £ � � � � � µ � � � � � � � � ¥ � � � � � � � �

� � � ¢ � � � � ¢ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ¥ � � � � � � � � � �

¢ � � � � � ¥ � � � � � � � � � � ¹ ¥ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ° ¢ � � �

� � � � � � � � ¥ � � � � � � � ¢ £ � � � � � � � � � � � � º ­ � � � £ � � � � � � ¢ � � �  

½ � � � � � � � � � � � � � � � � � � � ¿ � � £ � � � � � � £ � � � � � � � � � � � � � � � � � � � � � � � � � £ � ¢ �

� £ � ¢ � � � � � £ � � � � � � � � � � � � ¥ � £ � ¢ � � � � � � ¢ ± � � � � ¢ � � � � � � � � � �� £ � � ² � £ � � � � � � � � � � � � � � � � � �   À � � � � � � � � � � � � � � � � � � � � � � � º � � � ¥ � � �

� � � £ � � � � � � � � ² � � � � � � � � � � � � � � � » � � � � � � � � � � � � � � � �  

Á � � � � � � � � ¢ £ � � � � � � � £ � � � � � � � � � � � � � � � � � � � � �   ¡ � � � � � � � � � �

� � � º � � � � � � � � � � � � � � � � £ � � � ¥ � � � � � � � � � � ¢ � � � � � � � � � � � £ � �

� � � � � � º ± � � ¥ � � � � � � ¢ £ � � � � � £ � � � ¥ � � � � � � � � � � ¢ � � � � � £ � � � � � ¢ ° ¢

� � � � � � � � � � � � � � � � �   ¤ ½ � � � � � � £ � � ¥ � � µ � � � � � � � � � � � � � �   ®

Page 3: Программирование: теоремы и задачи

¶ � � � � � � � � � � � � � £ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �   ³ �£ � � � � � � � � � � � � � � � � � � � � � � � � ¥ � � � � � � � � � � � � � � � � � � � � ­ � � ¤ � � � � � � � � ¥� � � � � � � � � � � � � � � � � � � � � � � ª © � ª « » � � � � � � � « ª � � � � � � � � ° � � � £ � �

� � � � � � � � � � � � � � � � � � � � ® � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � £ � �   ¶ � � � � ° � � ¥ ¡ � £ ¢ � � ² � � � ³ � � � � � � � � � � � �

� � � ± � � � � � � � � � ¥ � � � � � � � � � � £ � � � ¢ � � �  

� � � � � � � � � � � � � � � � £ � � � � � � � � � � � � � ¥ � � � ¢ � � � � � � ¥ � � � � � � º � � � � � � � � �  ¤ ¶ � � � � � � � � � � � £ � � � � � ¢ � � � � � � � � � £ � � � � � � � � � � � � � � � � � � � ¢ º � � � � ¢ � � �

� � � � � � � � � � � � � � � � � � �   � �   � � � ° � � � � � � � � � � � £ � � £ � � � � � � � � � � � � � � � �   ®¶ � � � � � � � � � � � � � £ � � � � � ¥ � � � � � � � � � � � � � � � ¢ � � � � � � � � � � � � �¤ � � � � � � � � £ � � � � � � � � � � � � � ® � � � � ° � � � � � � � � � � � � � � � � � � · � � � � �  

� � � � � � � � � � � � � � � � � � � � ¢ � � � � � £ � � � � � � � � � � � ¥ � � � � � � � � � � � � � � � � �� � � £ � � � � � £ � � � � � � � £ � � � � � � � ¥ � � � � � � ± � � � � � � � � � � � � º  

³ � � � � � � � � � � � � � � � ¹ ¿ � � � � � » � � � ¢ � � � � � � � � � � £ � � ¢ � � � � � � � � � � ·

� � � � � � � � � � � � � � � � � � � � ¢ ° £ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ¥ � � � � �

µ � � � � � � � � �   ¶ � � � � � � � � � � � � � µ � � � � � � � � � � � � � � � � � � � � £ � �

� � � � � � � � � � � � � � � �   ¼ £ � � � � � � � � � � ¿ � � � � � � � � � � � � � ¥ � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � ¢ � � � � � ¢ � � � � � ¢ � � � � � � � � � � � � � � � � � £ � � � � � £ � � �

� � £ � � � � � � � � ¢ � � � � � ª � © ! �  

� � � � � £ � � � � � � � � � � � � � � � � � � � � � � � � � � � ¤ � � � � £ � � � � � � � � � � � � � � � � � � ®

� � � � � � " # $ % & � ¥ � � � º � � º ± � � � � � � � � � � » � � � � ¥ � � � � � � � � � � � ��   ¡   ' � � � � � � � � ¤ � �   ( ) * + , , ( ) * � � � � � � � � � , * � - , ) � . , ®  

/ � � £ � � ¢ � � º � � � � � � £ � � � � � � � � � ¥ � � � � � � � � � � � � � � � � � ¢ £ � � � � � � ¥� � � � � £ � � � � � � � � � � � � � � � � � � � � ¥ � � � � � � � � � � ¥ � � � � � � � £ � ¢ � ¢ º � � � � � � ¢

� � � � � � £ � ¹ ¥ � � � � ° � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �¤ � � � ­ � � � � � � � � � � � £ � � � � � � � ´ 0   ¶   ¡ � � � � � � � � � ¢ ®   ¼ � � � � � � � � £ � � � �¶   ¬ ¢ � � � � � � � � � � � � � � � � � � · � � � � � ¥ � � � � ° � � � £ � � � � � � � � � ¡ 1 ¦ ¡ ³

� � � � ¥ � � � � � � � � � � � � � � � � � � £ � � � � � � � � � � ° � � ¤ � � � � � � � � � � � � � � � ®

� � � � � � � � � � � � � � � � � � ­ � � � � � £ � � � � � � � � � � � � � � � ² � � � � � � ¤ � � � º � � � � � � � �

� � � � � ° � � ®   2 � � � � £ � � º � � � ° � 3 � � � � � ¢ � � � � � � � � � � � £ � � � � � » � � � � ­ � �

4 ¼ ¦ ¥ ¼ � � � � � � � � � � � � � � � � � � � � � � � � � � ± � � � � � ¤ » � � £ � � � � ± �

� � � � � ¢ � � � 4 ® ¥ » � � £ � � � � � � ¥ ¢ � � � � � � � � � � �   2 � � £ � ¥ » � � £ � ¢ � � � ¢ � � � �

� � � ­ � � � � � � ¹ ¥ » � � £ � 5 � 6 7 � ¤ ¬ � � ­ � � ® ¥ 8 7 9 5 ¤ : � � � ­ � � ® ¥ ; � � « ! 7 � � < ª « !¤ ' � � � ¥ : � � � ­ � � ® ¥ = 6 > ¤ ¡ � � � � � � ¥ : � � � ­ � � ® ¥ ¢ � � � � � � � � � � �   Á � � � � � �¤ ¬ � � ­ � � ® ¥ 4 � � � � � � � � � » � � £ » ¢ � £ � � � � � � � � � � � � � � � � £ � � � � � � ¤ � � � � � �

? ² ? @ ² ? ? @ ¦ 1 ¦ 3 � ¥ ? A ² ? @ ² ? ? B � � � £ � ¢ � � � ® ¥ � � � � ° � � � � � � � � £ £ � � ° � �

� � ¢ � � � � � � � � � � � � � � � � £ � � � � 4 : ¤ � � � � � ¦ ¬ ² A � C   ? ? A   @ ® � � � � £ £ � � ° � ¢  ¶ � � � � � � � � � � � £ � � ° � � � � � � � � � � � � � ° � � � � � � � ¢ � � � � � � � � � � � � ¥ � �� � � � � � � � � � � � � ¢ � � � � � � � � � � � � � � � � ­ � � � � � ­ �

� � � � � � � � � � � � � � � � � � � � ¢ � ¤ � � � � � � � � ®  

D E F G H I J K L G

n � M � N � O � � � P � Q R P N S T � � U V Q W N U X R S U R S � U V Y

� � � � Z � " � # � [ � ! � � ' ' � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � ) � ' ' � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � \

� � \� ] � " _ � � � � � ` � _ a � � b ( � q � c� � d � � � � � _ � e � � � � � � � \ �

f � M g N g T h � � U � i g O j U � S k g N � P l g j m � i k g R k f

� � � n � ! � � o � � � p ' ( � � � � � � � � p � � � � � � � � � � � � � � � � � � �

� � � � � � ' � � � � � _ � � � � � � � � � � � � � � � � � � � � � � � � � � � \

� \� � � " � � � q � ' � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � n � ! [ � � � � p � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � " c � � p � � � � . � � � # � � ! � " � # � � � � � � � � � � � � � � � �

� �� + � ' _ � . p _ � ! � � � # � � � r � � � � � � � � � � � � � � � � � � � � � �

� �� � � " ' # s � _ � . � # � ' � � � � � � � � � � � � � � � � � � � � � � � � � � �

t � u j l g h h � N � R S � M � N � j g N X R g v R N S k S O U h j

\ � � � w � � ! � & � � [ p x o � � " � � " � � � � � [ y � " " � � � � � ( � ! � a � r � � � � �

\ � � , [ y � " " � � � � � � " � � � y ! � " � # � y � � � � � � � � � � � � � � � � �

k � z g N k U N g R i S m n

� � � � � � " � � � � # � � � . � � � � � � � � � � � � � � � � � � � � � � � � � �

� � q . � � � � � � ( � � p " _ � { |} ~ { � � � � � � � � � � � � � � � � � � �

� \� � � � � � � � � � p ' � � � � � � � _ �� � � � � � � � � � � � � � � � � � � � � �

� � + � q � � � � a � � _ � " . p # � ' . � ' � � � � � � � r ( � � ' � � � � � � � _ � � � � � -

� � � n � " ' � � � � � � ' � � � � � � � _ � ! � " � # � � � � � � � � � � � � � � � � �

h � � g � � � � P � S R k g O S k P U g j N S j g k i S k � i X k g R Y j

� � � � � � ' � � � � � ' � � � � . & _ � � � � � � � � � � � � � (� � � � � � � � � � � � �

� � � � � � " # � ' � . � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

Page 4: Программирование: теоремы и задачи

� � � � � � � � � � �

� U W P h S � � P l j h

� � � � � � � _ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � , # � � � " � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � -

� � \� ) � � q � ' � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � -

� � � n � ! � � ! � " � # � � � � � � � � � � � � � � � � � � � � � � � � � � � �

m � � � i � N X U V n n

�� � � � � � � � � � � _ � ' � � � y ( � � � � � � � � � � � � � � � � � � � � � � � �

�� � n � _ � ' � � � � p � [ � � [ � � _ � " � � � � p � � � � � � � � � � � � � � � � � � � �

�� \� � � � � q " � � � � _ � � [ � � � � � � � y � [ � _ � � � & ( � � � [ � � � � � � � � �

�� � � � � � � ( � � � � � � � � p � � _ � ' � � � � � � � � � � � � � � � � � � � � �

Y � � S i g j g � k U X � j � v N � i � N X U U n t k

� � � � � � [ . � a � ! � � # � � � r b " � � � � � # � ' _ � � ( � � � � � � � � � � � � � � � e � � � \

� � � � � � _ � � . � q � � � y ! � " � � � r � � � � � � � � � � � � � � � � � � � \ �

� � \� � . � � ' . � q � � ' . # � � � � _ � ' � � � � � � � � � � � � � � � � � �

j � � S v � P � S � � g N U k O P � S � N S � S l n k h

� � � � � � � � # � r d � � ( � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � p ! � � _ � � ( � � � � � & ( � � ' _ � � . [ � � � d � � � � � � � � � � �

n l � z g W g X k S R � � � U � X g j N S v � g O n h h

� -� � � � � � ' � � r d � r ( � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� -� � � � � � � � � � � p � � [ � � ! a � $ � ' � � # � � _ ( � � [ . � � � � � � � � � � � � �

� -� \ � � ' ( � � � � � � � . p � � � � � � q " � � � p � � � � � � � � � � � � � � � � � � -

� -� � q . � � � � � � � � � � � ) � � � � ' � � � � � � � � � � � � � � � � � � � � � � -

� -� � � q . � � � � � � � r � � � � ) � � � � � � � � � � � � � � � � � � � � � � � \

� -� � � q . � � � � � � n � [ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� -� �� � . � � ' . � q � � � [ � � ! a � � � � � � � � � � � � � � � � � � � � � � � �

� -� � � � ` � _ ' � � " � � � � p p � � � � � � � � � � � � � � � � � � � � � � � �

n n � � � S � U v U � N n Y m

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � * � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � \ � � # � ' . � � � � a � � � ( � . � r � [ y � " � � � � � � � � � � � � � � � � � �

� � � � q . p �% [ � � �% ( � � a � " � � � � � � � � � � � � � � � � � � � � � � � - -

� � � � � n � � � � ' ( � _ � � � � r � � � . � ! � � � � � � � � � � � � � � � � � � � -

� � � � � � � � � � �

n f � u W k U O S � � � g � i g h U N g R S � U � f l

� � � � � � " � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � - �

� � � + � � � � � � ' � � � � � � � � � ) � _ � � . . � � � � � � � � � � � � � � � � - �

� � \ � � � " � � ` � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � " � � � � � � � � w � � � � � � � � � � � � � � � � � � � � � � � � � � \

n t � M N � h X k S R � � � U � O � g T � X k R � � � � U N g R S � U � f n m

� \� � � � � d � � � � � � � � ' � � _ � � � r � " � � ' � a � � r � � � � � � � � � � � � � �

� \� � � � d � � � � � � � � ' � ' ( � ' _ � � � � � � � � � � � � � � � � � � � � � � -

n k � � � N � R � V � z j S � S � X U N g R S � � P � h � N � R � V f f

� � � � � � � " ' � � � . � � � � � � � q � ' � � ' ( � � � o p x " � � � � p � � � � � � � � � �

� � � � [ � . � � ' � � � � � � � � " � � � � p p � � � � � � � � � � � � � � � � � � \

n h � � g � k � i X k � g i X R g j g h � P � � N S O O S k U i U f k h

� � � � � , [ o � r � . � � � � � � � � ! [ � � � � � � � � � � � � � � � � � � � � � � �

� � � � ) � � � " � � _ � ' � � � � � � ' ( ' _ � � � � � � � � � � � � � � � � � � � �

� � � \ � q . � � � � � � � � ! [ � � � " . p � � b � e % � � � � � � � � _ � � � � � � � � � � �

n � z U � k S i X U � � X i U � N S v j g N X � � R S � S W N S R g � � � � f m l

� �� � � � �% ( � � a � ' ' � � � � � � � � � � � � � � � � � � � � � � � � � � � -

� �� � � � b - e % � � � � � � � � _ � � � � � � � � � � � � � � � � � � � � � � � � � �

� �� \ � � � � b � e % � � � � � � � � _ � � � � � � � � � � � � � � � � � � � � � � � �

� �� � � � b � e % � � � � � � � � _ � & � � � b � e % � � � � � � � � _ � � � � � � � � � � � \

� �� � � , [ o � � ! � � � # � � � p � � � ! � y � � � � " � y � � ! [ � � � � � � � � � � � � �

� � � � � " . p # � � � � p � �

� � � " � � � � r _ � ! � � � . p � �

� _ � ! � � � . p � � s � � �

Page 5: Программирование: теоремы и задачи

��

� � � � � � � � � ��

� � � � � � �

� � � D � � �

�� �� � z � z | ~ � s y { z t t ~ � v �

n � n � n � � � � " � � a � . � ( � � � � � � � � � & � � � � ' � � � � � p � � � � � � � ( � �%

� � � � � & ( � ' . � � ' ( � . � � � � p _ � � � � � � � ! � � # � � � p ( � � � � � � � y ( � � � � p . � ' p

[ � � ' � � � � b � � � � � ! � � # � � � � � � � � � � ' � � � � � ! � � # � � � x � � � � � [ � � � � e �

� � � � � U � � � � � " s � " � ( � . � � � � . p � x a � . x ( � � � � � � � x � �

) + � � �

� + � - �

- + � ) � �

� � ( � _ � � [ � r � � ' p [ � ! " � ( � . � � � � . p � � r ( � � � � � � � � r & � � ( � ' � �

� + � - �

- + � � �

� � ( � � � � " � � _ a � . � b [ � ! � � ! � � � � � � � � � # � � � � � ' p � � # � . p � � � ! � � # � � � � ( �%

� � � � � � � r � e �

n � n � f � n � d � � p ( � � " " o x ! � " � # & � � � ' ( � . p ! p " � ( � . � � � � . p � y

( � � � � � � � y b � ( � � " ( � . � � � p & # � � ! � � # � � � p � � a � . y ( � � � � � � � y � � � �

[ � p ( � � � ! � � . p � � a � . � # � ' . � e �

� � � � � U � � + � # � . p � � ! � � # � � � p � � � � [ � ! � � # � � � � & � � �

� + � � � - � � � � � � � - �� - � - � �

- + � � � - � � � � � � � - �� - � � � �

� + � � � - � � � � - �� - � � � � �

�� �� � � � ! � " � # $ � % % � & � & �

n � n � t � � � � � a � . � � # � ' . � ' � � � � � � . p � � � b a � . � � � � � � � � a � � � . p � � � e

# � ' . � (� � # � ' . � � p � ( � � � � � � � ' . � � � � � & � � � [ y � " � � � ' � ' � � � � � p ( � �%

� � � � � & ( � � � ' ( � . � � � � � _ � � � � � r ! � � # � � � p ( � � � � � � � y ' � ( � � � � � p x �%

' p & � ! � � # � � � � � � _ � � � � � r " � � � r ( � � � � � � � � r b � � ( � � � � � & � e ' � � � � � � � ' p

� � � � � � ( � b � � � ) � � � � � ! � � d � � � ' p � ' ( � . p ! � � � � p � " � � � � ( � � � � � � � �� e

� � � � � U � � � � � " s � a � . x ( � � � � � � � x * & _ � � � � � p � � � p � � ' p � � � " � ( &( � � # s � ( � " " � � q � � � � � ' p � � _ � � ' � � r ' � � � � � + � * e �

, + � � � - + � - �� - � � . / 0 1 2 1 3 4 , �

5 � 6 � � , 7 8 � � 9 - � : 6 �, + � , � - �

- + � - ; � �

� � � � �

� � � � � � � d � � � � � � r q � ! � " � # � �

, + � � � - + � - �� � . / 0 1 2 1 3 4 � � - ; < � . / 0 1 2 1 3 4 , = �

5 � 6 � � , 7 8 � � 9 - � : 6 �, + � , � - �

- + � - ; � �

� � � �

n � n � k � n � d � � p ( � � " " o x ! � " � # & � ' . � � � � [ � � ' p & # � � [ # � ' . �

" � r ' � � � r b � ( � . � p � � y � ( � � � � � � � � ( � � ' � � � � � � � p e [ . � ( � � p " _ � |} ~ (

b � � � ' � p � � ( � � � � ' y � " � . � [ > |} ~ ( " . p � � _ � � � � � r _ � � ' � � � � > ? |} ~ ( $

) � � ' � � ( � � p & � _ � � � � x � q � � � � ! � � ' � � & # � � [ ( � . # � � p ( e �

� � � � � U � � � � � ' s � � � _ � � � � � � ! � � � � � � p � � � � � � � � � ! ( � � " . � q � � � y

� � d � � � r ( � � " " o � r ! � " � # � �

, + � � � - + � - � � + � � �� � . / 0 1 2 1 3 4 � � - ; < � . / 0 1 2 1 3 4 , = �

5 � 6 � � , 7 8 � � 9 - � : 6 �6 ( , � 9 � @ � � ) � � � - � : 6 �, + � , � 6 A @ �

� + � � ; � �

� � � � � � � - � : 6 �, + � , � - �

- + � - ; � �

� � � �� � � �

Page 6: Программирование: теоремы и задачи

� - �� � � � � $ � � � � � � & � � � � � � � � � � � � % & � � & � � � �

� � q " r � � � � � r � � ! b � � � � q � e [ " � � � ( � . � p � p ' p ( � � � r � � � � � � �

� ( � � � � � � � � [ � � � b � ' . � * � � # s � � � & � � ( � ' . � � # � � � � � p � " � � � a ' � �%

� � � � � ' p # s � � � e & � � _ # � � ! � " � � a � _ . � � � . � # � � � * � � � p d � � � ' p ( �

_ � � r � � r � � � � � " � � �� �

n � n � h � � � � � � � � � . p � � # � ' . � ' & �� � # � ' . � � p ( � � � ! � � " � � � � � · � &� ' ( � . p ! p � ( � � � � � � � � . � d p � ( � � � a � � � & � & � & � �

� � � � � U � �, + � � � � + � � �� 4 3 . � 4 3 0 + � � � ; , �

5 � 6 � � , 7 8 - � 9 - � : 6 �, + � , � - �

� + � � � � �

� � � �� � � � ; , 4 , � -� / � 1 � . 0 1 � � 3 � � � � � ; - �

n � n � � � � � � � � � � . p � � # � ' . � ' � �� � # � ' . � � p � y ' � � ' � ��] ' ( � . p ! � � � � p � ( � � � � � � ( � � ' � � � � � � � p . � d p � � " �

〈 ( � � � � � � � � p �〉 � � 〈 ( � � � � � � � � p 〉 �

〈 ( � � � � � � � � p〉 � � 〈 # � ' . �〉 �

〈 ( � � � � � � � � p �〉 � � 〈 ( � � � � � � � � p 〉 � ��

� � � � � U � �� � �� 4 3 . � 4 3 0 + � � � � , �

� � � �

n � n � m � � � � � � � � � � . p � � � b a � . � � � � � � � � a � � � . p � � � e # � ' . � ' � a � . � �

( � . � q � � � . p � � � # � ' . � �� � # � ' . � � p # � ' � � � � � � � ' � � � � _ � ( � � " � . � � � � '

� � � & � � � ' ( � . p ! p � ( � � � a � r � � � � � � ��

� � � � � U � � � � � . � ' � � � ( � � " � . � � � x & � + � · � � � & �

6

� � ��

� � 8 � � � � 8 � �

� + � � � � + � � �� 4 3 . � 4 3 0 + � � � ; � � �� � 7 � � �

5 � 6 � � � 9 ) < � 7 � = � 9 - � : 6 �� � 8 � � �

� + � � � � � � � 8 � � �

� + � � � - �

� � � � �

�� �� � � � ! � " � # $ � % % � & � & � �

n � n � Y � � � � � � � � � � . p � � � ( & � # � ' . � � p ( � b � � + � & ( � + ( · b ( − � e � e � �

n � n � j � � � ' . � " � � � � � . p � � ' � p w � [ � � � # # � � ( � � " � . p � � ' p � � _ � � � + � &� � + � & � * + � * � � � � * � � ( � � *

>

�� � � � � ( & � # � ' . � � p � ( � �

n � n � n l � � � q � ! � " � # � & � ' . � � � � [ � � ' p & # � � [ # � ' . � � ( � � � a � r [ . �

( � � ( � � a � � � � . p � � |} ~ (� b � � � � � � � � � " � . q � [ � p a � . � # � ' . � � � � �� e

� _ � ! � � � �� � � � � ' � ' � " � � y # � ' � . w � [ � � � # # � ( � . # � � � ' p � ! ( � � " " %

o � r � � � q � � � � � � � � � � � � a

∥∥∥∥

� �

� �

∥∥∥∥

$ � � _ # � � ! � " � # � ' � � " � � ' p _ � � ! � � " � � � x � � � � � a � ' � � ( � � p (� ! � �

� � q � � ' " � . � � p ! � > |} ~ ( " � r ' � � � r � � � q � ' ( � ' � [ � � & # � � � " . p # �%

' � .� "

n � n � n n � � � � � � � � � � . p � � � ( & � # � ' . � � p

�� � �

�� �

�� � �

��

( � � �

n � n � n f � � � q � & � ' . � � � � [ � � ' p & # � � [ _ � . � # � ' � � � � ( � � � a � r b � ( � .%

� � � � y _ � � � � " ( � � ' � � � � � � � p e [ . � [ ( � � p " _ � ( b � � [ � . � � > ( " . p � �%

_ � � � � � r _ � � ' � � � � > e �

� � � � � U � � ] � � � � � � � � � # $ � + � % � � �� � �

� � % * � & & � # � + � % * � b � � q � � � �

� # � ' . p � p ! � � � � � _ � q " r � � ! * � e � �

n � n � n t � � � � " � � � � � � � . p � y # � ' . � � � � & � � � � � � � � . x � " � �%

� � � � � � � �� � # � ' . � � p ' ( ) * � + � , $ � � � [ � . p d � r � [ o � r " � . � � � . p ' � � �

� � � � � U � � � � � � � � � � �

6 ( � 8 - ) � � � - � : 6 �, + � � �

� � � � � � � - � : 6 �, + � - �

� � � �� , � � � . < �� - = �

� 4 3 . � 4 3 0 + 3 4 - - � 1 . 4 / � � � / � � � 0 1 1 ,� 3 1 1 . � 1 1 0 / 1

� / 2 4 3 1 � 4 0 1 � 1 3 �

5 � 6 � � � 9 ) < < � � 9 � , � � = � � � < - � 9 � , � � = = � 9 - � : 6 �, + � , � - �

� � � �� , � � / 2 4 4 1 � 4 0 1 � �� / � � � 0 4 1 � 3 1 0 �

Page 7: Программирование: теоремы и задачи

� �� � � � � $ � � � � � � & � � � � � � � � � � � � % & � � & � � � �

� � � � � � � b � . � � � � � � � � _ . � " � e � " � � ' # � � � � p & # � � ' ( ) * �+� , � � �

� � � " � ' ( ) * � + � , + ' ( ) * � � � + � , + ' ( ) * � + � � � , ? ' ( ) * � + � , + ' ( ) * �+ � , + �

" . p � ' � y � � �

>

��

� + � � � � + � - �� 4 3 . � 4 3 0 + � � � < �� - = � � � � < �� � = � �� � 8 � � �

5 � 6 � � � 9 ) < < � � � = 9 � < � � � = = � 9 - � : 6 �6 ( � 8 � � ) � � � - � : 6 �

� + � � � � �

� � � � � � � - � : 6 �

� + � � � � �

� � � �� � � �� � � � 4 � 4 � � � �

6 ( � � � ) � � � - � : 6 �, + � � �

� � � � � � � - � : 6 � � � � � �

, + � � �

� � � � �

n � n � n k � + � ( � ' � � p � � " � � a � � � � � � � r � � � � � � � � . � � � � � � � � � _ . � " � &� ' ( � . p ! x o � r ' � � � � � d � � � p ' ( ) * � + � , + ' ( ) * � � � � �+

� , ( � � �

>

� &

' ( ) * � + � , + ' ( ) * � + � � � � � , ( � � �

>

�� �

n � n � n h � � � � � � � � � . p � � � � � & � � � � � � � � � " � � � � � � � � � �� + � r � �

� + ' ( ) * � + � , � � � _ � � a � . � � � � & # � � � + � · � � � · ��

� � � � � U � � � � [ � � � � � � . � � � � � � � � _ . � " � ( � � � � � � � � � & � & � & # � � ( �%

d � � � � � � � � � � � � ' . � � � p � � � � � � � � � ? ( � � � � � # � � �

� + � � � � + � - � * + � - � � + � � � � + � � � � + � - �� 4 3 . � 4 3 0 + � � � < �� - = � � � � < �� � = � �� � 8 � �

� � * ; � � � ; - � � � � ; � � � ; - � �5 � 6 � � � 9 ) < < � � � = 9 � < � � � = = � 9 - � : 6 �6 ( � 8 � � ) � � � - � : 6 �

� + � � � � � * + � * � � � � + � � � � �

� � � � � � � - � : 6 �

� + � � � � � � + � � � * � � + � � � � �

� � � �� � � �6 ( � � � ) � � � - � : 6 �, + � � � . + � � � � + � � �

� � � � � � � - � : 6 �, + � � � . + � * � � + � � �

� � � � �

�� �� � � � ! � " � # $ � % % � & � & � \

n � n � n � n � d � � p ( � � " " o x ! � " � # & � ' ( � . p ! p � � . � � � � � � � � � _ . �%

" � " � . � � � � ' � ' � � � _ � � � �

n � n � n m � b ! � � � r _ ' � � � e � � [ � � � � � � . � � � � � � � � _ . � " � " � ( � . � � � � . p � �

( � � � � � � � � $ & � & �

� + � � � � + � - � � + � - � A + � � �� 4 3 . � 4 3 0 + � � � < �� - = � � � � < �� � = � �� � 8 � � �

5 � 6 � � � 9 ) < < � � � = 9 � < � � � = = � 9 - � : 6 �6 ( � 8 � � ) � � � - � : 6 �

� + � � � � � A + � A � � �

� � � � � � � - � : 6 �

� + � � � � � � + � � � A �

� � � �� � � �6 ( � � � ) � � � - � : 6 �

+ � A �

� � � � � � � - � : 6 � � � � � �

+ � � �

� � � �

� � _ � ! � � p & # � � ( � ' . � � ' ( � . � � � � p � . � � � � � � � ! � � # � � � � � � � � � " � � � � � �%

� � � � � � � p d � � � [ o � � _ � � � � � � # � ' � . � & � � + � · ' ( � * � + � , �

� � � � � U � � Z � � � � � � & # � � � � . � # � � � � · $ � ( · � � � � � � p � � ' p � y � " � � %

( � . � � � � p � . � � � � � � �� , ' � � s � ' p � � ' ( � . p ! � � � � p ' p � � � & # � � � � � # � . � � � �

� � � � � � � � � # � � ' ( ) b � � � e · ' ( � b � � � e + � �� �

n � n � n Y � + � ( � ' � � p � � � � � � � � . � � � � � � � � � _ . � " � & � ' ( � . p ! x o � r ' � � �%

� � d � � � p

' ( ) b � � � � � e + � · ' ( ) b � � � e�

' ( ) b � � � � e + ' ( ) b � � � e ( � � � � # s � � � � ��

� � � _ . x # � x o � r " � . � � � p ' � ' � � � _ � � & � � ' ( � . p ! x o � r . � d p " � . � � � � � � �

� ( � � � � � _ # s � � � ' � �� b � � ' . � " � r ' � � � r " � . q � � [ � p ( � � p " _ � |} ~ * " . p

� ' y � " � y " � � � y & � � ( � � � � ' y � " p o � y *� e

� � � � � U � �

� + � � � � + � - � � + � - �� � � � < �� - = � � ; � � � < �� � = �

5 � 6 � � � 9 ) < < � � � = 9 � < � � � = = � 9 - � : 6 �6 ( < � � 9 � @ � � = � � � < � � 9 � @ � � = ) � � � - � : 6 �

� + � � ; @ � � + � � � 6 A @ � � + � � � 6 A @ �

� � � � � � � 6 ( < � � 9 � @ � � = � � � < � � 9 � @ � - = ) � � � - � : 6 �

Page 8: Программирование: теоремы и задачи

� �� � � � � $ � � � � � � & � � � � � � � � � � � � % & � � & � � � �

� + � � � 6 A @ �

� � � � � � � 6 ( < � � 9 � @ � - = � � � < � � 9 � @ � � = ) � � � - � : 6 �

� + � � � 6 A @ �

� � � � � � � 6 ( < � � 9 � @ � - = � � � < � � 9 � @ � - = � � � < � 8 � � = ) � � � - � : 6 �

� + � � � � �

� � � � � � � 6 ( < � � 9 � @ � - = � � � < � � 9 � @ � - = � � � < � 7 � � = ) � � � - � : 6 �

� + � � � � �

� � � �� � � �� � � � � 8 � 0 . 1 0 � � ; � � � � � � 8 � 0 . 1 0 � � ; � �

, a � � _ � # � ' . � " � r ' � � � r � _ � q " � � � � � � � � " � r ' � � � � " � . � � y � � p [ � " � � � !

# � ' � . � � ( ( � ( � . � � � �

n � n � n j � � � ( � . � � � p � . � � � � � � ( � � " " o � r ! � " � # � ( � � ' _ � � � � � & " . p

_ � � � � y � � � � � + ' ( ) b � � � e �

� � � � � U � � b ] " � p ' � � [ o � � � � � Z � � � _ � � � � e � � � q " � � ' � � � ! � � � � � � &# � � � " � � � � � � � � � � � " � . � � � � � � � ( � ( � . � � � � � � � p � � � ' _ � � y � � �� � �%) � � � � � q � � ' # � � � � p & # � � ' ' � � � � � � � # � . � � " � � � ! # � ' � . � � � � � # s � � ��b ! � � ' � � r ' � � � [ " � � ' � y � � � p � p ' p � " � . � �� e

� � ( � � p ( � ( � � � � ' p & _ � _ � � � � p d � & y � � � � � p � � _ � � # � ' . � � � � � � � # & # � �

� + � � � � � �

( + � � � � #�

� � � [ . � � � � � � � & # � � ( � � " � . � � � � & ' _ � q � � & � � � � � � " � � � ! " � . � � p �

� � � � � & � � � � ( � � � ' � � � � [ � p a � . � � b � ' � � � � " � � � # � �% � � a � � � � . p%

� � � e � � � � � # � �% � � a � � � � . p � � � # � ' . � � ' � � ' � � � � � � y � � � � � p � � � " � ( � �

〈 # � ' . � � � . p & ( � _ � ! � � � . p ' � � ( � � � " � � r _ � � ! � � � � � � � � . �〉� � � � � � � � ( �%

. # � � � � � � � " � _ � � [ � � � a � � � � � ' " � � � # � �% � � a � � � � . p � � � _ � ) ` �%

a � � � � � � �� ] � � � ' . � � � � � & � � � � � �

��� + � � � � �

" . p � � _ � � � � y a � . y � � � � � � � � � . p � � � � � � � � � " � . � � p & � ' . � � � � �

� ' . � � � � # s � � & � � � � � � � q � � ' � _ � � � � � p� � ' . � ) � � � � � � _ & ( � . � q � � � �

� � q � � � ' ( � � � � � p ( � � � [ � � ! � � � � � � �� � + � � ��

� � + � − �

b � � � � � � � � p � � � � � � � e � � [ � " � � ' p � ) � � � � + � ( � � � � � & # � � � ' # � � � � � &# � � � " � � � ! # � ' � . � � � � � # s � � �� � ' � p ) � � [ " � � �� � ' . � ( � � ) � � �

�� �� � � � ! � " � # $ � % % � & � & � �

� # s � � � & � � � � " � . q � � [ � p # s � � � b � � � # � � � � � � [ " � � � � # s � � � e �q ( � � � � # s � � � � � � # � � � � � � � ! � � � � � � # s � � � � � � " � . � � � � # s � � � � �

n � n � f l � � � ' � � � � � p ( � � � � � � � & ( � # � � � x o x _ � � " � � � � ' � y � � � %

� � . p � y # � ' � . � � � " � ! � " � � � � � � � � � � � . p � � � � (�

� � � � � U � �, + � � �

5 � 6 ) � � � < , ; , = �� 4 3 . � 4 3 0 + , 7 � �� 3 2 1 . 0 3 � . / 1

- . � 0 � � , . - � � . 4 0 1 � � 3 � �

5 � 6 � � � 9 ) < , � � = � 9 - � : 6 �, + � , � - �

5 � 6 ) � � � < , ; , = �

� � � � �

n � n � f n � � � q � ! � " � # � & � � � � ! � � d � � � ' p � ' ( � . p ! � � � � p � ! � � � � � � � # �%

' _ � y � ( � � � a � r . � d p ' . � q � � � � � � # � � � � � � & ( � � # s � � [ o � � # � ' . � " � r%

' � � � r " � . q � � [ � p ( � � p " _ � (�

� � � � � U � � � � � " s � ( � � � � � � � x * # � $ � � � b # � $ � � � $ _ � � " � � � e & ' � p ! � �%� x ' * ' � � � � � d � � � � � * # � $ � � � + * � �

, + � � � ,� � � � � � � + � � �

5 � 6 ) � � � < ,� � � � � � � = �

5 � 6 � � � 9 ) < , � � = � 9 - � : 6 �, + � , � - �� ,� � � � � � � � < , � - = ; < , � - = � , ; , � @ ; , � - �

,� � � � � � � + � ,� � � � � � � � , � , � - �

5 � 6 ) � � � < ,� � � � � � � = �

� � � � �� S O � � S � U � � ) � q � � � [ � r � � ' p [ � ! � # � � � � � p ' ( � � � o p x � � _ � r y � �%

� � ' � � �5 � 6 � � � 9 ) < , � � = � 9 - � : 6 �,

� � � � � � � + � ,� � � � � � � � , �� ,

� � � � � � � � , ; , � , �

, + � , � - �� ,� � � � � � � � < , � - = ; < , � - = � < , � - = � , ; , � , �,

� � � � � � � + � ,� � � � � � � � , �

� � � �

n � n � f f � � � ' � � � � � p ( � � � � � � � & ( � # � � � x o x � � ! . � q � � � � � � ( � � ' � �

� � � q � � � . � ! � " � � � � � � � � � � � . p � � � � # � ' . � ( � � b " � � � � � ' . � � � � � & � � �%[ � � ' p ( � # � � � � p � � . p _ � ( � � ' � � # � ' . � � ( � � � ! � � " � � � � � � ( � # � � � � � y

# � ' � . " � . q � � [ � p � � � � � ( ? � ' . � ( + � & ( � # � � � � p � � # � � � � � � � " � e �

Page 9: Программирование: теоремы и задачи

� � �� � � � � $ � � � � � � & � � � � � � � � � � � � % & � � & � � � �

� � � � � U � � � � � � � � � � �

, + � � �� 4 3 . � 4 3 0 + 2 � � 4 � . 1 1 3 4 1 3 2 1 . 0 3 3 � � . 4 / 1 � 4 , � . 3 �

�� 3 2 1 . 0 3 � 0 � � � - � 2 � � / 0 � 1 . 4 / � �

5 � 6 � � � 9 ) < , � - = � 9 - � : 6 �

� + � @ �� 4 3 . � 4 3 0 + , 3 1 4 3 1 1 0 1 � 4 0 1 � 1 4 . 4 3 0 1 � . � 1 < - � � = �

5 � 6 � � , � 9 � � 7 8 � � 9 - � : 6 �

� + � � � - �

� � � �� � � 3 4 3 1 3 � 0 4 4 1 � 4 0 1 � � ,� / � � � 0 4 4 - � / � 1 � . 0 1 � � 3 � �2 � � / 0 � 4 �

5 � 6 ) � � � < � = �, + � , � 6 A � �

� � � �� � � � � � � �

, + � � � � + � @ �� 2 � � 4 � . 1 1 3 4 1 , 4 3 2 1 . 0 3 3 � � . 4 / 1 � � . 3 � � � 3 2 1 . 0 3 3 � 1

. 4 / � 2 � � / 0 � � , 3 1 4 3 1 1 0 1 � 4 0 1 � 1 4� 3 1 3 � 0 4 � � �

5 � 6 � � � 9 ) < , � - = � 9 - � : 6 �6 ( , � 9 � � � � ) � � � - � : 6 �� , 1 � 4 0 / 1 3 � 4 3 1 4 3 1 1 0 1 � 4 0 1 � 1 4�

3 1 3 � 0 4 � � � � 3 . 4 0� � 2 � � / 0 � �, + � , � 6 A � �

5 � 6 ) � � � < � = �

� � � � � � � - � : 6 �� , 3 1 1 � 4 0 / 1 3 � �

� + � � � - �

� � � �� � � � �

n � n � f t � � � ' � � � � � p ( � � � � � � � � � d � � � p ( � � " " o � r ! � " � # � & � ' ( � . p%! x o x � � � � _ � & # � � ' � ' � � � � � � # � ' . � � � � � � " � . � � � . p & � � ( � � � � ' y �%

" p o � r _ � � " � � � � � � � _ � � � p � ! ) � � � � # � ' . ��

� � � � � U � � � � � � � � � � � � � � � � � � � � d � � � p � � � ' � � & � � & � � � � q � � � � ( �%

' � � p

6 ( � ; � 8 , ) � � � - � : 6 �

� + � , �

� � � � � � � - � : 6 �

� + � � � - �

� � � � �

�� �� � � � ! � " � # $ � % % � & � & � �

n � n � f k � � � � � � � � � p & p � . p � � ' p . � ! � " � � � � � � � � � � . p � � � # � ' . � ( � �

( � � ' � � � �

n � n � f h � b � . p ! � � _ � � y ' � ' � � � � � � � . � � [ � e � � � � a � . � � � � ' ' � � �

# � ' . � ( � � � b ( � � � � " . � q � o � �

Z

� " e �

b�e � � � � � � � � p & p � . p � � ' p . � � � � ( � � ' � � b �

Z

� " e �

b [ e + � ( � # � � � � p � � � � � ! . � q � � � � � � ( � � ' � � b �

Z

� " e � � � q � � � . � � �

n � n � f � n � ! � � d � � ( � � � � � p � p _ � � � � " � � � � � * � , . � d p ( � � � + ��

��

�� � � � �

� � � � ' � � � � � p ( � � � � � � � & ( � # � � � x o x " � ' p � � # � x ! � ( � ' p ! � " � �%

� � � � � � � � � . p � � � � # � ' . � ( � � � b � . # � r ( + � p � � . ' p [ � � _ � � � � � � '%

_ . x # � � � � � & � � _ _ � _ � [ # � � � . � � � � # � . � # � ' . � � � ( � # � � � x � ' p & � " . p

( + � $ ( � # � � � x � ' p� e

� � � � � U � �- � � � + � - �� - � � � � / 0 1 2 1 3 � - � � 3 1 2 � 1 . � / � � 1 2 1 � �

5 � 6 � � - � ; - � � � 7 � � � 9 - � : 6 �

- � � � + � - � � � ; - � �

� � � �� - � � � � 3 - / 4 3 � � 3 1 / 0 1 2 1 3 � - � � 3 1 2 � 1 . � / � � 1 2 1 � �, + � � �� 4 3 . � 4 3 0 + � / 0 � � / � 3 2 1 . 0 0 � , / 0 1 3 � 1 . 4 / � � 3

� 3 - � .� . 0 � . - � � � � - � � � � - � � � � � � �5 � 6 � � - � � � 7 8 - � 9 - � : 6 �

5 � 6 ) � < , � 6 A - � � � = �, + � , � 9 � - � � � �

- � � � + � - � � � � 6 A - � �

� � � �� - � � � � - � � / 0 � � / � 3 2 1 . 0 0 � � 3 � � 3 . 3 � 1 . 4 / � � , �

5 � 6 ) � < , = � �

� � ( � # � � p � d � [ _ � ( � � � � d � � � � ) � � r ! � " � # � � � � ( � � � � . p � � � [ � � [ � � %

� � x � ' p # � ' . � ' � . p � � ( � ' � � � " � � �� � � � � � " s � � r � � � � � � � � � " � ( ' _ � � �

' . # � r & _ � � " � * � � � # � ? � ) � � � ' . # � � ( � # � � � � � � * � � # � � � � � ' p ' � ' � � �%

d � y � . � r�

n � n � f m � � � q � ' � � � � & � � � � " � � � ( � # � � � � p " � ' p � � # � x ! � ( � ' p � � [ � � �%

� � � ( � � p " _ �� b � . p ( + � � � � � " � � � ( � # � � � � p � � � � e

� � � � � U � �

, + � � �� 4 3 . � 4 3 0 + � / 0 � � / � 3 2 1 . 0 0 � , . � / � 0 3 � 3 2 � � 1 - 1 �

Page 10: Программирование: теоремы и задачи

� � �� � � � � $ � � � � � � & � � � � � � � � � � � � % & � � & � � � �

5 � 6 � � , 7 8 � � 9 - � : 6 �

5 � 6 ) � < , � 9 � - � = �, + � , � 6 A - � �

� � � � �

n � n � f Y � � � � � � � � � � . p � � � (� � � " ' # � � � � p _ � . � # � ' � � � � � d � � � r � � � �%

� � � ' � � � �� � �� � ( � � � � � � . p � y b � � � � � � a � � � . p � y a � . y e # � ' . � y & � �� ' ( � . p ! p " � r ' � � � r ' � � o � ' � � � � � � � # � ' . � � ��

� � � � � U � �, + � � � � + � � �� 4 3 . � 4 3 0 + � � - � � 4 . 1 / 0 . � � 1 0 1 3 4 4 3 1 � . 1 3 / 0 .

. ; . � � ; � 7 � � . 7 , �

5 � 6 � � , ; , 7 � � 9 - � : 6 �

� � �� ) � . 4 / � � � 1 0 1 3 4 4 3 1 � . 1 3 / 0 . , ; , � � ; � 7 �

/ � 8 � � < 2 � 4 3 3 � 3 , = �

, + � , � - �

� + � � � ) �

� � � �� , ; , 8 � �� 2 � � 0 � 3 � � � - � � 4 . 1 / 0 . � . / 1 � � 1 0 1 3 4 4

3 1 � . 1 3 / 0 . �

Z "

Page 11: Программирование: теоремы и задачи

- �� � � � � $ � � � � � � & � � � � � � � � � � � � % & � � & � � � �

, a � � _ � # � ' . � " � r ' � � � r � # � � � " � � � ' � � # � . � � " � � q � � ' p � � � � y � � [ � . � �

# � � � � √ ( d � � � � & � ! � � � � � � � ! � � ( � � � � $ � _ � q " x ' � � � � � � � [ � . � �

# � � � � √ ( d � � � �� �

n � n � t l � � � � � � � � � . p � � # � ' . � ( � * & ( � � � + � ( � # � � � � p * " � ' p%

� � # � y ! � � _ � � # � ' . � � % (� b � � � � � . � # � � " � y " � ' p � � # � y � � ! . � q � � � r

� [ � � � � � ' p � � � ! � � y & _ � � � � � � � � ' � " � � q � � " � � p � _ � � ( � � � � " �� e � � �%

� � � � � � " � . q � � � ' ( � . p ! � � � � p � � . p _ � a � . � ( � � � � � � � ��

� � � � � U � � � " � � � � � " � ' p � � # � � r ! � ( � ' � # � ' . � � % ( ! � ( p � x � � * � � ' �

� ( � � � � & ( � . # � � # � ' . � � � * % (� + � � � � " � � � ( � # � � � � p � � � a � . x # � ' � p &

� � � ' � p � � ! " � . � � p � � * � � ( � � a � . �� � � � � " � � � � r ' ( � ' � [ � � � [ � � � '%

( � . p ! � � � � � p [ � . p d � y ( � � � . � # � � � # � ' � . & _ � � � � � � � � � � r � � ! �

� � � � � a " � � ( � ! � � � ( � � " ' � � � � � y # � ' � .� � � ) � � � � ' " � . � � � � � � # �

b ' . � " p � [ # � � � � � � � " � " � . � � � p � � . _ � � � e � [ " � � y � � � � � p � � ' � �%

� � _ � � �� + � � � � + � - �� 4 3 . � + 3 2 1 . 0 3 � � � � � 1 � . - , �� � / 0 � � / � 3 2 1 . 0 0 �, � � � � � 1 � . � � / 4 � , � �

5 � 6 � � � 7 8 , � 9 - � : 6 �

5 � 6 ) � < < - � ; � = � 6 A � = �

� + � < - � ; � = � 9 � � �

� + � � � - �

� � � � �

n � n � t n � � � � � � � � � � . p � � � # � ' . � ( � � � , ( � � " � . � � p " . � � ( � � � � " �

" � ' p � � # � � r ! � ( � ' � " � � [ � � % (�

� � � � � U � � � � � � � " " � � [ � � � � � � ( � � � � " � ( � ' . � " � � � � � . p � � ' � � � ' � � � %

_ � � b " � _ � q � � � ) � � ? � # � ' � � � ' � � & � � " � " � _ � ! � � p & # � � � � � � � � q � � [ � p

� � � p d � e � � � � � � � � � � & � ) � � r ( � ' . � " � � � � � . p � � ' � � � ' � ( � � � � " � # � ' _ � ( �%

� � � � p x o � � ' p # . � � � � ! . � # � & � ( � � " ( � � � � " � � � � � " . � � � � [ � . � � (�� � ) � � � " � ' � � � � # � � � � r � � b ( � � e % r # . � � ( � ' . � " � � � � � . p � � ' � � � ' � � � %

_ � � � ! � � � � � � � � � � . p � � � * & ( � � _ � � � � � � b ( � � � * e % r # . � � ' � � ( � " � � �

' b ( � � e % � �� + � � � � + � - �� 4 3 . � 4 3 0 + � , � � � 1 � � � � 0 0 � 0 / � / � . 3 4 1 � � 3 - � . . - , � �

5 � 6 � � � 7 8 � � - � 9 - � : 6 �

� + � < - � ; � = � 9 � � �

� + � � � - �

� � � �� + � � �

�� �� � � � ! � " � # $ � % % � & � & �

� � � < � � - = � � 4 . � 1 3 2 � / � 1 � . 0 1 � � 3 � / 0 4 � / 0 0 - � . �

� + � < - � ; � = � 9 � � �, + � - �� � � < � � , � - = � � 4 . � 1 3 2 � / � 1 � . 0 1 � � 3 � / 0 4 � / 0 0 - � . �

5 � 6 � � � 7 8 � � 9 - � : 6 �

� + � < - � ; � = � 9 � � �, + � , � - �

� � � � �

n � n � t f � b � � � [ o � . � � � � ) � � � p ' � � � # e � � � � ` � _ a � p � � { �� � �

�} →→ { �

� � �

�} + � r � � ( � � � � " ( � ' . � " � � � � � . p � � ' � � ��� * � ,

�� * � * � ,

� � � �

� �%

. � # � ' � � � " � r ' � � � r " � . q � � [ � p ( � � ( � � a � � � � . p � � ' � � � � � � r " . � � �

( � � " ( � � � � " � � ( � � � � " � b ) � � ' � � � � � q � � [ � p ' o � ' � � � � � � � � � p%

d � � e �� � � � � U � � � ' . � � � [ � � ' � � p � � # � . p � r _ ' � _ & ( � ' . � " � � � � � . p � � ' � p ( �%� � � " � # � � & ( � � # s � � ' � # . � � ( � � � � " � � � ! . � # � �

� � / � � 3 . 1 3 4 1 + ( � �� - � � ( < ( < � � � ( < - = � � � = = < � � � = �, + � - � � + � ( < - = � - + � ( < ( < - = = �� � � ( � ,� - � � - � ( � @ ,� - � �

5 � 6 � � � 7 8 - � 9 - � : 6 �, + � , � - � � + � ( < � = � - + � ( < ( < - = = �

� � � �� � � ( � ,� - � � ( � @ ,� - � � ( � ,� - � . � � 4 0 . 2 1 � 4 � 4 . 1 / - � � . / 0 � �

� + � - � - + � ( < � = �� - � ( � , � � � - � � ( � ,� - � � � � � � ( � , � � � - � - � � � � 4 . 3 � �

5 � 6 � � � 7 8 - � 9 - � : 6 �

� + � � � - � - + � ( < - = �

� � � �� 2 1 � 4 � � . 1 3 � �

n � n � t t � b ! � � � r _ ' � � � e w � _ a � p � ' � � � � � . p � � � � � � � � � � � � � �

! � � # � � � p � � � ( � � " � . � � � � � _ � � b � e + � & � b � e + � & � b � ( e + � b ( e & � b � ( � � e +

+ � b ( e � � b ( � � e � � � ' � � � � � p ( � � � � � � � � # � ' . � � � p � b ( e ( � ! � " � � � �%

� ( & � � � [ x o x ( � � p " _ � |} ~ ( � ( � � � a � r�

� � � � � U � �

, + � � � � + � - � - + � � �� 4 3 . � 4 3 0 + � 7 � ,� ( < � = � � ; ( < , = � - ; ( < , � - = �

5 � 6 � � , 7 8 � � 9 - � : 6 �6 ( , � 9 � @ � � ) � � � - � : 6 �

� + � , � 6 A @ �

Page 12: Программирование: теоремы и задачи

�� � � � � $ � � � � � � & � � � � � � � � � � � � % & � � & � � � �

� , � @ � � ( < , = � ( < � = � ( < , � - = � ( < @ � � - = � ( < � = � ( < � � - = �( < � = � � ; ( < , = � - ; ( < , � - = � < � � - = ; ( < � = � - ; ( < � � - = �

� + � � � - � , + � � �

� � � � � � � - � : 6 �

� + � , � 6 A @ �� , � @ � � - � ( < , = � ( < � = � ( < � � - = �( < , � - = � ( < @ � � @ = � ( < � � - = �( < � = � � ; ( < , = � - ; ( < , � - = � � ; ( < � = � < � � - = ; ( < � � - = �

- + � � � - � , + � � �

� � � �� � � �� , � �� ( < � = � � ; ( < � = � - ; ( < - = � -� . 0 � 4 0 � 1 / � . � � / � � �

n � n � t k � � � q � & � ' . � � b � e + � � & � b � e + � � & � b � e + � � & � b � e + � � & � b � ( e +

+ � � � b ( e � � � � b ( � � e & � b � ( � � e + � � � b ( e � � � � � b ( � � e ( � � (

>

� �

� _ � ! � � � �� � � � � � � p _ � ) ` � a � � � � � � � � q � � � � � b ( e # � � � ! � � � ' �%

' � " � � y # � ' . �� "

n � n � t h � � � � � � � � � . p � � # � ' . � ' � � & ( � � # s � � � � � + � r � � # � ' � %

� � � � � ' � � � � _ ( � � " � . � � � � � � � � & � ( � � � � p . � d p ' a � . � � # � ' . � � �

� � � � ' ( � . p ! p � ( � � � a � � � � � � � � � & ! � � ' _ . x # � � � � � " � . � � � p � � � # s � %

� y # � ' � . ? # � ' . � d � � � � � � " � . q � � ( � � � � ' y � " � � p > � |} ~ b � % � e � > � " . p

� � _ � � � � y _ � � ' � � � � > � � > � �

� � � � � U � �- - + � - �

5 � 6 � � - - 7 � � � 9 - � : 6 �

- - + � - - ; @ �

� � � �� - - 8 �� - - � - ; < 3 1 - � 0 � � 1 / 0 1 2 1 3 � @ = �

� + � � � � + � � �� 4 3 . � 4 3 0 + �� � � . / 0 3 � 1 4 � / 0 0 � - 2 � 4 1 � 1 3 4 4 � 3 - - �- - � - ; < 3 1 - � 0 � � 1 / 0 1 2 1 3 � @ = �

5 � 6 � � - - 7 8 - � 9 - � : 6 �

- - + � - - � 6 A @ � � + � � ; @ �� � � - - ; � � �� � 7 � �� � 7 @ ; - - �

6 ( � 8 � - - ) � � � - � : 6 �

� + � � � - - �

� + � � � - �

� � � �� � � �� �� � � . / 0 3 � 1 4 � / 0 0 � - 2 � 4 1 � 1 3 4 4 � 3 - �

�� �� � � % % � & � \

�� �� � z t t ~ � �

� ' . � " x o � y ! � " � # � y ( � � � � � � � � � � � � ( � � " ( � . � � � x � ' p � ( � ' � � � %

� � _ � _ � � � � � � �� �

( � � � ( � � � � b � " � ( $ � � _ � � � � � � � � � � � . p � � � # � ' . � &[ � . p d � � � e & � ' . � � � � � � � � � � � � � � � � p � � ��

n � f � n � Z � ( � . � � � p � � ' ' � � � � . p � �� b ! � � � ! � � # � � � & # � � � q � � ' � ' � �%

� � � p � � � � � � � ( � � � � � � � & ( � ' . � � ( � . � � � � p _ � � � � � � � � ' � ! � � # � � � p

� � � � �

� � ( � � � � p . � ' p [ � . x & � � ! � � � ' � � � � � � � # � . p � � � � ! � � # � � � p ( �%� � � � � � � r �� e

� � � � � U � �6 + � � �� 4 3 . � 4 3 0 + 2 1 � . � 1 6 � 3 . 1 3 4 4 . � - � � � . � 6 � � . 3 � � �

5 � 6 � � 6 7 8 � � 9 - � : 6 �6 + � 6 � - �� . � - � � � . � 6 � - � � � �

. � 6 � + � � �

� � � � �

n � f � f � � � " ' # � � � � p _ � . � # � ' � � � � . � r � � � ' ' � � � �� b � � ' � � � � � p � � �%

� � � � ( � � � � � � � & � � � � � p x o � r ! � � # � � � p � & ( � ' . � � ' ( � . � � � � p _ � � � � � � �

! � � # � � � � � � _ � � � � � r a � . � r ( � � � � � � � � r * � � � � p . � ' p [ # � ' . � . � r ' � � " �

_ � � ( � � � � � � � ' ' � � � �� e

� � � � � U � �� � �� 4 3 . � 4 3 0 + , � . 4 / � � 3 � � 1 4 / � 1 4 . � - � � � � . � 6 � �

� � � �

n � f � t � + � � ' ( � . p ! p � ( � � � � � � � ( � � ' � � � � � � � p " . p � � ' ' � � � � & ' � ' � �%

� � � p � � � � � � � ( � � � � � � � & ) _ � � � � . � � � � r � ( � � � � � � � � � ��

� � � � � U � �6 + � � �� 4 3 . � 4 3 0 + � 3 . 1 3 4 1 � 3 1 4 � 3 1 3 4 � � / �� . � � � � � � � � 2 � 4 � 7 � 6 �

5 � 6 � � 6 7 8 � � 9 - � : 6 �6 + � 6 � - �

. � 6 � + � � � 6 � �

� � � � �

n � f � k � + � r � � � � _ ' � � � � ! � � � � �

� � ( �

Page 13: Программирование: теоремы и задачи

�� � � � � $ � � � � � � & � � � � � � � � � � � � % & � � & � � � �

� � � � � U � �6 + � - � � � . + � . � - � �� 4 3 . � 4 3 0 + � � . � 3 - / 4 3 � 3 4 � . � - � � � . � 6 � �

5 � 6 � � 6 7 8 � � 9 - � : 6 �6 + � 6 � - �� � � . � 3 - / 4 3 � 3 4 � . � - � � � . � 6 � - � �6 ( . � 6 � 8 � � . ) � � � - � : 6 �

� � . + � . � 6 � �

� � � �� � � � �

n � f � h � � � � � � ' ' � � � � � � � � � � �� �

( � � � ( � � � � & ( � � # s � � ! � � ' � � � & # � �

� � �

6

� � �

6 � � � 6

� � ( � + � r � � _ � . � # � ' � � � � � ! . � # � y # � ' � . ' � � " � ) . �%

� � � � � � ) � � � � � � ' ' � � ��

� � � � � U � � � � � � � � � � �

6 + � - � , + � - �� 4 3 . � 4 3 0 + , � - � � 4 . 1 / 0 . � � � � 4 . 3 � � / � 1 4 . � - � � � . � 6 � �

5 � 6 � � 6 7 8 � � 9 - � : 6 �6 + � 6 � - �6 ( . � 6 � 7 8 . � 6 � - � ) � � � - � : 6 �, + � , � - �

� � � �� � � �

� � � � � � � � ] ' _ � � � � # � ' . � � � � [ � . p d � _ � . � # � ' � � � � � y # � ' � . � � !

�� �

( � � & " . p _ � � � � y � � � � � � � � � � � � � � � �

, + � - �

( 9 � 6 + � - ) 9 � � - � 9 - � : 6 �6 ( . � 6 � 7 8 . � 6 � - � ) � � � - � : 6 �, + � , � - �

� � � �� � � � �

n � f � � � � � � � ' ' � � � � � � � � � � �� �

( � � � ( � � � �� + � r � � _ � . � # � ' � � �

� � ! . � # � y # � ' � . ' � � " � ) . � � � � � � � ) � � � � � � ' ' � � �� b � � ' . � " � r ' � � � r " � .%

q � � [ � p ( � � p " _ � ( � � e �

n � f � m � � � q � ! � " � # � & � ' . � � � � [ � � ' p & # � � [ _ � . � # � ' � � � " � r ' � � � r

[ . � ( � � p " _ � ( | } ~ (�

� _ � ! � � � �� � � � � � � � . � � b � � � � � � � � _ � e � "

�� �� � � % % � & � �

n � f � Y � � � q � ! � " � # � & � ' . � � ! � � ' � � � & # � � � ' � ) . � � � � � � � ' ' � � � $

# � ' . � � � � " � * � # � ' . � " � r ' � � � r " � . q � � [ � p ( � � p " _ � ( � *� �

n � f � j � b � � � [ o � . q � �� � " � � e � � p � � � � . p � � � ( � . � �× ( � � ! [ � � � � �

� ( _ � � " � � � � y _ . � � � _� + � _ � � � � � _ . � � _ � ( � _ � � d � � � # s � � r a � � � �] ! � � ' � � � & # � � � ' � # s � � � _ . � � _ � � � � � [ � p � � ! [ � � � � � � ' _ � . p _ �

� � ( � � � ' � _ � x o � y ' p � � � � � � x o � y � [ o � y � � � d � � # s � � y ( � p � � � � . p%

� � _ � �� � # � � � p & # � � a � � � � _ . � � � _ " � � � � � " � � � ' ' � � � � � ( �

� � � � � � �� �

� � � � � � � � � �� �

( � � � � � & � � ( �

( � " ' # � � � � p # � ' . � # s � � y ( � p � � � � . p � � _ � � & � _ � � � � y d . � � � # p� � � ' . �

" � r ' � � � r " � . q � � [ � p ( � � p " _ � � (�

� � � � � U � � � � ' . � ( � p � � � � . p � � _ � � � � � � � # � ' . � y . � � y � � � y � � y

� . � �� � � . p � � ' p . � _ . � � _ � � � � y � � � � . � � & � � q � � ! � � � p & ( � ' � � � � � �

� � � s a � � � & � � � _ q � a � � � � � � y � � � � � . � � � � � ' � ' � " � r� b + � ! � [ " p � � & # � �

� y � � q � � � � [ � p & � ' . � _ . � � _ � ' _ � � x � e �

n � f � n l � � � � � � ' ' � � � � � � �

� � ( a � . y # � ' � .� + � � ' ( � . p ! p " � � � y

� � ' ' � � � � & ( � � � ' � � � � � p ) . � � � � � � � ' ' � � � � � [ � � � � � � ( � � p " _ ��

� � � � � U � � ! . � � � � � � � � � � � ( � � � � � q � � ( � � � � p � p � � ' � � � � " . p

� ' � y � & " . p _ � � � � y � � ( � � − � & � � � ' � p � � � ( � � ⇔ � �

6

( ⇔⇔ �

6

( � � � � �

( 9 � 6 + � - ) 9 � � 6 A @ � 9 - � : 6 �

� � � 2 � 3 1 3 1 0 � 3 1 / 0 3 4 . � 6 � 4 . � � � - � 6 � �

� � � � �n � f � n n � b ] ! _ � � � � � � c � � ' � e � � � � � ' ' � � a � . y # � ' � . � � � � �

� � � � ( &� � ' ' � � � � � � � � � r _ � _ ' � � " � � � � � � " � y � � � � � � � ! _ � � � � � # � . � � � � � �

� � �

" . � � � � _ � � a � � � � � � � �

� � � � ( " . � � (� + � � ' ( � . p ! p " � ( � . � � � � . p%

� y � � ' ' � � � � & ( � � � ' � � � � � p � � # � . � � _ � � � a� b � � ' . � " � r ' � � � r ( � � p " _ �

� � (� e� � � � � U � � � � � � � � � � � � � � � � � � � s � b � � ' ( � . � q � � � � [ � � � � � � ( � � p " _ � e

� � " � . p � � � � # � . � � _ � � � a � � ' ' � � � & � ! � � � � ( � � � � � � � s � � � ' p � � ' ' � � _ � _

� " � � � � a � . � ��� � � � � � � � b q � c � � d � � � � � _ � e n � ' ' � � � � � � � p � � ' ' � � ! � ( � ' � � � � ( �

_ � � & � � " � � & # � � � � � [ � � � � " � r ' � � � � $ ( � � � � � � _ � � �� � � _ � ! � � ' � � � &( � � � � � � � ' � p _ � � ( � ! � a � p " � y � ' � � y ' � � � � � � � r�� � � � � � � \� n � ' ' � � � � � � [ � . � � � [ o x ! � " � # $ � [ � � � " � y # � ' �%

_ � � � � ' ' � � � � � � � � � �

� � � � � � � � � � �

� � � � � � � " ( � . � q � � & # � � " . � � �

. � � � � � # � ' � _ � b � � ! � � s � � � � � e � � [ � . p d � " . � � ( � � � � � � b � � ! � � s �

Page 14: Программирование: теоремы и задачи

� �� � � � � $ � � � � � � & � � � � � � � � � � � � % & � � & � � � �

� � � � e � � " � . � � � � � � # � . � � � r q � " . � � & # � � � � & � � ! � � s � � � � � � &� � ' � � � � _ � � � b � � _ # � � � + � � � � � & � ' . � � [ � ! � � # � � p ( . x ' � � ( � � ( �%

' � � � � � � � ' ' � � � � " � � _ " � � � e + � � � � " � � ! � � � � � � � ( � . # � � p

� � � � � � �� ) � � p p � � ' � � � � # � ' � _ � � � � � $ � � � � � � x � � " � � � _ � � x

" . � � & � ' " � . � � p ) � � . � � _ � & $ ( � . # � � � � � � � � � � & � � ' � � . � ' p ( � � �%� p � p � � ' � � � � � � � � � � � � ' � � � � ' � � . � " � . � _ ( � � � ' � � � � � _ � " � y

� � � � ! _ � � � � � p d � r " . � � � ] � � _ & ( � . # � � � � � _ x ' y � � ( � � � � � � � �

* + � � � � + � � � � + � � � � �� 4 3 . � 4 3 0 + � / 0 � � / � 2 1 � 1 / 0 . 4 0 � . � * � - � � � � � . � � � - � � � � �

5 � 6 � � < * 7 8 � = � � � < � 7 8 � = � 9 - � : 6 �� � / � . / 0 - 3 1 2 � / 0 � �6 ( < � � * = 7 � < � � � = ) � � � - � : 6 �

� � 2 1 � 1 / 0 . 4 0 � . � * � - � � � . � � � 4 . � � � - � � � . � � � < � � * = �

* � � 5 + � � � � � � 5 + � � � < � � * = �

* + � * � � 5 � � + � � � � 5 �

� � � � � � � - � : 6 �

� � 2 1 � 1 / 0 . 4 0 � . � � � < � � � = � - � � � . � � � 4 . � � � - � � � . � � �

� � � 5 + � � � < � � � = � � � � 5 + � � �

� + � � � � 5 � � + � � � � 5 �

� � � �� � � �

, a � � _ � � � � � � � � � � [ � � � � � � # � � � " � � � d � � � � ' � � � d � r ' p " . p � [ � � [ � � _ �

# � ' � � _ ' � � � � � � � ' p _ � � � # � � � " . � � � ? # � ' . � " � r ' � � � r ( � � ) � � � � � _ q �

( � � ( � � a � � � � . p � � " . � � � �� �

n � f � n f � � � ) ` � a � � � � � � � � � # . � � � . � q � � � � � ' ' � � � � � � � � � � � �� �

(

� � � ( � � � � b ( $ � � � � � . p � � � # � ' . � & ' � � ( � � p � � � � � # . � � � e � � # � ' . � � p

! � � # � � � � ) � � � � � � � � � # . � � � � � � # _ � � & � � � ' � p � � ( �( �

� � �

� � � � � �

� � � � �� � � � � U � � b , ( � ' � � � � r � . � � � � � � � � ! � � � � ' p ' y � � � r c � � � � � �� e

, + � � � � + � � � � � �� 4 3 . � 4 3 0 + � 7 � , 7 � ��

� � � � � � ; < . . / 0 1 2 1 3 4 , = � � � � � � � � � - � ; < . . / 0 1 2 1 3 4 , � - = � � � � �� � � � � , � ; < . . / 0 1 2 1 3 4 � = �

5 � 6 � � , 7 8 � � 9 - � : 6 �, + � , � - �

� + � � ; . � � � � � , � �

� � � � �

n � f � n t � b � . p ! � � _ � � y ' � ' � � � � � � � � � . � ! � ? ' � � [ o � . q � c � � d � �%

� � � _ � e � � ( � . � � � p � . � � � � � � � # � ' . � � � p ! � � # � � � p � � � � � # . � � � � ! � " � �%

�� �� � � % % � & � �

� � r � � # _ � ( � ' y � � � c � � � � � � � # � ' . � � � � � ! � � # � � � p � � � ( � � � ! � � " � � r

� � � r q � � � # _ ��

� � � � � U � � � � [ � � . � � � � � � � � � � _ � ) ` � a � � � � � ' � � � � � � ' � � � � ( � � � y � "

� � � � � � � # . � � � � b � e _ � � � � � # . � � � � b � e � �� � � � ( � � � ! � � " � � p � � � #%

_ � � � � � � � � �′

b � e � � b � e � b ! � � � � d � � � � � [ . � " � � � ! � [ � � � � ' � � r ' � � � � �

� � � � " � ! � � � p ! � � � � � � ' � � ( � � p � � � � � # . � � �� � ' . � � � � [ � � � � p � ( � . � � � � p

) � � � � ' . � � � p & " � � o s ( � � ' � � p � # � ' . p � p � � . p _ � ! � � # � � � � ( � � � ! � � "%

� � r & � � ( � � � � � p � ' � � � � � � � � � # . � � � & ( � . # � � � ' p � � � � _ � p ^ q ( � � ' � � p

! � " � # �� e �

, [ o � � � � � � q " � � � � � ' . � q � � ' � � � # � ' . � � � p ( � � � ! � � " � y � � _ � � � �

n � f � n k � b � � � � & w � � � � � ' ' � � e � � � � ( � � � � � � � � � # � ' . � � � p ! � � # �%

� � p � � _ � � � � � � � � � � � � # . � � � � b � � � � � � � � �e & ' � " � � q � o � p � � . p _ � _ � � � � "

( � � ' � � � � � � � p� ] y ( � � � � # � ' � � $ � � � q � � � p & ' � " � � q � o � � ' . � q � � � � &� � � q � � � � & _ � � ' � � � � & ( � � � � � � � � � � � � � � � � � � � � � � � � ' � � � # � � d � � ' p

b � . � � � r # � ' � � e ( � � � � � � � �� � � _ � ! � � p & # � � ' o � ' � � � � ( � � � � � � � � � � � �

q � � � ( � & � # � ' . p x o � p � ' � { ( � � � ! � � " � y � � % � � � � � � � �� � % � � � & ( � � # s �� [ o � � # � ' . � � � � � � � � # � ' _ � y � ( � � � a � r � � [ � . � � # � � � > � � ! ( � � � � ' y �%

" � � # � ' . � � � � � � � � # � ' _ � y � ( � � � a � r � � ' y � " � � r ( � � � � � � � �� � � � ' � � �%

� � > � � ! � � � ' � � � � { �

� _ � ! � � � �� ) � q � � ' # � � � � p & # � � _ � q " � p _ � � � � " � $ ' . � q � � � � " � y

# � ' � . & � � � q � � � � " � y # � ' � . � . � � � � q � � � � � � _ � � ' � � � � � ] ' ( � . p ! �%

� � � p � � " _ a � x ( � # � ' . _ � � � � " & ( � � � � � p p � � " _ � � � � � � ( � � " ( � . � q � � � �

_ ( � � � � � � � � & ( � . # � x o � r ' p � � [ � � ' � � � � � � � � � � _ � � � � " � "

n � f � n h � � � � ' ' � � � y � � � � � � � � �� �

* � � � ( � � � � � � � � � � � � � �� �

& � �

� ( � � � � y � � � p � ' p _ � ) ` � a � � � � " � y � � � � � # . � � � � ' � � ( � � � r * � & � � �%

� � ' � � � p � � � ' ' � � � � � � � � � � �� �

� � � � ( � � � � _ � ) ` � a � � � � � y ( � �%

� ! � � " � � � p� b � � ' . � *�

&� � $ � � � � � . p � � & � + * � & ? ) . � � � � � � � ' ' � � �

' � � " � _ ' � � � ' � " � � q � � _ � ) ` � a � � � � ( � � ' � � ( � � � � � e

� � � � � U � �( 9 � 6 + � � ) 9 � � 9 - � : 6 �

� � 6 � + � � �

� � � �( 9 � 6 + � � ) 9 , � 9 - � : 6 �

( 9 � + � � ) 9 � � 9 - � : 6 �

� � 6 � � + � � � 6 � � � � � 6 � ; - � � �

� � � �� � � � �

Page 15: Программирование: теоремы и задачи

� �� � � � � $ � � � � � � & � � � � � � � � � � � � % & � � & � � � �

n � f � n � � � � " . � q � � � r � d � � . � � � � � � ( � � � � � � q � � � p � � � � � # . � � � �

� � � [ � � ( � � p " _ � { � " � r ' � � � r " . p ( � � � � � � q � � � p " � y � � � � � # . � � � � ' � �%

( � � � { � � � � " � � � p [ � . � � ) ` � _ � � � � r b " . p [ � . p d � � y { e � . � � � � � � & _ �%

� � � � � " � ' � � � � # � � ( � � p " _ � { � � � � � � � � � " � r ' � � � r�

� _ � ! � � � �� � � � " ' � � � � � ' � [ � & # � � � � " � ( � � � � � � q � � p " � � � � � � � # . � � �

' � � ( � � � � � ] y � � q � � ( � � " ' � � � � � p � � � " �

� b � e �� � � b � e > b � e �� � b � e �

� � � � ! � � " � � � � � y � � � � �

� b � e > b � e �� � � b � b � e b � e � � b � e > b � e e �� � � b � e b � e �

� ' � � ' � � � � � r ' ( � ' � [ � # � ' . � � � p � > & � � � > & � � � � [ � � # � � %

� s y � � � q � � � r � � � � � # . � � � � ' � � ( � � � � & � " � � _ � � y _ � . � # � ' � � � � � q � �

' � _ � � � � � p " � � � s y ' ( � � � o p x � � _ � r y � � � � ' � � � � # � ' . � � p � > & �

� b � � � e b > � e & � ! � � � � ! � � � � � � p & # � � � � � > + b � � � e b > � e −− � > − � � "

n � f � n m � � � � " � � � � ! � � ' � � x o � y � � ' ' � � � � � � � � � � � �� �

* � � � ( � � � �

� � � � � � � � � �� �

& � � � ( � � � � � + � r � � _ � . � # � ' � � � � [ o � y ) . � � � � � � �

� ) � � y � � ' ' � � � y & � � � ' � p _ � . � # � ' � � � � � y a � . y � & " . p _ � � � � y � +

+ � � � + � � � " . p � � _ � � � � y � � � � b � � ' . � " � r ' � � � r ( � � p " _ � * � &� e

� � � � � U � �, - + � � � � - + � � � � + � � �� 4 3 . � 4 3 0 + � 7 � , - 7 � , � � 7 � � - 7 � � �

4 / - � 3 � 4 � 0 . 1 0 � � � - � � 4 . 1 / 0 . � � / 2 4 �

� � 1 3 1 3 0 � . . . � , - � - � � � � . � , � 4 � � � - � - � � � � � � � � �5 � 6 � � < , - 7 8 , = � � � < � - 7 8 � = � 9 - � : 6 �6 ( . � , - � - � 7 � � � - � - � ) � � � - � : 6 �, - + � , - � - �

� � � � � � � 6 ( . � , - � - � 8 � � � - � - � ) � � � - � : 6 �

� - + � � - � - �

� � � � � � � - � : 6 � � . � , - � - � � � � � - � - � �, - + � , - � - �

� - + � � - � - �

� + � � � - �

� � � �� � � �� , - � , 4 � 4 � - � � � 2 � � 0 � 3 � � 3 � 4 � 3 3 � � 1 / 0 .� � 2 � 3 1 3 � 0 � �

. 4 3 . � 4 3 0 1 � 2 � / 0 � � � � . 3 � 4 / - � 3 � 3 � � 0 . 1 0 � � �

�� �� � � % % � & � �

� S O � � S � U � � � � � � � p � r � . p � � � � � � � � � " � ' � � � � # � � [ . � [ � � . � # �%

� � � p � " � � ! ( � � � � � � � y * � & & � ? � � � � � p " � [ � � . � � � " . p ' � � � � � � � ��

n � f � n Y � n � d � � p ( � � " " o x ! � " � # & � ' . � ( � � � � ' ' � � � ! � � ' � � �

. � d p & # � � � � �

6 � � � 6

� � * � � � �

6 � � � 6

� � & b � � ! � � ' � � � � � ! � � � � �%

� � � � [ � � � � � � e �

� � � � � U � � � ' . � � � � � � ! � � ' � � � � p [ . � � ' ( � . p ! � � � � � � � � � � p � r � . p � � �%

� � � � � � � [ � � � � ' " � � � � * � � & � � � � & � � � � ' � � � � � � p d � . � � � �

_ � . � # � ' � � � � [ o � y ) . � � � � � � � � � � * � � � � � � � � * � � � & � � � � � � � � & � � �%

( � � p ) � � ( � � " s � ' p " � . � � p ' . � q � � ��

� � �� � � � � � � - � : 6 � � . � , - � - � � � � � - � - � �

) + � . � , - � - � �

5 � 6 � � < , - 7 , = � � � < . � , - � - � � ) = � 9 - � : 6 �, - + � , - � - �

� � � �5 � 6 � � < � - 7 � = � � � < . � � - � - � � ) = � 9 - � : 6 �

� - + � � - � - �

� � � �� + � � � - �

� � � � �

� S O � � S � U � � ! � � ( � � � � � � � � � � � � � " � � _ � � ( � � ( � � � � � _ � ' . � � � p

* * � � * , � ( � * � � * � � � � � ,

b � . � � � � � � � � & � � � . � � � # � � � � e ( � � . � q � � r ( � � � � r ' _ � [ _ � � � � � � p � _ � q � �%

' p [ � ' ' � ' . � � � � r b � � " � _ ' � r " � � ! � � � � � � a � � ' ' � � � e � � � ! � � _ � � �

� d � [ _ �� + � _ � � � � � � � � ' � � ( � ' _ � . p & � # � ' . p p � � ( � & ' � � # � . � � # �%' . p x � � � ( � � . � q � � � � � � � # � ' . p x � � b � � _ � � " s � ' � [ p & � � ( � � � � � &' � ' � � � � � � � � } � � � o � | � � � ' � � � � - $ � � � � \ � - � e � � � " � � ( � ' � � � � p � d � [ _ �

� � � � ! � � _ � � � �+ � � ' . � � � � y � � � � ( � . � � � � p ' p � � � � _ � � ' � � r ' � � � � ' ( � . p ! � � � r

� � � � � � � . � ! � a � � ( � ' _ � . p b � � ( � � " ' � � � � � � � � � � � � � � � � � � � + � � � �%

� � � e & � � � � q � � ( � ' � ( � � p � � _� � � � " s � " � ( � . � � � � . p � x ( � � � � � � � x

� � � � � & � � ( � � � ( � d � � �

6 ( , - 7 , ) � � � - + � < . � , - � - � � ) = � � � � - + � ( � � � � �� - � < , - 7 , = � � � < . � , - � - � � ) = �

5 � 6 � � - � 9 - � : 6 �, - + � , - � - �6 ( , - 7 , ) � � � - + � < . � , - � - � � ) = � � � � - + � ( � � � � �

� � � �

Page 16: Программирование: теоремы и задачи

\ - �� � � � � $ � � � � � � & � � � � � � � � � � � � % & � � & � � � �

) � q � � � � _ q � ' " � . � � p � � � # � �

� � � � � � � - � : 6 � � . � , - � - � � � � � - � - � �6 ( , - � - � , ) � � � - � : 6 �, - + � , - � - �

� + � � � - �

� � � � � � � 6 ( . � , - � - � � . � , - � @ � ) � � � - � : 6 �, - + � , - � - �

� � � � � � � - � : 6 �, - + � , - � - �

� + � � � - �

� � � �� � � �

� � _ [ " � � _ � � � # � & y � � p � � � � � ' � � � � � � � # � ��+ � _ � � � a & � � q � � � � . � # � � p � � ! � � � � � ' ' � � � � � � � � ( � ' � � � � & � _ . x # � �

� � � � � � _ � � � � � ) . � � � � � �

n � f � n j � � � � " � � � � [ � � x o � y � � ' ' � � � � � � � � � � � �� �

* � � � ( � � � �

� � � � � � � � � �� �

& � � � ( � � � � � + � r � � # � ' . � � � ! . � # � y ) . � � � � � � � ' � � " �

� � � � � � � � � � * � � � � � � � � � � � & � b � � ' . � " � r ' � � � r ( � � p " _ � * � & � e �

n � f � f l � � � � " � � � � ' ' � � � � � �

6 � � � 6

� � * � � � �

6 � � � 6

� � & � �

� �%

� " � � � � p � � y � � � ' ' � � � �

6 � � � 6

� � b � + * � & ? _ � q " r ) . � � � � � " � .%

q � � � y � " � � p � � � ' ' � � ' � � . p _ � � � ! & ' _ � . p _ � � � ! � � � y � " � � � � [ o � r

' . � q � � ' � � � � � ' ' � � � � � e � � � ' . � " � r ' � � � r ( � � p " _ � ��

� � � � � U � �, - + � � � � - + � � �� 4 3 . � 4 3 0 + � 0 . 1 0 2 � � � . 4 0 / 1� 1 / � 4 - � - � � � � , - � � - � � / . 4 0 �

/ 2 � . / � 1 4 3 1 3 4 1 3 / / 4 . � . . � , - � - � � � . � , � 4 � � � - � - � � � � � � � �5 � 6 � � < , - 7 8 , = 9 � < � - 7 8 � = � 9 - � : 6 �6 ( , - � , ) � � � - � : 6 �� � - 7 � �

� - + � � - � - �

� , - � � - � + � � � � - � �

� � � � � � � 6 ( � - � � ) � � � - � : 6 �� , - 7 , �

, - + � , - � - �

� , - � � - � + � . � , - � �

� � � � � � � 6 ( . � , - � - � 7 � � � � - � - � ) � � � - � : 6 �, - + � , - � - �

� , - � � - � + � . � , - � �

� � � � � � � 6 ( . � , - � - � 8 � � � � - � - � ) � � � - � : 6 �

�� �� � � % % � & � \ �

� - + � � - � - �

� , - � � - � + � � � � - � �

� � � � � � � - � : 6 �� 0 - � � 3 1 / � . 1 0 �

� � � �� � � �� , - � ,� � - � � � 3 / / 4 . � / � 1 4 3 1 3 � � �

! � � � ( � � a � ' ' � � q � � ( � p ' � � � p � � _� � ' � p � � ' � ' � p " � � ' � � ( _ � _ � �%

� � # � _ & � � ' � � � � � � � � � � y ( � � . � � � � � ) ' � � " � � p � � � y � � " � ' � � (%

_ & � [ � � � p _ � q " r � � ! � � ! � � � y � � y _ � � � � # � _ � [ � � y ' � � ( � _ & _ � � � � � p

� " s � � � � p d � � � . � � � � � � � ( � � p " _ �� � ' . � � � " � � r ' � � ( _ � _ � � � � # _ � _ � �%

# � . � ' p & [ � � s � � y � ! " � � � r ' � � ( _ ��

n � f � f n � � � � " � � � � ' ' � � � � � �

6 � � � 6

� � * � � � �

6 � � � 6

� � & � + � r%

� � � y � ( � � � ' � # � � � � � & � � � ' � p � � ' ' � � � �

6 � � � 6

� � & ' � " � � q � o � r � y

� [ o � � ) . � � � � � & ( � � # s � _ � � � � � ' � p _ � q " � � � ) . � � � � � � � � � ' ' � � � � � �%

� p � � ' p � � � � � � � ! � � � _ � � � � � ' � � r � � � ' ' � � � y � � �� � � ' . � " � r ' � � � r

( � � p " _ � * � & � �

n � f � f f � � � � " � � � � ' ' � � � � � �

6 � � � 6

� � * � � � �

6 � � � 6

� � & � # �%

' . � �� + � r � � ' � � � � " � � � � � � � � & � � � [ � . � � [ . � ! _ x _ # � ' . �� b � �%

' . � " � r ' � � � r ( � � p " _ � * � & & " � ( � . � � � � . p � � p ( � � p � p $ � _ ' � � � � � � � � �

# � ' . � a � . y ( � � � � � � � y & ' � � � � � ' ' � � � � � p � p � � � � ! � � d � � � ' p� e

� _ � ! � � � �� + � " � � � r � � � � � � � � . p � � � � � ' ' � � p � � � � � q " ) . � � � � � �%

� � � � � 6 � � � 6

� � * � �− � � &

6 � � � 6

�− � � � & # � � � � � � " � � ' " � . � � p

� y � " � � y ' . � p � � p � � " � � b � � � [ � � q � � � r e � � ' ' � �� "

�n � f � f t � b � ! _ � � � � � � c � � ' � e + � _ � � � � � � # � ' . � ' � " � � q � � ' p � _ � q%

" � � � ! � � s y a � . � # � ' . � � � y � � [ � � x o � y � � ' ' � � � � � � �

6 � � � 6

� � � &

� � �

6 � � � 6

� � � & � �

6 � � � 6

� � � + � r � � � " � � � ! � � _ � y # � ' � .� � � ' . �

" � r ' � � � r " � . q � � [ � p ( � � p " _ � � � � � ��

� � � � � U � �* - + � - � � - � - � � - + � - �� 4 3 . � 4 3 0 + . � * - � � � . � * � � � � � - � � � � � � � � � � - � � � � � �/ � 1 � � 0 � / 2 4 4

� � 1 3 1 3 0 �

5 � 6 � � � 9 ) < < . � * - � � � � � - � = � � � < � � � - � � � � - � = = � 9 - � : 6 �6 ( . � * - � 7 � � � - � ) � � � - � : 6 �

* - + � * - � - �

� � � � � � � 6 ( � � � - � 7 � � - � ) � � � - � : 6 �

� - + � � - � - �

� � � � � � � 6 ( � � - � 7 . � * - � ) � � � - � : 6 �

Page 17: Программирование: теоремы и задачи

\ �� � � � � $ � � � � � � & � � � � � � � � � � � � % & � � & � � � �

� - + � � - � - �

� � � � � � � - � : 6 �� 0 - 3 1 / � . 1 0 �

� � � �� � � �� . � * - � � � � � - � � � � - � �

5 � 6 ) � � � < . � * - � = � �

n � f � f k � � � q � ! � " � # � & � � . p _ � ! � � � � � � � � � ! � � ' � � � & ' o � ' � � � � . �

� [ o � r ) . � � � � � � � � s y � � [ � � x o � y � � ' ' � � � y � � � � [ � � ' p ) � � � p ' � � � p

b � � � r � � � " � � � ! � [ o � y ) . � � � � � � � & � ' . � � � � � ' � p e � �

n � f � f h � ! . � � � � � � � � � � ' ' � � � � � �� �

( p � . p x � ' p � � [ � � x o � � � � ' ' �%

� � �� �

� a � . y # � ' � . �

� � � � � � � � �� �

( � � � � � � � � �� �

� � � � ( � � � � �

� � � � �

6 � � � 6

� � � � � � � � � � � � ( � �

6 � � � 6

� � ( � � �

] ! � � ' � � � & # � � ' o � ' � � � � # � ' . � & � y � " p o � � � � � ' � � � ' ' � � � � � b ' %

o � ' � � � � � � _ � � � & # � � " . p � ' p _ � � � � � ! �� �

( � � r " s � ' p � � ! �� �

� & " . p

_ � � � � � � � � � � � � + � e � + � r � � � " � � � ! � � _ � y # � ' � . ��

� � � � � U � � � � � " s � � � ' ' � � � � � � � �

� � ( & � � � � # � x o � r � � # � . � � � ' � � x%

o � r ' p # � ' � � � � � ' ' � � � � � � � � � � � � � � ( �

( 9 � , + � - ) 9 � � 9 - � : 6 �

- � , � + � - �

� � � �� � + � ) � � � �

( 9 � , + � @ ) 9 � � 9 - � : 6 �

� � + � � � � � � < � � - � � - � - � � � � � , � � - � , � � = �

� � � �� 4 3 . � 4 3 0 + � / 0 . 0 4 1 / 1 . / 0 4 2 1 � 1 / 1 - � 0 / 1� 0 � 1 � / � 2 1 / 0 . � 1 0

0 - � 1 �� . 0 � � 1 . / 1 - � � 6 4 � � - � � � � 3 4 0 / 1 4 � � - � � � � �3 1 3 1 3 � 0 1 1 - � 6 � � � 1 - � 0 � � � � � � 6 � � � � � � � � 7 � 8 2 1 � . � 1

� � 1 3 1 3 0 � � / 0 . 0 4 � / 1 . / 0 1 4 � . 3 � �

5 � 6 � � � 9 ) � � � 9 - � : 6 �

� + � - � , + � - �� � � � � � - � � � � � 3 4 3 4 3 � � 3 � 1 / � 1 4 � � - � � - � - � � � � � � , � � - � , � � �

5 � 6 � � , 7 8 � � 9 - � : 6 �, + � , � - �6 ( � � , � � - � , � � 7 � � � � � - � � � � ) � � � - � : 6 �

� + � , �

� � � �

�� �� � � % % � & � \ \

� � � �� � � � � � - � � � � � 3 4 3 4 3 � � 3 � 1 / � 1 4 � � - � � - � - � � � � � � � � � - � � � � �

- � � � + � - � � � � - �

( 9 � , + � @ ) 9 � � 9 - � : 6 �

� � + � � � � � � < � � - � � - � - � � � � � , � � - � , � � = �

� � � �� � � �

5 � 6 ) � � � < � � - � � - � - � � = � �

n � f � f � � � � � � " s � � � � � � d � � � � ( � � " " o � r ! � " � # � � � � [ � � ( � � p " _ �

� ( � " � r ' � � � r� � � � " � � � p ' ( � ' � [ ' # � ' . � � " � r ' � � � r ( � � p " _ � � (�

� _ � ! � � � �� � � � " s � ' p ( � q � � � � � � � � p ' � � � � � � � � r � � [ � � � p � " � � !

' � � � _ ! � � ' � � � � x � � � � � � p ' p ( � � ' � � � � � r ' � � � _ � & ( � " " � � q � � � � � � � _ � �

' � � � � � d � � � � � � � � ' � y � ' � � . p � y ' � � � _ � y � � � � # � � � � _ ' � � � . p � r ) . �%

� � � � & � � ( � � � � ' y � " p o � r � � _ o � � � ) . � � � � � � � ' � � � � � r ' � � � _ �� "

n � f � f m � b � � � � # � r ( � � ' _ e � � � � ( � ' . � " � � � � � . p � � ' � p � � �

6 � � � 6

� � (

a � . y # � ' � . � # � ' . � �� � p ' � � � p & ' � " � � q � � ' p . � � � ) � � r ( � ' . � " � � �%

� � . p � � ' � � & � � � ' � p ' o � ' � � � � . � � � ! �� �

( & " . p _ � � � � � � � � � � + ��b � � . � # � ' � � � " � r ' � � � r ( � � p " _ � |} ~ (� e

� � � � � U � � b � � � " ( � . � � � � � & # � � ( � � � e

� + � - � � + � � � - �� � 8 � � 1 / � 4 � 1 / 0 � . � � / 2 1 � 0 � 1 / 0 � 4 / � 1 4 . � � � � � . � � � - � �

5 � 6 � � � � � 7 8 - � 9 - � : 6 �

� + � � � < � � � = � 6 A @ �� � 7 � 7 � �

6 ( . � � � 7 � � ) � � � - � : 6 �

� + � � �

� � � � � � � - � : 6 � � . � � � 8 � �

� + � � �

� � � �� � � �

b , [ � � � � � � � � � � � � � � & # � � � � ' . # � � � � � + � � � � � � � � � � � � � � � d �%

� � ' p� e� � q " r � � ! � − & � � � p d � � � ' p ( � � � � � � � � " � � � & � � _ " � � � � � _ � � �

� � � [ � � � p � a � � _ � # � ' . � " � r ' � � � r� �

� S O � � S � U � �& � * � � & , � � � � + b � & � b � − & e e � � � � + b � � & e � � � �

Page 18: Программирование: теоремы и задачи

\ �� � � � � $ � � � � � � & � � � � � � � � � � � � % & � � & � � � �

� ) � � r ! � " � # � ' o � ' � � � � � � & # � � � � ' ' � � ( � � p " � # � � $ ( � � ' _ � � � ( �%

� p " � # � � � � � � � ' ' � � � � � � [ � � � � � � � � � & ( � � ( � � a � � � � . p � � � � " . � � � � � ' ' �%

� �� b � � � [ [ � " � � p ' p & # � � _ � _ � � �% � � # � ' . � � � � � � � ' ' � � � & � � " � ( � � ' � �%� � � � p � ' � � � � ) . � � � � � � e

n � f � f Y � b ] ! _ � � � � � � c � � ' � e ] � � � � ' p � � ' ' � � � � � � � � � � �� �

( � �

� � � � � � �� �

� � � � ( � � � � & ( � � p " � # � � � r ( � ' � � � _ � � � ( � ' � � . [ a � � �

� � � � �

6

� � � � � � � �

� � � � �

6

� � � � � � � �

� # � ' . � �� � � � [ � � ' p � p ' � � � p & � ' � � � # � � � ' p . � � ' � � " � � � � � � �

� � � � � U � � � � � " ' � � � . p p ' � [ � � � ' ' � � � _ � _ � � � � � a b ( � p � � � � . p � � _ &! � ( � . � � � � r # � ' . � � � e & � � [ � � � � ( � p � � � � . p � � _ & � _ � � � � � � � � . p _ �

� � � q � � ' � " � � q � � p ' p � & � [ " � � � � � ' q � � p� � � p � � � � . p � � _ ) � � � [ " � �

' � " � � q � � p � � � � � ( � � �

6

6

& � *

6

6

(&

� * �

b " � ( ' _ � x � ' p ( ' � � ( � p � � � � . p � � _ � ( � � & + � � * + � � � e �

� + � � � , + � - �� � 8 � � � , 7 � � � - � 1 / � 4 � 1 / 0 �� 0 � . � 2 4 / 3 3 � 3 2 � 1 3 � � � � � 3 4 - 1 �

5 � 6 � � < � 8 � = � � � < , 7 � � - = � � � < . � � � � , � 7 8 � = � 9 - � : 6 �6 ( . � � � � , � 7 � ) � � � - � : 6 �, + � , � - � � � 1 . � 4 / 0 � � / 1 � 3 1 / � 1 � � 4 0 �� � � 1 1 3 1 � �

� � � � � � � - � : 6 � � . � � � � , � 8 � �

� + � � � - � � 3 4 � 3 1 1 / 0 � � - 3 1 / � 1 � � 4 0 �� � � 1 1 3 1 �

� � � �� � � �� . � � � � , � � � 4 � 4 2 � 1 3 � � � � � 3 4 - 2 � / 0 �

� � � 5 � � + � < � 8 � = � � � < , 7 � � - = �

� S O � � S � U � � Z " � ' p � � q � � d � [ _ � � � � & � * � � q � � � _ � ! � � p ' p � � � ( � �%

" � . s � � � � b � s � ' ( � � � . � � � � ( � � " � ' � � � . p � � ' p # � � � � � . x� e �

�� �� � � % % � & � \ �

n � f � f j � b ) � ' _ � � ' _ � p � . � � ( � � " � ( � ( � � � � � � � � � � � � � � x e � � � � � [ %

� � x o � r � � ' ' � � ( � . � q � � � . p � y a � . y # � ' � . � � �

6

� � �

6 � � � 6� � ( �+ � r � � � � � � � � p d � � a � . � � ( � . � q � � � . p � � � # � ' . � & � � ( � � " ' � � � � � � � � � � " �' � � � � ' _ � . p _ � y ) . � � � � � � � ) � � � � � � ' ' � � � b _ � q " r ) . � � � � � � � ' ' � � �

� � q � � [ � p � ' ( � . p ! � � � � � � [ � . � � � " � � � � � � ! � e � � � ' . � " � r ' � � � r ( � � p "%

_ � (�� � � � � U � � � ' � p � ! � � ' � � � & # � � # � ' . � & ( � � " ' � � � � � � � � � " � ' � �

) . � � � � � � � � � � � � � � � � � * & ! � ( � . � p x � � � � � ! � _ � � � " � � � _ � � � � � � � �� � '%

. � � � * � � � � � � & � � � � � � [ " � � � � � � � � . p � � # � ' . � � & � � ( � � " ' � � � � � �� � � " � ' � � ) . � � � � � � � � � ' ' � � � � � � � � � � � ( � � ' . � q � � � * � �

6

� � � &� � # � ' . � & ( � � " ' � � � � � � � � � " � ' � � ) . � � � � � � � � � � � � � � � * � � & ! � ( � .%� p x � � � � � ! � _ � � � " � � � � � * � � �

, + � � � � + � � �� 4 3 . � 4 3 0 + . 4 / � � 2 � 1 / 0 . 4 3 � 1 . . 4 1 / � 3 3 � � � 1 3 1 3 0 � .

3 / / 4 . � � - � � � � � , � � � 2 � � 3 1 � 0 � 0 � 1 � � - - � � � �

5 � 6 � � < , 7 8 � = � � � < � � , � - � 7 � � � - = � 9 - � : 6 �

� + � � � � � , � - � �, + � , � - �

� � � �� < , � � = 4 � 4 < � � , � - � 8 � � - = � . � / � 4 � / � � . 1 � � 0 . 1 0 � � - �

5 � 6 ) � � � < � � - = �

b � � � � � � � � q � " � � _ � � � ' . � � � � a � _ . � ( � � . � q � � � ( � � � � � ' . � � � �

� � � � � � � � � ( � � " � . � � �� e �

n � f � t l � b � . p ! � � _ � � y ' � ' � � � � � � � . � � [ � e � a � . � # � ' . � � � � � � � '%

' � � � � � � � � � � � ( y � � � � � ' p ( � � � ' � � � � � _ � # � ' � . �� � � ( b _ � q " � � � ! # � ' � .

� ' � � � # � � � ' p ( � � " � � � � � ! e �b � e , ( � � " � . � � p # s � � � ' � p ( � � � ' � � � � � _ � � b ] � b � e & � � b [ e _ � . � # � ' � � �

" � r ' � � � r ( � � p " _ � { � eb [ e + � � ' ( � . p ! p " � � � y � � ' ' � � � � & ! � � � � � � p ( � � � ' � � � � � _ � � � [ � � �%

� x b � ' . � " � � � [ � � ( � � � � � � � � � � + � & � � ( � ' . � " � . q � � [ � p � � � +

+ � e � � _ � ! � � � �� b � e � s � � � ' � p ( � � � ' � � � � � _ � � ( � � " � . p � � ' p _ � . � # � ' � � � � a �%

_ . � �� � � � [ � � . � # � � p ^ q � ( � � r " � � � � a � _ . & � y ) . � � � � � � � � � q � � &� � ( � � � � � & � � � p � p ! � � _� b [ e , [ � � o � � � � ( � � � ! � � " � � ( � a � _ . � � � "

n � f � t n � � � � � � ' ' � � � � �� �

( � # � ' . � � � � � � � ' � � � � � p # � ' . � � � � ' ' � � �

� � _ � � � [ � � ! � � & # � � [ ' . � � � � � � � _ � � � � � r � � � � � a ' � � p . � # � ' . � & � � � p%

d � � � . � � � � � � � & � ' ( � � � � � � � � � � � a $ [ � � . p d � � � . � � � � � � �� � � ' . �

" � r ' � � � r ( � � p " _ � (�

Page 19: Программирование: теоремы и задачи

\ � �� � � � � $ � � � � � � & � � � � � � � � � � � � % & � � & � � � �

� � � � � U � �� + � � � � + � � �� 4 3 . � 4 3 0 + � � - � � � � � � � 7 � - � � � � � - � � � � � � � 8 � - �

5 � 6 � � � 7 8 � � 9 - � : 6 �6 ( � � � � - � 7 � - ) � � � - � : 6 �

� + � � � - �

� � � � � � � 6 ( � � � � 8 � - ) � � � - � : 6 �

� + � � � - �

� � � � � � � - � : 6 � � � � � � - � 8 - � � � � � 7 - �

� � 2 � 3 1 3 1 0 � � � � � - � 4 � � � �

� + � � � - � � + � � � - �

� � � �� � � � �

n � f � t f � � � q � ! � " � # � & � � � � � [ � � ' p & # � � [ ' � � # � . � d . � ) . � � � � � &� � � p d � � � & ! � � � � � � � � � � & � . � d p ! � � � � [ � � . p d � � � �

� � � � � U � � � � ( � � p ( � � � � [ x � ' p � � � � � � � � a � " � ( � � � � r [ " � � " � �

) . � � � � � & � � � p d � � � & � � ( � � � � r " � � � � � � r $ � � � � � � & ! � � � � � � � !%

� � ' � � � _ � _ � � " � � � � � p � r & � ( � ' . � � � � � p � r $ [ � � . p d � � � � b � . � � ' � �%

� � � � � # � � � � � d � � � � � ' ( � . p ! � � � . � [ # � � � � � � � � � a & � � � � p " . � � � � �' � � � � ' � � # � e � _ � # � ' � � � � # � � � " � � � � � � ' ' � � � � � � � � � � � � ) . � � � � � � [ � � s �

) . � � � � � ' ( � � � � � � ' � � " � � r � � � � � a �

� + � � � � + � � � � + � � �� 4 3 . � 4 3 0 + � � - � � � � 7 - � � � � � - � � � � � - � � � � � - � � � � � � � 8 - �

5 � 6 � � � 7 8 � � 9 - � : 6 �6 ( � � � � - � � - ) � � � - � : 6 �

� + � � � - �

� � � � � � � 6 ( � � � � - � 8 - ) � � � - � : 6 �

� � � / 3 1 3 1 0 � � � � � - � 4 � � � �

� + � � � - �

� � � � � � � - � : 6 � � � � � � - � 7 - �

� � � / 3 1 3 1 0 � � � � � - � 4 � � � � - �

� + � � � - � � + � � � - �

� � � �� � � � �

n � f � t t � b � � � � � � � ( � � " " o � r ! � " � # � & � � ! � � � � r � _ � � � � � � r _ ' � �

� � � � � � � � � � � � � � � � � � �� e � � � ' ' � � � " . � � ( ' � � p � # � ' . � � & � � � �� � � � ' � � � � � p � y � ( � � p " _ � � � ! � � ' � � � � p & � ' . � � " � � ' � � � � � � r � � ! � � d s � � � r

� ( � � � a � � r b ( � � � � � # � � � � p e � � " � � ' ' � � � � p � . p � � ' p ( � � � ' � � � � � _ � " � y

) . � � � � � � �� � � ' . � " � r ' � � � r ( � � p " _ � (� �

�� �� � � � � � � & � � � � � � � � � b � � �� �� � � � � � � � � � e \ �

n � f � t k � � � � � � ' ' � � � � �� �

( � # � ' . � �

6

(� � . p _ � q " � � � # � ' � _ � � !

� ' � � p o � y � p " � � # . � � � � b � � _ � y # � ' � _ � � & � # � � � " � � & ( − � � � e � # �%

' . � � p � � � ' � � � , [ o � � # � ' . � " � r ' � � � r " � . q � � [ � p ( � � p " _ � (�

� � � � � U � � � � � � y � " p � � # � ' � _ � _ ' � ' � " � � � & � " � [ � � . p � � � " � �

# . � � & � " � � � r � # � � � � � � �

n � f � t h � � � � � _ � � " � � � � � p � � [ . � a � � � �� �

( � �� �

( � # � ' . � �

6

(� � . p

_ � q " � � � _ � � " � � � � �× � � ) � � r � � [ . � a � � # � ' . � � p ' � � ' � � p o � y � � s �

# � ' � .� , [ o � � # � ' . � " � r ' � � � r ( � � p " _ � ( � �

� � � � � U � � � � � # � . � " . p _ � q " � � � � � � � ! � � � � . p � � � � ( � p � � � � . p � � _ �

� � ! � � � � � � × � � # � ' . p � � ' � � ' � � p o � y � � s � # � ' � .� b � � � ' " � � � �

� � _ � � � ( � p � � � � . p � � _ � ( � � � � � ! � � � � . � � � � � q � � " � [ � � � � p � " � � # �%

' . � � � " � � � # � ' � p� e Z � � � � & � ' ( � . p ! p ) � � ' � � & � # � ' . p � � ' � �

� _ � � " � � � � y� b � � � ' " � � � � _ � � " � � � � ( � � � � � � _ � . � " � [ � � . p � � ' p ( � . � ' _ � &� " � � � p ( � . � ' _ � [ � � . p � � ' p� e �

n � f � t � � � � ' ' � � � � � � � � � � � ( � ' � � � # � x � ' p ( � � " � � � � � ! � ' � a �%

. � # � ' . � � � � " � ( & _ � � � � � " � � � �� + � r � � ( � � ( o � � � � � # � ' . � ! � � � � � p

( � � p " _ � ( � ' _ � � � # � � r " � ( � . � � � � . p � � r ( � � p � p x�

� _ � ! � � � �� � . � q � � p � ' � # � ' . � � � � ' ' � � �� "

�� �� � } � � u � ~ � } � s � � } u � ~ ~ � � v �� �� � � � } ~ � s } u v

� ' � p ! $ � � _ � � � � � � � � � q � ' � � �� w � _ a � p � & � � � � � � � � � � _ � � � � � r

p � . p x � ' p ( � ' . � " � � � � � . p � � ' � � ) . � � � � � � � � � � q � ' � � � ! & � ! � � # � � � p � � $

) . � � � � � � � _ � � � � � � � � � � q � ' � � � � & � � ! � � � � ' p � � � � � � & � � " & � ' . � � s! � � # � � � � � � ( � ' . � " � � � � � . p � � ' � � � � � � � � � � ( � � q � � � � ' ' � � � � � � � p ( �

� s ! � � # � � � x � � ( � ' . � " � � � � � . p � � ' � � � � � � � � � � ( � � � ( � � � ( & � � � ' � p

� ' . � ' o � ' � � � � ` � _ a � p # � � × !→ � & " . p _ � � � � � r

� b〈 � � � � � � � � � � ( 〉 e + # b � b〈 � � � � � � � � � � ( � � 〉

e� � � ( e �

+ � ( � � � � � & ` � _ a � p # $ � b ' � � � � ' � y # . � � � � ( � ' . � " � � � � � . p � � ' � � e � �%

" _ � � � � � & ( � ' _ � . p _ � # � � � " � � r # . � � ( � ' . � " � � � � � . p � � ' � � ( � � [ � � . p � � ' p

_ � s ' � � � �# $ � b〈 � � � � � � � � � � ( 〉 e + # $ � b〈 � � � � � � � � � � ( � � 〉

e � � � ( �

� � � � r ( � � � � � � � " _ � � � � � r ` � _ a � � $ " . � � � ( � ' . � " � � � � � . p � � ' � ��� ) � � � ' . # � � # b ( � � e + ( � � �

Page 20: Программирование: теоремы и задачи

\ � �� � � � � $ � � � � � � & � � � � � � � � � � � � % & � � & � � � �

+ � ( � � � � � & ' � � " � � � � � � � � � � # � ' _ � � � � p � . p � � ' p � � " _ � � � � � r ` � _%

a � � r � � ' . � � ! � � � � ' � � " � � � � � � � � � � # � ' _ � � � � _ � � � � � r ( � ' . � " � � � � � . p%

� � ' � � & � � � � ! � � � � � s " . � � & � � � � � � q � � ( � � " ' _ � ! � � p & _ � _ � � ' � �%

� � � ' � � " � � � � � � � � � � # � ' _ � � ( � ' . � " � ( � ' � � � � p � � _ � � � � � � � b � ! � � ' � � �%

� � � � � e # � ' . ��

� y � � � � . � � � � � � � � # � ' . � � � p � � " _ � � � � � r ` � _ a � � �

, + � � � ( + � ( � �� 4 3 . � 4 3 0 + ( � � 3 . 1 3 4 1 � � 3 - � 4 4 3 7 . � - � � � � � � . � , � 8 �

5 � 6 � � , 7 8 � � 9 - � : 6 �, + � , � - �

( + � � < ( � . � , � = �

� � � �

Z " � ' p � � $ ! � � # � � � � ` � _ a � � � � ( ' � � r ( � ' . � " � � � � � . p � � ' � � b ( �%

' . � " � � � � � . p � � ' � � " . � � � e � � ' . � ` � _ a � p � � ( � � " � . � � � � � . p _ � � � � �%

( ' � y ( � ' . � " � � � � � . p � � ' � p y & � � ( � � � � p ' � � � _ � ! � � � � p � � ' p � �

, + � - � ( + � ( < 7 . � - � 8 = �

� ' . � ` � _ a � p � � � p � . p � � ' p � � " _ � � � � � r & ( � . � ! � � � ' _ � � p � s � � � � �

� � & � � � � � % � � � � � � � $ � � _ x � � " _ � � � � x ` � _ a � x & ! � � # � � � p _ � � �%

� � r � ( � � " � . p x � ! � � # � � � p � b ) � � ! � � # � � & # � � ' o � ' � � � � � � _ � p ` � _%

a � p � & # � �

� b〈 � � � � � � � � ( 〉 e + � b b〈 � � � � � � � � ( 〉 e e

( � � � ' � y 〈 � � � � � � � � ( 〉 e � ) � q � � " � _ � ! � � p & # � � ' � � " � � ' � y � � " _ � � �%

� y � � ' d � � � � � r ' o � ' � � � � � � � � � � . p � � � � � ' d � � � � � � # b � � � � � � . p%

� � ' � p � ! � � # � � � & # � � " . p . x [ � � � � � " _ � � � � � � � � � ' d � � � � � p ! � � # � � � p #

� ( � � " � . p x � ' p ! � � # � � � p � � e �

n � t � n � � _ � ! � � p � � " _ � � � � � � � ' d � � � � � p " . p ' . � " x o � y ` � _ a � r �

b � e ' � � " � � � � � � � � � � # � ' _ � � ( � ' . � " � � � � � . p � � ' � � � � o � ' � � � � � y # �%

' � . ?b [ e # � ' . � ) . � � � � � � � ( � ' . � " � � � � � . p � � ' � � a � . y # � ' � . & � � � � y � s � � _%

' � � � . p � � � ) . � � � � � ?

b � e � � � � � r ( � � � . � # � � � ) . � � � � � ( � ' . � " � � � � � . p � � ' � � a � . y # � ' � .

b � � � & _ � � � � r [ " � � � � � � � & � ' . � ( � � � ' � � � � � p # . � � � � � [ � � x o � �

( � � p " _ � e ?b � e � � _ ' � � � . p � � � # � ' . � � " o � y ( � " � p " � " � � � _ � � y ) . � � � � � � � ?

b " e � � _ ' � � � . p � � p " . � � � � � � � � � � � � � � b � � [ � � x o � � � � . � � � � � ! � �%

' � � x o � � � e # � ' � _ � � ! � " o � y ( � " � p " ) . � � � � � � � � ( � ' . � " � � � � � . p � � ' � �

a � . y # � ' � . ?

�� �� � � � � � � & � � � � � � � � � b � � �� �� � � � � � � � � � e \ �

b � e # � ' . � � � ( ( � ! � " � � � a & � � ! " � . s � � y � . p � � b � ( � ' . � " � � � � � . p � �%

' � � � . � r � � " � � � a e �

� � � � � U � �b � e 〈 ' � � � � ' � y # . � � � � ( � ' . � " � � � � � . p � � ' � � ? " . � � �〉 ?b [ e 〈 # � ' . � ) . � � � � � � � & � � � � y � � _ ' � � � . p � � � ? ! � � # � � � � � � _ ' � � � . p%

� � � �〉 ?b � e 〈 � � � [ � . p d � r ) . � � � � � ( � ' . � " � � � � � . p � � ' � � ? � � � � � r ( � � � . � # � � �

) . � � � � �〉 ?b � e 〈 � � _ ' � � � . p � � � # � ' . � � " o � y ( � " � p " � " � � � _ � � y ) . � � � � � � � ? # �%

' . � � " o � y ( � " � p " � " � � � _ � � y ) . � � � � � � � � _ � � a � ( � ' . � " � � � � � . p � � ' � � ?

( � ' . � " � � r ) . � � � � � ( � ' . � " � � � � � . p � � ' � �〉 ?b " e 〈 � � _ ' � � � . p � � p " . � � � � � � � � � � � � � � # � ' � _ � ? � � _ ' � � � . p � � p " . �%

� � � � [ � � x o � � � # � ' � _ � � _ � � a � ( � ' . � " � � � � � . p � � ' � � ? � � _ ' � � � . p � � p

" . � � � � � � � ! � � ' � � x o � � � # � ' � _ � � _ � � a � ( � ' . � " � � � � � . p � � ' � � ? ( � ' . � "%

� � r # . � � ( � ' . � " � � � � � . p � � ' � �〉 ?b � e 〈 # � ' . � � � ( ( � ! � " � � � a & ( � ' . � " � � r # . � �〉� �

n � t � f � b � � � [ o � . � � � � � � � ' � � � p � � e � � � " � � ( � ' . � " � � � � � . p � � ' � �

a � . y # � ' � . � � � � � � � � ( � � � � � � � � � * � � p ' � � � p & p � . p � � ' p . � � � � � � p

( � ' . � " � � � � � . p � � ' � p ( � " ( � ' . � " � � � � � . p � � ' � p x ( � � � � r & � � � ' � p � � q � � . �

� ! ( � � � � r � # � � _ � � p � � _ � � � � � # . � � � � _ & # � � [ � ' � � . � ' p � � � � � p�

� � ' . � " � r ' � � � r ( � � p " _ � ( � *�

� � � � � U � � � � � � � � � � � " � � ' � � " � � p ! � " � # _ ! � " � # � � � � p d � � � � � !%

� � � ��� - + � � �, - + � , �� 4 3 . � 4 3 0 + 4 / - � 3 � 4 � 0 . 1 0 7 � 8 . � � 3 � � 3 � / 0 � 4 � . � - � � � . � � - �

2 � � � . 4 0 � � � - � � � � � , - � �

5 � 6 � � < � - 8 � = � � � < , - 8 � = � 9 - � : 6 �6 ( . � � - � � � � , - � ) � � � - � : 6 �

� - + � � - � - �, - + � , - � - �

� � � � � � � - � : 6 �

� - + � � - � - �

� � � �� � � �� � - � � 4 � 4 , - � � � 1 / � 4 , - � � � 0 � � 0 . 1 0 � � 1 / � 4 , - 7 8 �< 4 � - � � = � 0 � � 0 . 1 0 � 3 1 0 �

� � � 5 � � + � < , - � � = �

) � ' ( � . p ! � � � . � � � & # � � � ' . � � � ( � + � � * � � � � � � � � � � * � $ ( � "%

Page 21: Программирование: теоремы и задачи

- �� � � � � $ � � � � � � & � � � � � � � � � � � � % & � � & � � � �

( � ' . � " � � � � � . p � � ' � p � � � � � � � � ( � & � � � � � � � � � � * � � � $ ( � " ( � ' . � " � � �%

� � . p � � ' � p � � � � � � � � ( � � � �� � � � � � � � w � _ a � p 〈 � � �

� � � � � ( � 〉 7→ � � _ ' � � � . p � � � * � & " . p _ � � �%

� � � � � � � � � � � � * � � ' � p ( � " ( � ' . � " � � � � � . p � � ' � p � � � � � � � � ( � " � � " _%

� � � � �� �

n � t � t � � � � " � � ( � ' . � " � � � � � . p � � ' � � � � � � � � � � ( � � � � � � � � � *

a � . y # � ' � .� + � r � � � � _ ' � � � . p � x " . � � ( � ' . � " � � � � � . p � � ' � � & p � . p%

x o � r ' p ( � " ( � ' . � " � � � � � . p � � ' � p x � [ � � y ( � ' . � " � � � � � . p � � ' � � r � � � . � # �%

' � � � � ( � � � a � r ( � � p " _ � ( · *�

n � d � � � � b ' � � [ o � � � ) � + � � � r � a � � r � � � & q � ) � � � � � � � � � � � � e � , [ �%

! � � # � � # � � � ! � b � � � e � � _ ' � � � . p � x " . � � � [ o � r ( � " ( � ' . � " � � � � � . p � �%

' � � ( � ' . � " � � � � � . p � � ' � � r � � � � � � � � � � � � � � � � � � � � � � � " �

� � � 6 + � � � ⇒ � b � � � e + � � � b � b � � �− � e�� b �− �

� � e e ?

� � � + � � � ⇒ � b � � � e + � � � b � b � � �− � e�� b �− �

� � e � � b �− �� �− � e � � e ?

b � � ' _ � . p _ � b � − �� � − � e � �

>

� b � � � − � e�� b � − �

� � e & � � � � � � � � ' . %

# � � � � _ ' � � � � � s y # � ' � . � � q � � ! � � � � � � p � � � � � � p � � ! � � y� e � � ) � � �

� � q � � ! � ( � . � p � p � � [ . � a ! � � # � � � r ` � _ a � � � & � � � x o x � � ! � � � ( · *�) � q � � � [ � r � � ' p � ( � � p � p x ( � � p " _ � * b � . � ( e & � ' . � � � " _ � � � � � b ( � � e

� # � ' . p � p 〈 � * � +� ,

� � � � �� * � +* , 〉 b _ � _ ` � _ a � p � � � ) � � � � � [ � � � � " _%

� � � � � e � �

n � t � k � b � ! _ � � � � � � c � � ' � e � � � � ( � ' . � " � � � � � . p � � ' � p a � . y # � ' � .

� � � � � � � � � � ( � + � r � � � � _ ' � � � . p � x " . � � � s � � ! � � ' � � x o � r ( � " ( � ' . �%

" � � � � � . p � � ' � � b # � ' . � " � r ' � � � r ( � � p " _ � ( |} ~ ( e �

� � � � � U � � ] ' _ � � � p ` � _ a � p � � � � " _ � � � � � & � � � � � � � ' . � " x o � �

� � " _ � � � � � � � � ' d � � � � � � � � � � � � � y � " p � ( � � � � � � � _ ' � � � . p � � r " . � �

� � ! � � ' � � x o � r ( � " ( � ' . � " � � � � � . p � � ' � � b � [ � ! � � # � � � s * e � � _ q � � # � ' . �

$ � � � � � � � $ � * & � " � $ � � $ � � � � � � . p � r � ! ( � ' . � " � � y # . � � � � � � ! � � ' � �%

x o � y ( � " ( � ' . � " � � � � � . p � � ' � � r " . � � � � , # � � � " � � & $ � �

6 � � � 6

$ � * �� � � " � [ � � . � � � � � � � � � � # . � � � � � ! � � # � � � p $ � * _ � � � � _ � � � x � ' p�

� - + � - � , + � - � � � - � + � . � - � �� 4 3 . � 4 3 0 + , 4 � / � � 0 . 1 0 / 0 . � � 0 3 3 � 3 � . � 0 1 � 2 4 / 3 4 � �

5 � 6 � � � - 7 8 � � 9 - � : 6 �

� - + � � - � - �

� � �� 6 � 3 4 / � � � 0 1 1 4 � 0 1 � . 4 / 1 � � 0 � 1 � - - � � ,� � 1

- � 0 � � � � � � 6 � 7 . � � - � � 1 / � 4 0 - 4 � 3 1 0� 0 � 6 � � �

6 ( 6 � , ) � � � - � : 6 �

�� �� � � � � � � & � � � � � � � � � b � � �� �� � � � � � � � � � e �

, + � , � - �

� � , � - � + � . � � - � �

� � � � � � � - � : 6 � � 6 7 ,� � � 6 � 7 . � � - � 7 � � � 6 � - � �

� � 6 � - � + � . � � - � �

� � � �� � � �

w � � � � � � �� � �

� ' ( � . p ! � � � " � x " � � � # � � � � ( � � ' _ � ? � � � � � � � � � � � ' . � �%

� � ( � . � � � � � $ � � � � � � � � � � ' [ � ' _ � � � # � � ' � � & � $ � * � � $ ( . x ' [ � '%

_ � � � # � � ' � �� + � d � a � . p � $ � � � � � ( �

6

$ � � � � �

6 + � � � + � , � - �� � � 6 � 7 . � � - � 7 � � � � � 8 6 �

5 � 6 � � < � 6 = 7 8 - � 9 - � : 6 �

� + � 6 � < � 6 = � 6 A @ � � 6 7 � 7 �

6 ( . � � - � 7 � � � � � ) � � � - � : 6 �

+ � � �

� � � � � � � - � : 6 � � � � � � 7 . � � - � �6 + � � �

� � � �� � � �� � � 6 � 7 . � � - � 7 � � � � � � 6 � - �

� S O � � S � U � � � . � � ( � � ' � � � b � � � � � � � � � � . p � � � e � � " _ � � � � � � � � '%

d � � � � � � ( � . # � � ' p & � ' . � " . p _ � q " � � � � y � � � � � p � � _ ' � � � . p � x " . � �

� � ! � � ' � � x o � r ( � " ( � ' . � " � � � � � . p � � ' � � & � _ � � # � � � x o � r ' p � � � � � � ! � �

� � ' d � � � � � � ( � � � � " � � _ � . � � � � � � ' # � ' . � � " � r ' � � � r ( � � p " _ � ( � � � ' � p

� " � � � r � ! p o � r � . � � � � � � ' _ � � " � � � � # � � � � � � � � � � � � [ � � b ' �%

� [ o � . ) � � � � p x � � � e � � � r � � � � _ ' � � � . p � x � [ o x ( � " ( � ' . � " � � � � � . p%

� � ' � p � ' y � " � � r ( � ' . � " � � � � � . p � � ' � � � � � ' � � � � � � � � � � � r ( � ' . � " � � � � � . p%

� � ' � � ' ( � � � o p x ( � � " " o � r ! � " � # ��

n � t � h � � � _ � � � ! � � � � � � p � q � � � � � ' � � � � � d � � � � ( � � " " o � r ! �%

" � # � & � ' . � � � " � � ' _ � � p � � _ ' � � � . p � x � � � � � � � � � � ( � ' . � " � � � � � . p%

� � ' � p �

Page 22: Программирование: теоремы и задачи

��

� � � � � � � � � �

� � � � � � � � � � � �

� � � � � � �

Z " � ' p ' � [ � � � ! � " � # � & � _ � � � � y � � � [ � � ' p ( � . # � � p � " � � ! � " � %

� � � � ' � ) . � � � � � � � _ � � � � � � � � � � q � ' � � ��

�� �� � z y { s s } ~ � t � v � � v � s } ~ � { ~

f � n � n � + � ( � # � � � � p � ' � ( � ' . � " � � � � � . p � � ' � � " . � � * � ! # � ' � . �� �

(�

� � � � � U � � " � � ( � # � � � � p � y � . � _ ' � _ � � � � � # � ' _ � � ( � � p " _ � b ( � ' . �%

" � � � � � . p � � ' � p � ( � � " d � ' � � � � ( � ' . � " � � � � � . p � � ' � � � & � ' . � " . p � � _ � � �%

� � � � # � y � � # � . p � � � � � � ! _ � " . � � # � � � � & � * # � � , % r # . � � ( � ' . � " �%

� � � � . p � � ' � � � � � � p d � e � � � � � � r [ " � � ( � ' . � " � � � � � . p � � ' � p � �+

�+ � � � +

� &( � ' . � " � � r $ ( � ' . � " � � � � � . p � � ' � p � ( + ( + � � � + ( � " � � y � � � � � p ( � ' . � " %

� x x � � ( � # � � � � � x ( � ' . � " � � � � � . p � � ' � p � � � ' ' � � � � � � � �

� � * �

� � � . � - � � � � . � , � 2 � � � � 4 0 � � . 3 � 3 4 -

� � � 3 2 1 . 0 0 � .

� � � � � � ) � - � � � � � � � ) � , � 2 � � � � 4 0 � � . 3 � 3 �� 3 2 1 . 0 3 � . / 1 � . . - � � . 4 0 1 � � 3 � �

5 � 6 � � . 7 8 � � � ) � 9 - � : 6 �

� � � . + � / � 1 � � 2 1 � . 2 � / � 1 � . 0 1 � � 3 � / 0 �

� � � 3 2 1 . 0 0 � .

� � � �

, ( � d � � & _ � _ � � q � � ( � � � r � � � � � _ ' . � " x o � r ( � ' . � " � � � � � . p � � ' � � �� � � . � ' � � � ( � � " � . � � � x & ' . � " x o � r ( � ' . � " � � � � � . p � � ' � � ( � � � � # # . �%

� � � " � . q � [ � p � � _ � � � q � & � * # � � , % r $ [ � . p d �� ! � � � � ! � � q � � & � '%. � � � # � � � � � p d � (� � � � " � � � _ � y # � q � � � [ � � � p � � � [ � . p d � � b � � � # �

( � . # � � � � p ( � ' . � " � � � � � . p � � ' � p � � [ " � � � � ( � ' � � " ' � � � � � � ' . � " x o � r e �

�� �� � � � � % � � � � & � � \

� � � � � � � ' � � x o � � � � # � � � q � � � � . � # � � p � � � � ] � � _ & � � " � & " � � � � p ' p

' _ � � a � ( � ' . � " � � � � � . p � � ' � � & � � r � � ' � � r ( � � � r # . � � & � � � p d � r ( b � �

� � r " s � ' p & � � _� ( � ( � � " ( � . � q � � � x � � & � # � e & � � . � # � � p � � � � � � & � � " %

o � � ! � � � � # . � � ( � . � q � � p � � � � � � � �

* + � , �5 � 6 � � � 9 ) < . � * � 7 � = � 9 - � : 6 �

* + � * � - �

� � � �� . � * � 7 �� . � * � - � � � � � � . � , � � � �

. � * � + � . � * � � - �

( 9 � 6 + � * � - ) 9 , � 9 - � : 6 �

. � 6 � + � - �

� � � � �

� S O � � S � U � � � ' . � # . � � � � � ( � ' . � " � � � � � . p � � ' � � ' # � � � � p # � ' . � � � � � �

" � ( & � � � � " � ( � � & � � ( � � � y � " _ ' . � " x o � � ' � � � � � � ' � � � � ( � � [ � � . �%

� � x � " � � � a � (% � # � � r ' � ' � � � � ' # � ' . � � � p�

f � n � f � � ( � � " . � q � � � � � � . � � � � � � � � ' ( � . p ! � � ' p ' � � � � � � � � " � y � � '%

' � � � � b � � & � # � e � � ' � � � � � � p � � � & " � [ � � � � [ . � � ' _ x ( � � � � � � � x &

� � _ . x # � � � � � � � � � � � � ' � � � � � d � � � �

& ⇔ ( � ' . � " � � � � � . p � � ' � p � $ ( � ' . � " � p p� �

f � n � t � + � ( � # � � � � p � ' � ( � " � � � q � ' � � � � � � q � ' � � � { �� � �

*}�

� � � � � U � � � � " � � � q � ' � � � � � y � " p � ' p � � � ! � � � � � � " � � ! � � # � � � ' � � �%

� � � ' � � � � ' ( � ' . � " � � � � � . p � � ' � p � � � . � r � � " � � � a " . � � *� �

f � n � k � + � ( � # � � � � p � ' � ( � ' . � " � � � � � . p � � ' � � ( � . � q � � � . p � y a � . y

# � ' � . " . � � * & ^ _ � � � � y � % r # . � � � � ( � � � � ' y � " � � � � �

�� �� s � s t � z } v � u ~

f � f � n � + � ( � # � � � � p � ' � ( � � � ' � � � � � _ � # � ' � . �� �

( b � � � ' � p ( � ' . � " � � �%

� � . p � � ' � � " . � � ( & � _ � � � � � _ � q " � � � ! ) � � y # � ' � . � y � " � � ( � � " � � �

� � ! e �� � � � � U � � � � � � ' � � � � � _ � [ " � � y � � � � � p � � � ' ' � � � � � � � �

� � ( � ( �%

# � � � � p � . � _ ' � _ � � � � � # � ' _ � � ( � � p " _ �� b � � � � � r ( � � ) � � � [ " � � ( � � �%

' � � � � � _ � 〈 � �� � � (〉 & ( � ' . � " � � r $ 〈 ( � � �

� �〉� � . p ' � ' � � � . � � � p � . � � � � � � �

( � � � y � " � _ ' . � " x o � r ( � � � ' � � � � � _ � ! � " � " � � ' p � � ( � � ' � � � � _ � _ � � ' . %

# � � *% r # . � � ( � � � ' � � � � � _ � � � q � � � � . � # � � p & � � � � � p p ( � � " " o � y �

Page 23: Программирование: теоремы и задачи

�� � � � � � � � � � � � � $ " � � � � � � � � � � " � � � � � &

, � � � � � � ' . � � � � � � p d � _ � _ � � �% . � [ � � ! ' . � " x o � y # . � � � � b � � �� # . � � � �

' � � � � � � � � [ � . p d � * e � ) " � . q � � � r � � � � � [ � . p d � � * & ( � � _ � � � � � �

) � � � � _ & � � �� � � _ � � * & # � �� � * � � � * � � �� � �

� � � (

� � ' . � ) � � � � ! � � # � � � � � � * � q � � � � . � # � � p � � � � � � . p � � � � ! � � q%

� � ' ( � ' � [ � � & � � �� � � r � � ' � � " � � � * � � � �

� � ( � � � � � � p d � � # � ' . � & [ � . p%

d � � � � �� � � � � � p � � � * ' � � � & � ' � � s � ' p � � ' ( � . � q � � p # � ' . � ' � � � � � � � �

* � �� �

( � � _ & # � � [ ( � � � ' � � � � � _ � [ . � � � � � � � p d � r & � � �� � � � ! � � ' � � x%

o � � ( � � p " _ �� ! � � � [ . � � # � � � ' p � � � & # � � � � � ^ q � � � ' ( � . � q � � � [ � �%

x o � � ( � � p " _ ��q . � � � � � � ( � � � y � " � _ ' . � " x o � r ( � � � ' � � � � � _ � �

� 7 . � - � � � � . � � � 8 7 8 7 � � � � @ � - 8 �

, + � � � - �� 2 � / � 1 � . 0 1 � � 3 � / 0 � / 2 � . � 0 , � / � . � 2 1 + . � , � - � 8 � � � 8 . � � � �5 � 6 � � . � , � 8 . � , � - � � 9 - � : 6 �, + � , � - �

� � � �� . � , � 7 . � , � - � 8 � � � 8 . � � � �

) + � , � - �� ) 7 � �� . / 1 . � 1 3 � � 0 � 1 � - . � , � - � 8 � � � 8 . � ) � / � � � 0 1 . � , � �

5 � 6 � � < ) 7 � = � � � < . � ) � - � 8 . � , � = � 9 - � : 6 �

) + � ) � - �

� � � �� . � , � - � 8 � � � 8 . � ) � 8 . � , � 8 . � ) � - � 8 � � � 8 . � � � �

� � � � / 3 1 3 1 0 � . � , � 4 . � ) �� . � , � - � 8 � � � 8 . � � � �

� � � 2 1 � 1 / 0 . 4 0 � � . / 0 � - . � , � - � � � � . � � � . � / � 0 3 � 3 2 � � 1 - 1 �

� S O � � S � U � � � � � � � � � � � � � � � � ! � � _ � � r " � � _ � � � ' . � � � ( & � � � � � � �

� � � ( � � " � . � � ��

�� �� v � { } v � s t � � z

f � t � n � � . p ! � " � � � y ( � * b *

6

( e ( � � � # � ' . � � p � ' � *% ) . � � � � � � � ( � " %

� � � q � ' � � � � � � q � ' � � � { �� �

(}�

� � � � � U � � " � � ( � � " ' � � � . p � p _ � q " � � ( � " � � � q � ' � � � ( � ' . � " � � � � � . p%

� � ' � p x � � � � �

� � ( � . � r � � " � � � a " . � � ( & � _ � � � � � r � � � � � * � " �%

� � a� b � � � � r ' ( � ' � [ ( � � " ' � � � . � � � p � � ! [ � � s � ( � ! q �� e � � _ � � ( � ' . � " � � �%

� � . p � � ' � � ( � � p " � # � � . � _ ' � _ � � � � � # � ' _ � b ' � � � d � e � , # � � � " � r ' ( �%

' � [ � � d � � � p ! � " � # � $ ( � � � [ � � � � p � ' � ( � ' . � " � � � � � . p � � ' � � _ � _ � � � p d � &

�� �� � � � $ � � � � % � & � �

� ! � � � � � � [ � � � � p ' � � " � � � y � � & ^ _ � � � � y * � " � � � a $ � � � [ � � ' � � &' # � � � p � � � � � ) _ � � � � � # � � b # � ' . � ( � ' . � " � � � � � . p � � ' � � r ' * � " � � � a � � �

� � q � � [ � p � � � � � � � � p d � # � ' . � � ' � y ( � ' . � " � � � � � . p � � ' � � r e � " � � � '%

_ � � p � � _ � r � . � � � � � � & # � � [ ( � . # � � � � � # � � � " � � r ( � ' . � " � � � � � . p � � ' � �

� � � [ � � � . � � � [ � . � � �· ( " � r ' � � � r�� _ � _ � � ' . # � � # % r # . � � ( � ' . � " � � � � � . p � � ' � � � � q � � � � . � # � � p & � �� � � p p ( � � " " o � � � � ' . � � � # � � � p � � ' p ' � � � � & � � " . p ' � y � � � � � � p

� [ o � � � # � ' . � � " � � � a � q � � ' ( � � � � � � �� # ! � � � � � � p � � � �� � . p ) � �%

� � � � " � & # � � [ ' ( � � � � � � � � # � " � � � a [ . �� � ' . � � y � � � � ( � � � r � �

_ � � � � � � � � � � � � ' . � " x o � � & � � � � # " � . q � � [ � p � � � � � � � � �

� . s � & ! � _ � � � � � ' � � p � � " � � � a � � � � _ � � � " � � p & # � � �� # � � � � b � � � # �

�� # � � ( � � � r e � � � _ � � � [ � � ! � � � � " � � ' _ � � p � � � [ � . p d � � # & " . p _ � � �%

� � � � �� # � � & � � # � � � � �

� �↑#

�� �

� �� �

Z � �� # � � � � � � � " � � � o s � � ' _ � . p _ � � " � � � a & � ( � ' . � � � y � � ' _ � . p _ � � %. � r� Z � � � � � � �

� # � � � & � � " � � [ � � � p � " o � � ! � � � � # . � � � � _ & # � � [ ( � ' . � " � � � � � . p � � ' � p [ . � [ � � � � � � . p � � ' � � # _ � ! � � � � p � � d � � � ( � � p "%

_ � & � � �� # � � [ ' � � # � . � d . � � . � & � ( � � � � � " � � � a � � � � # � � ( � . # � � � ' p �

( � � � � p ( � ' . � " � � � � � . p � � ' � p � �� �

� �� �

� b ( � * � . � r & * � " � � � a e ?

( � ' . � " � p p ( � ' . � " � � � � � . p � � ' � p � �� �

� �� �

� b * � " � � � a & ( � * � . � r e ?

� . � � � � � � ( � � � y � " � _ ' . � " x o � r ! � �� � � �

� � ( ( � ' . � " � � � � � . p � � ' � �

b ( � � " ( � . � � � � � & # � � � � � � ' � p e �

� + � � � - �

5 � 6 � � � 9 ) < < . � � � � � = � � � < . � � � - � � - = = � 9 - � : 6 �

� + � � � - �

� � � �� � � . � 1 3� 2 � � 1 � 2 4 4 4 � 3 1 3 1 3 4 � / � 3 - �

� � � + � � �

( 9 � , + � � ) 9 � � 9 - � : 6 �

� � � + � � � � � . � , � �

� � � �� � � � � . 4 / � � 1 4 3 4 � 3 � . / 0 - 1 . � � � � � � . � � � � . 4 / � � 3 � � 1 4

� . 3 � < � 4 3 � . 4 / � � 1 4 3 4 � = � 0 � 1 � < � � � � - = � � � � �

. � � � + � - �

( 9 � , + � � � - ) 9 � � � � � � - � 9 - � : 6 �

. � , � + � � �

� � � �

Page 24: Программирование: теоремы и задачи

� �� � � � � � � � � � � � � $ " � � � � � � � � � � " � � � � � &

� � / 0 � � / � 2 � 3 1 / 0 4 0 � � � � � - 1 4 3 4 � . - � 3 � 1 �

( 9 � , + � � � � � � � @ ) 9 � � 9 - � : 6 �

. � , � + � - �

� � � � �

� � � � r ' ( � ' � [ ( � � " ' � � � . � � � p ( � " � � � q � ' � � $ ) � � ( � � � # � ' . � � � � � y

) . � � � � � � �� � � � [ _ � q " � � ( � " � � � q � ' � � � � � � . � � � � � � � " � � ( � � " ' � � � . �%

� � � & " � � � � � � � � ' p ( � � � # � ' . p � p ) . � � � � � � � � ! � � ' � � x o � � ( � � p " _ �� � � �%

y � " � � _ � � _ � r ! � " � # ��

f � t � f � � � � � # � ' . � � p � ' � � � ! � � ' � � x o � � ( � ' . � " � � � � � . p � � ' � � " . � � *

� ! # � ' � . �� �

( � . � _ ' � _ � � � � � # � ' _ � � ( � � p " _ �� b � � � � � � � ( � � ( � � & * � �

( � . # � � � � � � � � � � � � � � � � � � � � � � � �� e

� � � � � U � � ) � � � � � . p � � r [ " � � ( � ' . � " � � � � � . p � � ' � p 〈 � �� � �

*〉 ? � � _ ' �%

� � . p � � r $ 〈 * ( � * � � ,� � �

* ( � � , (〉� � _ � _ � � ' . # � � # % r # . � � ( � ' . � " � � �%

� � . p � � ' � � � � q � � � � . � # � � p � , � � � � � � ' . � � � � � � p d � ( � * � # � � � ' . � � �%

. � # � � � p # % � � ) . � � � � � � � ' � ' . � " x o � � " � . q � � � ! � � ' � � � p ' d � � � � � �� � . # � � � � � _ � r � . � � � � � � ( � � � y � " � _ ' . � " x o � � �

� + � � �5 � 6 � � � 9 ) < . � � � 7 � � , � � = � 9 - � : 6 �

� + � � � - �

� � � �� � � 3 � 3 1 � � � 1 3 1 3 0 � 2 � � 1 � 2 1 � � . 1 � 4 . 1 3 4 � � �

. � � � + � . � � � � - �

( 9 � 6 + � � � - ) 9 � � 9 - � : 6 �

. � 6 � + � . � 6 � - � � - �

� � � � �

f � t � t � � ' � p � � � d � . � ( � � " ' � � � . p � p *% ) . � � � � � � � ( � " � � � q � ' � � �

� � � q � ' � � � { �� �

(} [ � � x o � � � ( � ' . � " � � � � � . p � � ' � p � � " . � � * & ( �%

� p " � # � � � � � ( �% ( � � q � � � . � _ ' � _ � � � � � # � ' _ � � b � � � � � � � � � � � � � � �

� � � � � � � � � � � �� e � � _ � � . p " � � � � � " � � . � � � � � � ( � � � y � " � _ ' . � " %

x o � r �u k R � k � ] o � � � � � [ � . p d � � # & " . p _ � � � � � � � �� # � � � � � � � # � b � ' . �

� � _ � � � # � � � & ( � . � � � � � # � � � e � � � . � # � � � � # � � � � � & _ . � " s � � ' � � . p � �

� � � � � � . p � � � � ! � � q � � � b � � � � * � � � � " . p � # e � �

f � t � k � n � d � � p " � � ( � � " " o � � ! � " � # � & ! � � � � � � . � _ ' � _ � � � � � # �%

' _ � r ( � � p " � _ � � � [ � � � � r b � � � p d � � " � � � & _ � � � � � [ � . p d � � . � _%

' � _ � � � � � # � ' _ � � ( � � p " _ � e � �

f � t � h � � � � � # � ' . � � p � ' � � . � q � � � p b ` � _ a � � & ( � � � � � " p o � � � � ! � �

) . � � � � � � � � ! � � e � � � q � ' � � � { �� �

*} � { �� �

(} b ( � � " ( � . � � � � � ' p & # � �

�� �� � � # " � � � � � �

*

6

( e � � � � � q " � � � � � # � � � " � � � � ) . � � � � � � " � . q � � � � � [ � � � � p � � [ � . � �

� · * " � r ' � � � r�

� _ � ! � � � �� ! � � ! � " � # � � � q � � [ � p ' � � " � � � _ ( � � � # � ' . � � � x ( � " � � �%

q � ' � � � ( � � � ' � � � � � � _ ) . � � � � � � � _ � q " � � � ( � " � � � q � ' � � �� "

�� �� � z y � ~ s } ~ �

f � k � n � � � � � # � ' . � � p � ' � � � ! [ � � � � p a � . � � � ( � . � q � � � . p � � � � # � ' . � (

� � a � . � ( � . � q � � � . p � � ' . � � � � � � b � � ! [ � � � � p & � � . � # � x o � � ' p . � d p

( � � p " _ � � ' . � � � � � y & ' # � � � x � ' p ! � � " � � e � b � � � � � � � ( � � & � � ! [ � � � � p

� � � � � � � & � � � � � & � � � & � � � & �� e

� � � � � U � � � � � � � � � � � ' p & # � � b � e � � � ! [ � � � � p y ' . � � � � � � � " � � � �%

� � ! � � ' � � x o � � ( � � p " _ � & b e ' � � � � � ! [ � � � � p � ( � � � # � ' . p � � � . � _ ' � _ �%

� � � � # � ' _ � � ( � � p " _ �� n � ! [ � � � � � y � � � � � � � � # � . � � � ' ' � � � � � � � �

� � ( &( � � ) � � � _ � . � # � ' � � � � y � " p o � y � � � � � # � ' � . � [ � ! � � # � � *� � � � # � . �

� � � �� � �

� � � ( � � & * � ( & � _ � � a � � � � � ( & * � � �� _ � _ � � ' . # � � � � # � � q � � � � . � # � � p & � � � � � p p ( � � " " o � y �

� �% ( � � � y & " � . q � � [ � p � � # � � � � # � . � # � � � � �% � � � � y & # " � . q%

� � [ � p � � ( � ' . � " � � � ) . � � � � � � � b � � . � # � � � � # � � " � _ � � ( � � ' � � � � � � p

� � � p d � � � � � ' . � " x o � y e � � � � . � # � � # & � ' � ' . � " x o � � ) . � � � � � � � " �

� ! p � p � � � � � � . p � � � � ! � � q � � ��

� + � , � - �

5 � 6 � � � 9 ) < < � � - = 9 � < . � � � - � 8 . � � � = = � 9 - � : 6 �

� + � � � - �

� � � �� � � 2 � � 1 � 2 1 1 � . 1 � 4 . 1 3 4 � / � 1 3 � 1 �

. � � � + � . � � � � - �

� � � + � � �

( 9 � 6 + � � � - ) 9 , � 9 - � : 6 �

� � � + � � � � � . � 6 � �

� � � �� � � � � / � 3 3 . � 1 3 � .� / 0 � 1 . 0 4 � 2 � / � 1 . � � � �

( 9 � 6 + � - ) 9 � � � � - � 9 - � : 6 �

. � � � 6 � + � - �

� � � �, + � � � � � � � - � �

f � k � f � � � � " ' � � � . p p ( �% ( � � q � � � � � ! [ � � � � p _ � _ � � � � ! � � ' � � x o � � ( �%

' . � " � � � � � . p � � ' � � & ( � � � # � ' . � � p � y � ( � � p " _ � & � [ � � � � � � . � _ ' � _ � � � � �%

# � ' _ � � b " . p ( � � & � � ( � � � � � & " � . q � � [ � p � & � � � & � � � & � � � � � & � � � � � � � e �

Page 25: Программирование: теоремы и задачи

� �� � � � � � � � � � � � � $ " � � � � � � � � � � " � � � � � &

� _ � ! � � � �� � � � � p d � � p � � q � � ( � � � r ' ( � � � � # . � � & � � � � � � r � ? � � r%

" p � � � & � � � p d � � � � � & � ' . � " x o � � � � ! p � s � � � _ ' � � � . p � � � � ! � � q � %

� � b � � � � � � � � & ( � _ � y � � � � � � ' � � & � ( � ' . � " � � r $ ' _ � . p _ � � ' � �%

� � � ' p e � "

f � k � t � � � � " ' � � � . p p � � ! [ � � � � p _ � _ � � [ � � x o � � ( � ' . � " � � � � � . p � �%

' � � & ( � � � # � ' . � � p � y � . � _ ' � _ � � � � � # � ' _ � � ( � � p " _ �� � � � � � � " . p ( � � �

� � � � � � � & � � � � � & � � � & � � � & ��

� _ � ! � � � �� � � ' . � " � � r # . � � � � . � # � � p � � . p ! p & � ( � � " ( � ' . � " � � r $

� � q � � ? � ' . � ( � ' . � � � . � # � � � p � � � ( � � " ( � ' . � " � � � � # . � � � ! � ' # s � ( �%

' . � " � � � � � � � d � � ' p � � ! � � ' � � � � � & � � � ! " � y # . � � � � � � " � ' " � . � � p � " � � &� ' . � � � � & � � ( � ' . � " � � r # . � � � � " � � � ! [ � � p � � ' . � � � � � � & � � � � � ( � � " %" o � � & � � ' � � � � _ & � � � � � p d � r � � �� "

f � k � k � � � � " ' � � � . p p � � ! [ � � � � p _ � _ � � [ � � x o � � ( � ' . � " � � � � � . p � �%

' � � & ( � � � # � ' . � � p � y � ( � � p " _ � & � [ � � � � � � . � _ ' � _ � � � � � # � ' _ � � � � � �%

� � � " . p ( � � � �+

� � �+

� � � +

� � � � �+

� � � � � � � �

� _ � ! � � � �� � � � [ ) . � � � � � � � # � � q � � [ . � � � � p d � � p & � � � [ y � " �%

� � & # � � [ # � � � . � � � # � � � � � # � � ' . � � � # � � ( � ' . � " � � r & � � ) � � � � � " �%' � � � � # � �� � ' . � � � ( � ' . � " � � r & � � � q � � & # � � [ � � # � �

6 b � � # � �c � . �

# � � � b Z " � ' p b �c � [ � ! � � # � � � a � . x # � ' � p � � e " �

�� �� � v � � � � s � ~ z } z w v � ~ | } � s y z � z | ~

] � � � " � [ � � � � ( � . � ! � � ( � � � # � ' . p � p � [ � _ � � � � _ � � ( � � p " _ � & # � �%[ _ � q " r ' . � " x o � r � � � � � � . p � � � � . � # � . ' p � � ( � � " " o � � �� n � '%

' � � � � � � � � ' _ � . p _ � ! � " � # � � _ � � � � � " ��

f � h � n � � � � � # � ' . � � p � ' � ( � ' . � " � � � � � . p � � ' � � " . � � ( � ! # � ' � . �� �

*

� � � _ � � ( � � p " _ � & # � � [ _ � q " � p ' . � " x o � p � � . � # � . � ' p � � ( � � " " o � r

� � " � � ' � � � � � � r a � � � & ( � � # s � � � [ � . � � & # � � � � � �

� � � � � U � � n � ' ' � � � � � � ( � p � � � � . p � x " � ' _ d � � � � ( � � ' � � *�+ � _ � q " � r � � � � � _ � . � [ " � � ' � � p � p d � d _ �� � � _ � � � [ � � ! � � & ( � . � q �%

� � p d � d � _ ' � � � � � � ' � � x � ( � ' . � " � � � � � . p � � ' � p � � ! # � ' � . �� �

* " . � � (

b # % r # . � � ( � ' . � " � � � � � . p � � ' � � ' � � � � � � ' � � � � � ' � � � d � d _ � � � # % � r

� � � � � _ � . � e � + � _ � q " � r d � d _ � � � � � ' � � ' � � � . � # _ & _ � � � � � p � � q � �

[ � p � � ( � � � . � � � � � � � y � . � � � � ! � � � � # � . � � ' � d � d _ � ( � ' � � � � � � � � � q%

� x x � � � � ! � � � � . p ' � � � . � # _ � r � � � � y� � � . � � " � � � � � � d � d _ � ( � � � _ � �

( � � � � . � � � r " p ' � � x ( � � � x d � d _ & _ � � � � x � � q � � ( � " � � � � p � � �%

( � � � . � � � � b � � � � ' � � � � � � r � � � � r e ' � � � . _ � & " � � � � � � � s � � � " � _ . � � _

�� �� � � � � � � � � � � � � � � � � ! � � � # � � � ! � �

� ) � � � � � ( � � � . � � � � & � � ' � ' � � p o � � ( � � � � � � � s d � d _ � b � � � ( s � . � ' p

� _ � � r e � � ! � � � � # � � � � � _ � � � � � �

� ' � � & # � � � � _ � q " � � d � � � � � . p _ � � " � � d � d _ � ' " � � � � � � ' p & � � �� � " � �

# . � � ( � ' . � " � � � � � . p � � ' � � � � � p � � ' p � � � � � � _ � q � � � � " _ a � � r ( � ( & # � �

( � � y � " p � ' p � ' � ( � ' . � " � � � � � . p � � ' � � � ! # � ' � . �� �

*� � . # � r ( � � � # � � � " � ��� ' � p ( � � � ' � y � " ( � " � . � � � � � � & � " � " � � � � � � ' p ( � ' . � " � p p d � d _ � &� � � & � " � " � � � � � � ' p � � ( � ' . � " � p p� � � � � � � � � ' . # � � ( � ' . � " � p p d � d _ �

' � � � � ' � � � & � � � s ( � � � � � # � � � � � & � � _ # � � ! � _ � q " � y � " � � � � � � �%

� � � � ( � ' . � " � � * � � y � " � � ( � � � � � � � � ( � & ! � � � � � p _ � � � � y ( � ' . � " � p p

d � d _ � ( � [ � � � � � � � ' � y _ . � � _ � y� � ' . � � � � ( � � p ! � [ " � � � ( � ' . � "%

� � r d � d _ � & � � " � � q � � � p ( � � � y ( � � ( � ( � � " ( � . � q � � � x � � " _ a � � ( � �%

[ � � � x � � ' � ( � ' . � " � � � � � . p � � ' � � " . � � ( � � ( � � " � � � � � ! ? " � � q � � � p

q � ( � ' . � " � � r d � d _ � � ! _ � q " � r ( � ' . � " � � � � � . p � � ' � � " . � � ( � � " � . � x �

* ( � ' . � " � � � � � . p � � ' � � r " . � � (�� ( � � � � � � � � & ( � � � � � ( � ' . � " � � � � � . p � � ' � � � � � � �

� � ( & [ " � � y � �%

� � � p � � ' ' � � � � � � �

� � ( � ! # � ' � . � � � � � b � � ' � � � � � � ' � � � � ' � � � . _ �

� � � � y & � � $ ' � � � . _ � � � � ! e �+ � # � . p � � � ' � ' � � p � � � � � � � �� � �

� � � ( � � ? � � � �� � �

� � � ( � � �� � � � � " s � � . � � � � � � ( � � � y � " � _ ' . � " x o � r ( � ' . � " � � � � � . p � � ' � � b � "%

� � � � � � � � � � � p ' � p � � ' p & � � ! � � q � � . � ( � � � y � " $ � � � � � ' � � � � � � � ' p ! � �%

# � � � � � [ . � � ' _ � r ( � � � � � � � � r � e �

� 1 / � 4 3 � � 3 �� / 1 � 0 � 0 4 2 � � � � 4 0 � * + � ) � � � � 1 / � 4 3 1 0�2 � � � � 4 0 � * + � ( � � � � �

6 + � � �

5 � 6 � � < 6 8 - = � � �< < < � � 6 � � - = � � � < . � 6 � � � = = 9 � < < � � 6 � � � - = � � � < . � 6 � � - = = =

� 9 - � : 6 �6 + � 6 � - �

� � � �6 ( < � � 6 � � - � � � . � 6 � � � = 9 � < � � 6 � � � - � � � . � 6 � � - = ) � � � - � : 6 �

* + � ( � � � � �

� � � � � � � - � : 6 �

* + � ) � � � �

. � 6 � + � . � 6 � � � � 6 � �

( 9 � + � 6 � - ) 9 � � 9 - � : 6 �

� � � + � � � � � �

� � � �� � � � �

� S O � � S � U � � � . p ( � ' . � " � � � � � . p � � ' � � r � . � r � � " � � � a � � ! � � q � � " � %

� � � � � d � � � � & � ' ( � . p ! x o � � " � � � # � x ' � ' � � � � b ] � � � � � � � � ' � p ! � �%

� � ' p � [ # � � ' � � ! � � � � � � � _ � " c � � p � � e

Page 26: Программирование: теоремы и задачи

� - �� � � � � � � � � � � � � $ " � � � � � � � � � � " � � � � � &

Z � ( � d � � ( � " � p " � ' � # � ' . � � � - " � � − � � " � � � # � � r ' � ' � � � �� + �%

( � � � � � & " . p { + \ � � ( � d � � �

- - - - - � - � - - � � � - - � - � � � - � � �

Z � � � � _ � q " � � � ! # � ' � . ( � " � � � � � � � ( � � � [ � � ! � � � � � x & ! � � � � � � _ � q " x

a � � & _ � � � � ( � � � � r & � � � s ' � � ' ( � � " " o � r a � � � r b ( � � � " . x e �] � � � ' . � � � � � & # � ' . � � � � � � � � � � � � � ( � � � [ � � ! � � � � � � � � � � � � � � � � � � � � �

� � � � � �−

� � � �b ' � � � ( � � � " . x e � � . p { + \ ( � . # � � �

- - - - - � - � � - � - � � - � � � � - � � - -

� � � _ � ( � � � � � � � p & # � � � ( � ' � � � � � ( � � � [ � � ! � � � � � � # � ' � . � [ � � � � � �

b � � � � ' � � � " � s � � ' � ( � ' . � " � � � � � . p � � ' � � ( � � " � � � � � ! e � � � � � � � �%

� � & " � � � # � � ! � ( � ' � ' � ' � " � � y # � ' � . � � . � # � x � ' p ! � � � � � r _ � � a � - � � � � ��

� � _ � � � a � - - � � �- & # � � $ ( � ' . � ( � � � [ � � ! � � � � � p $ ( � � � � " � � _ � ! � � � �%

� � x � " � � ' � � � � � � r a � � �

� � � � � � � � � � � � � � � �� � ' � p � ' � p � � � o � x o � p ' p � ' p & � � y � � � �

( � ' � � � � � p " � � # � _ � . � ( � � � � � � � ) � � r � ' �� + � ' � " � � � � � ' p [ � � � [ � � &� _ � � ' � � ( � . � � � � [ � � � [ � � � � # s � � r a � � � & ( � . � � � � � [ � . r � ' � �%

� � � � � � � � ) . � � � � � � + � � � � � y � " � [ " � � � ( � . � � � � � ' . # � � � - & � �

( � . � � � � � � b � � �� � � ! � � � p � � � � . � ' � � # � � ' � p x " � � � - � e �n � ! � s � � _ � [ � � � [ � � � �

- �

← ' _ . � � � p [ � _ �

� " � . � � � p " � � " � � x " � � � q _ � ! " � y # s � � y � [ � . y # � ' � � r � ( �%

' � � � � � � � � � � r � � � ) . � � � � � & ( � . # � � � � � ! � � q � � ' � p � ! � � � � � p � � . '

� � # � � ' � p x " � � -◦ �

- - � �

- � - �

� " � . � � � � � � p x &

- - - - � � � �

- - � � - - � �

- � - � - � - �

�� �� � � � � � � � � � � � � � � � � ! � � � # � � � ! � � �

� � ! � � � � � � � . ' � � # � � ' � p x " � �◦ � � � " � ! � � � " � p � � � � � & � " � � _ � & � �%" � ' � � � � _ � � � � � � � � ( � � � ' � # � � � p � � � � � a ' � � ! � � ' _ � . p _ � � � � ) . � � � � � � �

� � � p x � ' � � � � . & � � ' . � ) � � � ! � � � � � � p ( � � � ! � r " � � � ' � � ' � � � " � � � � �%

� � � � � & � � _ � _ � �% � � � � � � p ( � _ � ! � � � p � � � ) . � � � � � � � [ " � [ � ' ' � ' . � �%

� � �� � � " c � � p ( � ! � � . p x � � ! [ � q � � p ) � � r � ( � ' � � ' � �� � " � . � � � � � _ &# � � [ � � _ � q " � � d � � � � � � p . � ' p ( � _ � ! � � � � . � d p � " � � � � � � � ) . � � � � � �

b � � � � # � ' . � � � � ( � ' . � " � � � & ( � ' . � a � . � � � � [ � � � � � e �

- - - - � � � �

- - � � � � - -

- � � - - � � -

+ � ( � ' � � � � p � � � � � � � . � ( � ! � � . p � � . � � _ � ( � � � [ � � ! � � � � p " � � � �

� � � � � ) . � � � � � � � � " � � � # � r _ � " � . � ( � � � � � � ��Z � � � � � � � � _ q � & # � � � � � � � � � � # � ' _ � ' o � ' � � � � � � � � _ � " � c � � p � ! � �%

# � � � � � . � # � � � � � � � . p � � � � � � a � _ . � � � {% � � � � � � _ [ � b � � ! � � q � � ' � p

� [ � r � � � ' � � � � d � � _ [ � ( � � � ! & " � � � � p ' p ( � � s [ � � � & � � � � � � p ' p

� � ' y � " � x � � � d � � e �

f � h � f � + � ( � # � � � � p � ' � ( � � � ' � � � � � _ � # � ' � . �� �

( � � _ & # � � [ _ � q " � p

' . � " x o � p ( � . # � . � ' p � ! ( � � " " o � r ( � � � ' � � � � � _ � r b � � � � ' ( � ! � a � � r e

" � y ' � ' � " � � y # � ' � .� + � ( � � � � � & ( � � ( � � " � ( ' � � � � � _ � r ( � � p " � _ �

��

� � → � ��

� → ��

� � → � ��

� → ��

� � → � � �

b � � q " ( � � � ' � � � . p � � � � # � ' . � � � � ' � � � . � � � � # _ � e �

� � � � � U � � + � � p " ' � � � q � ' � � � � ( � � � ' � � � � � � _ � � ' ' � � � � � � � � � q �%

' � � � ( � ' . � " � � � � � . p � � ' � � r � � � � �

� � ( a � . y � � � � � � a � � � . p � y # � ' � . &" . p _ � � � � y � � �

6

� & � � � & � � (

6

( � � � � � s � ' � � . p _ � q � ) . � � � � � � � &' _ � . p _ � � � � � q � ' � � � � ' � y ( � � � ' � � � � � � _ & � � ' � r # � ' ' � � � � � � � � � q "

� � � � � ! � � � � � � " � � ! � � # � � � ' � � � � � � ' � � � �� ] � � � � � & _ � q " � r ( � � � ' � � � � � _ �

( � ' � � � � � � ' � � � � � � ' � � � � ( � ' . � " � � � � � . p � � ' � p � � � � �

� � ( & � " � � � � $

_ � . � # � ' � � � # � ' � . & � � � p d � y � � ' � � p o � y . � � � � � � ) � � r ( � � � ' � � � � � _ ��� ! � � � � � p � " � � ! � � # � � ' � p � � � _ � � � � ! � � _ � � � ! � � � # � � � p� � � � � ' � � � � �%

_ � # � ' � . �� �

( ( � . # � � � ' p � ! ( � � � ' � � � � � _ � # � ' � . �� �

( � � " � [ � � . � � � � �

# � ' . � ( & _ � � � � � � � � q � � � ' � � � � � p � � . x [ � � � ! ( � � ' � � � � � ) � � � _ ' � ( �%

' � � � . p � � � r ' � � r ( � ' . � " � � � � � . p � � ' � � " � [ � � . p � � ' p � o s � " � � # . � � & ( � �%

� � � � x o � r ! � � # � � � p � � � " � ( � � & � ( � � " " o � � # . � � � � � � � p x � ' p�� � � ) � � � � _ � ! � � � � ' p & # � � � ! � � � � � � � � � � " � � � a � " � � � � � ! # . � � � � ( �%

' . � " � � � � � . p � � ' � � � ' � � � � � � ' � � � � � � � � ' ( � ! � a � � " � y ' � ' � " � � y # � ' � . &

Page 27: Программирование: теоремы и задачи

� �� � � � � � � � � � � � � $ " � � � � � � � � � � " � � � � � &

� ' . � � ' � ' . � " x o � � # � ' . � ( � ' . � " � � � � � . p � � ' � � � ( � � � � � � x � � � _ ' � � � . p%

� � � . � � � � � � � . p � � � � ! � � q � � " . p � � y ! � � # � � � p� ] � � � � � & � � . � # � � � �

� � � � � � ' � � � � � � ' � � � � � � � � ' ( � ! � a � � # � ' . � � ' � � � ( � � � � ' � ' � " � � &� � � � p d � � � � $ ' . � � � �

� � ( � � p � ' ( � � � � � � � d � � � � ! � " � # � � ( � � � # � ' . � � � � � ' � y ( � ' . � " � � �%

� � . p � � ' � � r & � � _ � q " � � d � � � _ � � � � � � � � " � � # . � � � � � p � � ' p � � � " � � � a �Z � � � � � � ( � p � � � � . p � x " � ' _ " � ' _ � r � � � � � . � ' � � � a b � ' � � � � % � r

� � � � � _ � . � � � � � � � e � " � � � � p d � d _ � ( � � � � q � ( � � � � . � � & � ( � � � # �%

' . � � � ' � ( � ' . � " � � � � � . p � � ' � � � & ( � � # s � � % r # . � � [ " � � � � � p � p ' p _ � _

� � ! � � . p _ � � ' . � � ' � ' . � " x o � � d � d _ � ' � � p � ^ _ � � p� + � " � � o s � � � p

( � � � . . � . p � � ' � ! � � � � � � � � � _ � � � � _ � � � � � � � p ( � � � ' � � � � � _ � , # � � � " � r

' ( � ' � [ � � � [ � � � � ' _ � � � p � � � r # � ' . � � ? ) � � � � q � � � [ . � � # � � p & � ' . �( � � � � � ' � � � r ( � � � ' � � � � � _ � y � � � � � p ` � _ a � x

� 7→ ( � ! � a � p # � ' . � � � ( � � � ' � � � � � _ � &

� � �� � [ � � � � � � _ ( � � � ' � � � � � _ � � � � [ � � q � � � � & � ' � � � � � � ' � � x o � � � [ � �%

! � � � s _ � � � � _ � � � � � � � p� � � � _ � _ � p ( � . # � � � ' p ( � � � � � � � � �

* � 9 : � � � ) � � ) �

� 9 � � ) � � � � � �A � �. + � � � � � � - � � � � 9 ( - � � � � � 2 1 � 1 / 0 3 � . - �6 � A� . + � � � � � � - � � � � 9 ( - � � � � � � / � 0 3 1 2 1 � 1 / 0 3 � . - �

� + � � � � � � - � � � � 9 ( 6 � ) � : � � � � � � 6 � 7 6 �

� + � � � � � � - � � � � 9 ( � - � � - � � 3 2 � . � 1 3 4 1 �

- + - 9 9 � � � � �

* � 9 � � � � � � * � 6 � ) � . �

A � � 6 + 6 � ) � : � � �

- � : 6 �( 9 � 6 + � - ) 9 � � 9 - � : 6 �

5 � 6 ) � < . � 6 � � � � = �

� � � �5 � 6 ) � � � �

� � � �* � 9 � � � � � � � � ) � ( 6 � � ) � � 2 1 � . 1 + � � 6 � � � 2 � 4 . / 1 � 6 �

A � � 6 + 6 � ) � : � � �

- � : 6 �( 9 � 6 + � - ) 9 � � 9 - � : 6 �

. � 6 � + � � � - � 6 �6 � A � . � 6 � + � � � - � 6 �

� � 6 � + � � �

�� �� � � � � � � � � � � � � � � � � ! � � � # � � � ! � � \

� � 6 � + � - �

� � � �� � � �

* � 9 � � � � � � � 9 A � < A � � � 9 � � + - 9 9 � � � � = �

A � � 6 � � * 9 � - � * 9 � @� A � � - � A � � @ � ) � * + 6 � ) � : � � �

- � : 6 �6 + � � �

5 � 6 � � < 6 8 - = � � � < < < � � 6 � � - = � � � < � � 6 � � 6 � - = = 9 �< < � � 6 � � � - = � � � < � � 6 � � � = = = � 9 - � : 6 �6 + � 6 � - �

� � � �� 9 � � + � < 6 8 - = � �� 2 � � 2 1 3 4 1 + 2 1 � . � 4 . � 1 3 3 1 � � � 1 3 1 3 1 0 � �6 ( � 9 � � ) � � � - � : 6 �

� � 6 � + � � � 6 � � � � 6 � �

( 9 � + � 6 � - ) 9 � � 9 - � : 6 �

� � � + � � � � � �

� � � �* 9 � - + � 6 � A� . � 6 � �

A � � - + � 6 �

* 9 � @ + � * 9 � - � � � 6 � �

A � � @ + � . � * 9 � @ � �� * 9 � - � * 9 � @ � 3 � 3 1 � 2 1 � 1 / 0 . � 1 1 3 � � � � 1 3 1 3 0 � . �

A � � - � A � � @ � 4 � � 3 . 1 3 4 1 � A � � @ 7 A � � - �

) � * + � . � * 9 � - � �

. � * 9 � - � + � . � * 9 � @ � �

. � * 9 � @ � + � ) � * �

) � * + � 6 � A� . � A � � - � �6 � A � . � A � � - � + � 6 � A� . � A � � @ � �6 � A � . � A � � @ � + � ) � * �

� � � �� � � �

- � : 6 �� � ) � ( 6 � � ) �

* � 6 � ) � . �

- + � ) � � � �� 3 2 1 . 0 3 � . / 1 2 1 � 1 / 0 3 � . - 4 � 0 1 - � 2 1 4 . - � � . 4 0 1 � � 3 � �

1 / � 4 - � � � 3 � � 0 � 0 1 - � 2 1 � 2 � / � 1 3 1 1 �

5 � 6 � � - � 9 - � : 6 �

� 9 A � < - = �6 ( - ) � � � * � 6 � ) � . �

� � � �� � � � �

Page 28: Программирование: теоремы и задачи

� �� � � � � � � � � � � � � $ " � � � � � � � � � � " � � � � � &

�� �� r s t u v w x u v y z { s | z } ~ �

� � ' � � � � � � � o s � � ! � � � ' ( � . p ! � � � � � � � � � � ( � � s � � � � � # � . � " �%

� � . � ' p � � d � � p ! � " � # ( � � � _ � r ' y � � � � � ( � � " � . p � � ( � � p " � _ � � ( � " . �%

q � o � y ( � � � # � ' . � � � x � [ � _ � � y � p � � � � ( � ' � � � � ( � � a � " � ( � � � y � " �

� � " � � � � � � � [ � _ � � _ ' . � " x o � � b � ' � ' . � ) � � � � ( � � p " _ � e � � ! � " � # �

� _ � " � y c � � p ( � � � � [ � � � . � ' p y � � � � � p & ( � � � � � � � _ o � � � � [ � _ � � & � � � _ �%� � � x " � ( � . � � � � . p � x � � � � � � a � x b � � ( � � � . � � � p ' � � � . � _ e � + � _ � � � a &� ! � " � # � � ( � � � # � ' . � � � � ( � � � ' � � � � � � _ b � � _ � q " � � d � � � " � ( ' � � � � � " � �

� � � � ' ( � ! � a � p e � ( � � � � � � . � � � _ � r ( � � s � � ' � � � � � � . � � ! � � � � � � " � �%

! � � # � � � ' � � � � � � ' � � � � � � q " ( � � � # � ' . p � � � � � � q � ' � � � � � " � � � � & [ �%

. � � ( � � ' � � ' � � � � � � � � � � _ � y ' � � � � � � ' � � � r � _ � � [ � � � � � � � _ � � ! � � ' � � �

� � � � �� ) ( � � � � " s � � � ' _ � . p _ � ! � " � # & ' � p ! � � � y ' � � _ � � ! � � � � � �

� # � ' . � � � � � � � . � � � � �

f � � n � � � � � # � ' . � � p � ' � ( � ' . � " � � � � � . p � � ' � � " . � � � ( & ' � ' � � � . � � � �� ! ( � " � � � a � ( � � � ' � " � � � a & ^ _ � � � � y ' � � � . x [ � � � � � # � . p � � � � � � %

� � ! _ � � � � � � � a � � � . p � � & � � �� # � ' . � � � � ' � " � � � a � � s � � � ( � � � � ' y � " � �

# � ' . � � " � � � a� b � � ' . � � � _ � y ( � ' . � " � � � � � . p � � ' � � r � � ! � � x � ! � % � � $ � � �

� � � � � � ? � � � . " . p # � ' � . � � � � . � � � ' � � � ' . � " x o � � � � ! " � . �� e

� � � � � U � � ] ! � [ � � q � p � " � � � a � � _ � � � � � * �+

� , & � � � � ' � " � � � a � � _%

� � � � � * �+ � � , & � � q � � ' _ � ! � � p & # � � � � o � � ( � � � ! � � # _ � * �+� , � � � #%

_ * ( + � , & � � � ( ' _ � x o � � ' p � � q � � ' � � [ ' a � ' '� " � � ( � � � # � ' . p � p ( � ' . � " � � � � � . p � � ' � � � . � _ ' � _ � � � � � # � ' _ � � ( �%

� p " _ � & ' # � � � p & # � � � � ( � � " d � ' � � � � � � � � � � � r ( � ' . � " � � � � � . p � � ' � p x

[ " � � � ( � . � �

�+ � �+

�+ � �+ � � �

� ( � ' . � " � � r $ � � � � _ � ��

+

�+

�+ � � � +

�+ � �+ � �+ � � � + � ��

� � _ ( � � � r � � � � ( � ' . � " � � � � � . p � � ' � � _ ' . � " x o � r � � � � � _ � � � � � � �

� � ' � � � � � " � . q � ' � � ( � " � � p & � ! � � � � � � " � ! � � � � � � p � � � � � � ) � ' � � ! �%

� � � " � . q � � [ � p � � ' ( � . � q � � � _ � _ � � q � � ( � � � � �� + � ! � � � � p � p � � � � �

� � q � � � � . p _ � � � � � ' . # � � & � ' . � ' ( � � � � � � � � s � ' � p � " � � � a � b _ � � � � x

� � q � � ! � � � � � � p � � � � e � � � ' . � ! � � � � � � � � � � ( � � y � " � � _ � � _ � r ! �%

" � # � � � _ ' � � � � � � � � # � . p � r _ ' � _ ( � ' . � " � � � � � . p � � ' � � & � � " � � � r � � � �%� � � � . p � � � ( � � " � . q � � � �� � s � � d � � � � � � � " � ( � � ( � ' � � � p � � & � ' . � ) � � � �� � � d � � ' . � � � p � � � � � � a � � � . p � � ' � � & � � � � # � ( � � ( � ' � � � p � � � � . # � � �

�� �� � � % � � � � � � # � $ � ! � � � " � �

� � _ x ( � � � � � � � �

� � �) � * � � � � � � @ � � � � � � � � - � � @ � � 9 ( 6 � ) � : � � �

� � �* � 9 � � � � � � : � ) � � � . ) < A � � � + � � � � � @ � � A � � � � � ) + � 9 9 � � � � = �� . � 2 � 3 1 2 1 0 / 1 / � 1 � � 2 1 2 � / � 1 � . 0 1 � � 3 � / 0 �� 1 / � 4 �� � 3 1 / 0 � < 2 � 4 � 0 � 3 � � � ) + � ( � � � � = � 4 3 . 1 � � � ) + � ) � � � �

A � � ,� 6 � � � � + 6 � ) � : � � �

- � : 6 �, + � @ ; � �� 4 3 . � 4 3 0 + . � � , � - � � @ � � 0 � � � - � 3 4 3 � / 1 4 3 4 � � �

5 � 6 � � � � , � � � - � 9 - � : 6 � , + � , � - � � � � �� , � 3 - / 4 3 � � 3 � 1 / � 1 4 0 1 �� � 1 - � 0 � � � � � � , � � - �

5 � 6 � � < , 8 � = � � � < � � , � � - = � 9 - � : 6 � , + � , � - � � � � �� � � , � � / 3 1 2 � . 1 � - � � - � 0 � � � 4 1 / 0 � - �

1 / � 4 0 - 4 � 3 1 0� 0 � , � � �6 ( , � � ) � � � - � : 6 �

� � � ) + � ) � � � �

� � � � � � � - � : 6 �

� � � ) + � ( � � � � �6 + � � � � � � + � � �� � � � � � � - � � � � � � � � 6 � �5 � 6 � � 6 7 8 , � 9 - � : 6 �6 + � 6 � - � � � � + � � � � � � � 6 � �

� � � �� � � � � � � - � � � � � � � � , � � � � , � � � - �

� � , � + � - � � � � + � � � � � @ �� . 2 � � 0 � � � � , � . / 4 � 3 1 3 1 3 � � � � � � � � - � � � � � � � � , � �5 � 6 � � , 7 8 @ ; � � 9 - � : 6 �, + � , � - �6 ( � � � 8 � ) � � � - � : 6 �

� � , � + � � -

� � � � � � � - � : 6 �

� � , � + � - �

� � � �� � � + � � � � � � � , � �

� � � �� , � @ �� � � � � � � - � � � � � � � @ � � � � �

� � � �� � � � �

f � � f � � � � � # � ' . � � p � ' � � � ' ' � � � � � _ � ' _ � [ � _ � ( � � � ! � � " � � � � ( ' � � � �%

q � � � . � r� � � � p " � _ ' � � � � q � � � . � r � � � � � p � � ' p & ' _ � [ _ � ( � . � � ' � p x � ( � �%

Page 29: Программирование: теоремы и задачи

� � �� � � � � � � � � � � � � $ " � � � � � � � � � � " � � � � � &

" � . p x � ( � � p " � _ " � r ' � � � r� + � ( � � � � � & " . p ( � � � ' � p � � � ' ' � � � � � � _ �

* * � � , � , �+

* � * � � , , �+

* � � , * � � ,+ � * * � � , � ,+ � * � * � � , ,�

� _ � ! � � � �� � � q " � � ( � � p " _ " � r ' � � � r ' � � � � � � ' � � � � ( � ' . � " � � �%

� � . p � � ' � p _ � � � � " ' � � _ � � � � � _ � . p _ . p � � � � & � ( � ' � � � � � � � � '� � \� "

f � � t � + � � _ � q � � ' � � ! � " � � � � ( � � # � _ & ( � � � � � � � � � � � y � � � " � � (�� � � � # � ' . � � p � ' � ' ( � ' � [ ( � � � � ' � � ( � � ( � � � ' � _ � x o � y ' p y � � " ' � � � d �%

� � � � � ) � � y � � # _ � y� �

f � � k � � � � � # � ' . � � p � ' � ' ( � ' � [ � � ! � � ! � � p (% � � . p � � _ � � � � � � � . p%

� � _ � & ( � � � � " p ( � � � � � " � � � � � � . � � �

b ) � � � � s � ' p _ � � ! � � ! � � � x � � � � � � � . p � � _ � � � � ! " � . � � " � � � � � # �%

' _ � � ( � � � � � � � � � � � � � � � & '� � \ �� e

� o s � " � � _ . � ' ' ! � " � # � � ( � � � # � ' . � � � � � ' � y ) . � � � � � � � ! � " � � � � � �

� � � q � ' � � � � � � ' ' � � � � � � � � q � & � [ ' q " � p � � � � " ( � � ' _ � ' � � ! � � � � � � �

b � � o � � � � o � �� ~ e �

�� �� v � t | � � u v w ~ | s t � �

] � � � " � � � q � � � � r � � _ � . � # � ' � � � � [ � _ � � � ' � � � � . � � � � ' � � r%

' � � � � & � � ( � � � # � ' . p p � y� � . � ' ' � # � ' _ � r ( � � � � � � > �� $ # � ' . � � ' � y �% ) . �%

� � � � � y ( � " � � � q � ' � � {% ) . � � � � � � � � � � � � q � ' � � � $ � � q � � � � r � � & ! �%( � . � p p � � [ . � a ( � � � � . � �

> �� + > �� + � b {

>

� e

> �� + > �− ��−

� � > ��−

b { � ��- � � � { e

� . � ( � � � � . �

> �� + { �

� � ·

b { − � e � �

b � � � � r ' ( � ' � [ ) ` � _ � � � � � � & � ' . � � � " � � # � ' . � � p � � � � � ! � � # � � � r > �� � e� � � � � " s � " � � � � ( � � � � � �

f � m � n � b � � ' . � � � ! [ � � � � r ? ( � � " . � � � . � ' p � � � ' � ' � x ! � � r � . � � ( � � " � ( �

( � � � � � � � � � � � � � � x � � � � � � " � e � ' � p � b { e $ # � ' . � � � ! [ � � � � r a � . � � �

( � . � q � � � . p � � � � { � � a � . � ( � . � q � � � . p � � ' . � � � � � � b [ � ! # s � � ( �%

� p " _ � & � � � � � $ � " � � � � � q � � � ! [ � � � � � e � � � � { + - ( � . � q � �

� b { e + � b � " � � ' � � � � � � � � � ! [ � � � � � � � ' � " � � q � � ' . � � � � � y e � � � ' � � � � � p

� . � � � � � � � # � ' . � � � p � b { e " . p ! � " � � � � � � { �

�� �� � � � % ! � � � � � � ! � % � & � �

� � � � � U � � ) � q � � " � _ � ! � � p b ) � � � � � � � � � � . p � � e � � _ x � � � . " . p

� b { e �� b { e + � b {− � e � � b {− e− � b {− � e− � b {− � e � � b {− � e � � b {− � � e �

� � �

b ! � � _ � ^ ( � � # . � � � � # � � � " x � ' p & � # � � � � � � � � " � � r ( � � � � � � �

b \ � − e % � b \ � � e % ? ' � � � _ � � � # � � $ � ' # � � � � � & # � � � b � e + -

( � � � � - e �, " � � _ � � [ � ! � s � ' ( � . p ! � � � � � p � � q � � ( � � " � � � p ' ( � ' � [ � # � ' . �%

� � p � b { e & _ � � � � r ' o � ' � � � � � � ) ` � _ � � � � � � ( � � � [ � � � � ( � " ' # s � � � ' � y

� � ! [ � � � � r�, [ � ! � � # � � # � � � ! b { � � e b " . p {

>

- & �

>- e # � ' . � � � ! [ � � � � r { � � a �%

. � ( � . � q � � � . p � � ' . � � � � � � & � � ( � � � � ' y � " p o � � � � b � � � ) � � � b -�

� e

' # � � � � � � � � � � � " . p � ' � y �

>

-� e , # � � � " � � & � b { e + b { � { e � � ' � � � ! [ �%

� � � p { � � ' . � � � � � � & � � ( � � � � ' y � " p o � � � & � � ! � [ p s � � � � � ( ( � ! � � �%

' � � � ' � � � � � � _ ' � � � . p � � � � ' . � � � � � � � � b � [ � ! � � # � � � � � � e � � � ' . � b { � � e

� � � � � ' � � � b ( � � ' � � � � � � " � � e _ � . � # � ' � � � � ! [ � � � � r ' � ' . � � � � � � �

� � [ � . p d � � � � � _ ' � � � . p � � ' . � � � � � � & � � � � � � � q � � ! [ � � � � p { � �

' . � � � � � � � � [ � . � � � ' ( � � � � ' . � � � � � � & � � � � � � & ( � ' o � ' � � ( � � "%

' � � � . p x � ' � [ � r � � ! [ � � � � p { − � � � ' . � � � � � � & � � ( � � � � ' y � " p o � � � b ( � �

6

� e � � � _ # � � b { � � e +

� � � b { − �

�� e ( � � �

6

{ �

b { � � e + b { � { e ( � � �

>

{ �

# � � ( � ! � � . p � � ! � ( � . � p � p � � [ . � a ! � � # � � � r ` � _ a � � � �f � m � f � b � # � ' � . � � � [ � . � � ? ( � � " . � � � . � ' p � � � ' � ' � x ! � � r � . � � ( � �%

" � ( � ( � � � � � � � � � � � � � � x � � � � � � " �� e � � ' . � " � � � � � . p � � ' � p � ! { a � �

b _ � q " � p a � � � � � - " � � e � � ! � � � � ' p ' # � ' � . � � � [ � . � � � � & � ' . � ' � � �

( � � � y { a � � � � � � � ' � � � ( � ' . � " � � y { a � � � + � r � � # � ' . � ' # � ' � . � � y

( � ' . � " � � � � � . p � � ' � � r " � � � � r " . � � �

� � � � � U � � b � � � [ o � � � � " � � � � ! # � ' � � � _ � � � . � � ( � � " ? _ ' � q � . � � � x &� � � � � _ � ! � � p � � � . � x & � � _ _ � _ � � [ � � ( � � � � � p . � ' p ! � d � � � � � �%

� � �� e n � ' ' � � � � � � [ � . � � � [ o x ! � " � # � � � r � � # � ' . � ( � ' . � " � � � � � . p%

� � ' � � r & � " � � � ! � � a � � � q " ' � � � r ( � � � y { a � � � ' � � � r ( � ' . � "%

� � y { a � � � � � � � � b � + − � { � � � � �� { e � � ' � p b { � � e $ # � ' . � � � _ � y

( � ' . � " � � � � � . p � � ' � � r�n � ! � [ p s � � � � q � ' � � � � � _ � y ( � ' . � " � � � � � . p � � ' � � r � � _ . � ' ' � ! � � � ' �%

� � ' � � � � � � ! � � a � � q " ( � � � � r � ( � ' . � " � � r a � � � � � � � ' . � ) � � � � ! � �%

a � � � � � � � & � � � � ! � � a � � � q " ' � � � � � � � ( ( � ! � ' � � � d � y ' p { − � a � �

Page 30: Программирование: теоремы и задачи

� � �� � � � � � � � � � � � � $ " � � � � � � � � � � " � � � � � &

� � � � � � − �� � # � � � � p & # � � ( � � a � � ' � � ! � � ' � p x � [ � � � � � - − | �| & ( �%

. # � � � � � � . b { � � e +

� �− �b � - − | �| e b { − �

�� − � e

b + � _ � � � � � ' . � � � � � � � � � � � � ' � ' � � � � � � p & � � _ _ � _ � − � � � q � � [ � p

' . � d _ � � � � . � _ �� e �

� � � _ � � � � y ' . # � p y � � � � � " � s � ' p ( � . # � � p � � � " � p � � � r � � � . �

f � m � t � � � _ � ! � � p & # � � # � ' . � � � � � . � � � b _ � . � # � ' � � � ( � ' . � " � � � � � . p � �%

' � � r " . � � { � ! { � " � � � a � { � � � ' � " � � � a & � . x [ � � � � # � . p � � � � � � � !%

_ � _ � � � � y � � � � � p d � � " � � � a & # � � � � � ' � " � � � a e � � � � � > �� �

% b { � � e �

� _ � ! � � � �� � � ' . � � � � � . � � � � ' � p # � ' . � . � � � � y & � " o � y � ! b -�- e

� b { � - e d � � � � � b ��� e � b �

� − � e & � � � ( ' _ � x o � y ' p � � � q � x x ( � . ( . � '%

_ � ' � p & � � �� � � ! � � ' � p # � ' . � � ' � y . � � � � y b _ � � � � � � � ' � p > �� �

e � # � ' . � . �%

� � � y & � ( ' _ � x o � y ' p � � � q � x x ( � . ( . � ' _ � ' � p� � � ' . � " � � � � � q � � � ( �%

' � � p � � _ q � _ � _ . � � � � � & ( � � � ' � _ � x o � � ( � p � x � + − � � , � � � ! � � � y _ %

' � _ ' ( � � � � � � ' � � � r ( � � � � r � � # _ � ( � � � ' � # � � � p � � � � ' � � � . p � � _ � ! � � � � r

( � p � � r & � ' � � � � � � � � ! � � � � � � " � � ! � � # � � � ' � � � � � � ' � � � � � � q " � � � �

� . � � � � � � � ! b -�- e � b { �− e � , ' � � s � ' p ( � � � � � � � p & # � � > �� � − > � � �� � +

+ > �� �

% b { � � e � "

��

� � � � � � � � � �

� � � � � � � D � � � � � �

�� �� � s � y ~ � } s � x � ~ s � � � � � � � � z

v � v � � s � s � z � v y ~ � ~ �

� ( � � " " o � r � . � � � � � � ' ' � � � � � � � . � � � ' _ � . p _ � ! � " � # � " � � � � � � �%

� � q � � � ( � � � ( � � � # � ' . � � p � ' � ) . � � � � � � � _ � � � � � � � � � � q � ' � � � �� � � y �%

� � � � d � � � p [ . � � � _ � � � � � � � � � q � ' � � � � � � � " � . ' p ( � � p " � _ � � ( � ' %

� � . � ' p ( � � a � " � � ( � � � y � " � � � ( � � � ! � � . p � � � � ) . � � � � � � � � � q � ' � � � �

_ ' . � " x o � � ! � � � � b � ) � � � ( � � p " _ � e � � � _ x ' y � � � � � ' � � " � " � s � ' p

� � � . � ! � � � � p � � ( � ' � � " ' � � � � � � & � � ) � � r � . � � � � � � ' ' � � � � � � " � � � r

( � . � ! � r ( � � s � ( � � � # � ' . � � � p � ' � y ) . � � � � � � � � � _ � � � � � � � � � � q � ' � � ��

� � � � � ! � � x � � ( � � ' _ ' � � ! � � � � � � � � & � � � � � " � � � � � r � � � � � � a � & �� � o �%

� � � o � � � ~ � � + � � � d � ! � . p " & � � � [ � . � � � � # � � � � � ! � � � � � ) � � � � � � � � " � $

� [ y � " " � � � � ��

t � n � n � � � � � # � ' . � � p � ' � ' ( � ' � [ � � ' ' � � � � � _ � { � � ! � r � � d � y � � �%

� � r " � ' _ � { × { & ( � � _ � � � � y � � � � � [ p x � " � � " � � ��

� � � � � U � � , # � � � " � � & � � _ � q " � r � ! { � � � � ! � � � � . � r " � . q � � ' � � p � p

( � � � ! x� " � � � � ! � � � p �% ( � ! � a � � r b " . p � + - � � � � � � � { e ( � � � ! � � . p%

� x � � ' ' � � � � � _ � � � ! � r � � � � � q � � y � � � � ! � � � � . p y b � � ! � � � � �

[ � � p " � � " � � � e � + � � � ' � � � " � � � � � ( � ! � a � r � � � � � _ � � � � � [ " � � � " � �%

' � � � � � � p -% ( � ! � a � p & � � ! _ � q " � r �% ( � ! � a � � � y � " � � { ' � � � . � _ � � � � y

� b � � � e % ( � ! � a � �� ! � � { ( � ! � a � r � � . � # � x � ' p ( � . � q � � � � � � � ! p � �

b � � � e % � r � � � � ! � � � � . � � " � � ' # � � � � p & # � � � � ' ( � . � q � � � � � y � � � � ' �%

_ � ' � � � � � � ' � � � � ( � . � q � � � x ) � � � � � � ! p � . � � � � � � ( � ! � a � p & � _ � � � � � r

� � ! p � � ' ( � . � q � � . � � � ��� � � " � ( � ! � a � r ) � � � � " � � � � � � � � � � " � � � � [ � � � p � � {% ( � ! � a � � & � _ �%

� � � y � � ! � � � [ p x � " � � " � � �� � � � � � � � � � [ " � � � � [ y � " � � p " � � � � � �

� � ' _ � � p � y� � � � [ � � " � . � � p . � d � � r � � [ � � & ! � � � � � � � � � # � � � � ' . �

� _ � _ � r% � � �% ( � ! � a � � � � ! � [ p x � " � � " � � � & � � ' � � � � � p " � . p � � r d � y

Page 31: Программирование: теоремы и задачи

� - �� � " � � � � � � � & �� � � � � " � � % & � # & � � � � $ �

PPPPPi�����1

@@I ��� @@I ���

••

••

••

••

• •

� � � � � � ( � ! � a � r " . p { +

� � ! � r ' � ' . � � � � � � � ) � � � & � [ � � � q � � ) � � & � [ " � � ( � � _ � � o � � p

( � ' � � � � � � � " � � � � � � ) � � � � � ( � � � . � � � � �

� � # � � � & � � ! � � s � �% ( � ! � a � x " � ( ' � � � � r & � ' . � � � � � � � � � � � � � � � � �

� � � � � � � � ' � � � d � � ' p � � [ p x � " � � " � � �� + � d � ( � � � � � � � � [ " � �

� � ' ' � � � � � � � � p � � . p _ � " � ( ' � � � � ( � ! � a � ��

PPPPPPPPi

��������16

BBBM

����6

BBBM

����6

BBBM

����6

BBBM

����6

BBBM

����6

••

•••

•••

••••

•••

•••

••

••

••

••

••

••

••

••

••

• • •

� � � � � � " � ( ' � � � y ( � ! � a � r " . p { + \

n � ! � [ p s � ! � " � # � � " � � # � ' � � � b � e � [ y � " ( � � � ! � � . p � � � � " � � � � � � b e

� � � . � ! � a � x " � � � � � " � ( ' � � � y ( � ! � a � r�

�� � � � � � # � � � � " � � � � � � � � � � � � � � " � � � � � � � & � � � # � � � " � �

� � � � . � � � � ! � " � # � [ y � " � ( � � � ! � � . p � � � � " � � � � �� " � � ' # �%

� � � p & # � � � � ' � � � � � ' p n � [ � � & _ � � � � r � _ � q " r � � � � � � � � y � " � � ' p

� � " � � r � ! � � � d � � " � � � � � b � � � d � � � ! � [ � � q � � � � � � ' � _ � _ � q � #%

_ � � � e � , � � � � � � ( � . � p � p _ � � � � " �

• � � � � ' � � � � b � " � � ( � ' � � � r . � � � r � ! � y � " p o � y � � � � y ' � � � . � _ e

• � ' � � b ( � � � r � � � ' � ' � " � x x ' ( � � � � � � � d � � e• � � b ' ( ' � � � p ' p � � � ! � � � " � � � � � � � p e

b + � � � ' � _ � y ' � � � . _ � � � ( � _ � ! � � � & _ � _ � � ( � � � � � o � � � p ' � � � � � � ' � � x �

) � � � _ � � � � " � � � e

� � � � ' � � � �q@@@

���

q qAAA

���

q���

���

q q q q@@@

AAA

���

q q

q q q q

@@@I

AAAK 6

@@@I

� ' � �q@@@

���

q qAAA

���

q���

���

q q q q@@@

AAA

���

q q

q q q q

- -

- - - -

- - -

� �q

q@

@@R

q

?

q�

��

qAAAU

q

?

q�

���

q

?

q�

���

q�

��

q@

@@R

qAAAU

q

?

q�

���

@@@

���

AAA

���

���

���

@@@AAA

���

� � � � � � � � � & � � � ( � � � � � n � [ � � � � y � " p � ( � � � � � _ � b ' � � � � � � ' � � x o � �

� � ! � � q � � ' � � � ( � . � � � p _ � q " x � ! _ � � � � " e �

• � � � � � � � � � ?

• � � � � � ' � ' ?

• � � � � � � � ?

b ( � ' . � " � p p ( � � � � � _ � � ' � � � � � � ' x " & _ � � � � _ � � � p e � , [ � � � � � � � � � � � � � � &# � � _ � � � � " � � ' � � ( � ! � � . p � � ( � � � r � � . � d p _ � � � " � � � [ � � � � & � � � �

_ � " � � x � � " � � � � �

q@@@I

����

qAAAK

����

qAAAK

����

q q q q-��@@ � � _ _ � � � � " �

� ' � �

� � " � r ' � � � � �

" � � ' # � � � � p & # � � ^ n � [ � � � � ' � p _ � � � � " � � � ' � � � ' � � � # � � � � �

! � " � # � $ � [ � � [ � � � � p � ' � . � ' � p p b � � � d � � & � ! _ � � � � y � � � ' � � � . � _

Page 32: Программирование: теоремы и задачи

� �� � " � � � � � � � & �� � � � � " � � % & � # & � � � � $ �

� � � � y & � � � ' � p � " � ' . � � � � � � � � � � � � � . � q � � e � � . p � � d � r d � y � � � � � r

! � " � # � _ � � � � " � � � ' � � � ' � � [ " � � ' � � � � � � ' � � � � � � p ( � � � � � _ � � ( � # � � p

( � ! � a � � � � ! � r�� � _ � ! � � � . p ' � � � ( � � � � . p � � ' � � ( � � � � " � � � r " � . � � ( � � � � � � � � '%

( � . p ! � � � � _ � � � ( � � " � . � � � p� � ' � p � _ ' � � � � � � � ( � . � q � � � � n � [ � � �

� � " � � r � ! � � � d � � " � � � � �� � � � " � � ' � . � ' � p p " � � � � � � � ! [ � � � x � ' p � �

� � � _ � � � � � � � � � � � � n � [ � � � � & � � & � � n � [ � � � � � � � & � � n � [ � � �� b � � p � !

_ � � � p � . � ' � � � q � � ( � � y � " � � p # � � � ! � � � d � � ' n � [ � � � � & ' � � � � # � � � � p� . � � � & � � " � y � " p " � � � s � ' � � � � # � � � � p � ( � � � � & � � " � y � " p " � � � s� e � � � � !

* ( � , � [ � ! � � # � � ' . � � � � � � [ � � [ � � � � � ' � . � ' � p p . � � � � n � [ � � � � & � # � � � !

* ( � ' , $ ' . � � � � � � [ � � [ � � � � � ' � . � ' � p p . � � � � � � � " n � [ � � � � � �

q

q

@@

@@

@@

��

��

��

����

@@@

���

� � � � � + � " � � � � � �

+ � � ( � � � " � [ � � ' p � � _ � p ( � � a � " � � �

* � 9 � � � � � � . . 1 � �� � � � 2 � � � 4� � / � / � 0 0 � �� 3 � + < � � = � 3 � + < � � � = �

- � : 6 �� 4 3 . � 4 3 0 + � � �

5 � 6 � � 1 / 0 �� / . 1 � � � � 9 - � : 6 �

. . 1 � � � 3 � 1 . � �

� � �� � �� � � / � 0 . � 4 / 0 1 �

� / � / � 0 0 � �� � � � �

� � � �

, ' � � � � � r � . � � � � � � �

3 � + � � / � 0 . - � � 3 1 � � 4 / 0 � 1 3 1 � / � / � 0 3 �

3 � + � � / � 0 . - � � 3 1 � � 4 / 0 � 1 � / � / � 0 3 �

� � � �. . 1 � � � � � � 2 � � � 4� � / � / � 0 0 � �� 4 3 . � 4 3 0 + � � � �

�� � � � � � # � � � � " � � � � � � � � � � � � � � " � � � � � � � & � � � # � � � " � \

5 � 6 � � 1 / 0 �� / 3 4 � � � 9 - � : 6 �6 ( 1 / 0 � � / 2 � . ) � � � - � : 6 � � � � �� 1 / 0 � / 2 � . �

. 2 � . � �� � � �. . 1 � � � � � � 2 � � � 4� � / � / � 0 0 � �

� � � � � � � - � : 6 �� � � �� 3 1 1 / 0 �� / 2 � . � 1 / 0 �� / 3 4 � � �

. 3 4 � �

� � � �� � � �� � � �� � � / � 0 . - � � 3 1 � 8 . / 1 � 4 / 0 � 1 � / � / � 0 3 � �

, ' � � . � ' p � � ' ( � . p ! � � � � p ' p ' . � " x o � � � ' � � r ' � � � � � _ � � � � " n � [ � � �

b � _ � q " � r ' � � � _ � � ( � � � � r � � � � � r ' _ � [ _ � ! � ( � ' � � ' . � � � p & � _ � � �%

� y � ( � . � p � � ' p _ � � � � " � & � � � � � � � r $ � � � � q " � � � p � � � ! . p � � � � � s

� ( � . � � � � p e �

b � e { , � & � � � � � � � � � � �} � � ' � � � ' � � { , � +}b e { , � & � � � � � � � � �} � � � � ' � � � � { , �}b \ e { � � � � � ' � ' & , � +} � ' � � { , �}b e { � � � � � � � ' � ' & � � � � � � � & , � +} � � { , � +}

t � n � f � � � _ � ! � � p & # � � ( � � � � " s � � � p ( � � � � � � � � ! � � � � d � � � � � [ � � b � �

. x [ � � _ � � � # � � � " � � � � � e �

� � � � � U � � � � � a � " � � � � � � � � � � ' � � ' � � � ' � � ! � � � � d � � � � � [ �%

� b � ' � � � n � [ � � � � � � � q � � � � . � # � � � � p ' p [ � ' _ � � � # � � e � � ' . � ( � � � � � �%

� � � � [ � � � � � [ � ' _ � � � # � � & � � & ( � ' _ � . p _ . � ' � p p � � � [ � � [ � � � � x � ' p ( �%

� � � � � � & � � # � � � p ' � � _ � � � � � � � � � � � � � � � � � " � � . � ' � � � � [ � � [ � � � � � � ' p�q ) � � � � ! � � q � � & � � . p _ � � ' . � n � [ � � � ' s � � � � p ' ( ' _ � � � ' p � � � !� � � � � �%

� � � � # � �� b , [ � a � � _ � # � ' . � " � r ' � � � r ' � � " � . � �� e �t � n � t � � � _ � ! � � p ( � � � � . p � � ' � p ' . � " x o � r ( � � � � � � � � [ y � " � " � � � � � �

A � � � ) � ) � + < � � � � � � = �

� ) � ) � + � � � �

5 � 6 � � 1 / 0 �� / 3 4 � � 9 � < � ) � ) � 7 8 � � � = � 9 - � : 6 �6 ( < � ) � ) � � � � = � � � 1 / 0 �� / . 1 � � � ) � � � - � : 6 �

. . 1 � � � 3 � 1 . � �

� � � � � � � 6 ( < � ) � ) � � � � = � � � � 9 ) 1 / 0 �� / . 1 � � � ) � � � - � : 6 �

� / � / � 0 0 � � � ) � ) � + � � � � �

� � � � � � � 6 ( < � ) � ) � � � � � = � � � 1 / 0 � � / 2 � . ) � � � - � : 6 �

. 2 � . � � � ) � ) � + � � � �

� � � � � � � - � : 6 � � � ) � ) � � � � �� � 9 ) 1 / 0 �� / 2 � . � 1 / 0 �� / 3 4 � � �

. 3 4 � �

� � � �� � � �

Page 33: Программирование: теоремы и задачи

� �� � " � � � � � � � & �� � � � � " � � % & � # & � � � � $ �

� � � � � U � � ] � � � � � � � � a � _ . � �# � � � � + � �⇒ ( �

# � � � � + � � �⇒ ( � '

� � _ � ! � � � . p ' � � � ! � � � � d � � � p � � [ � � � ( � � � y � " � ! ' � ' � � p � � p ( � � ( � '

� � ! � � q � � � � . p _ � ( � � � [ � � [ � � _ � � � � d � � & ( � ) � � � � ' . � ( � � � � � � � � � �%

[ � � � � � [ � ' _ � � � # � � & � � ' � � _ � � � � � � � � � � � � � � ! � � # � � � � # � � � � � � � � � p%

� � ' p & # � � � � � � ! � � q � �� �

t � n � k � + � ( � ' � � p ( � � � � � � � � [ y � " � " � � � � � & � ' ( � . p ! x o x ( � � a � " %

� ( � � � y � " � � ' . � " x o � r . � ' � b ' � y � " � � ( � � � � � � � � � & ' � � [ o � x o � � &" � . � ' p . � ) � � ' " � . � � p � . � . � ' � � _ � ! � . ' p ( � ' . � " � � � e � �

t � n � h � n � d � � p ! � " � # � [ � [ y � " � " � � � � � & � ' . � � y � � � � & # � � [ � [%

� � [ � � � � . � ' p � ' � � � � d � � b � � � � . p _ � . � ' � p p e �

� � � � � U � � � ' � p � $ � � _ � � � � � p � � � d � � �� � � � " � . x [ � p � � � d � � � �

� � � � ' � � ' p _ � " � � r � ! # � � � s y _ � � � � � � � r� n � ' ' � � � � � � ( � p � ! _ � � � p

� � � , � � � q � � �

b � e [ � p # � ' � p x ( � � � ! _ � � � p � � b � � � � � � e ?

b [ e ' � � � � � p � � . � � � ' ( � � � � b � � � & � � � e ?

b � e ( � � r � � # � � � ! � b � � � � � e ?

b � e ' � � � � � p � � ( � � � � ' ( � � � � b � � � � & � � � e ?

� # � ' � � � ' � � & ' � � � � � � d � � � � � � � � ' � � ' p _ _ � � � � � � � � b � e � � ' . � � � p � � ( � � p

[ " � � � _ � � � �

* ( ' � , � [ � � [ � � � � � ' � � � � d � � � � q � � . � � � � ?

* ( ' � ' , � [ � � [ � � � � � ' � � � � d � � � � q � & . � � � � � � � " �

� � � _ � _ [ " � � � � . p " � � p ( � � � � � � � � �

* � 9 � � � � � � . . 1 � �� � � � 2 � � � 4� � / � / � 0 0 � �� 3 � + < � � � = � 3 � + < � � � � = �

- � : 6 �� 4 3 . � 4 3 0 + � � � �

5 � 6 � � 1 / 0 �� / . 1 � � � � 9 - � : 6 �

� / � / � 0 0 � �

. . 1 � � � 3 � 1 . � �

� � �� � � �� � � / � 0 . � 4 / 0 1 �

� / � / � 0 0 � �� � � � � �

� � � �

�� � � � � � # � � � � " � � � � � � � � � � � � � � " � � � � � � � & � � � # � � � " � �

, ' � � � � � r � . � � � � � � �

3 � + � � / � 0 . - � � 3 1 � 3 4 . 1 � 3 1 � / � / � 0 3 �

3 � + � � / � 0 . - � � 3 1 � . / 1 . 1 � 0 4 3 � � / � / � 0 3 �

� � � � �. . 1 � � � � � � 2 � � � 4� � / � / � 0 0 � �� 4 3 . � 4 3 0 + � � � � �

5 � 6 � � 1 / 0 �� / 3 4 � � � 9 - � : 6 �6 ( 1 / 0 � � / 2 � . ) � � � - � : 6 � � � � � �� 1 / 0 � / 2 � . �

. 2 � . � �� � � � �. . 1 � � � � � � 2 � � � 4� � / � / � 0 0 � �

� � � � � � � - � : 6 �� � � �� 3 1 1 / 0 �� / 2 � . � 1 / 0 �� / 3 4 � � �

. 3 4 � �

� � � �� � � �� � � � �� � � / � 0 . - � � 3 1 � 8 . / 1 . 1 � 0 4 3 � � / � / � 0 3 � � �

t � n � � � � � � � " s � � � p � � . p _ � # � � ( � � � � � � � � � [ � � [ � � � � � � � � � d � �

" � � � � � & _ � _ � [ � � [ � � � � . x [ � r � ! � s ( � � � � _ � �� � � _ � ! � � � � � p ( � � � � � �%

� & # � � [ _ � q " � p � � � d � � � & � � p � . p x o � p ' p . � ' � � � & � [ � � [ � � � � . � ' p

" � � q " � � " � � � � ! " � & � " � � � r � � ! ( � ' . � � ' � y ' � � � y ( � � � � _ � � � b � �%

' � p p ( �% ( � � q � � � � [ � � [ � � � � x � ' p ( � � � ! � e

� � � � � U � � � � " � � [ � � [ � � � � � � � q � � . � � � � � [ " � � ( � � � � � � p � � � q � � [%

� � [ � � � � � ( � � � ! & ' . � � � � [ � � [ � � � � � ( � . � � ' � p x b . � ' � p p ( � � � ! & � ' � � . p%� � ( � " � � e � � � � " � � [ � � [ � � � � � � � q � & . � � � � � � � " � [ " � � ( � � � � � � p � � � q �

� [ � � [ � � � � � ( � � � ! & . � � � � � � � " $ ( � . � � ' � p x � �� � � � � � � � � [ " � � � � _ � r �

* � 9 � � � � � � . . 1 � �� � � � 2 � � � 4� � / � / � 0 0 � �� 3 � + < � � � = � 3 � + < � � � � = �

- � : 6 �� 4 3 . � 4 3 0 + � � � �

5 � 6 � � 1 / 0 �� / . 1 � � � � 9 - � : 6 �

� / � / � 0 0 � �

. . 1 � � � 3 � 1 . � �

� � �� � � �� � � / � 0 . � 4 / 0 1 �

� / � / � 0 0 � �� � � � � �

� � � �

Page 34: Программирование: теоремы и задачи

� � �� � " � � � � � � � & �� � � � � " � � % & � # & � � � � $ �

, ' � � � � � r � . � � � � � � �

3 � + � � / � 0 . - � � 3 1 � 3 4 . 1 � 3 1 � / � / � 0 3 �

3 � + � � / � 0 . - � � 3 1 � . / 1 . 1 � 0 4 3 � � / � / � 0 3 �

� � � � �. . 1 � � � � � � 2 � � � 4� � / � / � 0 0 � �� 4 3 . � 4 3 0 + � � � � �

5 � 6 � � 1 / 0 �� / 3 4 � � � 9 - � : 6 �6 ( 1 / 0 � � / 2 � . ) � � � - � : 6 � � � � � �� 1 / 0 � / 2 � . �

. 2 � . � �� � � � �. . 1 � � � � � � 2 � � � 4� � / � / � 0 0 � �

� � � � � � � - � : 6 �� � � �� 3 1 1 / 0 �� / 2 � . � 1 / 0 �� / 3 4 � � �

. 3 4 � �� / � / � 0 0 � �

� � � �� � � �� � � � �� � � / � 0 . - � � 3 1 � 8 . / 1 . 1 � 0 4 3 � � / � / � 0 3 � 2 � � 3 � / 0 � � � �

t � n � m � � � _ � ! � � p & # � � # � ' . � � ( � � � a � r � ) � � r ( � � � � � � � � ( � ( � � p " %

_ � � � � � # � ' . � � � d � � " � � � � �� b � � _ � � " � � � y ( � � � � � � � � y & _ � � � � �

� � . � # � x � ' p � � ) � � r . � d p ( � � ( ' _ � � � � _ � � � � y _ � � � � " � � ' � � � ' � �� e

� _ � ! � � � �� � � � � � � � � _ � q " � � � � � � � � " � r ' � � � � ( � � � ' ( � . � � � � � ) � � r

( � � � � � � � $ � [ � � [ � � _ � � � � d � � & � _ � q " � p � � � d � � � � [ � � [ � � � � � � ' p

� � _ ' � � � " � � q " � "

� � � � s � ' p � � ( � � p _ � � d � r ! � " � # � � � � ! p y b � " � � ! � ' � y ( � � � � � � �

� [ � � [ � � _ � " � � � � � ( � � � " � [ � � ' p . � d p ( � � � � p & ' � � � p ( � � ' � � p e � n � � . � ! %

� � � ( � � � a � � ' " � � � � � � ( � ! � a � r� � � ! � a � x [ " � � ( � � " ' � � � . p � p ' ( �%

� � o p x ( � � � � � � � � r * � �� �

( b # � ' . � � � ! � r e � � � ' ' � � � � � � � � � � � �� �

(

� � �� �

( b � � � $ _ � � � " � � � � � � ! p � � �% � r � � � � ! � � � � . � ? ( � � � � *

! � � # � � � � � � � � � . � � � � � � � � � e � � � � " ( � . � � � � � ' p & # � � � ' � ( � ! � a � �

" � ( ' � � � b � ' . � [ � � � p � � � y � � � � � � ! p & � ' � � . p � � � � [ p x � " � �

" � � � e �* � 9 : � � � � � � � � � �

� 9 � � ) � � � � � �A � � , + � � � � �

� + � � � � � � - � � � � 9 ( - � � � �

�� � � � � � # � � � � " � � � � � � � � � � � � � � " � � � � � � � & � � � # � � � " � �

* � 9 � � � � � � - � : 6 �� 5 9 � , � � 3 . 0 � � / � 0 � �

- � : 6 �, + � � �

� � � �( � � � ) 6 9 � � � � : � � + - 9 9 � � � � � � . 1 � � 3 4 4 � 1 � � � 2 � / � 1 3 �

A � � - + - 9 9 � � � � � 6 + 6 � ) � : � � �

- � : 6 �6 ( , 7 � - ) � � � - � : 6 �

� � � : � � + � ( � � � � �

� � � � � � � - � : 6 �

- + � ( � � � � �6 + � - �� - 7 � 8 . 1 � � 3 4 4 � 1 � � � 2 � / � 1 3 � 1 � � 1 4 / 3 � 3 1 � 3 4 7 6 �

5 � 6 � � 6 7 8 , � 9 - � : 6 �

- + � - 9 � < � � 6 � � � � , � = � . 1 � 0 4 - � � �

9 � < � - � < � � 6 � � � � , � = = � � - � < 6 � , = = � � 4 � 3 � � �6 + � 6 � - �

� � � �� � � : � � + � - �

� � � �� � � �

( � � � ) 6 9 � 6 � � � * + - 9 9 � � � � � � 1 / 0 �� / . 1 � � � �

- � : 6 �6 � � � * + � < , 7 � = � � � � 9 ) � � � : � � �

� � � �( � � � ) 6 9 � 6 � � � 6 : � ) + - 9 9 � � � � � � 1 / 0 �� / 2 � . �

- � : 6 �6 � � � 6 : � ) + � < , 8 � = � � � < � � , � 7 � = �

� � � �� . � � 3 � � 3 � 0 4 / - + 2 � 4 , � � 3 1 � 2 � 1 1 � 1 3 � � � , � �

( � � � ) 6 9 � 6 � � � 9 5 � + - 9 9 � � � � � � 1 / 0 �� / 3 4 � � �

- � : 6 �6 � � � 9 5 � + � < , 8 � = �

� � � �* � 9 � � � � � � � * � � . . 1 � � � 3 � 1 . � �

- � : 6 � � , 7 �� � 9 ) � � � : � � �, + � , � - �

� � , � + � - �

� � � �

Page 35: Программирование: теоремы и задачи

� � �� � " � � � � � � � & �� � � � � " � � % & � # & � � � � $ �

* � 9 � � � � � � � 6 : � ) � � . 2 � . � �

- � : 6 � � , 8 � � � � , � 7 � �

� � , � + � � � , � � - �

� � � �* � 9 � � � � � � � 9 5 � � � . 3 4 � �

- � : 6 � � , 8 � �

, + � , � - �

� � � �* � 9 � � � � � � 5 9 � , � � � / � / � 0 0 � �

A � � 6 + 6 � ) � : � � �

- � : 6 �6 ( < , � � = � � � � 9 ) � � � : � � ) � � � - � : 6 �

( 9 � 6 + � - ) 9 � � 9 - � : 6 �

5 � 6 ) � < � 7 � � 6� � � � � � � 6 � � � 8 � = �

� � � �5 � 6 ) � � � �

� � � �� � � �

* � 9 � � � � � � � � � � . . 1 � �� � � � 2 � � � 4� � / � / � 0 0 � �

- � : 6 �5 � 6 � � 6 � � � * � 9 - � : 6 �

� * �� � �

5 9 � , �

� � � �- � : 6 �

- � : 6 �� 5 9 � , �

� � �5 � 6 � � 6 � � � 9 5 � � 9 - � : 6 �6 ( 6 � � � 6 : � ) ) � � � - � : 6 �

� 6 : � ) �� � �

� � � � � � � - � : 6 �

� 9 5 � �

� � � �� � � �

� � � � �

�� �� � " � � � � � � � & � & � � � � � # � � � ! � � � �

t � n � Y � � � � � � " s � � � p ( � � � � � � � � � � � � � � " � � � . p � � � � � � � � � � � � � � � �

� ( � . � � � � � ( � � � � � _ � � � � � � � � � � b ( � � � � � _ � & � � y � " � � ' p . � � � � y � � r

� � ! p ( � " [ � � � & � � � [ � � # � ' . � " � r ' � � � r ( � � p " _ � ( e � ] ! � � � � � p � �%

� . � ! � a � x � ( � � � a � r ' " � � � � � � ( � ! � a � r � � _ & # � � [ � ' � � � � ( � � � � �%

_ � � � � � � � � � � � � ' � ' � � � � � ' � � � � � � ' � � x o � � _ � � � � " � � � [ � � �%

. � [ _ � . � # � ' � � � " � r ' � � � r & � � � � � � # � � � � � � � � ! � � � ' p o � r � � ( _ � � '%

� � � � � r�� � � � � U � � � . p _ � q " � r � � � � � _ � . � & _ � q " � r � � ' y � " p o � r � _ � q " � r � � '%

y � " p o � r " � � � � � � . � [ " � � y � � � � � p [ . � � ' _ � � ! � � # � � � � $ ' � � " � � � p �

� � � & � � y � " � � ' p . � � � ) � � r . � � � � � � ! p b � � � y � � r � � ! p � � # � � � � � �%

' p e � b Z � � � � � � & # � � � ' � . " � ( ' � � � � ' � � ( � ! � a � � � � _ � q " � r � ! . � � � r

� � q � � [ � p � � [ � . � � � " � � � � � � ! p� e �

�� �� � � v � � s � s � z � � � � � ~ y z � z | z

t � f � n � ] ' ( � . p ! � � � � p � � � � " � [ y � " � " � � � � � " . p � � d � � � p ' . � " x o � r

! � " � # � � " � � � � ' ' � � � ! ( a � . y ( � . � q � � � . p � y # � ' � . � � � � � � � � ( � # �%

' . � # ? � � � [ � � ' p ! � � � p & � � q � � . � # � ' . � # [ � p ( � � " ' � � � . � � � _ � _ ' � � �

� � _ � � � � y � ! # � ' � . � � ' ' � � � �� b � � q " � � # � ' . � � � q � � � ' ( � . p ! � � � � p � �

[ � . � � # � � ( � � " � � � � � ! � e

� � � � � U � � " � � ! � " � � � � p *% ( � ! � a � x ( � ' . � " � � � � � . p � � ' � p x � ! * [ %

. � � ' _ � y ! � � # � � � r & � ( � � " � . p x o � y & � y � " p � . � � ' � � # � ' . � � � � � � � � � *

� . � � � � y � " p � � � � ! � a � p " � ( ' � � � � & � ' . � � s ' � � � � � ( � � � � ' y � " � � # � �� S O � � S � U � � � � ' � � � � � � � x ' ( � . � � ( � � � [ � � � � � ' � y � ( ( � " � � � q � ' � �

� � � ' � p � � _ � � � � r � � � � d� ) � q � � � � _ q � ( � � " � � � � � � . p � � � � ' � � � �%

� � � � � p � � ' ' � � � � [ � � x o � � ( � � p " _ � & � � � _ q � ' # � � � � p � � " � ( ' � �%

� � � � � ( � ! � a � � & � _ � � � � y ' � � � � � [ � � d � � � y # . � � � � [ � . p d � & # � �

� � ! � � ' � p ' � � � ' � y # . � � � � � # � � � ' . � " � � r ( � � s � � � ! � � x � � � � � � " � �

� � � � � r � � � � � � a � � + � ( � � � a � ( � � . p � � � � . # d � � � p ( � ' � � � � � � � x ' ( � .%

� � ( � � � [ � � � � � � � � ( � . # � � � ' p b ) � � ! � " � # � & _ � _ � � � � � p � & � �% ( � .%

� � & ( � " � � [ � � ' � � ' � � � _ � � � � q y � & � � ( _ � � � � � � . p � � � � �

� � ' � � � � � � �

� � � � . � ! � # � ' . � � � . p � y � . � � � � � � � � � & ) � � & � � � � & � � � _ q � � _ � � � � c )%

� � � � q � � ' � � � �

� # � ' . � � � . p � � � � d � � � � � " � � � � d � � � � ! � " � # � � &) � � & � � � e � � � � " � a � � � � � � � � ! � � � � � ) � � r ! � " � # � $ � ! � " � # � � � x _ ! �%

_ � � b � x _ ! � _ � [ o � r � � ! � ( � " s � � � ' � p x # � q � � ( � _ � � � � p ( � " ! � � p !%

_ & � � ' ( � . � � � p ( � � " � � � � � � � � ' � � � � � � � � � ( e � � � � � � _ q � � � . � � � � b � � _

� [ � r � � ' p [ � ! � � _ � ' � � e � . � � � � � � � s � � d � � � p & ( � . � � � � � � . p � r ( � ( � #

b � ' ( � . p ! x o � r � " � � � � � # � ' _ � � ( � � � � � � � � � � � � � � � � e �

Page 36: Программирование: теоремы и задачи

� - �� � " � � � � � � � & �� � � � � " � � % & � # & � � � � $ �

t � f � f � � � � � # � ' . � � p � ' � ( � ' . � " � � � � � . p � � ' � � � ! { � . � r & � " � � � a �

" � � � _ & � _ � � � � y � � _ � _ � p � � ( ( � a � � � � ( � � � � � p � � ' p " � � � � ! � ( � " � p "

b � � � _ ' _ � � � " � � � e � �

t � f � t � q � � . � � � # � � p ! � " � # � " . p ( � ' . � " � � � � � . p � � ' � � r � . � r � � " � � � a &� _ � � � � y � � _ � _ � p � � ( ( � a � � � � ( � � � � � p � � ' p � � � � � ! � ( � " � p " b � � �

_ ' _ � � � " � � � � e � �

� ) � � r q � _ � � � � � � � � � � � � ' p � ' p ! � " � # � � � ( � � � � q � � . � ' . � q � � p

" � � � x � � � � ! ( � � � � � � � � � � � � ( � " � [ � �� � � � y � � q � � � � . � � ' �%

_ � � o � � � � ( � � � [ � � � b � � � � � � p � � ' ( � ! � � � p & # � � � � � x o � � ' p � � ' ( � . � q � � � �

� � � � _ ^ q � ( � � � � � � � � # � � � � � [ � � � � � p � & � ( � ) � � r � � � � � ( � � ' _ � � ( � �%" � . q � � p e �

��

D � � � � � � �

�� �� � � z � � z � ~ | } � s z w � v � ~ � { �

k � n � n � � ' � p � � � � � � � � � � ( $ a � . � # � ' . �� � � � [ � � ' p ( � ' � � � � � p

� � ' ' � � � � � � � � � �

� � ( & ' � " � � q � o � r � � q � # � ' . � & " . p _ � � � � � � � � � �

6 � � �

� � � 6

� � ( �� S O � � S � U � � � � � " � # � ' � . � � �

� � � � � ( � � � � [ � p � � � � �� � � � [ � �%

' p & # � � [ _ � q " � � a � . � � # � ' . � � y � " � . � � � � � � � �

� � ( ' � � . p _ � q � � � ! &' _ � . p _ � � � � � � � � � � � ( �

� � � � � U � � � " � [ � � ' # � � � � p & # � � # � ' . � � � � � � � � � ( � � � � � � �

� � (

( � � " ' � � � . p x � ' � [ � r � � # � . p � � � � _ � � � # � � � ! � � # � � � p � � ' ' � � � �� � � � [ �%

� � � � � � � � � ' � " � � q � � � " � � � � � q � # � ' . � � [ " � � ! � � � " � � � � ( � . � � � � &� ' . � � ( � � a � ' ' � � � [ � � � � � � � � � # � � ' p ( � � � ' � � � � � _ � � � ) . � � � � � � � ��

, + � � �� , 3 4 3 1 3 � 0 4 � � � 1 3 1 3 0 � . 3 / / 4 . � / 0 3 � . � 1 3 � 3 / . � 4 3 1 / 0 �

5 � 6 � � , 7 8 � � 9 - � : 6 �

� + � , � - � ) + � , � - �� . � � � � 3 4 3 1 3 � 0 4 4 / � 1 4 . � , � - � � � � . � ) � �

5 � 6 � � ) 7 8 � � 9 - � : 6 �

) + � ) � - �6 ( . � ) � 7 . � � � ) � � � - � : 6 �

� + � ) �

� � � �� � � �� . � � � � 3 4 3 1 3 � 0 4 4 / � 1 4 . � , � - � � � . � � � �

� � � 2 1 � 1 / 0 . 4 0 � . � � � 4 . � , � - � �, + � , � - �

� � � � �

k � n � f � � � � p " � � � � � � d � � � � ! � " � # � ' � � � � � � � _ � & � ' ( � . p ! x o � � � �%

� � � � � � � � ( � � � � * ) . � � � � � � � ( � � p " � # � � � b � � �

6 � � � 6

� � * e �

Page 37: Программирование: теоремы и задачи

� �� � � � � � � � & � �

� � � � � U � �, + � - �� 2 1 � . � 1 , � � 1 3 1 3 0 � . � 2 � � 1 � . 1 3 � �

5 � 6 � � , 7 8 � � 9 - � : 6 �

) + � , � - �� , � - � � 4� � 1 3 1 3 0 2 � � . 4 1 0 / 1 - 3 . � �� 2 � - 3 1 � 4 3 0

3 � 1 � 2 1 � 3 1 / 0 � ) � 1 � 0 1 - � 2 4 4 3 � 3 1 � �

5 � 6 � � < ) 8 - = � � � < . � ) � 7 . � ) � - � = � 9 - � : 6 �

� � � 2 � 3 1 3 1 0 � . � ) � - � 4 . � ) � �

) + � ) � - �

� � � �� � � � �

� S O � � S � U � � � � � _ � ( � � � � � � � � ( � � . � q � � � � � � q � � � � * � � , ( � �%

� � � _ � � � � � � � � � � � � � [ � � � � ' o � ' � � x o � � � ! � � # � � � p � � � �

, [ � ( � � " . � q � � � y � � d � � � p � � � [ x � # � ' . � " � r ' � � � r & ( � � ( � � a � �%

� � . p � � � � ( � � � o � ' � � x � [ � . � � ) ` � _ � � � � � � . � � � � � � �

�� �� � w � v � ~ � { � � v � � � u z � �� � �

k � f � n � � � � " . � q � � p � . � � � � � � ' � � � � � � � _ � ! � � � � � p { |} ~ { b # � ' . � � ( �%

� � a � r ( � � ' � � � � � � � _ � { ) . � � � � � � � � � [ � . p d � > { | } ~ { " . p � � _ � � � � � � � >

� " . p � ' � y { e �

) ( � � " . � q � � " � � � � d � � � p�

� � � � � U � n b ' � � � � � � � _ � ' . � p � � � � e �� ' � p * $ ( � . � q � � � . p � � � a � . � � # � ' . �� n � ! � [ p s � � � ' ' � � � � � � � � � � (

� � � � � � ! _ � " . � � *� b � � � � r $ � � � � � � � � * & ! � � � � � � * � � � � � � � � *

� � � _ " � . � �� e � � ' . � " � � r � � � � ! � _ [ " � � � � ( � . � � & � ' . � ( � � " � . � � ' p

� � *� + � ! � � s � � � ' ' � � *% ( � � p " � # � � � � & � ' . � _ � q " r � ! ) � � y � � � � ! _ � �

� � � " � . p � � ' � � ( � � p " � # � � � � x [ � r � � ' ' � � � % ( � � p " � # � �� � ' . � � � ' ' � �

*% ( � � p " � # � � � (

6

* & � � � � ( � � p " � # � ��) � ( � d � � & _ � _ ( � � � [ � � ! � � � � p *% ( � � p " � # � � � r � � ' ' � � � � *% ( �%

� p " � # � � � r b � ! � � y q � ) . � � � � � � � e � � ( � � � o p x ) � � � � ( � � � [ � � ! � � � � � p

� . � � � � � � ! � ( � ' � � � � ' p � � _ �

, + � - �� 3 / / 4 . . 1 . � 1 1 0 / 1 , � � 2 � � 1 � . 1 3 3 � 3 �

5 � 6 � � , 7 � � 9 - � : 6 �

� � � 2 � 1 � / � � � . 0 � , � � 2 � � 1 � . 1 3 3 � 4 3 / / 4 . . @ , � � 2 � � 1 � . 1 3 3 � 4 �, + � @ ; , �

� � � �

�� �� � � � � � � � $ � � � � � � � � { |} ~ { � \

� � � [ � � � � ( � � � [ � � ! � � � � � � ' � ' � � � � � � � � & # � � � � � � � � _ � � � � � � ' . �%

� � � � � " � � ( � � p " � # � � � y � � � � ! _ � " . � � � � [ � . p d � * � � " � � ( � � p " �%

# � � � r � � � � ! � _� � ' � p ( � � a � " � �

� � � � * � + � + � � � ( � � � � ,

( � � �

6

6

� ' . � � � � � � � � � ! _ � � � � � � � � � � � � � � � � � � � � � � � � � ( � � p%

" � # � � � r � � � � ! � _ � � � � � � � � � � � b � � ! � � � � � � � � p " � � � y # � ' � � r � � ' ' �%

� � � e �

( � � p " � # � � � r ( � � p " � # � � � r

( � � p " � # � � � r

� � �

� � � " � ( � � � [ � � ! � � � � � � *% ( � � p " � # � � � � � � � � ' ' � � � � � *% ( � � p " � # � � � r

� ' o � ' � � . p � � ' p � � _ �

) + � � �� ) - � 0 3 � @ , 4 � 4 ) � �� . � - � � � . � ) � 1 . � 1 1 0 / 1@ , � � 2 � � 1 � . 1 3 3 � 3 � � / 0 0 � - 3 / / 4 . . 3 1 4 � 3 1 3 4 � / 1 �

5 � 6 � � ) � , 7 � � 9 - � : 6 �

* + � ) �

� + � ) � , �

� + � � 6 � < ) � @ ; ,� � = �� � 6 � < �� - = � 3 4 3 4 3 � 3 4 � � 4 - �

/ � 4 1 3 4 1 < *� �� � = �

) + � � �

� � � �

� . � p � � � � � � [ � � � ' ( � � � � � � � . p � � � � � � ' ' � � � " . p ! � ( � ' � � � ! . p � � � � �

' . � p � � p $ � [ � ! � � # � � � � � �� � � � � ! � � � � � � [ � ! � � # � � � � � � � � ( � ' . � " � � y

) . � � � � � � � # � ' � _ � � & ( � " � � � � d � y ' p ' . � p � � x & # � $ ( � ' . � " � � r ! � ( � ' � �%

� r � � � ' ' � � � ) . � � � � � � + � _ � q " � � d � � � ' . � p � � p ( � � � ! � � " � � ' p � " � �

� ! " � y " � r ' � � � r �

- � � � � - � + � . � * � � - � �

* � + � * � � - �

� � + � � � � - �

Page 38: Программирование: теоремы и задачи

� �� � � � � � � � & � �

� . �

- � � � � - � + � . � � � � - � �

� � + � � � � - �

� � + � � � � - �

b � x [ � � � . � p ! _ � � � � ( � ' � . � [ � ) � � � ' . # � � � � � � # � � � � � � � � �

� � � � # � � � � � � � � � e� � � � � � " � r ' � � � � b � ! p � � � ) . � � � � � � � ! ( � � � � � � � � � � ! _ � e � � q � � ( � �%

� ! � � " � � p ' p ( � � � " � � � � � � � � � � � � ( � . � � � � � " � y ' . � � � r �

b � e ( � � � r � � � � ! � _ � � _ � � # � . ' p b � � � � e ?

b e � � � � � r � � � � ! � _ _ � � # � . ' p b � � + � e � . � � � _ � � # � . ' p & � � ) . �%

� � � � � � s � � � � � � p d � � # � � � " � � � � ) . � � � � � � ( � � � � � � � � � � ! _ � b � � � � e

� b � � � � � �

6

� � � � � � e " �q � � . � � � # � � " . p � � � � � � � " � r ' � � � p� ] � � _ & ( � . # � � �

* � + � * � � � + � � � � � + � * �

5 � 6 � � < * � 7 8 � = 9 � < � � 7 8 � = � 9 - � : 6 �6 ( < * � 7 � = � � � < < � � � � = 9 � < < � � 7 � = � � �< . � * � � - � 7 � . � � � � - � = = = ) � � � - � : 6 �

- � � � � - � + � . � * � � - � �

* � + � * � � - �

� � + � � � � - �

� � � � � � � - � : 6 �� < � � 7 � = � � � < < * � � � = 9 � < < * � 7 � = � � �< . � * � � - � 8 � . � � � � - � = = = �

- � � � � - � + � . � � � � - � �

� � + � � � � - �

� � + � � � � - �

� � � �� � � �

b � ' . � � [ � � � � � ! _ � � � _ � � # � � � ( � � � � � � � [ � � � � � ) . � � � � � � � � y

� � � � & � � " � ( ' � � � � [ � " � r ' � � � p ? � ( � � � � � � � � � [ � � � � ( � � � � �� e, ' � � s � ' p . � d p ( � � � ( � ' � � p � � ! . p � � � ' . � p � � p � [ � � � � � � � � ' ' � � ��b � � � " ( � � q " � � � �� � ' . � � [ � � � � � � _ � ( � � � � � � � � � ( � . � p � � ' p � � � ( � � a �%

" � ' . � p � � p & � � � � ! � [ " p � � ( � � ( � ' . � " � � r � � � � ! � _� e� � � � � � � � � � � � � � ( � � � # � r " � � _ � � � [ � � o � � � � _ � � ' o � ' � � x%

o � � ) . � � � � � � � � � ' ' � � � ( � � � # � ' . � � � � [ . � � ' _ � y � � � q � � � r �

� � � � � U � f b ' � � � � � � � _ � " � � � � � � e �+ � � � ' � � � ( � . � � � " � � � # � � � " � � � � � � $ _ � � � � � _ & � _ � � � � � r ' � � !

� " � � _ � q � _ & � ! � � � � � y � " p � ' � � � . _ � � " � � " � � � y & � ! _ � q " � � � $

�� �� � � � � � � � $ � � � � � � � � { |} ~ { � �

� " � � " � � � y � � � _ " � . � � �

rr r

r r r rr r r r r r r r

@@@I

����

AAAK

����

AAAK

����

BBM���

BBM���

BBM���

BBM���

" � � � � � � � � � p & # � � ' � � � . _ � � � " � � � � � � a � � _ ' � � � p p � � � ^ _ � q " �%

� � _ � q _ � " � � ' � � � � " � � � � � a b � ' . � _ � q � _ � � � ' � � � � � � � y � . �

� � ! e � � � � " ( � . � q � � " . p ( � � ' � � � & # � � _ � . � # � ' � � � ( � " . � q � o � y ' � � � �%

� � � _ � # � ' � . � ' � p ' � � ( � � p " � � r _ � & � � � � � � � � ! � ( � . � � � p � " � � � ! � p " � �

a � . � _ � � � Z � ( � d � � � y � " �� Z � � � � ! � ( � . � � � # � ' � p " � � � � � ( � " � � � � ( �

( � � � � . �# � ' . � � _ � q _ � + � � � � � � � ! # � ' � . � _ � q _ � � y% ' � � � p p y

� � � ' � � � � _ � � � � " � � � � � b � � q � � � _ � q _ � e [ " � � ! � ( � ' � � � � � � � � � . p%

� � � # � ' . � � � � ' s � � � ' ' � � ��] ! � � � � ! ' � � � � � � � � � � � � ' ' � � � � � � � � � . p � r ) . � � � � � � � . p ) � � � �

� � � � � " � � � � # � . � � � r � � � ! � � � � q � � ' " � . � � p & � " p � � _ � � � p � � � � � a � ( �%

� � y � " � � _ � � � ' � & � " � ! � ( � ' � � � � � q � # � ' . �� ] ! p � � � � � � � . p � r

) . � � � � � & ! � � � � � � � � � ' � � � � . � � �∞ � ' _ � � � � _ � � � � � [ � . � � � � ! _ � � p � %

' b " . p ) � � � � � � " � ' � � � � ( � � r � � ( � p _ _ � � � x e � � � � ) � � � ' # � � � � � & # � �

� � � b ��

�∞

e + �� � � � " � � _ � � � � ( � p � � � ' p � � � � � r ( � � � . � # � � � ) . � � � � � & �

� ! � � � � � � � & ! � � � � p p [ � ' _ � � � # � � ' � p x � _ � � � � _ � � � p " � � � � �� � � _ ( �%

' � � ( � � � � � � ! � � � � ' � ) . � � � � � � ( � � p " _ � � � ! � � ' � � � � p & ( � _ � � _ � � � �

� � � ' � � � � � ' p [ � ' _ � � � # � � ' � p�� � � ! � ( � ' � ) � � � � � . � � � � � � � ( � . � ! � � � � � � � � � � p _ � q _ � # � ' . � � �

��

�� � � � $ ( � � ) � � � ' � � � p p � � _ � q _ � � � � � � ( p � . p x � ' p _ � q _ � � (

� � ( � � � � � " � � [ � � � � ! . � q � � � � ) � � � � � . � � � � � � � � � ( ' � � � & ( � ' _ � . p _� � ! . � q � � [ � . � � ) ` � _ � � � � r � � � � � � � & � � � � � [ x o � r " � ( � . � � � � . p%

� � r ( � � p � � & _ � � � � _ � � � # � � � � # � ' . � ( � � � � � � � y b � " � ( � . � � � � � _ ' � �%

� � � � � � � � � ' ' � � e �) [ " � � ! � ( � ' � � � p ' � � � � � � � � # � ' . � � � � ' � y � � � d � � � y " �%

� � � � & � � � � � . p _ � � � � � � y � � � � � � � �� � ' � p � � � � � � � � ( $ � � '%

' � � & ( � " . � q � o � r ' � � � � � � � _ �� � � � d � � � � � " � � � � � [ " � # � ' . � � � �

" � ( ? � # � ' . � � � � � [ " � � � � � � � � � p _ � _ � # � ' . � & ' � � p o � � � � � �%

d � � � � � � ( � � a � ' ' � ' � � � � � � � _ � _ � . � # � ' � � � � � � d � � " � � � � � [ " � �

' � _ � � o � � p ' p� � � ' . � � � � d � � � � _ o � � � " � � � � � [ " � � y � � � � � p � ( �%

Page 39: Программирование: теоремы и задачи

� � �� � � � � � � � & � �

� � � � � � � r *� � � _ � � � [ � � ! � � & � ( � � a � ' ' � � � [ � � � . � � � � � � � � � ' ' � �

� � � � � � � � ( " � . � � ' p � � " � � # � ' � � � � � � � � � � � � * y � � � p � ' p # � ' . � � �

" � � � � � & � � � � * � � � � � � � ( y � � � � � ' p ^ q � � � ' � � � � � � � � � � � p � ( � � p " _ �

� � ! � � ' � � � � p # � ' � p � � ' ' � � � $ ) . � � � � � & ^ q � ! � � p � d � � ' � � s ! � _ � � � � �

� � ' � ��+ � _ � q " � � d � � � � . � � � � � � [ " � � � ! � � � p � � _ ' � � � . p � r ) . � � � � �

" � � � � � � ( � � � o � � p � � � � � � ' � � � � � � � � � � x # � ' � p & � � � ' � � [ � " � � d � � ' p

� � � ! . p � � � � ' � _ � � o � � � p " � � � � � � � ' � ��� � � � � � � � � ' p � � � � � � � � . � � � � � � � � d � � � � � " � � � � � ' # � � � x � ' p # � ' . �

� � � " � � � _ o � � � ! � � # � � � p ( � � � � � � � � r *� � _ � q " � r � � � d � � # � � � �

[ � p % � � � & � � � # � � # � � � � ' . � � [ � ) � � y # � ' . � [ � . p d � * & � � ' � � � � r � � � ?� � _ � p � � � d � � � � � ! � � � � ' p � � % � � $� � ' . � � # + * & � � � � � d � � � # � � � � �

� � � � � � " � � � � ' � � b � # e �� . p _ � q " � � � # � ! �� � �

* � � ' ' � � � � � � � ( � " " � � � � � � ' _ � � � � � � # � � � �

' � " � � q � � � � � d � � # � � ' � y � s ( � � � � _ � � b ' � � � � r & � � _ � � � � � _ " �%

. � � $ " � � � y ( � � & ( � _ � � � � � r " � � � ! � � � � ! _ � �� � �

* e � � � � d � � #

[ " � � � � ! � � � p � � � � � � � � � & � ' . � ' � � p o � � � � � r # � ' . � $ � � _ ' � � � . p%

� r ) . � � � � � # % ( � " " � � � � � ? # % ( � " " � � � � � � � ! � � s � � � � � � � � � $ & � ' . � � ' �

� � � � � � d � � � � � . p � � � b � # � ' � � � ' � � & . x [ � r . � ' � � [ � � ! � � � � � . p � %

� � � � " � � ) . � � � � � � � � ( � " " � � � � �� eZ � � � � � � & # � � � ' � � � � � ' � p � � � � q " � � � p � # % ( � " " � � � � � � � � . p � � � � ! �%

� � ' � � � � � � . p _ � � � # & � � � � � � _ o � � � ! � � # � � � p *�

� y � � � � . � � � � � � � � � _ � � � �

, + � �

� � � � 1 � 0 � - � 2 � 1 � 1 . � � 1 � � 1 � 3 � 3 �� . � - � � � � � . � , � 7 � . � , � - � 7 � � � 7 � . � � � � - � 2 � 1 � 1 . � � 1 � � 1 � 3 � �. . / 0 3 � / 0 4� . � - � � 3 - / 4 3 � � 3 � 4� � 1 3 1 3 0 / � 1 4 . � - � � � . � , � �

5 � 6 � � , 7 8 - � 9 - � : 6 �

� � � � / 3 1 3 1 0 � 3 1 / 0 3 4 . � - � 4 . � , � �, + � , � - �� . � - � � � . � , � - � 7 � . � , � 7 � � � � 7 � . � � � � - � 2 � 1 � 1 . �

� 1 � � 1 � 3 � . 1 � 1 � - � � 3 1 � . � � 3 � � 3 � � / 3 � � - � � 3 1 �

� � � . � / / 0 3 � . 4 0 � � 1 � � 1 � 3 � / 0 � - � 2 � 1 � 1 . . / � �

� � � �

� _ � # � ' � � � � ' ( � � � � � � � . p � � r ( � � a � " � � � � ( � � � " � [ � � ' p ( � � a � " � �

� � ' ' � � � � � . � � � p � � � . p � � � ' � � # % ( � " " � � � � � � _ � � � �� � � � � � � �

� � � 2 � 1 � 1 . � � 1 � � 1 � 3 � . 1 � 1� - � � 3 1 � . � � 3 � � 3 � � - � � 3 1 �

) + � � �� � � 2 � 1 � 1 . � � 1 � � 1 � 3 � . 1 � 1� - � � 3 1 � . � � 3 � � 3 � � . 1 � 0 4 3 � ) �

�� �� � � � � � � � $ � � � � � � � � { |} ~ { � �

5 � 6 � � < < @ ; ) � - 7 � , = � � � < . � @ ; ) � - � 8 . � ) � = = 9 �< < @ ; ) 7 � , = � � � < . � @ ; ) � 8 . � ) � = = � 9 - � : 6 �6 ( < @ ; ) � - 7 � , = � � � < . � @ ; ) � - � 8 � . � @ ; ) � = ) � � � - � : 6 �

� � � � / 3 1 3 1 0 � . � ) � 4 . � @ ; ) � - � �

) + � @ ; ) � - �

� � � � � � � - � : 6 �

� � � � / 3 1 3 1 0 � . � ) � 4 . � @ ; ) � �

) + � @ ; ) �

� � � �� � � �

� � � [ [ � " � � p ' p � ( � � � � . p � � ' � � ) � � r ( � � a � " � & ( � ' � � � � � � � �

� � s ( � � � � � � � � . p � � �� � ' � p � #% ( � " " � � � � � � ' � � � � d � � & _ � � � � � � ! � �

# � � � � � d � � � & � � � . p � � � n � ' ' � � � � � � ' � � � � r � � � d � � � � , � � � �%

� . p � � & � ( � � � � ' � " � � q � � � � � [ � . p d � � # � ' . � � ' � � � y ( � " " � � � � p p y�

� � _ � � � [ � � ! � � & � � � � . p � � � [ � . p d � � � # � ' . � � � % ( � " " � � � � � � � � � ( � �%

� � � " � � � � p # � ' . � � ' � � � r � � � d � � � � � # � ' . � � � s ' � � � p p y� b � ( � � � � �

' . # � � � � � d � � � � � � � . p � � � & � � ' s � ( � � p " _ �� e � ) � � y � � � � � � � y a � _ .

� � q � � ! � ( � ' � � p � � _ �

5 � 6 � � 3 4 / � � � 0 1 1 . 4 / � � 3 1 . ) � . � 3 � 3 4 � / � 3 � . 1 4 � 9 - � : 6 �6 ( � 3 � . 2 � . � 3 / � 3 1 ) � � � - � : 6 �

2 � 3 1 3 1 0 � ) / 1 2 � . � 3 / � 3 � 3 � ) + � 2 � . � 4 / � 3

� � � � � � � - � : 6 � � 3 4 / � � � 0 1 1 . 4 / � � � . � 1 . � 3 / � 3 1 �

2 � 3 1 3 1 0 � ) / 1 � 1 . � 3 / � 3 � 3 � ) + � � 1 . � 4 / � 3

� � �

� � �

� � ' . � � [ � � � � � � � d � � � � ' � � � � � � � ' p � � � . p � � � r b � � � s ( � ( � " � � � � � _ ' �%

� � . p � � � # � ' . � � % ( � " " � � � � � e � + � ( � � � p � d � r # � ' � � p � � [ � � � � ' � � ' � �%

s � ' p � � � . p � � � & � ( � � � p � d � r # � ' � � � � � q � � � � � [ � p � � � . p � � � �� � ' � � . p � y � � � d � � � y # % ( � " " � � � � � � � � ! � � � � . � ' p � � # � ' . � & � � ( � " " �%

� � � p p � y ( � � � � _ � � b � � ! � � # � � " � � ) . � � � � � � ( � " " � � � � � ( � � � ' � � � � . � ' p e &� � _ # � � � � � . p � � � ' � p � � � � � d � . � ' p�

! � � q � ( � � a � " � � � � q � � � ' ( � . p ! � � � � p ' p " . p � � � � & # � � [ ' " � . � � p

� % ( � " " � � � � � � � � . p � � � � � � � # � . p � � r ' � � " � � ' � � � � � � � _ � �

, + � � � � + � � �� . / 1 � � 2 � 1 � 1 . � 1 / � 8 � � 1 � � 1 � 3 � �

5 � 6 � � � 7 8 � � 9 - � : 6 �� � � 2 � 1 � 1 . � � 1 � � 1 � 3 � . 1 � 1� - � � 3 1 � � . 1 . 0 � - � � 3 1 �

� � � . � / / 0 3 � . 4 0 � � 1 � � 1 � 3 � / 0 � � � 2 � 1 � 1 . . - � � 3 1 �

� + � � � - �

� � � �

Page 40: Программирование: теоремы и задачи

� � �� � � � � � � � & � �

� � ( � � p ! � ( � d � � ( � � a � " � ' � � � � � � � _ � � � ( � ' _ � . � b ( � � " ( � . � � � p &# � � ( $ _ � � ' � � � � � & � � � � � � � � ( � � � + � � � � � � �� �

( � � � ( � � � � e �

* � 9 � � � � � � � 9 � ) < A � � . + � � � = �

A � � �� , + 6 � ) � : � � �

* � 9 � � � � � � � . � � � � : � < 6 � + 6 � ) � : � � = �

A � � ) � * + 6 � ) � : � � �

- � : 6 �) � * + � . � 6 � �

. � 6 � + � . � � �

. � � + � ) � * �

� � � �* � 9 � � � � � � � � � ) 9 � � < � + 6 � ) � : � � = �

A � � ) + 6 � ) � : � � �

- � : 6 �) + � � �

5 � 6 � � < < @ ; ) � - 7 � , = � � � < . � @ ; ) � - � 8 . � ) � = = 9 �< < @ ; ) 7 � , = � � � < . � @ ; ) � 8 . � ) � = = � 9 - � : 6 �6 ( < @ ; ) � - 7 � , = � � � < . � @ ; ) � - � 8 � . � @ ; ) � = ) � � � - � : 6 �

� . � � � � : � < ) � @ ; ) � - = �

) + � @ ; ) � - �

� � � � � � � - � : 6 �

� . � � � � : � < ) � @ ; ) = �

) + � @ ; ) �

� � � �� � � �

� � � �- � : 6 �, + � � �

� + � � �5 � 6 � � � 7 8 � � 9 - � : 6 �

� � � ) 9 � � < � = �

� + � � � - �

� � � �5 � 6 � � , 7 8 - � 9 - � : 6 �

� . � � � � : � < - � , = �, + � , � - �

� � � ) 9 � � < - = �

� � � �� � � � �

+ � ' _ � . p _ � ! � � � # � � � r �) � � � " & � ' ( � . p ! � � � � � r ( � � ' � � � � � � � _ � " � � � � � � & [ � � � � ( � . � ! � �

� " � � � y ' . # � p y� b � � � � � . � � � � b � � ( " � � � y e � [ � # � � � " � ' ( � � � � �%

� � � � � � � e

�� �� � � � $ � � � � � � % � � � � � � & � �� � �

� � � � � � � � _ � ' . � p � � � � y � � � d � � � � & # � � � � � � � � � � [ � � & # � � [ � � ' p

' � � � � � � � r � � ' ' � � ( � � � o � . ' p � � ( � � � � � � � � r ( � � p � � � ) � q � � ' � � # � . �

� � ' � � � � � � � � � p � � _ � � _ ' _ � & _ � � � � � ( � � � o � x � ' p � ( � � p � � b � � ( � � � � � &' ( � � � o p x " � � � � � e & � ! � � � � ' . � � � � p ( � . # � � � � � r . �

� o s � " � � ( � � _ � � # � ' _ � � � q � r � . � � � � � � ' � � � � � � � _ � b [ ' � � � p ' � �%

� � � � � _ � � � � � � e � � _ � � � # � � [ � � ' � � � � � � � � � p � � ' ' � � & � [ � � � � ' . # � r%

� r � � � ) . � � � � � � & � � � ! � [ p s � � � ' ' � � � � � � � # � ' � � � � � � p d � � � & � � � � � �

� [ � . p d � � �� b ! � � ! � " � # � ( � � � � " � � � � � . � � � � � e � � ( � � p � ' � � . � ' p � � ' � �%

� � � � � � � p ( � � � x � � � � � p x # � ' � � � ) � � " � . � � � ' p � � � q � ' ( � ' � [ � � � � � � � p

� � [ � � ) � � � � � . � � � � � � � $ ' . # � r � � p � � . � # � � � ? � � q � � " � _ � ! � � p & # � �

� ' � � " � � � � � � � [ � � � � � � � [ � . p d � > { |} ~ {� + � ( � � _ � � _ � $ � � � " � � � !

' � � y [ ' � � y� b ) � o s � � � � s � ' p _ � � � & ( � � � � " p � � � � � _ � ' � � � x

� � � � � _ � ' � � � x � � � . � ! � a � �� e+ � _ � � � a & � � � � � � � & # � � ' � � � � � � � _ � ! � � � � � p ( � � p " _ � > { |} ~ { � �%

q � � [ � p � ( � . � � � � ' ( � � � o p x � � y � � _ � ' [ � . � � ' � � � � � � � y " � � � � p � �

b ' � � � . � � � e & � " � � _ � ( � � � � � � � � � ' . � q � � � � _ � � ' � � � � � > " � � � . p � �

� � . � _ ���� �� � ~ { s } s } ~ � t v � � ~ � v � u ~ �

k � t � n � + � r � � _ � . � # � ' � � � � � ! . � # � y # � ' � . ' � � " � ) . � � � � � � � " � � � � � �

� � ' ' � � �� � � ' . � " � r ' � � � r ( � � p " _ � { | } ~ {� b ! � � ! � " � # � ^ q � [ . � � � . �%

� � � � e� � � � � U � � , � ' � � � � � � � � � p # � ' . � & � ! � � � � ( � ' # � � � � p _ � . � # � ' � � � � � !%

. � # � y & ( � � ' � � � � � � � p ) . � � � � � � � ' ' � � � ( � ( � � p " _ � �k � t � f � � � � � ( � � � � ! _ � � � � �

�� � � " � � ( � p � � r b � + �

� � � ( e � + � r � �

� � _ ' � � � . p � � � * & " . p _ � � � � � � � ' o � ' � � � � � � # _ � ( � p � � r & ( � _ � � � p

* � � � � ! _ � � � b� � � _ ' � � � . p � � � # � ' . � ' . � s � � e � � � ' . � " � r ' � � � r $ ( � � p "%

_ � ( |} ~ (�� � � � � U � � � ( � � p " � # � � � ' � . � � � � ( � � � � _ � � a � � � � ! _ � � � � � ' � �

b ( � � ) � � � . � � r _ � � � a ' # � � � � � ' p � � � p d � ( � � � � � � _ � � a � & � � ' ( � . � q � �%

� � � � � � � r q � � � # _ � ( � p � � r e � � � . � � " � � � � � � ' p ' . � � � � � ( � � � � & ' # � � � p# � ' . � ' . � s �� � ' � � � # � � � r . � � r _ � � � a � � . � # � � � � � # � ' . � ' . � s � � � � &( � � � r $ � � � p d � � � � , � � � � � � & # � � ( � � � _ � x o � � " � � _ " � � � �%

� � ! _ � � [ � � [ � � � � x � ' p ( � � � � . p � � � ' � � # � . � � " s � . � � r _ � � � a b ( � � � � � �

� � � � ! _ � e & � ! � � � � $ ( � � � r b . � � � � � � � � � ! _ � e � �

k � t � t � � � � � { � � # � _ � � ( . � ' _ � ' � �� � _ � ! � � p b { − � e % ! � � � � x � � ' �%

� � ( � � � ' � _ � x o x ' p � � ! � � _ � � x . � � � � x & ( � � y � " p o x # � � � ! � ' � ) � �

Page 41: Программирование: теоремы и задачи

� - �� � � � � � � � & � �

� � # _ �� b � � ' � " � � � � � � � ! _ � � . � � � � � r � � ! � � d � � � ' p . � q � � p � � � " � � r ( � p%

� � r � e � � ' . � " � r ' � � � r ( � � p " _ � { |} ~ { �

� � � � � U � � � ( � � p " � # � � � � # _ � ( � � % _ � � � " � � � � � & � ( � � � � � � y �% _ �%

� � " � � � � � y $ ( � �% _ � � � " � � � � �� � � � _ � � ( � � p " _ � � � � q � � ( � � � � " � � p

. � � � � x� �

k � t � k � � � q � ! � " � # � & � ' . � . � � � � � p " � . q � � [ � p ! � � _ � � � r�

� � � � � U � � � � ! p � s � ' � � x . � � x � � # _ b � � � ' � p � � # _ ' � � � � � � p d � r

�% _ � � � " � � � � � r e � ( � � � � " s � � ! � � s . # � � � � ' � � ' � � . p � � � � # _ � � � � ( � � p

( � � p " � # � � ) � � . # � ' � � ! � � � � y & � � � # _ � � � � " � � � . # � ( � � p " � # � � ( �

� � ' ' � � p � � x � � � � # � . � . # � b ) � � " � . � � � ' p " . p � ' � y . # � r & _ � � � � � � q � � � �

� � � � y � � � � e � � � � � � � p � y � " � � � ! � [ � � � � � r b ' � � � r . � � � r e � � # _ � ( �

� � q � � � . # & ! � � � � ( � � ' � � � ' � � . p � � . # � � b � � ( � ' � � � � � ( � � p " _ � e

� � � ! � � � o � � � ' p ( � � � � y � � � . # � �

k � t � h � � � � � { � � # � _ � � ( . � ' _ � ' � �� � � ' � � � � � p � y � ( _ . x � [ � . � #%

_ $ � � � � � � . p � x � ( _ . x � � � & � y ' � " � � q � o x� b n � ! � � � � � � _ �%

. � # _ � & � � � p � � � � � � � [ � � � � " � ' _ � � � ! " � $ � y � ( _ . � p � [ � . � # _ �� e

� � ' . � � ( � � � a � r � � [ � . � � { |} ~ { �

� _ � ! � � � �� � ( � � p " � # � � � � # _ � $ � � " � � ' p . x [ � r � ! ( � � p " _ � � & � '%

( � . p ! � � � � � y � " � y ( � � " " o � y ! � " � # � y� Z � � � � & � � ' ' � � � � � � � p � � #%

_ � ( � � # � � � " � & [ " � � ' � � � � � p � ( _ . x � [ � . � # _ ^ q � � � ' ' � � � � � �%

� y � � # � _� b � . p y � � � � � � p � ( _ . � r � [ � . � # _ � ( � . � ! � � � ' ( � . p ! � � � � p

" � _ & ' � � � . � � � � � ( � � # � � & ( � � ( � � p " � # � � � � � � # � _ ( � � . � � ) � � � !%

. � d � �� e " �

�� �� r ~ � } ~ s v � s } u ~ � w � | ~ t w z t � z � } s } ~ �

� � ~ t v � � ~ � v � u s

� ' � p � � � � � ' p { � � ! . � # � y ( � � � ' _ � � � � r � � � ' & _ � � � � � ( �%

! � � . p x � ! � � " � � � ! � � d � � � � � � � ( � � " � . � � p & _ � _ � r � ! " � y � [ � � � � y

� � � � _ � � � � r � p q � . � �� b � ( � � � � � � � � ' � ' _ � y � � � � � � � y � � � � � � � " �%

' � ( _ ` � _ a � � � � � � � � � * �+

� � �� �

( , � � � � & � � (� e + � " � ( � � p " � # � � p _ � �%

� � ( � � � ' & ' " � . � � _ � _ � � q � � � � � p d � � ! � � d � � � � � r b � ! � � � � ` � _ a � �

� � � � � � � e �n � ! � � � � ' p & # � ' . � � ! � � d � � � � � r ! � � � ' � � � � � � . p _ � � � � [ � � � � � � �

� � � � � . � � � � � � � & � � � � � � � � � & _ � _ � _ � ! � . � ' p � � ' ( � . � q � � _ � � � � � � . � q%

� � ' � p x � . � � � � � � � � � ! � � s � # � ' . � � ! � � d � � � � � r ( � � � � � � � � � � � � ' ( �%

. � q � � � � _ � � � � r �

�� �� � � � � � � � � � � � � � � � ! � % � � % � � & � � � � " � � � % � � � � � � & � � � �

k � k � n � � � _ � ! � � p & # � � ' . � q � � ' � p ( � � � ! � � . p � � � � � . � � � � � � � ' � � � � � � �%

_ � { _ � � � � r � � � � � p d � |} ~ � { � b � " � { � + � · · � � � · { e �

� � � � � U � � � ' � p � � � � � ' p � . � � � � � � ' . � q � � ' � � � � [ � . � � �� � . p _ � q " �%

� � � ! { � � � ! � � q � y � � ' ( � . � q � � � r _ � � � � r ! � ( � � � � _ � . � � � � � � ! . p � �%

� � ! � � d � � � � � r b � [ � � o � � � r _ ` � _ a � � � � � � � � � e ? � y � � q � � ! � ( � ' � � p

� � � " � ( � ' . � " � � � � � . p � � ' � � � ! � � [ � . � � # � � � � . � r � � " � � � a� � . p � " �%

� � � [ � � ! � p " � ( � . � � � ( � ' . � " � � � � � . p � � ' � p � . p � � & # � � [ � s " . � � � ' � � . �� � � � � r �� � � � ' � � � � � ' � � � � � ' p { � ( � ' . � " � � � � � . p � � ' � � r � ! � � . � r

� � " � � � a� � ' � ) � � ( � ' . � " � � � � � . p � � ' � � � � ! � � $ � � � # � � � d � . � � � � � �

" � . [ � " � � � _ � � � � � � � � " . p � � ! � y ( � � p " _ � � b � � " � � � ! � � � � � � �

[ . [ � � ( � � � � . p � � e � � � . # � � � & # � � �>

{ � $ # � � � � � � [ � � � . � ' p

" � _ � ! � � p� �

� � � � r ' ( � ' � [ � [ p ' � � � p � � q � ' � � � � $ � � ' ' � � � � � � p " � � � � � � � � �%

� � � � � & � � ! � � _ � x o � � � y � " � � ( � . � � � � p � . � � � � � � � & � ' � ' . � � p ' p � � � � &# � � " � � � � � � ' � � � � � � � q � � � � � � p [ � . � � � . � ' � p � ��+ � ' . � q � � ! � � � � � � p & # � � |} ~ � { �

>

� { |} ~ { ( � � ( � " y � " p o � � � � - & ( �%

' _ � . p _ � ' � � �|} ~ { � + |} ~ � � |} ~ � |} ~ \ �

� � �

� | } ~ {

� � � � � p ( � . � � � � � ' . � � � � � y � � � � � p d � |} ~ � b { % e + |} ~ � { − � _ � q " � ��

� � � ' � � � . x [ � r � . � � � � � � ' � � � � � � � _ � & � ' ( � . p ! x o � r � � . p _ � ' � � �%

� � � � p ) . � � � � � � � � � ' ' � � � � � y ( � � � ' � � � � � _ � & � � � [ � � � � � � � � � � { |} ~ {

" � r ' � � � r & � � _ # � � � � d � � . � � � � � � [ . � ! _ � _ � ( � � � � . p � � � , " � � _ � � .%

� � � � � � ' � � � � � � � _ � & � ' ( � . p ! x o � r " � � � � � ( � � � a � � & � � q � � " � r ' � � �%

� � � p � [ ' � � � �� � � � � " � � � ! ( � � � � � � ��

k � k � f � ] � � � � ' p � � ' ' � � a � . y # � ' � . � � � � � � � � ( & ( � � # s � � ' � # � ' . �

� � � � � � a � � � . p � � � � ( � � � � ' y � " p � �� , � ' � � � � � � � � � p ) � � � � � ' ' � � ? # � ' . �

" � r ' � � � r ( � � p " _ � � � (�

� � � � � U � � � . p _ � q " � � � # � ' . � � � � " � � ( � " ' # � � � � � � & ' _ � . p _ � � � !

� � � � ' � � � # � � � ' p � � � ' ' � � �� � � ' . � ) � � � � � ' y � " � r � � ' ' � � � � q � � ' � �%

� � � p � ! � ( � . � � � p ! � � � � � � ( � � p " _ � � � ! � � ' � � � � p & � ' ( � . p ! p ' � � " � � � p

� _ � � � � � ' � � _ � q " � � � # � ' . �� �

, � � � � � � & # � � ) � � � � . � � � � � � � � ( � � � ' � � � . p � � # � ' . � � � � ' ' � � � & _ � _

[ � . p d � � ' � � � " � � � y & � � ! � ( � ' � � � � � y � " � ! � � � � � � �

� ' � p � � _ q � � � � � " ' � � � � � � � _ � & � _ � � � � � � ( � ' . � " � � � � � . p � � ( � � � �%

" � � ' p � p " � # � ' � � # � y ' � � � � � � � � _ � ( � � � " � . p � � [ � � � � � + � # � s � ' � �%

_ � r ! � " � # � �

Page 42: Программирование: теоремы и задачи

� �� � � � � � � � & � �

k � k � t � � � � ' ' � � � � � � � � � � � ( a � . y # � ' � . ( � � � ' � � � � � p ) . � � � � �

� � _ & # � � [ # s � � � # � ' . � d . � ( � � � " � � # s � � � � b � � � � � p p � ! � � � � r

( � � p " � _ � _ � q " � r � ! � � ( ( e �

� � � � � U � � � � � # � . � ' ( � d � � b � � � ' ( � � � � � � � . p � r � � ' ' � � e � ' � # s � � � &� ( � � � � $ � ' � � � # s � � �� �

k � k � k � ] � � � � ' p � � ' ' � � � ! { # � ' � . � � - " � � − � & _ � q " � � � ! _ � � �%

� y � [ " � � � � ' ' � � � � � � � � p _ � _ �% [ � � � � � � ' . � � � � ! � . � r � � " � � � a�] ' ( � . p ! p ( � � � � � _ � �� % r [ � � � � � � � -� � �� % r [ � � � � � � � �� � � � ' � �

' � � � � � � � r & � � ' � � � � � � � � � p � ' � # � ' . � ! � � � � � p ( � � p " _ � { � �

� � � � � U � � , � ' � � � � � � � # � ' . � ( � ( � ' . � " � � � [ � � b ' � � ( � � " " o x

! � " � # e & ! � � � � ( � ( � � " ( � ' . � " � � � � � � _ " � . � �� � � � ! . p � � � � � � � [ " �

� � ' � � � � � � � � � � � ' � � � � " � . � & � � " _ a � � r ( � � . � � _ � " � _ � ! � � p & # � � ( � ' . �

� d � � � � . x [ � " � � # � ' . � & � � . � # � x o � � ' p � � . p _ � � � ( � ' . � " � � y [ � � � y &� " � � ( � � � � . p � � � ( � � p " _ �� b � � � � � � � � ( � ' . � � d � � � � � % [ � � � � � _ � � a

# � ' � . � " � � ( � � � � . p � � � ( � � p " _ �� e �

q � � . � � � # � r � . � � � � � � � � q � � [ � p ( � � � � � s � " . p �% � # � � r ' � ' � � �

' # � ' . � � � p � � � ' � � " � � � # � � r� � � � ) � � � ( � . � ! � � � � _ � p � ' ( � � � � � � � . p � � p

! � " � # � �k � k � h � � � � { # � ' � . � ` � _ a � p � & ( � � � � � � x o � p b � � � � y e ! � � # � � � p

�� � � � � � � � [ � � ' p ( � � � ' � � � � � p # � ' . � � � � _ � � ( � � p " _ � & # � � [ ! � � # � � � p

` � _ a � � � � � [ � � . � b ' � y � � � p p ( � � p " � _ " . p # � ' � . ' � � � � � � ! � � # �%

� � p � � � e � � � ' . � " � r ' � � � r ( � � p " _ � � � { �

� _ � ! � � � �� Z � � � ' � � � ' ( � ' _ � � ' � � � � � � r " . � � { b _ � _ ) � � ' " � . � � p &' � � � � � � � . � � � � � � � ( � y " � � � y e � ( � � � o � � p � � % r ' ( � ' � _ # � ' . � & " . p

_ � � � � y ! � � # � � � � ` � _ a � � � � � � � � � � � � � � � � � � ( � ' # � � � � p " . p � ' � y � &' _ � . p _ � � � � � � ' p # � ' � . � ' � b � e + � & ( � ' . � # � � � . � � _ � � ( � � " � . � � p & ' _ �%

_ � � � � � ' � � � q � � � � # � � � � p � � ! � � o � � p # � ' . � � ' � b � e + � � "

k � k � � � � � { a � . y # � ' � . � " � � ( � ! � � � � � � " � { � � � � _ � � ' � � � � � �%

� � � p � y & ' " � . � � ( � � p " _ � { " � r ' � � � r �

�� �� � v � t � � s } } � s t v � � ~ � v � u s y z � z | ~

k � h � n � � � _ � � � � � � � � � . p � � � � ! � � q � � p ' . � q � � ' � p b # � ' . � ' � � � � � � � r

� � � � y " d � � ' . # � � e � . � � � � � � � � � ' _ � � � p ' � � � � � � p q s . � � � � ! { _ � �%

� � r �

�� �� � � � % � & � � � � � % � � � � � � & � � # � � � ! � � \

� � � � � U � � , # � � � " � r � . � � � � � � ' � � � � � � � � � � � � � � r " � � ' � � r � p%

q s . r _ � � � � p ' � � " � ( � � � y �� � � � [ � � { − � ' � � � � � � � r � q . � � � � � � �

� � � p d � r ' . � q � � ' � � � � � � ! � � � � � _ � � � � ! ' . � " x o � � � [ � . � � ' � . p � � � �

� � � � q " � � � p� �

k � h � f � ! _ ' ( � � � y � # � � [ � " � � p ' " & # � � " � � � r _ � � � � p $ ' � � r � p%

q s . r ' � � " � { _ � � � � r & ' " � . � � � � � � � { − � � ! � � d � � � � � r� � � _ � ! � � p & # � �) � � � � � � ! � � q � �� b � � ' � _ � � � � r � � � ! � � ' � � ' " & � � � ! � � ' � � ) _ ' ( � � � � e

� � � � � U � � ] ! � [ � � ! � � _ � � � � � � # _ � � � & � � ! � � d � � � � � p $ . � � � p � �

� � q " � � � � � � � . # � � � � � ' { � � � d � � � � � � � � � � � # � � { − � � s [ � � � ��

� � _ � r � � � � � ' � p ! � � b " � [ � � . � � � � _ � q " � � � ' . � " x o � � � � � [ � � � � � p d �%

� � # � ' . � ' � p ! � y _ � � ( � � � � � � � [ � . � � # � � � � � e � � � ) � � � ' " � � # � � �

� � ! � � � � � � � � ' � � � . p � � ' � � � � � d � � � p � � ' � � _ � � � � r � � � ! . � # � y ' � p !%

� y _ � � ( � � � � � � y � � � q � � " � ( ' � � � p & # � � ' � � r � p q s . r _ � � � � p $

� . x [ � r � ! � � y� � . � � ( � � ' � � � � [ p ' � � � � � � [ " � � ' . � " � � p ! � � � � & ' _ � . p _ � _ � � � � r

_ " � � � � � � � � � � � � � � ( � � � � � � . � � b � � � ' � p � � � _ � ! � . � ' p . � � # � " � � � y e �� � � # � . � � y { ? ( � � _ � q " � � � ! � � d � � � � � � ( � � � � � � � � � � � . p _ � � " � � _ �%

� � � p & � � ' . � � ' � p " � � � � � ( � � � � � � � d � y � � _ � � & . x [ � r � ! � � y � � q � �

b ' � � # _ � ! � � � � p ' " � e � _ � ! � � p ' p ' � � � � p q s . � � �n � ! � � a � � � q " ) � � r ! � " � # � r � ( � � " " o � r � � ) � � r ! � " � # � � " � _ �%

! � � � � & # � � { − � ! � � d � � � � � r � � " � ' � � � � # � � � � � � . p _ � " . p � � y � q " �%

� � p ' � � � � � � p q s . � � � & � � " � q � " . p � � � � & # � � [ [ � " � � p ' p & # � � " � � � r

_ � � � � p p � . p � � ' p � � _ � � � $ � ' . � ( � � " ( � . � q � � � . p � r � � � � � � ! � � ' � � ��b � ' . # � � ' � � � � � � � _ � & ! � � p ( � � " ( � . � q � � � . p � r � � � � � & � � � q � � [ �%

" � � p ' p � � � � ( � � � � . p � � ' � � & ' " � . � � � ' � � � { − � ' � � � � � � � r $ _ � q " r

' � � � � � � � � � ' � ' . � " x o � � ( � � � ' � + � ( � � � � � & # � � ' � � � � � � � _ � � � � [ � �

� y " d � � ' . # � � ! � � # � � � . p � � [ � . p d � ' � � � � � � � r � e

k � h � t � � � _ � ! � � p & # � � � � q � � � � r � � ' � � r . s � _ � r � ' � � r � p q s . r

� ! { _ � � � � r b � " � � � � � � � � � � e & ' " � . � � \ { − � ! � � d � � � � � r �

� � � � � U � � n � ! � [ p s � _ � � � � ( � � � ! � � . p � � � [ � � ! � � � � { ( � � � ' � � �%

� � � _ � � � � � _ � q " � r ( � � � b { � ! � � d � � � � � r e � , � . � q � � � � " � . p � � � ( � [ �%

" � � � . � r � b [ � . � � � p q s . y � ' � � � r ( � � � e � � ( � � � � � � � d � y � b [ � . � � . s � _ � y e �

� ' � � & # � � ' � � r . s � _ � r _ � � � � p � � " � � ' _ � � p ' � � " � ( � � � � � � � d � y b { − �

' � � � � � � � r e & � ' � � r � p q s . r $ ' � � " � ( � [ � " � � � . � r b � o s { − � ' � � � � �%

� � r e � �

k � h � k � � � _ � ! � � p & # � � � � ' o � ' � � � � � . � � � � � � � & ( � ! � � . p x o � � � � �%

� � � � � � � � � � � � � � r � � ' � � r . s � _ � r � ' � � r � p q s . r ' � � " � { _ � � � � r

b � " � � � � � � � � � � e & ' " � . � � � � � � � \ { − � ! � � d � � � � � r�

Page 43: Программирование: теоремы и задачи

� �� � � � � � � � & � �

� � � � � U � � � ' � p � � _ � r � . � � � � � � ' o � ' � � � � � + � [ . x " � p ! � � � � ( � �%

� � � � � � � � _ _ � _ � r% � � � � ( ( � � ! { _ � � � � r & � [ " � � ' . � " � � p ! � # � � � p%

� p ( � � � � � � � � � � � q � � � � � � & � [ " � � ' � � � � � � p & ' _ � . p _ � _ � � � � r

b�e _ � � % � � ^ q � ( � � � � � � . � & � ^ _ � � �% � � ^ q � � � � � � . � ?

b � e _ � � % � � ^ q � ( � � � � � � . � & � � � o s � � ^ _ � � � � � � � � � � . � ?

b o e ^ _ � � �% � � ^ q � � � � � � . � & � � � o s � � _ � � � � ( � � � � � � . � ?

b � e � � ^ _ � � � � � � � � � � . � � � � _ � � � � ( � � � � � � . � b � � � ' � p � � ' _ � �

� � ' � � � � � � � . � ' p e �b + � ( � � � � � & # � � � � � � � � d � � � ' � � � � � � � � � ' # � � � � � [ � . � � � p q s . r

_ � � � � p� e � � � � � � � ( � b � e & � # � � � " � � & � � � � � � ^ q � � _ � ! � � p ' p � � ' � � � �

. s � _ � � � & � � ' � � � � � p q s . � � & _ � _ � � [ � � [ . � � � ! . p � � � " � . p%

� � r d � y ' � � � � � � � r� � x [ � r _ � � � � p � � ( � b � e � � � � � d � � ' � _ � ! � � p ' p ' �%

� � . s � _ � � b � ' � � � � " � . � & � � � � � q � � ( � � � ! � � . p � � � [ . � � # � � p & � � � � � p p

� � ! . p � � � � � ^ q � � ( � . � � � � y ' � � � � � � � r e & � � ^ q � � � � � q � � [ � p ' �%

� � � p q s . � ? " . p _ � � � � r � � ( � b o e � � � [ � � � � � + � _ � � � a & . x [ � r _ � � � � p

� � ( � b � e � � q � � [ � p � ' � � � . s � _ � � & � ' � � � � p q s . � �, [ � ! � � # � � # � � � ! � ��

� � �� _ � . � # � ' � � � _ � � � � r � ' � � � � � � ' � � x o � y _ �%

� � � � � � p y � ( � � ' . � " � � & _ � _ � � � p x � ' p ) � � ( � � � � � � � ( � � � # � � � " � � �

' � � � � � � � � b � ! � � � ' � � � ' � � � � � � � � & _ � � � � _ � _ � � � � � ( � ' � � � � � � � x � ' p

� ' _ � _ � � � � ! . p � � � � � e �

' � � � � � � � � � � � � � � � � b \ % e �

� � � - - - - -

� � � - - - - -

� � � � � − � - - − �

� � � - - - - -

� � � � � - − � - − �

� � � - � � - − � − � %

� � � - - � � − � − � %

�� � � − � - - − �

� � � - - - - -

� � � � − � − � - −

� � � - - � � − � − � %

� � � � � - - − � − \ %

� � � � � - − � - − �

� � � � � - - − � − \ %

� � � - � � - − � − � %

��� - � � � � − − �

� � ' . � " � � r ' � � . [ � a � � [ . � a ( � _ � ! � � � � & _ � _ � � � p � � ' p � � . � # � � � � +

+ � � � � b \ % e � b _ � � � � x � � q � � � � ' ' � � � � � � � � p � _ � # � ' � � � � � � � � ' � � �%

�� �� � � � % � & � � � � � % � � � � � � & � � # � � � ! � � �

d � r ' p � � [ � � � � _ � � � � p & ( � � _ � � � � r � � � ! � � ' � � � � � # � � � & ' � � # _ � ! � �%� � p ) � � r � � � � ( � . � � � � � � ! � ' . � q � � � _ � � � p & " . p _ � � � � � � � � ' � p � " � �%

' � � � � � � p p � a � � _ � e � ] ! � � # � . p � � � + \ { & � � _ � � a � � + b ( � � � ' � _ � � � � &

_ � � � � " � y & � ! � � ' � � � & # � � � � � � � � � ' p � ' p _ _ � � � � � � � � b�e e � ] ! � � [ . � a

� � " � � & # � � ( � � . x [ � � � ! � � d � � � � � � � ' � p � � � " � # � r � ' y � " � & ( � � _ � � �%

� � � � � � � p d � � � ' p � � [ � . � � # � � � � � " � � � a � � � _ � � � ' y � " " � r ' � � �%

� � . p � � � � ! � � q � b � � ( � � � � � � � � # � � � � ! . p � � � � � ( � � " " o � y � ! � � d �%

� � � � r e � ( � � ' � � � � � � � � �% _ � � � p � �% _ � � � p � � q � � � _ � ! � � p ' p & # � � �% _ � � � � p

� p q � . � � b � � � � � ' � � � � � � � � # � � ' � � � y ( � � " " o � � � � ! � � d � � � � � p � � e &� ( � � ' � � � � � � � � o # � ' � � � � �% _ � � � p � � ! . p � � � � � q � � [ � p . x [ � &( � ' _ � . p _ ( � � �% _ � � � � p � � # � � � � � � ! � � ' � � �� b � � � � � � � � � & � � q � � ! � � �%

� � � p & # � � � " � � � ! � ' y � " � � � ! � � d � � � � � p � � � � ! � � q � � & � � ) � � � ! � � d � � � � � �

� � � [ o � � ! . � d � � � � � � � � q � � � � " � . � � p� e q � ' . � � ' y � " � ' � y � ! � � d �%

� � � � r � � " � # � & � � � � � p d � � � � � ' \ { " � ( � � � � [ � � _ � _ � � � � � �

\ { − � ! � � d � � � � � r & # � � � � � � [ � � � . � ' p " � _ � ! � � p� �

k � h � h � � � � � { � � ! . � # � y ( � � � ' _ � � � � r � + � r � � ' � � r � p q s . r

� � � � � � r ( � � � ' _ � � � � & ' " � . � � � � [ � . � � { � d |} ~ � {e − � ! � � d � � � � � r

b

d |} ~ � {e $ � � � � � � p d � � a � . � � � & ( � � _ � � � � � � �

>

{ e �

� � � � � U � � � � � # � . � � � r " s � ( � [ � " � � � . p b ' � � r � p q s . r _ � � � � p e & �

( � � � � [ " � � � ' _ � � p � � � � � r ( � � � ' � � ' � � & # � � � � � � � � � � � q � � � ' _ � � p

. � d p ' � � " � � � y & _ � � ( � � � � � � . . � # � � ( � [ � " � � � . x b ( � � � � � � � d � � _ � � % � �

� o s . � � # � ' � � ! " � y _ � � � � r e � � ' . � � ( � � " � . p � p ( � [ � " � � � . p � � � � � � �

( � � . � � ( � r ' _ � r ' � ' � � � � b � ' � " � . p � ' p � � ( � � & ( � � � � � � � d � � � [ � �%

x � & ( � � � � ' � � � � " � . p � ' p � � ( � � � � � _ " � . � � e & � � " . p � # � ' � � � _ � �

( � � � " � [ � � ' p � � � � " � � & � " . p { # � ' � � � _ � � $ d |} ~ � {e � � � " � �� � _ �%

q " � r � � � � � � � � � � � [ � � � � � " � � # � ' � � � _ & ( � ) � � � � ' � � � [ " � � { − �

� � � " . p � ( � � " � . � � � p ( � [ � " � � � . p � � o s d |} ~ � {e − � � � � � � � � ! � � � � � � �

� � ' � � ' � � " � ( � � � � � � � d � y ( � [ � " � � � . x� �

k � h � � � � _ � ! � � p & # � � � � _ � _ � r � . � � � � � � � � y � q " � � � p ' � � � � � � p q s . �%

� � � � � � � � � � ( � � � ' ' � � " � { _ � � � � r � � � � q � � � � � � � � � � � � � � � � ' " � . � � p

) � � � � � � � # � � ! � { � d |} ~ � {e − � ! � � d � � � � � r�

� � � � � U � � � ' � p " � � � � _ � r � . � � � � � � � � _ � q " r � � � � � � � � � � ' ( � .%

� � � � p � � ' ' � � � � � � # � ' . � � � _ � � � � r% # � ' � � � _ � � & ( � � � � � � � d � y � � � � � � � �

� � �% ' � � � � � � � r � b � � ' � � � � � ( � � � � � d � $ � ' . � � ( � � � � � � . � & � � ( � �%

� � � � . � & $ � � # � � � � x � ' p� e � � � _ � ( � � p � p & # � � ' � � � � � ( � � ' � � �

� � � � � # � ' . � � � & � � _ _ � _ ( � ' . � _ � q " � r � � � � " � � � ! � � � � . � # � � � � � ' p

� � � " � � � a �� � ) � � � " � ' � � � � # � � ( � _ � ! � � p & # � � _ � _ � � [ � � [ . � . � � � � � � & ( � �

� � " � # � y " . p � � � � � � ! . p � � � � y � � � [ " � � � ( � . � � � � � � � � � � � ' � � �

Page 44: Программирование: теоремы и задачи

� � �� � � � � � � � & � �

� � � � � >

{ � d | } ~ � {e − � " � � � � ! � � � p � . � " � � � � � � � � y # � ' � � � _ � � &

_ � � � � � � o s � � _ � � � � ( � � � � � � . �� � � � # � . � � y { & � � _ � � a � � ' � � s � ' p

� � . p _ � � " � � . � " � � b ( � ' _ � . p _ . x [ � r � ! . � " � � � � � � q � � [ � p ( � [ � " � � �%

. � � e � � � ) � � � � � > { − � b � ' � � � � � _ � & _ � � � � � " � � � � & _ � � % � � ( � � � � � �%

. � e � , [ p ' � � � & _ � _ � � " � � [ � � � � p � � ! . p � � � � � � # � r & # � � [ " � [ � � p ' p� � � � � � � ' � � � � � > d |} ~ � {e − � n � ! . p � � � � ' � � � # � " � y � �% . � " � � � � � �%

q � � [ � p � [ � � � . x [ � � � ' . � . � " � � � ' � � � # � � � ' p ' � �% . � " � � � � & � � � %� � � � � � � . � " � � � � � � � ' � � � # � " � y . � " � � � � � � � � � � � � [ � . � � � ( � � r &� � � ' � p � � � & _ � � � � � � � . _ ) � � � � � � � � � [ � . p d � � � � b ( � � � � � � � ' � � � $

. x [ � r e �� � � [ " � _ � ! � � p & # � � � ) � � � ' . # � � � ( � . � � � � � ' _ � � � � � � � � � � � ' � � �

� � � � & � � � " s � � � � � d � � � p ( � " # � � � � � p & ' # � � � p ( � � ) � � � & # � � _ � q " r

� � � � _ � . x [ � r � � � � � � � � � ( � " # � � s � � � � � � � " � � � . � " � � � � � � # �%

. � _ � q " r ' � � ' � [ � . � " � � � ( � " # � � s � � � . p _ � ' � [ �� � � � � ' � � � # � . � " � � �

' � �% . � " � � � � b � . � " � y � �% . � " � � � � e ( � " # � � � � � � � � � � � p � � ' p ? ( � � � ' � � �%

# � " � y . � " � � � � ( � � � � � � � d � r � � ' � � � � ( � " # � � s � � � ( � � � ( � " # � � p x � ' p

� � � � � � d � � �

� � � _ � " � _ � ! � � p ( � � � " _ a � � & # � � � ' . � . � " � � � � � � � . � � � � & � � � � (%

( � � � � ( � " # � � s � � y b � _ . x # � p � � � ' � � � � � e ' � " � � q � � � � [ � . � � � # � . � � � _�� � � # � . � � + - � � � � � � � ( ( � � � . p _ � � � ' � � � � ' . � . � " � � � � � � � . � � � �

� ( � [ � q " � � � . � " � � � & � � � � � � d � � � � � [ � . � � � � � � & � � � _ � q " � r � ! � � ( (

� � [ � . � � � � � � � _ � � & � � � [ � " � � � � � � � � [ � . � � � � � � � � � _ � ��� . � " � � � � � . p � � & ( � � _ � � # � � � � � � � � � � . � " � � � � � � � . � � � � � � �

d |} ~ � {e � � � & ( � ' _ � . p _ � � � � � � ( ( � � ' � { � � � � _ � �� � ' � ( � [ � q " s � � �

� � & _ � � � � � � � � � � � ( � ' � . � � � � � _ � & ( � � � � � � . � � o s _ � � % � � b � � � # � ( �%

# � � � � � � � � & # � � � � � � � � � � � � ( � ' � . � � e � , � ' x " � � ( � . # � � � ' p

� � � [ � � � p � a � � _ � � � � � � �

k � h � m � � � _ � ! � � p & # � � � a � � _ � ( � � " " o � r ! � " � # � � ' � � s � ' p � ' � . � &� ' . � � � � [ � � ' p � � r � � . � d p � � � � � r ( � � � ' _ � � � � p & � ' � � r � p q s . r

� ' _ � � p � � � [ p ! � � � . p � ��

� _ � ! � � � �� � ' . � ( � � _ � � # � � � x � � � � � � � ( � � " � . � . ' p � � � � � r ( � ' �%

. � � � � � _ & � � � � _ � � % � � ( � � � � � � . b � � _ " � � ! � � � � � � � # � & # � � � � � �

( � � � r � e & � � � � ' � � � � ! � � ' � � � � ( � [ � " � � � . p� "

k � h � Y � � � � � { � � ! . � # � y ( � � � ' _ � � � � r � # � ' . � � b � � � " � { e � � � �%

[ � � ' p � � r � � �% r ( � � � ' _ � � � � p & ' " � . � � � � [ � . � � > { � ! � � d � � � � � r &� " � > $ � � _ � � � � � p _ � � ' � � � � � & � � ! � � � ' p o � p � � � � { �

� S O � � S � U � � � � � � � � � � _ � ( � ! � � . p � � ' " � . � � p ) � � ! � > { |} ~ { � ! � � d �%

� � � � r� � _ � ! � � � � _ ) � � r b � � " � � r e ! � " � # � ( � � � � " � � � � � . � � � ( � � � � _ � %

' � x� �

�� �� � � � % � & � � � � � % � � � � � � & � � # � � � ! � � �

� . � " x o � p ! � " � # � � � � � � � � � q � " � � � � ( � � ' � � � � � d � � � ��

k � h � j � ] � � � � ' p { � " � � � _ � � y � � � � " _ � � � � r & � � _ � � � � � � ! _ � � �%

� y � � ' � � � � " � . � � � ! . � # � ( � � � ' � ] � � � � ' p ( � � [ � � & ( � ! � � . p x o � r

( � " � � _ � � � p � � ( � � " � . � � p & � " � � � _ � � � � � � . � � � ! . � # � b � � � � � � � �%

� p o � r & _ � _ � r � p q � . � � e � ] ! � � ' � � � & # � � ' � � " � ) � � y _ � � � � r [ � . p d � � ' � � �

b [ � . � � { % e � " � � � _ � � y� � " � . � � � � [ � . � � { � ! � � d � � � � � r & � � r � � y � � p [

� " � � _ � � � � p � ! ) � � � � [ � . p d � � ' � � �� b � � � " � ' � � � � q � � � �� � ' . � " � � _ � � � p

� " � � � _ � � � & ) � � � � � � � � � � � � � � � y ( � � � � " . � q � � ' � � _ [ � . p d � � ' � � � e

� _ � ! � � � �� � ' . � � � r " � � " � � � � ! . � # � y _ � � � p & � � � y � [ � � � q � �

� [ � � ' � � p $ y � � p [ � " � � � ! � � y ( . � y � r � [ � . p d � � ' � � � � ' � � � � � ' p

[ � . p d � � ' � � � � � "

� � � � � U � � � � � � � � � � � ( � � ' � � � � � � � � � _ � � � � ( � � # � � � " � & y � � � p � ( �%

� � � � � � � r � # � ' . � ( � � ' � � � � � � � y _ � � � � r� b � # � � � � � _ � � � � ( � � � � � � �%

� � � � � � � � � " � (� e � � � � � � ) � � � � ( � � � � � � � � y � � � � � � � � � � � � � _ o � � �

_ � � " � " � � � � � � � � � � _ � � � � � ' � p � *� � � ' . ) � � y � � ! � � � � r � [ p ' � p � � ' p

� � � � � � � � � � � * � , �

� ' . � _ � � ( � � ' � � � � � � � � _ � � � p � b ' � � � � � � � � � � �� � � ( e " �%

[ � � � . � [ * _ � ( � r �% � � _ � � � p & � � � � � [ � . � � # � ' � � ' � � " � � � y[ . [ � � _ � r q � _ � � � � p & # � � � " . p � ' y � " � � � � � � ' ' � � ��

� � . # � � � � � _ x ( � � � � � � � �

, + � � � 6 + � � �� < � = �5 � 6 � � 6 7 8 � � 9 - � : 6 �6 ( , � � ) � � � - � : 6 �, + � - � � + � 6 � - � 6 + � 6 � - �

� � � � � � � 6 ( < 6 � - � � 4 - 3 1 3 � � 4 3 - � . / � � � 3 = ) � � � - � : 6 �6 + � 6 � - � , + � , � - �� � 3 1 3 1 1 3 3 0 1 � 4 � � 3 � 4 - 3 1 3 � 4 1 � � 3 � 3 �

� � � � � � � - � : 6 �6 + � 6 � - � , + � , � - �� . � - 4 � . 1 3 � 4 3 3 0 1 � 4 � � 3 � 4 4 � 4 3 4 1 � � 3 � 4 - 3 1 3 � �

� � � �� � � �

4 / - � 3 � 3 1 . � 1 1 0 / 1 � � � 4 - 3 1 3 �

� S O � � S � U � � � � ' _ � . p _ � � � ' � y � � s y � � � � � � � � y � [ � � � ' � � � � _ � � � � " �

� � � � � � & � s � � q � � � � � ' � � � � � q � �

Z � � � � � � � � _ q � & # � � ) � � ( � � � � � � � � � � � � � � � � � � � � ' _ � � � � � � � [ �%

. � � # � ' � � � � _ � � � p & . � d p � ' . � � � ' � ' � � � . p � � [ � . p d � � ' � � ��

Page 45: Программирование: теоремы и задачи

� � �� � � � � � � � & � �

� . � " x o � p ! � " � # � � � � � � � � � � ( � � � r � ! � . p " � � _ � _ � � � � � � � d � � � p

_ ' � � � � � � � _ ��

k � h � n l � ] � � � � ' p _ � � " � � � � � p � � [ . � a � � � �� �

( + �� �

( � ] ! � � ' � � � & # � �

" . p � � _ � � � � � � � � ' � � � _ � ' � � � � � � � � ! � ( � . � � � � � " � � � � � . p � � & � ' � � .%[ � a ' � � � � � � � � $ � " � � � � � " � � � a � � � b ! � � ' _ . x # � � � � � � y ( � � � ' � # � � � p

� � " � � � � � � . � & � " � ' � � � � � � � ! � � ' � � � # � � e � + � r � � � � _ � � � b � � � & � # � � � " � � &� " � � ' � � � � � � e � � � ' . � " � r ' � � � r ( � � p " _ � (� b Z � � � � � � & # � � ) � � ' o � ' � � � �%

� � � � � p d � # � ' . � ) . � � � � � � � � � � [ . � a �� e

� _ � ! � � � �� n � ' ' � � � � � � � � � � � � _ � _ � � ! . p � � � � ' � � � � � � � p � � ' �

� � ' ( � � � � � � & # � � ' � � r � p q s . r � ! ( _ � � � � r � � q � � [ � p � � r " � �

! � ( ' � � � � � � � r� b Z � � � � � � & # � � � � [ . � a � � � q � � � � [ � p � � � � � ! � � � �%

� � r � & � � � ' s � � � � � ( � � � ' � � � � � � � � � " � y ) . � � � � � � � � " � � � ! � � y � � %

( � " � � � � e " �

��

� � � � � � � � � � � � �

� � � � � � � � � � � D � �

�� �� � v t � z � } � s t ~ { � v w � � u v { { s } � z � ~ ~ ~ �� � �

h � n � n � � � � _ ' � � � � ! � � " � � � � � ' � � ( � � p � [ � ! � � # � . � ' p " � � p � " o � � �

( � " � p " ! � s ! " � # _ � � � � n � d � � � ! � � � � � � p ) � � � [ � ! � � # � � � � � � � b � � _ # � � &

_ ( � � � � � & � � � � ! � � � � � � ' p � � � � � e � � � _ ) � � ( � � o � � ' � � � ' " � . � � p � ] '%

y � " � r � � _ ' � # � � � � � ' p ' � � � � . ! � ' � � � � . � � & ( � . # � x o � r ' p � � _ ' � � � �%

[ � � ' p ( � # � � � � p ' � � � � . ! � ' � � � � . � � �

� � � � � U � � � _ � q " r � � � � � � ( � � � � � � � � � � y � " � � ' p � � " � � � � ! " � y

' � ' � � p � � r � � � ' � � � � � � � � � ( � ' . � � b ! � s ! " � # _ � e �

� � ' � � p � � � , # � � � " � � r + � � � � � � r ' � � � �

� y � " � � r ' � � � � . ' � ' � � p � � �

� ' � � � � � � � ( � ' . � � � �

� ' � � � � � � � 6 + � � ' � � � � � � ( � # � � � � p �

( � ' . � � � ' � � � � � � ( � # � � � � p �

( � ' . � � 6 + � � ' � � � � � � ( � # � � � � p � & �

� ' . � � _ � � a � � � _ ' � � ( � � � � � � � � � _ � ! � � � � ' p � ' � ' � � p � � � � ( � ' . � � & � �' . � " � � � � ( � # � � � � p ! � s ! " � # _ b � _ � � # � � p � � [ � � e � �

� S O � � S � U � � + � d � ( � � � � � � � � ! � � � � p � � � � � � � � � b � � � � � � � � e � � '%

. � � � � ! � " � # � � � � � � � � � � � � � . � " � � � . � r & _ � _ ) � � # � ' � � " � . � � � ' p $

( � � " ( � . � � � � � ' p & # � � ( � � � � � � � � � " � . q � � " � r ' � � � � � � p � � ! � � � � � � " � �%

� � � ' . # � � & ( � q � . r & ' � � r ( � � ' � � r ' ( � ' � [ � [ p ' � � � p & _ � _ ( � � � � � � � �

" � r ' � � � � $ ) � � � ( � ' � � p � s ' � ' � � p � � p � " � r ' � � � p � � � y�

h � n � f � + � ( � ' � � p ( � � � � � � � & " � . p x o x � ! � � _ ' � � � ' � ( � " ' . � � � � �%

" � � � � � �

Page 46: Программирование: теоремы и задачи

� - �� � � � � ! � � � � & � � $ � � � � � " � � " � � � � � � � % � � &

h � n � t � � ( � ' _ � . � _ � � � � � � � � � � ! � _ . x # � x � ' p � � � � � � ' _ � [ _ � �

- � : 6 � � 3 . � � � 4 - � �6 + � 6 � - � �� . 1 � 4 . 4 . 1 3 6 3 - �

+ � ( � ' � � p ( � � � � � � � & _ � � � � � p " � . p . � [ _ � � � � � � � � � � � � ' � � � . p . � [

� � � ' � � � ' _ . x # s � � � � � _ � � � � � � � � � p ( � � [ � . b # � � [ � � � � � � ( � � � � � � �%

. � ' p � � � � � & � � � � e �

� � � � � U � � � � � � � � � � � � � � � � " � � ' � ' � � p � � p � � � ' � � � � � � � � � � � � � � �

b _ � � � � � � � � � p e �

� � ' � � p � � � , # � � � " � � r + � � � � � � r ' � � � �

� y � " � � r ' � � � � . ' � ' � � p � � �

� ' � � � � � �

� � � � � � � �

� ' � � � � � � � 6 + � ' � � � � � � ( � # � � � � p �

� � � � � � � ' � � � � � � ( � # � � � � p ( � � [ � .

� � � � � � 6 + � � � � � � � �

� S O � � S � U � � ! � � ( � � � � � � � � � � � � ' ( � � � � � � � � � . � q � � � � _ � � � � � � �%

� � � � ' � � � _ � � � � " �

� � - � 3 3 1 3 0 � 4 4 . 3 � 0 � 4 � - � 3 3 1 3 0 � 4 1 �

( � � � � � � � � ' p �- � 3 3 1 3 0 � 4 1 �

b � � � # � . � ' � � p � " � � ( � � [ � . � e � , [ � � [ � � _ � � . � q � � � y _ � � � � � � � � � � �

_ � � � # � � � � � � � � � � � � � � � ! � � q � � b � q � � � ( � � � � � p # � ' . � ' _ � [ � _ � $

� ( � � � ! � � . p � � � � � � � � . p � � � # � ' . � � � ( � � � o � � � ' p � _ � � � # � x ( � � p � p e �

h � n � k � � ( � ' _ � . � � ' _ � y ( � � � � � � � � y [ � � x � � � _ q � ' � � � _ � & ! � _ . x # s �%

� � � _ � � # _ � � � ' . � � � � � � p ' _ � [ _ � � ' � � � # � � � ' p � � � � � ' � � � _ � & � �

� � � � � � ! � � # � � � � � # � . � � . � _ � � a � _ � � � � � � � � � p� � ' � � x � # � � � " p & _ � � #%

_ � � _ � � � � � � � � � � � � � ! � � # � � � � � # � . � � . � _ � � a � ' � � � _ �� � � _ � ! � � � � � p

( � � � � � � � & # � � [ ) � � # � ' � p �

� _ � ! � � � �� � � ' � � p � � r [ " � � � � � � � ' � � � � � � & � � � � � _ � � � � � � � � � p &� � � � � ' � � � _ �� "

h � n � h � � o s � " � � � � ! � � q � � ' � p � � � � � y � � � . � ! � a � r ( � ' _ � . p $ ) � �

_ � � � � � � � � � � � � " �

6 + � 6 � - � < ; � � � � 6 6 � 6 � � � � � � � � - � - ; =

( � � ) � � � ! � _ � � � x o � p ' _ � [ _ � " � . q � � ' � � � � � � ' � � � � � � p � � _ � � � x o � r

b � � � ' � p {� � �� , � � � � ! � � d � � � ' p e � � � _ " � . p � p � � _ � � _ � � � � � � � � � � �

�� �� � & � � ! � % � � � �

�� �� � � v � | ~ t s w

� ' � p " � ' p � � # � � p ! � ( � ' p # � ' . � ( � " � s � ' p � � � y � " ( � � � � � � � ' � � � � .

! � ' � � � � . � � � ) y � � � � � ( � � # � ' � p � ) � � # � ' . � b ( � � � ' � � � p � ( � � � � � � � x

� � ( � � � � & � � � ! � � # � � � � e � � � � � � � � � � & � � " � ' � � [ o � � p � [ � d � [ _ � & � ' . �# � ' . � ! � ( � ' � � � � � � � � � �� � . � � _ � � _ � � � � � & ( � � " ' � � � � � ' � [ � � � _ x ' � � � a � x� � � ' . � " � � � � � . p%

� � ' � p ' � � � � . � � � � � y � " � " � . � � ' p � � ( � � # � � � � � x � � ' � � � d x ' p # � ' � ��) � � q � � ( � . p ! � � � � p ' p ` � _ a � � r � � � � � � � � � & _ � � � � � p " � s � ( � � � r

' � � � � . � ' � � � d � r ' p # � ' � � & � � � _ q � ( � � a � " � � r ! � � � & _ � � � � � p ! � [ � � � � �

( � � � r ' � � � � . � ! � ' � � � d � r ' p # � ' � � & ( � � � � � " p � � � � _ � � � � � � � x ( � � # �%

� � � � y�

( � � # � � � � � � p # � ' � p � � � � � �

" � � � � ! � � � p " � ' p � � # � � r ! � ( � ' p x � � _ x ( � ' . � " � � � � � . p � � ' � p

' � � � � . � � �

〈 - � . � [ � . � � ( � � [ � . � �〉 〈 � � . � [ � . � � a � �〉 �

� � � _ q � � � _ x �〈 - � . � [ � . � � ( � � [ � . � �〉 〈 � � . � [ � . � � a � �〉 � 〈 � � . � [ � . � � a � �〉�

Z � � � � � � & # � � ' � � . � ' � � ) � � � � ( � � " � . � � � x

�� �

� �� �

� � ��

� � p � . p x � ' p " � ' p � � # � � � ! � ( � ' p � �� � � � � . � � � � � � ( � � p ! � " � #

� � # � � �h � f � n � � � � # � ' � p � ! � y � " � � r ' � � � _ � � � _ ' � � � . p � x # � ' � p & _ � � � � � p

� � q � � [ � p � � # � . � � " � ' p � � # � � r ! � ( � ' �� , ( � � " � . � � p & p � . p � � ' p . � ) � �

# � ' � p " � ' p � � # � � r ! � ( � ' p x � . � � � � �

� � � � � U � � Z � ( � d � � ( � � � � � � � � � ( � ' _ � . � b � ' ( � . p ! p � ( � � � # � ' . � � r

� � ( � " . p � � � . p " � � ' � � ! � ( � ' � � ( � � � � � � � � p # � � � � � � q � � ( � � � � � � � p � " � �

� ! ! � � # � � � r & _ � ! � � � y � ' _ � [ _ � y e �

A � � � ) � ) � +< � � � � * ) � � � � 9 �� � � 6 ) 6 � � � � � ) � � ) � � � 9 6 � ) � � � � � � � ) = �

� ) � ) � + � � � 6 ) 6 � � �

Page 47: Программирование: теоремы и задачи

� �� � � � � ! � � � � & � � $ � � � � � " � � " � � � � � � � % � � &

5 � 6 � � < � ) � ) � 7 8 � � � � * ) = 9 � < � ) � ) � 7 8 � � � 9 � = � 9 - � : 6 �6 ( � ) � ) � � � � 6 ) 6 � � ) � � � - � : 6 �6 ( � � . ) � � � ) � � � - � : 6 �

� ) � ) � + � � � 6 ) 6 � � � � 9 A � �

� � � � � � � 6 ( 6 : 6 ) < � � . ) = ) � � � - � : 6 �

� ) � ) � + � � � ) � � ) � � 2 � / � 1 3 . � � 1 � � 4 . / 0 4 �

� 9 A � �

� � � � � � � - � : 6 �

� ) � ) � + � � � � 9 � �

� � � �� � � � � � � 6 ( � ) � ) � � � � ) � � ) ) � � � - � : 6 �6 ( 6 : 6 ) < � � . ) = ) � � � - � : 6 �

� ) � ) � + � � � ) � � ) � � 9 A � �

� � � � � � � 6 ( � � . ) � � � � ) � � � - � : 6 �

� ) � ) � + � � � 9 6 � ) � � 2 � / � 1 1 / 1 0 4 . 3 � 4 0 � . - 4 �

� 9 A � �

� � � � � � � - � : 6 �

� ) � ) � + � � � � � * ) �

� � � �� � � � � � � 6 ( � ) � ) � � � � 9 6 � ) ) � � � - � : 6 �6 ( 6 : 6 ) < � � . ) = ) � � � - � : 6 �

� ) � ) � + � � � � � � � ) � � 9 A � �

� � � � � � � - � : 6 �

� ) � ) � + � � � � 9 � � � � � � 3 / � 0 � � � 0 � � 3 � 4 � � �

� � � �� � � � � � � 6 ( � ) � ) � � � � � � � � ) ) � � � - � : 6 �6 ( 6 : 6 ) < � � . ) = ) � � � - � : 6 �

� ) � ) � + � � � � � � � ) � � 9 A � �

� � � � � � � - � : 6 �

� ) � ) � + � � � � � * ) �

� � � �� � � � � � � 6 (

� 0 - � � / � 0 � 3 1 3 � � 1 0 �

� � � �� � � �

Z � � � � p � � & # � � ( � � ' � � � � � � � p # � � � � � � � � � � � � � # � � � � � � � � � � � � � ' � ( � �%

� � q " � x � ' p ' " � � � � � b ' � � � � . & _ � � � � r � � � � q � � [ � p # � ' � p x # � ' . � & � �! � [ � � � � � ' p e � �

� � � � � " s � � � p ( � � � � � � � � � � ! � ( � � � � � � � ! � � # � � � � ( � � # � � � � � � � � # � ' . ��

h � f � f � n � d � � p ( � � " " o x ! � " � # ' " � ( � . � � � � . p � � � � � [ � � � � � � � �

� ' . � ( � � # � � � � � r _ ' � _ p � . p � � ' p " � ' p � � # � � r ! � ( � ' p x & � � � ( � � � � � �%

� x � � & � � � � & ' . � " � � ( � � � ' � � � p � s ! � � # � � � ��

�� �� � & � � ! � % � � � \

� � � � � U � � � � � # � � � � � " � � [ � � r # � ' � � ( � � � � � � � � p # � � � y � � � � � � � �%

q � � � . p ( � � ' . � " x o � r " � ' p � � # � � r a � � ��

� ) � ) � + � � � 6 ) 6 � � � A � � + � � �

5 � 6 � � < � ) � ) � 7 8 � � � � * ) = 9 � < � ) � ) � 7 8 � � � 9 � = � 9 - � : 6 �6 ( � ) � ) � � � � 6 ) 6 � � ) � � � - � : 6 �6 ( � � . ) � � � ) � � � - � : 6 �

� ) � ) � + � � � 6 ) 6 � � � � 9 A � �

� � � � � � � 6 ( 6 : 6 ) < � � . ) = ) � � � - � : 6 �

� ) � ) � + � � � ) � � ) � � 2 � / � 1 3 . � � 1 � � 4 . / 0 4 �

A � � + � 6 : 6 ) � � � � � < � � . ) = � � 9 A � �

� � � � � � � - � : 6 �

� ) � ) � + � � � � 9 � �

� � � �� � � � � � � 6 ( � ) � ) � � � � ) � � ) ) � � � - � : 6 �6 ( 6 : 6 ) < � � . ) = ) � � � - � : 6 �

� ) � ) � + � � � ) � � ) � A � � + � - � ; A � � � 6 : 6 ) � � � < � � . ) = �

� 9 A � �

� � � � � � � 6 ( � � . ) � � � � ) � � � - � : 6 �

� ) � ) � + � � � 9 6 � ) � � 2 � / � 1 1 / 1 0 4 . 3 � 4 0 � . - 4 �

� ) � * + � � � - �

� 9 A � �

� � � � � � � - � : 6 �

� ) � ) � + � � � � � * ) �

� � � �� � � � � � � 6 ( � ) � ) � � � � 9 6 � ) ) � � � - � : 6 �6 ( 6 : 6 ) < � � . ) = ) � � � - � : 6 �

� ) � ) � + � � � � � � � ) �

A � � + � A � � � 6 : 6 ) � � � < � � . ) = ; � ) � * � � ) � * + � � ) � * , - � �

� 9 A � �

� � � � � � � - � : 6 �

� ) � ) � + � � � � 9 � � � � � � 3 / � 0 � � � 0 � � 3 � 4 � � �

� � � �� � � � � � � 6 ( � ) � ) � � � � � � � � ) ) � � � - � : 6 �6 ( 6 : 6 ) < � � . ) = ) � � � - � : 6 �

� ) � ) � + � � � � � � � ) �

A � � + � A � � � 6 : 6 ) � � � < � � . ) = ; � ) � * � � ) � * + � � ) � * , - � �

� 9 A � �

� � � � � � � - � : 6 �

� ) � ) � + � � � � � * ) �

� � � �� � � � � � � 6 (

� 0 - � � / � 0 � 3 1 3 � � 1 0 �

� � � �� � � � �

Page 48: Программирование: теоремы и задачи

� �� � � � � ! � � � � & � � $ � � � � � " � � " � � � � � � � % � � &

h � f � t � � � q � ! � " � # � & � ' . � ( � � � " # � ' . � � � � q � � ' � � p � p ! � � _ � � . �

! � � _ � b � � � q � � � � # � � � � � ' � � p � p e �

w � � � � � # � ' � . � ) � � r ! � " � # � � [ # � � � . . x ' � � � � x � � � _ � r _ � � � � �%

_ � r �

��

〈 a � � �〉

� 〈 a � � �〉-

- -

� �

h � f � k � � � q � ! � " � # � & � ' . � _ � � � q � ( � ' . � # � ' . � � � q � � ' � � p � p

( � _ � ! � � � . p ' � � ( � � � " � ' p � � & _ � _ � � � � � � � b + -� - � e � . � � ��

� � � � � �

b + � \ - - - - - - e � + � � � ' � � � � p ' � � � � � � ' � � x o x _ � � � � � _ � �

h � f � h � � � � � � " � � ! � � � � � p � ( � � � � " s � � � r � d � ( � � � � � � � � & # � � [

� � ! � � d � � p ( ' � � a � . x � " � � [ � x # � ' � � b _ � _ � � ��� & ��

� � � . � " �%

q � ��� $ ( � ' . � " � � � # � ' . � ' # � � � � � � � � � � � . x e �

) � � � � s � ' p _ _ � � � # � � � � � � � � � � � � � . � � � � - b � � � � � � � � � ' � [ � � !%

a � � e �

��

� � � � � � � � �

�� �� � � s u ~

� ' � p � $ � � _ � � � � r � � (� n � ' ' � � � � � � b � � ' � ' � � x o � r � ( � ' _ � . � e

� � ( � ' � � _ ) . � � � � � � � � � ( � � � � � � � ! � � # � � � p � � p � . p x � ' p ( � ' . � " � � � � � . p%

� � ' � � ! � � # � � � r � � ( � ��

, ( � � � a � � �

• � " � . � � p ( ' � � b � � � # � ' � � _ ) . � � � � � � � � � ( � � e

• � � [ � � � � p b � � � ? � � � # � ' � � _ ) . � � � � � � � � � ( � � e

• � ! p � p b � � � � � � ? � � � # � ' � � _ ) . � � � � � � � � � ( � � e

• � ' � b # � ' � � _ ) . � � � � � � � � � ( � � e � � � � & � � (

• � � � d � � � b # � ' � � _ ) . � � � � � � � � � ( � � e � �

b ) ( � . p ! � � ' p � [ � ! � � # � � � p � � & � � ( � � � � � x o � � � ( � ' _ � . p & y � � p �

( � ' _ � . � � � ( � � ' � � _ � � � � � e � � � a � " � � �

� " � . � � p ( ' � � � " � . � � � ' � � _ #

( ' � � � � � � a � " � � �

� � [ � � � � p � " � [ � � . p � � � � _ � � � a ( � ' . � " � � � � � . p � �%

' � � # � � � � a � " � � �

� ! p � p � ( � � � � � � � � & � ' . � ( � ' . � " � � � � � . p � � ' � p # � � ( %

' � � ? � � � ! � [ � � � � � � ! � � s ( � ' . � " � � r ) . � � � � � & _ � � � � r ' � � � � � � � ' p ! � �%

# � � � � � ( � � � � � � � � r � � � � � q � � � � �

� ' � b # e � � ' � � � � � & � ' . � ( � ' . � " � � �%

� � . p � � ' � p # ( ' � �� � � � q � � � � �

� � � d � � � b # e � � ( � � " � . � � � & � ' . � ( � ' . � " �%� � � � . p � � ' � p # � � ( ' � � & � � � � � � ( � ' . � " � � � ) . � � � � � ( � ' . � " � � � � � . p � �%

' � � # �) ( � _ � q � � & _ � _ � � " � . � � � � � � p ' � � _ � ( � ' _ � . � � " . p # � � � � � � � q � �

[ � p � q � ��

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

" � � ' # � � � � p & # � � _ � . � # � ' � � � ) . � � � � � � � � ' � � _ � � � ( � � � � ' y � " � �

� � _ � � � � � � � # � ' . � (� � � � " � ' � � _ � � q � � � � " � . � � � � � � p ' ( � � � o p x " � y

Page 49: Программирование: теоремы и задачи

� � �� � � � � � � � � � �

( � � � � � � � y �� � 1 � � 3 4 1 + � � � � � � - � � � � 9 ( � �

� � 4 3 + 6 � ) � : � � �

' # � � � p & # � � � ' � � _ � � � y � " p � ' p ) . � � � � �

� � 1 � � 3 4 1 � - � � � � � � � � 1 � � 3 4 1 � � � 4 3 � �

• � � � [ ' " � . � � p ' � � _ ( ' � � & " � ' � � � � # � � ( � . � q � � p

� � 4 3 + � �

• � � [ � � � � p ) . � � � � � � �

� � � 4 3 7 � �

� � 4 3 + � � � 4 3 � - �

� � 1 � � 3 4 1 � � � 4 3 � + � ) �

• � ! p � p ) . � � � � � � ( � � � � � � � x � �

� � � 4 3 8 � �

) + � � � 1 � � 3 4 1 � � � 4 3 � �

� � 4 3 + � � � 4 3 � - �

• � � � _ ( ' � & � ' . � ) � ' � ��

• � � � d � � � ' � � _ � � � � � � � � � � � ' � � ) � ' �

� � _ � � � [ � � ! � � & � � � ' � � ( � � � � � � � � r � � ( � ' � � _ � ( � � � � � � � � � � ( �%

' _ � . � � � q � � � ' ( � . p ! � � � � p " � � ( � � � � � � � � � � � � � ' � � ) � '� ) � q � �

� � _ q � � ( � � " � . � � p � � ( # � � � * & ! � ( � ' � �

� 9 � � ) � � � � �

) � * �� ) � � , � � � � 9 � �

� � 1 � � 3 4 1 + � � � � � � - � � � � 9 ( � �

� � 4 3 + 6 � ) � : � � �

� � � �

b ) ( � ! � � . p � � ' � [ � � ' ( � . p ! � � � � p � � � � � ( � � � � � � � y � ! � ' ' _ � y [ _ � &

y � � p � [ # � � ( � ' _ � . p ) � � � � � � . x [ � � � e � � ' . � ) � � � � � � � � [ � p $ � ' �%

� � � � � ' � � � � ' ( � � � � . � � � ( � ' _ � . p $ � ( � ' � � ( � � a � " � � � [ � � ' � ' � �%

_ � � � + � ( � � � � � & � � q � � � � ( � ' � � p

* � 9 � � � � � � � � / . 4 0 � < ) + � � A � � � + � ) � � , = �

- � : 6 �� � � � � 4 3 7 � �

� � � � 4 3 + � � � � � 4 3 � - �

� � � � 1 � � 3 4 1 � � � � � 4 3 � + � ) �

� � � �

�� �� � � � � � � �

� � � � � � � � � � � � � � �

" � � � � ' ' � � � � � � � � p ( � ' . � " � � � � � . p � � ' � � � � _ � � � x o � y ' p � ! � _ � %

� � x o � y ' p _ � � . y � _ � � " � � � � y ' _ � [ � _ * , � � � � � " � � ' � y � � _ � y

( � ' . � " � � � � � . p � � ' � � r � " � . � � ( � � � � . p � � $ � � & _ � � � � � � � � � [ � p

( � . # � � ( � � � _ � � ( � � � � . � � �

• ( ' � � p ( � ' . � " � � � � � . p � � ' � p ( � � � � . p � ��

• � ' . � � � � ( � � � � . p � & � � � � � ( � � � � . p � ��• � ' . � � ( � � � � . p � � & � � � � � * � , ( � � � � . p � �

M N U O � N � � � ' . � " � � � � � . p � � ' � � * , & � � & � * , � * , � ( � � � � . p � & �

( � ' . � " � � � � � . p � � ' � � & , * & * & * � , $ � � � �

� n � n � � � � � � � � � p ( � � � � . p � � ' � p ( � ' . � " � � � � � . p � � ' � � ! � � � � � p & � �

( � � � � ' y � " p o � � _ � � ' � � � � & � � � q � � � � r � � � s " . � � � � � � " ( � . � � � � � ' p &# � � # . � � ( � ' . � " � � � � � . p � � ' � � ! � _ � " � � � � � � # � ' . � � � �

* �

, − �

� � � � � U � � � ' � p � � � � � � � � ( $ ( � � � � � p � � � p ( � ' . � " � � � � � . p � � ' � p�n � ! � � d � � y � � � � � p � ' � � _ � � � _ � � � x o � � ' p _ � � . � � _ � � " � � � � � ' _ � [%

_ � b � � �� � � e �� � � # � . � ' � � _ " � . � � � ( ' � � � � � . � � ( � � ' � � � � � � � � � # . � � ( � ' . � " �%

� � � � . p � � ' � � ' . � � � � � ( � � � �� � ' � � � � � � � � _ � � � x o x ' p ' _ � [ _ b _ � � . x

� . � _ � � " � � � � x e & ( � � � o � � � � s � ' � � _� � ' � � � � � � ! � _ � � � x o x ' p & ( � �%� � � p � � & # � � � � � d � � � ' � � _ � $ ( � � � � p � r ' _ � [ _ � ? � ' . � ) � � � � � � _ & � �

� � q � � � � � � q " � � p & # � � ( � ' . � " � � � � � . p � � ' � p � � ( � � � � . p � � & � ' . � ' _ � [ _ �( � � � � p & � � ! � [ � � s � � s b � � � d � � e � ! ' � � _ �� � � ' . � " � � � � � . p � � ' � p ( � � � � . p%

� � & � ' . � � _ � � a � ' � � _ � _ � ! � � � � ' p ( ' � �

� 1 � 0 �� 2 � / 0 � 3 < � = �6 + � � � � / 3 � � � 1 3 � � 0 4 / - + � ( � � � � �� 2 � � . 4 0 3 � 6 / 4 3 . � � � . 2 � / � 1 � . 0 1 � � 3 � / 0 4 �

5 � 6 � � < 6 7 � = � � � � 9 ) � / 3 � � � 1 3 � � 0 4 / - � 9 - � : 6 �6 + � 6 � - �6 ( < � � 6 � � - = 9 � < � � 6 � � @ = ) � � � - � : 6 �

� � / . 4 0 � < � � 6 � � � = �

� � � � � � � - � : 6 � � � � 6 � � . 3 � � - 4 � 4 � @ �

Page 50: Программирование: теоремы и задачи

� � �� � � � � � � � � � �

6 ( � � / 0 < � = ) � � � - � : 6 �

� / 3 � � � 1 3 � � 0 4 / - + � ) � � � �

� � � � � � � - � : 6 �

� � 1 0 � < ) � � = �

� / 3 � � � 1 3 � � 0 4 / - + � < ) 7 8 � � � 6 � = �

� � � �� � � �

� � � �� � . 4 � � 3 � + � < � 9 ) � / 3 � � � 1 3 � � 0 4 / - = � � � � � / 0 < � = �

� [ � " � � ' p � ( � � � � . p � � ' � � ( � � � � � � � �b � e � ' . � ( � ' . � " � � � � � . p � � ' � p ( � ' � � � � � � ( � ( � � � � . � � & � � ( � � � � � � � �" � ' � � � � � � � " � � � ! � � . � � _ � " � _ � ! � � p � � " _ a � � r ( � ( � ' � � � � � � x ( � �%

� � . p � � r ( � ' . � " � � � � � . p � � ' � �� + � " � ( � � � � � � � p " . p ( ' � � r & " . p ( � ' . �%

" � � � � � . p � � ' � � � � � ( � � " ( � . � q � � � � & # � � " . p � � � ^ q � ( � � � � � � � � &� & � � _ � � � a & " . p ( � ' . � " � � � � � . p � � ' � � r � � � * � , $ � ( � � " ( � . � q � � � � &# � � " . p � ^ q � ( � � � � � � � �� � . p ( ' � � r � # � � � " � �� � . p � � " � r ' � � � p ( � �%

� � � � � ( � � � ' y � " p � _ � _ " . p � � _ � � # � x � ' p ' ( ' � � ' � � _ � � ? ! � � � � � ' s

( � � � ' y � " � � _ � _ " . p � � � . p � � ' � � # � . � ( � � � o � � � ' p � ' � � _ � � _ � � � x%

o � p _ � � " � � � � � p ' _ � [ _ � � ! � � � � � ' s � " s � _ � _ " . p � $ ' � � r � � ! � � a � r &# � � � � . [ � � � ' � � _ � . � q � � . � d � p p ' _ � [ _ �� � � � _ � � # � � � � � ' � � _ ' � �%

� � � � � ' p ( ' � � $ � ' . � � � ' # � � � � p ) � � r ' _ � [ _ � $ � ! � � � � � ' � � ' � �

( ' � � � q � � . � � � # � � " . p * � , �b e � � _ � q � � & # � � � ' . � ( � � � � � � � � ! � � � � d � � � � � [ � � ' � � � � � � � � " � � &� � ( � ' . � " � � � � � . p � � ' � p ( � � � � . p � �� n � ' ' q " � � � � � " _ a � � r ( � " . � � � ( �%

' . � " � � � � � . p � � ' � �� � � � ' . � " � � ! � ' � ' � � p � � � � ' � � _ � � ( � � a � ' ' � � � [ � �

( � � � � � � � � � ' . � � � � � � _ � � � � r ( � � � � q � � # � r � � � � � � ( ' � & � � ( �%' . � " � � � � � . p � � ' � p � � ! [ � � � � � ' p � � " � � # � ' � � & " . p _ � q " � r � ! _ � � � � y

( � � � � � � � � " � s � � � � � � � " � � ? � ' � � s � ' p � � ' ( � . p ! � � � � p ' p ( � � " ( � . � q � � � � �

� � " _ a � � � � ( � � " � . � � � � � ( � � � � . p � � ' � �� � ' � p ' � � _ � ' s � � � � p � � ( ' � �

! � � ! � � # � � & # � � ( � . � q � � � � p � � � � � � � ( � � � � � d � � � ' _ � [ _ � [ " � � � � %

� � . � d p � � ( � ' . � " � � � d � � �� � � � ' � � � & ( � � � r � ( � ' . � " � � r ' � � � � .

( � ' . � " � � � � � . p � � ' � � $ ) � � ( � � � � ' _ � [ _ � & � ( � ' . � " � � � � � . p � � ' � p � � � � �� � " * � , � . � � � & � � � [ � � � ( � � � � � � � b _ � � � � ( � � � � � � � ( � ' . � " � � � �

d � � � � e � � . � # � � � ' p � � � s � � [ � � � � � . � d p � � . � # � � � . � d � � r ' _ � [ _ �

� � " � � ' � � _ � b � � ! � s � � � � � � � x � & � � � � � _ � _ � � � . � p � � � � � � [ � � ( � �%

� � � � � e � � � � � � ' ' . � � � ' p � � ( � � " ( � . � q � � � � � � " _ a � � � � ( � � " � . � � � �

( � � � � . p � � ' � � � �

� n � f � � � _ ( � � ' � � � ' p ( � � � � � � � � & � ' . � � ! � � ' � � � & # � � � ( � ' . � " � � �%

� � . p � � ' � � � � � � [ � p � � . p _ � _ � � . � ' _ � [ _ � �

�� �� � � � � � � �

� � � � � U � � � ) � � � ' . # � � � � ' � � _ � � ' � � s � ' p . � d p � � � " . � � � & � � � _%

� � # � ' _ � ( � � y � " � � _ � � _ � � � � � � q " � � � x � ( � ' . � " � � � � � . p � � ' � p _ � �%

. y ' _ � [ � _ ( � � � � . p � � � � � " � � � � . p _ � � � � " � & _ � � " � � . x [ � � � s � � # � . p%

� � � � � � � ! _ � # � ' . � ! � _ � � � x o � y ' p ' _ � [ � _ � � ( � � � � ' y � " � � # � ' . � � � _ � %

� � x o � y ' p & � " . p � ' � r ( � ' . � " � � � � � . p � � ' � � ) � � # � ' . � � � � � � �

� n � t � n � � . � ! � � � � p ' ( � � � o p x � " � � � � � � ' ' � � � " � � ' � � _ � & ' � � � �%

� � � _ � . � # � ' � � � ) . � � � � � � � � _ � � � � y � � � � � � # � � � " . � � � r � � ' ' � � � ? � ' �

" � r ' � � � p ' � ' � � _ � � � " � . q � � ( � . � p � p ' p ! � � � � � p & � � � � � � # � � � � � _ � �%

' � � � � � r & � � ! � � � ' p o � r � � " . � � ' � � _ � ��

� � � � � U � � � � � _ � " � . q � � � ' � � ' _ � � a � � � � ' ' � � � � � � ' � � � # " � �

" � � � ( � � � r " � . q � � ! � � � � � � p � � ' � �

� � � � � ' � � � � � �

� � � � � ' � � ) � ' � �

� � � � � � r $� � � � � ' � � ( � � �� � � � � ' � � ( � ) � ' � � �

b � � � d � � � [ � � y ' � � _ � � ! � ( � ' � � ( � ' . � " � � � � e � �

� n � k � n � � . � ! � � � � p * ' � � _ � � ' ) . � � � � � � � � � � ( � � & � [ o � � _ � . � # � ' � � �

) . � � � � � � � � _ � � � � y � � ( � � � � ' y � " � � ( & ' � ' ( � . p ! � � � � � � � � � ' ' � � � � ' �%

� � � � � r " . � � > b ( � * e & ! � � � � # � � � p � � _ � q " � � " � r ' � � � � ' � ' � � _ � � �

b _ � � � � � � # � . p � y " � r ' � � � r & " � . � x o � y � ' � ' � � _ � ( ' � � � e � � � � p � �

[ � . � � � � _ � � � � � r _ � � ' � � � � >� b � � _ � � � � � p � & � [ o � p " . � � � � � ' ' � � � �

" � . q � � [ � p � b � � ( e & � � � � � p � � _ � q " x � ( � � � a � x $ � b � e � e

� � � � � U � � � � � � � � p � � r � � � � " � � ! � � � � ' p � ' ' . � # � � r � � � . � ! � a � � r � �, � � ' ( � . p ! � � � � � � � ' ' � � � �

� � 1 � � 3 4 1 + � � � � � � - � � � � 9 ( � �

� � 1 � � 2 4 4 + � � � � � � - � � � � 9 ( � � � � �

� 1 � 0 4 3 + � � � � � � - � � , � 9 ( � � � � �

� " � [ � � � ! � [ � � q � � p � � ' ' � � � � � � � ' � _ � _ ( p # � � _ ' � � � � � �%

� � �� � � ( & _ � q " � p � ! _ � � � � y ' � " � � q � � ) . � � � � � � � ( � �� ) � ' ' � �

� � � � � � � � � ! � [ � � ! � � � � � " � ' � � � . � _ & ( � � � � " p ' � � � . _ � ! � � � & � '%

. � � � � � � � � � � � � � � b � ' . � � � � � � � � � � � � � & ' � � � . � _ � ! � � � ( � � � �%

" � � � e � � " � � q � � � � # % � � ' � � _ � b # ∈ �� � �

* e y � � � � � ' p � � _ � � � � d � � � � � � � �

� � � � � ' � � � � � ' � # & � ' � � . p � � ) . � � � � � # % � � ' � � _ � � � q � � � � r � � &� " p ( � ' � � � . _ � � $ " � � � y ( � � & ( � _ � � � � � � _ � � # � � ' p� � � � ) � � �

b # % r ' � � _ ( ' � e ⇔ � � � ' � # � ��

Page 51: Программирование: теоремы и задачи

� - - �� � � � � � � � � � �

� � � � . � # � � � � � � _ � � � � � & � y � " p o � � � !

� � � ' � � � � � � �

� � � ' � *

b � ! � � y & _ � � � � � � � � � � � - e � � " � . q � ( � � � ' � _ � � p ' p� � � � � � � � � y &� � � ( � � � " � [ � � ' p � o s � " � � ' � � � . � # � � p � � � � _ � � � � p & ' � " � � q � o � p � ' � � �%

� ' ( � . p ! � � � � " � � � r � � � � � � p # � r _ � � � s � � # � . � � [ " � � y � � � � � p

� ( � � � � � � � � r � � � � � � ' � b � � � � � ' � � � � � � � � � ' � + � � ! � � # � � � & # � � ( ' � �%

� � � � ' � � � � � ' � � . � ' p e � � � � ( � � � � � �

� � � '

� � � � � ' �

� � � � � � ' �

� � � � # � � �

� � � � � ' � � � � � # � � �

� � � � � � � � � � � � � � � �

� � � ' � �

� � � � � � ' � + �

� � � _ � � � % r ' � " � � q � � � & � & � & � b � $ � � � d � � � e ? % � r ' � " � � q � � # & � b � $

� � � d � � � e �* � 9 � � � � � � � . 0 �� � / � 0 � � � � 1 � 1 0 . / 1 / 0 1 - 4 2 � / 0 � 3 4 �

A � � 6 + 6 � ) � : � � �

- � : 6 �( 9 � 6 + � - ) 9 , � 9 - � : 6 �

� 1 � 0 4 3 � 6 � + � � �

� � � �( 9 � 6 + � - ) 9 � � - � 9 - � : 6 �

� � 1 � � 2 4 4 � 6 � + � 6 � - �

� � � �� � 1 � � 2 4 4 � � � + � � �

�� � � � � � � � � - �

� . � / � 3 1 + � - �

� � � �( � � � ) 6 9 � � / 0 � � 3 1 / 0 � + - 9 9 � � � � �

- � : 6 �� / 0 �� 3 1 / 0 � + � < � . � / � 3 1 7 8 � = �

� � � �* � 9 � � � � � � � � / . 4 0 � < ) + � � � + 6 � ) � : � � = �� � � / . 4 0 � ) - � � 3 � / 0 1 - � �

A � � 6 + - � � � �

- � : 6 �� � / 0 �� 3 1 / 0 � �6 + � � . � / � 3 1 �

� . � / � 3 1 + � � � 1 � � 2 4 4 � 6 � �

� � 1 � � 2 4 4 � 6 � + � � 1 � 0 4 3 � � � �

� 1 � 0 4 3 � � � + � 6 �

� � 1 � � 3 4 1 � 6 � + � ) �

� � � �( � � � ) 6 9 � � � / 0 < � + 6 � ) � : � � = + - 9 9 � � � � �� � � � 4 / 0 1 - 2 � / 0 �

- � : 6 �� � / 0 + � < � 1 � 0 4 3 � � � � � = �

� � � �* � 9 � � � � � � � � 1 0 � < A � � ) + � � � + 6 � ) � : � � = �� . � 1 0 � 4 � � � � / 0 1 - . ) �

A � � 6 + - � � � �

- � : 6 �� � 9 ) � � / 0 < � = �6 + � � 1 � 0 4 3 � � � �

) + � � � 1 � � 3 4 1 � 6 � �

� 1 � 0 4 3 � � � + � � � 1 � � 2 4 4 � 6 � �

� � 1 � � 2 4 4 � 6 � + � � . � / � 3 1 �

� . � / � 3 1 + � 6 �

� � � �( � � � ) 6 9 � � 1 � 0 4 3 � / 0 1 - < � + 6 � ) � : � � = + � �� . 1 � 0 4 3 � � � / 0 1 - �

- � : 6 �� 1 � 0 4 3 � / 0 1 - + � � � 1 � � 3 4 1 � � 1 � 0 4 3 � � � � �

� � � � �

Page 52: Программирование: теоремы и задачи

� - �� � � � � � � � � � �

�� �� � | s � s � ~

Z � � # � � � p � � � � ( � � � # � � � " p ) . � � � � � � � � � ( � � � & _ � _ � " . p ' � � _ � � & p � . p%

x � ' p ( � ' . � " � � � � � . p � � ' � � ! � � # � � � r � � ( � �� n � ! � � a � ' � ' � � � � � � � � & # � �[ � � � ' p ) . � � � � � � � ' _ � � a � & � ' � � # � . � b � " � [ � � . p x � ' p ( �% ( � � q � � �

� _ � � � a e �, ( � � � a � � ' � # � � � " p � � �

• � " � . � � p ( ' � � r b � � � � � � # � � � " p ) . � � � � � � � � � ( � � e ?

• � � [ � � � � p b � � � & � � � � � � # � � � " p ) . � � � � � � � � � ( � � e ?

• � ! p � p b � � � � � � & � � � � � � # � � � " p ) . � � � � � � � � � ( � � e ?

• � ' � � b � � � # � � � " p ) . � � � � � � � � � ( � � e � � � � & � � ( ?

• , # � � � " � � r b � � � # � � � " p ) . � � � � � � � � � ( � � e � ��

� � � � ( � . � � � � � _ � � � � " �

� � [ � � � � p � _ � ! � � � r ) . � � � � � " � [ � � . p%

� � ' p � _ � � � a � # � � � " � � � � � � � " � �

� ! p � p � � ( � . � � � � & . � d p � ' . � � # � � � " p

� � ( ' � � & � ! � [ � � � � � � ! � � s ( � � � r b ( � . � q � � � r � " � � � � p d � � ' � y e ) . �%

� � � � & ( � � � o � p � � � � � � Z � � # � � � � � ` � _ a � � �

, # � � � " � � r � b � ( � � " � . s � � � r

" . p � � ( ' � � r � # � � � " � e p � . p � � ' p ( � � � r ) . � � � � � � # � � � " � �q � � . � r ' _ � � � � ! � � � � p ' � � _ � � $ � � � � �� � � � � � � � � b ( � ' . � " � � � � �%

d s . $ ( � � � � � d � . e & � � # � � � " � r $ � � � � � �� � � � � � � � � b ( � � � � � � d s . $

( � � � � � d � . e � � � _ � � o � � � p � � � � � & � � � � �

� � � � � � � � � � � � � � � � � � � �

� f � n � n � � . � ! � � � � p � ( � � � a � � ' � # � � � " p x � � � � � � # � � � � r " . � � � � _ &# � � [ _ � . � # � ' � � � " � r ' � � � r " . p _ � q " � r � ( � � � a � � [ . � � � � � � � # � � � _ � �%

' � � � � � r & � � ! � � � ' p o � r � � " . � � � # � � � " ��

� � � � � U � � " � � y � � � � � p ) . � � � � � � # � � � " � � ' � ' � " � � y ) . � � � � � � y

� � ' ' � � �� � � � " � � # � � � " p [ " � � ( � � � � ' � � � p ' ( � � � � � [ � � � p ' . � � �� � �%

' _ � . p _ ( � � ) � � � � � � � � q � � " � r � � " � _ � � p & ' � � � � s � � � ' ' � � � � _ � q%

� � ' � p�� � � " s � � � ' ' � �

� � 1 � � 3 � 1 + � � � � � � � � � � � - � 9 ( �

� ( � � � � � � � �� 1 � . � 4 + � � � � � - �

� � � 3 + � � � � �

�� �� � ! � � � � � � - \

� � � ) � � � ) . � � � � � � � � � # � � � " � [ " �

� � � � � ' � � � � � � �� � � � � ' � � � � � � � �

� � � � �

� � � � � ' � � � � � � � ) � ' � � �

� " � ' . � q � � � � � ( � . � p � � ' p ( � � � " . x (� b � � � " ( � � q " � � � �� � ' . � � � � ' � �

) � � � � � � � ' � � ( � � � � � � � � � � � � � � � � � � � � & ! � � # � � � p _ � � � � y $ � %

# � � ( � � � " . x ( & � � ( ' � � p � # � � � " p � � q � � [ � p ' ( � � � � ' � # � � � " p x

� ! ( ) . � � � � � � �� e

, ( � � � a � � � ( � . � p x � ' p � � _�

� " � . � � p ( ' � � r �

� � � 3 + � � �

� 1 � . � 4 + � � �

� [ � � � � p ) . � � � � � �

� � � � 3 7 � �

� � 1 � � 3 � 1 � < � 1 � . � 4 � � � � 3 = � 9 � � � + � � � 1 3 1 3 0 �

� � � 3 + � � � � 3 � - �

� ! p � p ) . � � � � � �

� � � � 3 8 � �

� � 1 3 1 3 0 + � � � 1 � � 3 � 1 � � 1 � . � 4 � �

� 1 � . � 4 + � < � 1 � . � 4 � - = � 9 � � �

� � � 3 + � � � � 3 � - �

� ' � � �� � � 3 � �

, # � � � " � � r �� � 1 � � 3 � 1 � � 1 � . � 4 �

� f � f � b � � � [ o � . q�

c�

d � � � � � _ � e � � � � � � � � � � � � � [ � � � � � � � � � � �

� � p � � � � � � � � � � � � o � x � � y � � � _ � � b � ` � _ � � � � � � � � � � � � � � � � � � � � � � � �

� y � � � � � e!

� � � ) � � � � � � � [ � � _ � ( � � � � � a � r � � � � � � � � x b � � � � � y "

_ � � � � � � � � � � � [ � � � � � � e � � � q � � � � � [ � � � � � � � � p � _ � � � � r � � � � r!

� # � # $ U # % ] � � � � � � � � & � � � � ' ( � � � � � � � � � � � � ) � ' ( � � � � � � � � � �

� � � *!b � � � � � � � � p p ) � � � � � � � � � � � � � � � _ � � � � [ � � + � � � � ) � � � � � � � � � �

� � � � � � � q " � � � � � � � � � p � � � � � ) � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � �!e � � � � " � � � � � [ � � � � � � � � � � � � � � p _ � � [ � � � � � � x _ � � � � � � +

Page 53: Программирование: теоремы и задачи

� � � �� � � � � � � � � � �

� � � _ � � " � � � � � � � _ � � � � � � � _ � � � � � � _ � � � � � � � [ � � y � � � _ � �! � � �

� � � y � � � � � + p � � ) � � � � � � " � � � � � � � � � � � p! � � � � � � � _ " � � � � � y � � � � � p

� � � � � � � � � � � � � " � � � � � " � � [ � � s � � + � � � � ) � � � � � �! � � � � � � � � � " � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ) � � � � � � � � � � � � � � � � _ � " � � �

� p p � � � p � � _ b ) � � � � � � � y � � � � � � � � � � [ � r � � � � � � � _ � � � � � � � � � + � � � _ �

� � � � _ e � � � � � � � � � � � _ � � � � � � � � � � x! � � � p � � � � � � � r � � � � r � � ) � � �

d � � � � � � � � � � � � � � � � _ � � � � � � � � r " � � � � � [ � � � � � � + � � � � � � � � � � � � � "

� � _ _ � _ _ � q � r ) � � � � � � � � � � � � � � � q � � ^ � � � � � � � � � � � ) � � � � � � a � � � �

� � [ � � � � � � � � � � � � + �! �

%�%�%

� _ � � � � + � � x � � � � _ � � "

� � � � � � x o x � � � � � � � � � � � _ &

_ � � � � � � + � [ � � � � � ) � � � � � � � � q � � � � [ � � y _ � � a � �! � _ � � � � � + � � � � �

� � _ � � � � � � � � � � � � � � � + � � � � � � [ � + � � � � � � � � � � _ " � � � [ _ � q � � p � � � � � �

a � p � � � [ � � � � � � � � � � � � � � � � � � � � � � � � � r � � � � r �

%�%�%

b � � � [ o � � �!

c!

d � � � � � _ �! e ] � � � � � p � � _ ) � � � � � � � � � � � � �

� _ � � � � � � � � � � � � � � � � � � � � y � � � � � � a � � � � � � � � �! � � � � � � � � � � � � � � � �

p � � � � � � _ � � � _ � � � � � � � � � � � ) � � � � � � � �! � � � � � � � � � � � � � � � � � " � � � � �

� � � � � � � � � p _ � � � � � r � � � _ � � � � � � � � � [ � � q � � � � � ) � � � � � � " � � y

� � � � � [ � � [ � � � � � r � + a � � y � � � � � � � � y!

� _ � + � � � �! b � e ! � � � � � � � � _ � � � q � � a � _ � � � � � _ � � � � � � � � � � p � � " + � �[ � � � p � � � � � � � _ � � a � � � � � � o � p � � � � � r! � � � � � ) � � � � " � � � � � � � � � � � _ �

q � d � � � � � � [ � � � � � � � � � � � � � � � � � " � � q � � � � � � � � � � s � � � � � � �! b e � _

� � � p � � " � � � d � � � � � � � r _ � � � � � � � � � � d � � � � � � � [ _ � _ � r � � � ) � � �

� � � � + � � � � � � � � � � � � � � � � � � � � � _ � " � � � � q � � [ � � [ � � � � � � � � � �

� q � � � � � � � � � � � � � � � � p � � � � � p! � � � � � � ) � � � � � � � � � � �! � � � � � ) � � � �

� � � � � � � � � � � � � � � ( � � � � � � � � � � � � � � � � � � � � � � � � � � ( � � � � � " � � � �

� � � � � � � + � � ) � � � � � � " � � � � � � � � � � � " � � � � � � � � � � � � + � � ) � � � � � �

� � � � + ( � � � � �!"

� � � � � � � � � � � � � � � � �

�%�%�%

� � � � � � � � � � � � � � � � � � � + � � � � � � � � � � � � � ( � � � � � � � � � � � � �

� � � " � � � + � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � "�"�!

� # � # $ � # % � � � � � � � � � � � � � � � � �" � � " � � " � � � � � � �� � � � � � � � � � � �

) � � � � � � " � � � � � � � � � "� � � � + � � � � � � � � � � � � � � � � � " � � � � � � � � �

� � � � � � � !

! � � � � � � � � � � � � � � � � �

" # 9 $ % & ' # % ( ) 1 . 0 0 *� �� + / , � 0 * < - . 6 / - % : % # = �

0 % : 6 /

�� �� � ! 1 2 1 3 4 � � �

5 # 6 - % 5 / < - = �

� + / , � 0 * < @ ; - � 6 @ = �

� + / , � 0 * < 7 ; - � 6 7 = �

� + / , � 0 * < 8 ; - � 6 8 = 9

% / & 9

� � � � � � � � � � � � � � � &

: : : / 1 � 0 * 6 @ � 6 7� 6 8 ) ; / 0 � 3 �

( ) 1 . 0 0 *� �� + / , � 0 * < - = 9

< . = - 9 > < ? . � @ � + ( ) A . B C ( ( � � D

> � ( , C E � C ( B . ( C ) A . C B C ( + , ) + E 1 F G A , + � E C @ B C ( � H < I � ( � I C J * ( � �

K J A ( + , ( ; L ( + + I ( + L A @ B , C 9 , + K A E A F H � � J A I A ( B M N , F , + A N

, B E + A � , ) H B A E + O + J * P � A ( C ) A K C B C ( ( M Q N ( + ( A ( C ) A K C B C ( ( M A N

E C @ ) + J + L A ( ( M A , , + R E C @ B C S T A I ) + E H F G A D

U V W 5 % < X Y / & Z 0 % [ W /

6 . = � W / \ + K A E A F ( + ] \ 6 _ N + K A E A F ( + ] \ 6 7 _ N + K A E A F ( + ] \ 6 8 _ _ 9

( C ) A K C B C B *` �` F +O C , � B * \ 6 _ 9

< . = < a b 9

: : : , R H B * 6 � R B A Q + K A E A F A ] N c F A + ( O M J + K A E A F ( M I 9

% / & 9

� � � � � � � � � � � � � � � � � � � � � � � �!

d � � � � � � � � � � � � � � � � � � � � + � � � � � � �

� � � � � � � e � � � � � � � � � � � � � � � � � f � � � � � e � � g!

h � � � � � � � � � � � � � � � � � � �

� � � � � � � + � � � � � "�"�" � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �!i � � �

� � � "

� � � � � � � � � � � � �!

i � � � � � " g � � � � � � � � � � � � � � � � + � � � � � � � � � "� � � �

� � � � � � � � � � " � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � "

� e � � � � � �

� � � � � � � � � � � � � � � � � � � �!

� � � � � � � � � " � � � � � � � � � � + j � � � � � � � � �

� � � � � � � � � � � � �!

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � e � � � � � � � �! �k � � � � l � � + � � � � � � � � + � � � � � � � m � � � � � � � � � � � � � � � � � � � � � � � � �

� � n � !� � � � � + � � � � � � � � � � � � � � � � � � � � � � " e � � � � � � � � � � � � � � � � + � � l �

o 1 2 p 4 q r s 4 " � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � "

� � + � � � � � 2 � t 2 r s 4!

e � � � � � � � � � � � � � � � � " � � � + � � � � u 2 4 1 q v 4 2 u o r q w

q x y � 2 r z!

� � � � 〈 { | 〉 � � + � � l � � � � � � � � q r } r � u s { � ~ u q � u s f � � � � � � �

� � � � � " � � � � � � o x � u3 4 v � + � � � � � � { � o � u 3 4 v � � � � � � � � !

! � � � �

� � � � � � � � � m � � + � � � � � � l � � � � � � � � " � �� � � � � � � � � � � � � � " � � � � � � �

� + � � � � � � � � � � � �!� � � � � � � � � � � � � � � � � � � � � � � � � � � + � � � � � � � � � � � �

� � � � � � � l � � � � � � � � � � � � � � � � � � � � � � � f � � + � � � � � � � � � " � � � � � � �

� � � � � � � � � � � � � � � � � � � � � �!�

�%�%�%

� + � � � � � � " � � � � � � � � � � � � � � � � � � � � m � � � + � � " �!

�! � + � l � � �

� � � � � � � � � � � � � � � � � � � l � � l � � � � � � � �!

� � � � � � � � " � + � � � � � �

Page 54: Программирование: теоремы и задачи

� � � �� � 4 � x 3 r q q x �

� � � � � � � � � � � � � � � � � � � � � � � � � � " � � � � � � � � � � � � �!

� � � + � � � " � � � � � �

� � � � � � � + � � � � � � � � � � � " � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � +!

k � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �!

� # � # $ � # %i � � � � � � � � � � � + � � � � � � � � � � � l � � � � � � � � + � � � � � � " � � � �

� � � � � � l � � � � � � � � � � � � � � � � � � � � � � � � � � � � m � � � � � � � � � � � � � � � � �

� � � � � � � � � " � � � � � � �� � � � � � � � � � � � � � � � � � � � !

k � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � + � � � " � � � � � � � � � � � � � � � � �!

� � � � � � � � + � � �

� + � � � � + � � � � � � � � � � � � � � � � � � � � � m � " � � � � � � � � � � � � � � � � � � � � � � � � � �

� �!

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � + � � � � � � � � � "

� + � � � � � � � � � � � � � � � + � � � l � � � � � � � � � � � � �!

� � � � � � + � � � � � � � � � � � + � � � � � � � � � � � � � � � � � � !

� � � � � � � � � � � �

� � � � � � � � � � � � � &

U V W 5 % R I A H , G J S K C A B ( A , @ A E � O E C & Z 0 % [ W /

W � � R c + J + , M , M Q + F � B ( A , Q + F H T A A , R I A S E A O E + - V % / 0 % [ W /

; F J � ( � B * R I A S � B � I E A O E + I

% / & % 5 � % 0 % [ W /

> c + J + , C R I A � , B + ] L A , A E P � ( A N K B + � Q , + @ B D

+ B E A R C B * G + ( A � Q , + @ B C � F + O C , � B * A c + G c + J + , A

> � R I A H + B G ; @ M , C A B G + ( A � Q , + @ B C � D

% / & 9

% / & 9

� � � � � � " � � � � � � � � � � � � � � � � �!

� � � � � � � � + � � � � � � � � � � � � � � � � � � " � � � � � � � � � � � � � l � � � � � � �

� � � � � � � � � � � + "

� � � � � � � � � � � � �!

h � � � � � � � l � � l � � � � � � � � � � � � �

� � � � � � � � � � � � � � " � � � � � � � � � � � � � " � � � � � � + � � � � � � � � � � � � � + � � � � � � "

� � � � � � � � � + � � � � � � � � � � � � � � " � � � � � � � � �!

� � i � � � � � � � � � � � � � � � � � � " � � e � � � � � � � � � � � � � � � � � � � � � � � � � � "

� � � � " � � � � � � � � � � � � � � � � � � � � � � � � � "

� � � � � � � � � � � � � � � � � � � l � � � � �!

� � � � � � � � � � � � � + � � � � � � � � � � � � � � � � � � � � � � � + � � � � � � � � � ! � � � � � + �

� � � � � " � � � � � � � � � � � + � � � � � � � � � � + � � � � � � � � � � � � � � � � � � � + � � � � � �

� � � � �!

e � � � � � � � � � � + � � � + � � � � � � � � � � � � " � � � + � � � � � � � � � � � � � � � � �

� � � � � �!i � � � � � � � � � � � � � � � + � � � � � � � � � � � � �!

� � � � � � � � � � � m � � � � � � � � � �

� � � � � � � � � � �! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � + � � � � � � � � " � � � � � � � � � � � �� � � � � � � � �"! ! ! "

� � � � � � � � � � � � � �

� � � � � � � � � " � � � � e � � � � � � � � � � � �!

� � � � � � � � � � � � � � � � � � + � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � �!

h � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � ' � � � � � � � � � � � + � � � � � � � � � + � � � � � � � � � � � � e � �

� � � � � � � � � � " � � � � � � � � � � � � � � � � � �! �

�� � � � } 1 2 1 3 4 � � �

�%�%

�%

� � � + � � � " � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � l � � � � � � � � � � � & � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � m � � � � � � � � � � � � � � � � � � � � � � � � � � "

� � � � � � � � � � � � � � � + � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� �! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � " � � � �

� � l � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � !!

" � � � + � � � �! d � � � � � � � � � � � � m " � � � � � � � � � � � � � � � � � � � � � l � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � − �!

# � � � � � � � � � � � " � � � � + � � � �

� � � $ � � � � � � � � � � � � � � � � � � % " � � � � $ � � � � � � � � � � � � � � � "

� % �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �! h � � � � � + � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � �!

& � � � " � � � � � � � � � � � � � � � � � � � � � "

� � � � � � � � � � � � � l � � � � � � � � � � � � � � � � � �!'

�%�%

(%

d � � � � + � � � � � ) � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � "

� � � � � � + � � � � � � � � � � � * ) � "+ � � � � � � � ! � � * ) � � � � � � � � � � � � � � � � �

� � � � � ! '" � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � " � � � � l � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � �!

� # � # $ � # % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � + � � � � � � � � � � & �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � " � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � l � � � + � � � � � � � � � � � � � � � � � � � � � � � � � � � " � � � + � � � � � � � � �

m � � � � � � � e � � � � � � � � � � � � � � � �!

� � � � � � � � " � � � � � � � � � � � � � � �

� � � � � � � + � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �!

� � � � � � � � � � � � " � � � � � � � � � � � � � � � � � � � � � � � � � � � � �!

i � � � � � � " � � �

� � � � � � � � � � � � � � � � � � m � � � � � � � � � � � � � � � � � e � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � + � � � � � � � � � � � � � � � � � �!

� + F A E L C ( � A . , # # , - . b : : / / Z � � 9

� J A F ; S T � ] . , # # , - . b : : / / Z � 0 : : / 9

� A E , M ] . , # # , - . b : : < / Z � 0 : : / 9

� + @ J A F ( � ] . , # # , - . b : : < / Z � 0 : : / 9

� , + O + F ( C H . 0 : : / 9

" # Z $ % & ' # % � F A J C B *` ) ; @ B M I 9

1 , # W . W / - % [ % # 9

0 % [ W /� Z # W . = b - Z / ? b & Z 0 % [ W /

� J A F ; S T � ] . W / . = W a b 9

% / & 9

� J A F ; S T � ] . / / . = 0 9

� , + O + F ( C H . = b 9

� Z # W . = b - Z < & Z 0 % [ W /

Page 55: Программирование: теоремы и задачи

� � � �� � 4 � x 3 r q q x �

� A E , M ] . W / . = 0 9

% / & 9

% / & 9

� ' / $ - W Z / � @ B *` I A @ B + .0 Z Z 5 % , / 9

0 % [ W /� @ B *` I A @ B + . =� , + O + F ( C H X Y 0 9

% / & 9

� ' / $ - W Z / � ; @ B C \ ( + I A E` + K A E A F � .W / - % [ % # _ . 0 Z Z 5 % , / 9

0 % [ W /� ; @ B C . = � A E , M ] . ( + I A E` + K A E A F � / = 0 9

% / & 9

" # Z $ % & ' # % � R H B * \ 1 , # - . � 9 ( + I A E` + K A E A F � .W / - % [ % # _ 9

1 , # ) A E , . W / - % [ % # 9

0 % [ W /> / Z - � ; @ B C \ ( + I A E` + K A E A F �_ D

) A E , . = � A E , M ] . ( + I A E ` + K A E A F � / 9

- . = � + F A E L C ( � A . ) A E , /

� A E , M ] . ( + I A E` + K A E A F � / . = � J A F ; S T � ] . ) A E , / 9

� J A F ; S T � ] . ) A E , / . = � , + O + F ( C H 9

� , + O + F ( C H . = ) A E , 9

% / & 9

" # Z $ % & ' # % � + O C , � B * \ - . � 9 ( + I A E ` + K A E A F � .W / - % [ % # _ 9

1 , # ( + , N ) + @ J . b : : / 9

0 % [ W /> � @ B *` I A @ B +D

( + , . = � , + O + F ( C H 9 � , + O + F ( C H . = � J A F ; S T � ] . � , + O + F ( C H / 9

> � R @ ) � @ G C @ , + O + F ( + c + I A @ B C � R � H B ( + I A E ( + , D

W � � ; @ B C \ ( + I A E` + K A E A F �_ - V % / 0 % [ W /

� A E , M ] . ( + I A E` + K A E A F � / . = ( + , 9

� + @ J A F ( � ] . ( + I A E` + K A E A F � / . = ( + , 9

� J A F ; S T � ] . ( + , / . = 0 9

� + F A E L C ( � A . ( + , / . = - 9

% / & % 5 � % 0 % [ W /

) + @ J . = � + @ J A F ( � ] . ( + I A E` + K A E A F � / 9> � J A F ; S T � ] . ) + @ J / = 0 D

� J A F ; S T � ] . ) + @ J / . = ( + , 9

� J A F ; S T � ] . ( + , / . = 0 9

� + F A E L C ( � A . ( + , / . = -

� + @ J A F ( � ] . ( + I A E` + K A E A F � / . = ( + , 9

% / & 9

% / & 9

�� � � � } 1 2 1 3 4 � � n

� ' / $ - W Z / � K A E A F ( + ] \ ( + I A E` + K A E A F � .W / - % [ % # _ . � 9

0 % [ W /� K A E A F ( + ] . = � + F A E L C ( � A . � A E , M ] . ( + I A E ` + K A E A F � / / 9

% / & 9 �

�%�%

�%

h � � � + � � � � � � � � � � � � � � � � � � � � � � � � � � �!

" � � � + � � � �! � � � � � � � � � � � � � � � � � � � � � � � � " � � e � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � + � � �!

� � � e � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � m � � � � � � � e � � � � � � " + � � � � � � � � � � � � � � � � " � � � � � � � � � � �

� � � � � � � � � � � � + � � � � � � + � � � � � � � � � � + � � � �!'

� � � � � l � � + � � � � � � � � � � � � � � + � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � �!

�%�%

� �%

� � � � � � � � � � + � � � � � � � � � � � " � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � + � � � � � � �!

k � � � � � � � � � � � � � � � � � "

� � � � � � � � � � � � � � � � � � � � � � � � � � " � � � � l � � � � � � � � � � � � � � � � � � � � � � "

+ � � � � � " + � � � � � � � � � � ! � ' � � � � � � � �!

� # � # $ � # %# � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � + �

� � � � � �!

� � � � � � � � + � � � " � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � + � � � � � � � � � � � � � � � � � � � � �!

� � � � � � � � � � � � � � � � � � + �

� � � � � � � � � � � � �!

! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � + � � � � � � �

� � � � � � � � � � � !

� � � � � � � � � � � � � � � � � � � � � � � " � � � � � � � � + � � � � � � � ��

� � � � � � � � � � " � � � � � l � � � � � � � � � � � � � � l � � � � �!

� � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � " � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � !

r

r

r

r

r r

r

r

��

��PPPPPP

AAAA�

�����BB

BBBB

HHHHHHH

�����������

# � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � �!

� � � e � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �!

� � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

Page 56: Программирование: теоремы и задачи

� � � �� � 4 � x 3 r q q x �

� � � � � � � � �!

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �!

� + � � � �

� u 3 � o u � v u s � � u 3 � u 3 � o u � v u s e � � � � � � � � � � " � � � � � � + � � � � � � � � � � �!

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � &

U V W 5 % ) + F + E + c A � R Q , + @ B C , ) + F ) + F Q , + @ B I M ) + , + E C K � , C A I

; ) + F Q , + @ B C , J A , + \ � , ) ; G J + @ B * � _ & Z 0 % [ W /

, M G � ( ; B * ) + F Q , + @ B � R F A G C

% / &

h � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �!� � � # � � $ � # % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � " � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � + � � � � � � � � � e � � � � � � � � � � � � � � " � � � � � � � � � � � � � � � � � � �

� � � � � !� � � � � � + � � � � � � � �!

� � � � � � � � � � � � � � � � � � � � � � & � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � �! �! � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �◦ � � � � � � � � � " � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �!

� � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � " � � �

� � � � � � � � � � � � � � � � + � � � � � � � � � � � � � � �! �

��

��

� � � � � �

� � � � � � � � � � � � � � � � � �!

k � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � e � � � � � � � � � � � � � f � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � �!

� � � � � � � � � � � � � � � � � { � �}

�%�%

�%

� � � � � � + � � � � � � � � � � � � "+ � � � � � � � � � � � � � � � l � ' " � � � � � � �

� � � � � � � � � � � � � � � � � � � � � { � �}!

! � � � � � � � � � � � � � � � � � � � �

k � � � � � � � � � � � ! �

� � � � � � � � � � � � � � � � � � � � � � � !

� � � � � � � !

� � � � � � � !

� � � � � � � � � � e � � � � � � ! �

� � � � � � � � � � � � � � ! �

� # � # $ � # %� � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � �! �

� � � � q u ! 1 � v o r � � �

�%�%�%

h � � � " � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � + � � � � � �!!

� # � # $ � # %� � � � � � � � � � � � � � � � � � � � � � � � � � � � � e � � � � � � � �! �

�%�%�%

h � � � � � � � � � � � l � � � � � � � � � � � � � � � � � � � � � � � � � � � � � &

! � � � � � � � � � � � � � � � � � � � �

k � � � � � � � � � � � ! �

� � � � � � � � � � � � � � � � � � � � � � � !

� � � � � � � !

� � � � � � � ! �

� � � � � � � � � � e � � � � � � !

� � � � � � � � � � � � � � !

� # � # $ � # %

� � � � � � � � � � � � � � � � � � � � � � � � � � � � e � � � � � � � � � � � �

� � � �! �

�%�%�%

h � � � � � � � � � � � l � � � � � � � � � � � � � � � � � � � � � � � � � � � � � &

! � � � � � � � � � � � � � � � � � � � �

k � � � � � � � � � � � ! �

� � � � � � � � � � � � � � � � � � � � � � � !

� � � � � � � ! �

� � � � � � � !

� � � � � � � � � � e � � � � � � !

� � � � � � � � � � � � � � !

� # � # $ � # %� � � � � � � � � � � � � � � � " � � � � � � � � � � � � � � � � � l � � � � � � �

� � � � � � � � � � � � � � �! �" # $ � % & ' ( ) * % + , - . / & % +

� � � � � l � � + � � � � � � � � � � � � � � e � � � � � � � � � � � � � � � � � � � � � � � � � � � � � "

� � � � � � � � � � � � � � � � � � � � � � � � � � � �!

�%�%�%

� � � � � � ! �!

! � � � � � � � � � � � � � � � � � � � �

k � � � � � � � � � � � !

� � � � � e � � � � � � � � !

� � � � � � � � � � � � � � � � � � � � � � � ! �

� � � � � � � � � � � � + � � � � � � � � � � � � � � � � l � � � !

� � � � � � � ! �

� � � � � � � � � � e � � � � � � ! �

+ � � � � � � � � � � � e � � � � � � !

Page 57: Программирование: теоремы и задачи

� � �� � 4 � x 3 r q q x �

� # � # $ � # %

� � � � � � � � � � � � � � � � � � � � � � � � � � � l � � � � � � � � �

, . , # # , - . b : : / / Z � W / - % [ % # N < . 0 : : / 9

� � � � � � � � � � � � � � � � � � e � � � � � � � � � � � � | � � � � f � � � � � � � � + � � � � !

� �

� � � � � � � � � � � � � � e � � � � � � � � � � � � � � � � � � � � � � � � + � � � � � � � � � � � !

k � � � � � � � � � � � � � � � � � � � � � � � � � � � � + � � � � � � � � � � � � � � l � � � � � �

� � � � �! �

�%�%�%

� � � � � � ! �!

! � � � � � � � � � � � � � � � � � � � �

k � � � � � � � � � � � !

� � � � � � � � � � � � � � � � !

� � � � � � � � � � � � � � � � � � � � � � � ! �� � �

� � � � � � � ! �

� � � � � � � ! �

� � � � � � � � � � e � � � � � � !

� # � # $ � # %k �! � � � � � � � � � � � � � � � + � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � + � � � � � � �

� � � � � � � � �! �

� � � � � l � � + � � � � � � � � � + � � � � � � � � � � � � � � � � � + � � � � � � � � !

�%�%

�%

� � � � � � + � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � " � ���

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � m � " � � � � � � � � � + � � � � �� � � � � �

� � � �!� � � � � � � � � � � � � � � � � � � � � �! � � � � � � � � � � � � � ! ·

� � � � �

� � � � � � � � � � " � � � � � � � � + � � � � � � � � � � � � � ��!

� # � # $ � # %� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � " + � � � � ��!�!

� �

� � � � � � � � � � � � � � � � � � � � � � " � � � � � � � � + �"

� � � � � � � � � �|

| � � � � � � � � � � � � � � � � � � � � � " � � � � � � � � � �� � � � � + � � � � � � �!

" # Z $ % & ' # % � + @ B ; ) ( M A \ W . W / - % [ % # _ 9

> ( C ) A K C B C B * , @ A , A E P � ( M N F + @ B ; ) ( M A � R W N , G J S K C H W D

1 , # � . ) + F I ( + L A @ B , + b : : / 9

� . ) + F I ( + L A @ B , + b : : / 9

N 1 N U . b : : / 9

< . W / - % [ % # 9

0 % [ W /: : : @ F A J C B * � N � ) ; @ B M I � 9

U # W - % 5 / \ W _ 9

: : : F + O C , � B * W G � N � 9

� � � � q u ! 1 � v o r � � �

> \ b _ � = I ( + L A @ B , + ( C ) A K C B C ( ( M Q , A E P � ( 9 � @ + F A E L � B W 9

\ _ ( C ) A K C B C ( M B + J * G + F + @ B ; ) ( M A � R W , A E P � ( M 9

\ 7 _ � ? ) + F I ( + L A @ B , + � 9

\ _ , @ A ( C ) A K C B C ( ( M A , A E P � ( M N � R G + B + E M Q , M Q + F � B

E A O E + , ( A ( C ) A K C B C ( ( ; S , A E P � ( ; N ) E � ( C F J A L C B � D

U V W 5 % � ( A ) ; @ B + & Z 0 % [ W /

: : : , R H B * G C G +] ? ( � O ; F * � J A I A ( B � , 1 9

� Z # < . = b - Z / ' � . 1 / & Z 0 % [ W /

U . = Z ' - . 1 / . < / 9

W � U ( A ) E � ( C F J A L � B � - V % / 0 % [ W /

U # W - % 5 / \ U _ 9

F + O C , � B * U , � 9

F + O C , � B * U , � 9

% / & 9

% / & 9

% / & 9

% / & 9

k � � � � � � � � � � � � � � � � � � � � � � " � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � �!

k � � � � � � � � � & � � + � � � � � " � � � � � � � � � � � "

� � e � � � � � � � � � � � � �!

k � � � � � � � � � � � � � � � � � �!

k � � � � � � � � � � & � � � � � � �

� � � + � e � � � � � � � " � � � � � � � � � � � " � � � � � + � � � � � � � � � � " � � � � � e � � �

� � � � � � � � � ! �

�%�%

(%

� � � � + � � � " � � � � � � � � � � � � � � + � � � � � � � � � � � � � � � � � � � � � & � �

� � � � � � � � � � � � � � � � � f � ⊂ � f � � � � � � � � � � � � � � � � � � � � � � � � " � � � � � � �

� � � + � � � � � � � � � � � � � � � � � � � � � � � �! �

! � � � � � � � � � � � � � � � � � !i � � � � � � " � � � � + j � � � � +� e � � � � � � � � � � �

� � � � � � � � � � � � � � � l � � � " � � � � � � � � � � � � � � � � � + j � � � �� � " � � � � � � � � � � � �

� � " � � � � � � � � + � � � � � � � � � � � � � � �"

� � � � � � � � l � � � � � � � � � e � � � � � � � �

� + �!

� � e � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � " � � � � � � � + � " � � � e � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � + " � � � � � � � � + � � � � � � � � � � � � �� � � �!

� � � � � � � � � � � � � � + � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � l

� � �! � � � � " � � � � � � � � � � � � � � � � � � � �! �

�%�%

�%

d � � � � � � � � � � � � l + � � � � � " � � � � � � �� � � � � � " � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � & � � � � � � � + � � � � � � � � � � � � � � "

� � � � � � � � � � � � � " � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �! �!

" � � � + � � � �! h � � � � � � � � � � � " � � � � � � � � � � + � � � � � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � � � � � & � � � � � � � � � � � � � � � � � � � )" � �� � � � � � � � �

� � � � � � � � " � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � )"

� � � � � � � � � � � � � � � � � � � � � � � � � � � " � � � �� � � � � � � � � � � )!

'

Page 58: Программирование: теоремы и задачи

� � � �� � 4 � x 3 r q q x �

# � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � + � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � !

��

��

� � � � � � � � � �

�%�%

�%

d � � � � + � � � � � � � � � � � � � � � � � � � " � � � � � � � � � � � � � � � � � � � � � � �

� � � � � " � � � � � � � � � � � � � � " � � � � � � �

• � � � � � � � � � � � � f

• � � � � � � � � � �� l � � � � � � � � � � � g f

• � + � � � � " � � � � � � � � � �� �� � � � � � � f

� � � � � � � � � � � � � l

• � � � + � � � � � � � � � � � � � � � � � � � � e � � � � � � �

� � � � � � � " � � � � � � � + � � � � � � � � � � e � � � � � � � � ��

� � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � ! �� � � " � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � �

� # � # $ � # %� � � � � � + � � � � � � � � � � " � + � � � � � � � � � � + � � � � � � � � � � � � � � �

� � � � � � ��

� � � � � � " � � � � � � � � � � � � e � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � " � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � e � � � � � �

� � � � � � � � � � � � � � l � � � � � � � � � � � ��

� � � � � � � � � � � � e � � � � � m � � � � � � � "

� � � � � � � � � � � � � � � � � � � � � � � � + � � � � � � � � � � � � � � � � � � e � � � � � � � � � � �

� � l � � � � � � � m � � � � � � � � � � � � � � � � � � � � � �� �

�%�%�%

� � � � � � � � � � � � � � � � � � � � e � � � � � � � � � " � � � � � � �� � � � � � � � � � "

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � " � � � �

� � � � � � ��

# � � � � � � � � � " � � � � � � � � � � � � � � � � � � � � � � + � � � � � � � � � � � � � � �

� � � � � � � � � � � � j � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � "

� � � � � + � � � � � + � � � � � � � � � � � � � � � � e � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � + � � � � � e � � � � � � � � � �d � � � � + � � � � � � � � � � � � � � � � l � � � � � � �

� � � " � � �� � � � � � � � � � � + � � � � e � � � � � � � � � � � � � � � � � � � � � � m � � � � � � � � �

� � � � � � � � � � � � � � � � � � + � � � � � � � � � � � ��

� # � # $ � # %k � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � "� � � � � � � + � � � � � e � � � � � � � � � � � � � � � � � � � � � g � � � � � �"

� � � � � � � � � � � � � � � � � � � � � � � � & g � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � g � � � � � g � � � � � " � � � � � � � � � � � � � � � � � l � � � "

� � � �

� � � � � � � � � � " � � � � � � e � � � � � � � � � � � � � � � � � � � � � � � � ��� k � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � " � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � 〈 e � � � � � � | � � � � � � � � �〉�� � � � � � � � � e � � � � � � � � � � � � � � � � � � � e � � � �

� � � � r � q x 1 � r 3 r } 4 � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � e � � � � � � � � � � � � ��

� � � � � � � � � � � � � � &

- . = ( + I A E F + O C , J A ( ( + c + � J A I A ( B C

> � ( , C E � C ( B . , F A E A , A J S O + ] ) E A F + G ) E � + E � B A B ( A A ) + B + I G C N

A @ J � � B + B ) + B + I + G ? ( A - D

U V W 5 % - ? ( A G + E A ( * � - @ B C E P A @ , + A c + + B � C & Z 0 % [ W /

) + I A ( H B * - @ A c + + B � + I

% / & 9� � � � � � � � � � � � � � � + � l � � � � � � � � � " � � � � � � � � � � � � � � � � � � � � � "

�� �� + � � � � � � � � � � � � � � � " � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � " � �

� � � � e � � � � � � � � � � � � � � � � � & � � � � � � � � � � � " � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � " � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � �� �

� � � # � � $ � # % � � � � � � � � � � � l � � � � � � � � � � � � � � � � � � � � � � � � + � � � � � � � �

� � � � � � � � � � � � � � � � � � � � l � � � � � � � � � � � � � � � � � � � ��

� � � e � � � e � � � � � �

� � � � � � � � � e � � � � � � � � � � � � � � � � " � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � " � � � � � � � � � � � � + �� � � ��

Page 59: Программирование: теоремы и задачи

��

� � � � � � � �

� � � � � � � � � � � � � � � � � � � �

� � � � � � � � + � � � � � � � � � � � � � � � � � � � � � � + � � � � � � " � � � � � � � � " � � �

� � � � � � � &

� � � � � � � � � � � � � � � � � � + � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � " � � � � + � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � " � � � � � � � � � � � � � � � � � � � � � � � � �

+ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � " � � � � � � � � � � � " � � � � + � � � � � � � l

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � " � e � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � + � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � l � � � � � � � � � � � � � � � � � � � � � � e � � � " � � � � � � � � � � � � � � � � � � � � � � � � ��� � � � � � � � + � � � � � � " � � � � � � � � � � � � l � " � � � � � � � � � � � � � � � � � � �

� + � � � � + � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � " � e � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � ��

�%

�%

�%

� � � � � � � � � � � � � � � � � l � � � � � � � � � � � � � � � � � � m � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � �� �� � � � � + � � � � � � � � · · · · � " � � � + � � �

� � � � � � � � � � �

� # � # $ � # %� � � � � � + � � � � � � � � � � � � � � + �" � � + � � − � � � · ��

" # Z $ % & ' # % � , $ - Z # W , 5 \ / . W / - % [ % # 9 1 , # � , $ - . W / - % [ % # _ 9

> ) + J + L � B * � , $ - E C , ( M I � C G B + E � C J ; K � @ J C / D

0 % [ W /W � / = b - V % / 0 % [ W /

� , $ - . = b 9

% / & % 5 � % 0 % [ W / > / Y b D

� , $ - Z # W , 5 \ / ? b N � , $ - _ 9

> � , $ - = \ / ? b _ � D

� , $ - . = � , $ - � / 9

% / & 9

% / & 9 �

�� � � � 2 4 s 1 2 x 2 1 ~ � 2 � 4 o q x � � 2 u 2 r s s � � �

k � � � � � � + � � � � � � � � ! � � � � � ! � m � � � � � � � � � � � � � � � � � � � � � � &

� ' / $ - W Z / � , $ - Z # W , 5 \ / . W / - % [ % # _ . W / - % [ % # 9

0 % [ W /W � / = b - V % / 0 % [ W /

� , $ - Z # W , 5 . = b 9

% / & % 5 � % 0 % [ W / > / Y b D

� , $ - Z # W , 5 . = � , $ - Z # W , 5 \ / ? b _ � / 9

% / & 9

% / & 9

! � ! � � � � � � � � � � � � � � � � � � � � � ! � l � � � � � � � � � � � � � � � � � � � � + � � � � � � � � � �

� � � � " � � � � � � � � � � ! � � � � � � � � � m � � � � � � & � � � � � � + � � � � � � � � � � � ! � � � � �

� � l " � � � � � + � � � � � l ! � � � ! � � � � � m � � � � � l� � � � � � � l " � � � � � � � � � �

� � � � � � ! � + � � � � l � � � � � � � � � � � � � � � � � � � � � � " � � � � � � � m � � � � � � � � �

� � + � � ! � � � � ! � � " � � � � � � � � � � � � � � ��� k � � � � � ! � � � � " � � � ! � � � � � � � � �

� � � � � � � � � � � � � + � � � � � � " � � � � � � � � ! � ! � � ! � � � � � � � � � � � � � � � � � � � � "

� � � � � � � � � � � + � � � + � � � � � � � � � ! � � � � � � � " � � � � � � � � � � ! � e � � � � � � � �

� � � � � ! � � � ! � � � � � � + � ���

�%

�%�%

! � � � � m � � � � ! � � � � � ! � � � � � l � � � � � � � � � " � � � � � � " � � � � � + ��

� + � � � � � � � ! � � ! � � � � � � � � � � � � � � � � �� �

�%

�%�%

� � � � � � � ! � � � ! � � � � � l � ! � � ! � � � � � � + � � � � � � � � � � � � l � � � �

� ! � � � � � � � � � l � � � � � � �� �

�%

�%�%

h � � � " � � � � � ! � � � � � � � " � � � � � � � � � � � ! � � � ! � � � � � � ! � � � � � � �

� � � � ! �� � � " � � � � � � � � � + � � � � � � � � � � � ��

� # � # $ � # %

� ' / $ - W Z / " Z U % # \ , N / . W / - % [ % # _ . W / - % [ % # 9

0 % [ W /W � / = 0 - V % / 0 % [ W /

" Z U % # . = b 9

% / & % 5 � % W � / # Z & ^ = 0 - V % / 0 % [ W /

" Z U % # . = " Z U % # \ , � , N / & W 1 _ 9

% / & % 5 � % 0 % [ W /

" Z U % # . = " Z U % # \ , N / ? b _ � , 9

% / & 9

% / & 9 �

Page 60: Программирование: теоремы и задачи

� � � �� � 1 ~ � 2 � 4 �

�%

�%�%

� � � � � � � � " � � � � � + � � � � � � � ! � � ! � � � � " � ! � � � � � � � � l � ! � � � � � �

� ! � � � � � � + � � � � � " + � � � � � � � � ! � � �

" Z U % # . = " Z U % # \ , � , N / & W 1 _

� �

" Z U % # . = " Z U % # \ , N / & W 1 _ � " Z U % # \ , N / & W 1 _ �

� # � # $ � # %� ! � � ! � � � � � � � � � � � � � � ! � � � � � � � �� ! � � � � � � � � � � � � � � ! � �

� � � � � � � � � � � � � � �� � � � � � � � � " � � � � � � � ! � � + � � � � � � � � � ! � � � � � � � �

� + � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � + � � � � � � � ! �

! � � � � � � � � � � � � � � ! � � � ! � � �� � ! � � ! � � � � � � � � ! � � � � � � � � � � � � � � � ! � m �

� � � � � � � l � � � � � � � ! � � � ! � � � " � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � ! � m � � � � � � � � ��� � � � � � � � � � � � � � � � � � � � � � ! � � � � � " � � � � � � �

- . = " Z U % # \ , N / & W 1 _ 9

" Z U % # . = - � - 9

� � � � � � � � � � + � � � � � � � � m � � � � � � � � � + � � � � � � � � � � � � ! � � � � � � � � �

�%

�%�%

� � � � � � + � � � � � � � � � � � � � � g � � � � � � ! � g + � n" � � � � � � � �

! � � � ! � � � � � l � ! � � ! � � � � � � � � � � � � � � � � � � � � + � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � ��

� # � # $ � # %i � � � � � � � � � � + � � � � � � ! � � � ! � � � � � � � � � � � � � � + � � � � ! � � � � �

� � � � � � � � � " � � � � � m ! � � � � � � � � � � � � � � � � � � � " � � � � � � � � � � � � �

� � � � � � � ��" # Z $ % & ' # % " # W / - \ / . W / - % [ % # _ 9 > / Y 0 D

0 % [ W /W � / X b 0 - V % / 0 % [ W /

U # W - % \ / _ 9

% / & % 5 � % 0 % [ W /

" # W / - \ / & W 1 b 0 _ 9

U # W - % \ / # Z & b 0 _ 9

% / & 9

% / & 9 �

�%

�%

�%

� � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � l � ��

� � � � � ! �

� � � ! � � �� � � � ! � � � + � � � � � � � � � � � ! � � � � � � � + � � � � � � � � � � � � � �

� � � � � � � � � + � " � � � � � � � � � � ! � � � � h ! � � � � � � � � � ! � � � � � � � � � � � � � � � � ! � �

� � � � � � ! � � � �� d � + ! � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � ! � � � � � � � � ! �

� � � � " � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � + �� k � � � � � � � �

� ! � � ! � � " � � � + � � l � l � ! � � � � � � � � � � � ��

�� �� � 1 ~ � 2 � 4 o q r � u t 2 r t u v ~ r 3 1 2 1 o � 1 o � � n

� # � # $ � # % � � � � � ! � � � ! � � � � � l � ! � � � � ! � � � ! � � � � � � � � � ! � � � �

� � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � l � � � � � � � �

� � � � � ! � + � ! � � � � � � � � � � � � ! � � � � � � + � � � � � � � � �

" # Z $ % & ' # % # Z 1 % \ W N # N / . W / - % [ % # _ 9

1 , # � . W / - % [ % # 9

0 % [ W /W � W = b - V % / 0 % [ W /

U # W - % 5 / \ � @ F A J C B * Q + F � N # N � ? Y � N / _ 9

% / & % 5 � % 0 % [ W /

� . = ? # ? / 9 > � ? B E A B � ] @ B A E L A ( * . @ ; I I C ( + I A E + , E C , ( C D

# Z 1 % \ W ? b N # N � _ 9

U # W - % 5 / \ � @ F A J C B * Q + F � N # N � ? Y � N / _ 9

# Z 1 % \ W ? b N � N / _ 9

% / & 9

% / & 9

� k � � � � � � � � ! � � � � � � � � � � ! � � � � � + � � � � � � � � � � � ! � � � l � � � � � � �� � � �

� � � e � � � � �� � � � � � � � � � � � � � � � � � � � � " � � � � � � � � � � ! � � � � � � � � � � � � �

� � ��

! � � � � � � � � � � � � � � � � � � � � � � � ! � � � ���

�%

�%

(%

� � � � � � � ! � � � ! � � � � � l � ! � � ! � � � � � ! � � � � � � � � � � � �

� � � � � � � � �� � � � � � � � � � � � ��

" � � � + � � � �� d � � � ! � � � � � � � ! � � � � � � � m � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � ! � � � ! � � � � � � � � � � � � � � e � � � � � � ��'

��

� � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � ! � � � � � + � � � � � � � � ! � � � � � � ! � � � � � � � &

k

k k

k k

k

AA

AA

AA

��

��

� � � � � � � ! � � � � � � + � � � � � � ~ u 2 q 1 s�

� + � � � � � � � ! � � � � � � � � �

� � � � � � � � � & � � � � � � � � ! � � � � ! � � � � � � ! �� � ! � � � " � � � � � � � � � � "

Page 61: Программирование: теоремы и задачи

� � �� � 1 ~ � 2 � 4 �

� � + � � l � � � � 1 o x s 4 � 2 r o x s � x q u o � � s 4 � � � � � � � � � ! � � � � � ! � � � �

� � � � � � � � � � � � � � � � � " � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � ! � � � � � �� ! � � � � � � � � � � � � � � � � � � � � � � � � " � � e � � � � � � � �

� � + � � � � � � � 4 � v u s�

� � � � � $ � � � � � � � � � � � ! � � � � � � � � � � � � � ! � � �� ! � � � � � � � � � �

� � � � � � � � " � � � � � � " � ! � � � � � � � � �� � � � ! � + � � � � � � ! � � � � � � ! � �

� $ � � u 3 3 1 2 1 o u � u v u s ~ u o $�

� � � � l � � + � � � � � � ! � � � � � � � � " � � � � � ! � � � � ! � � � � ! � � � �

� ! � � � � � � � � � � � � � � � � � � � � � � � � � � " � ! � � � � � � ! � � � � � � � ! �

� � � ! � + � � � � � � � � � � � � " � � � � � � ! � � ! � � � ! � � � � � � � � � ! � � � � � �

� � � �� � � � � � � � " � � � � � l � � � � � � � � � � �

5 N # . , # # , - . b : : � / Z � W / - % [ % #

� � � � � � � ! � � � � � � � ! � � � � � � � ! � � � � l � � � � � � � � � � � � � � �

� � � ! � � � � � � � � � � � � � � � � � ! � � � � � � � � ! � � � � � � � � � � � � � � � � � �

� ! � � � � � � � � � " � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � ��

� � � � ! � � � � �

� ! � + � � � � � � ! � � ! � � � � � � � + � � � � � � � � � � � � � � � � � � � � � � � " ! � � �

� � l � � � l��

i � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � ! � � � � � � � ! �

� � � � � � ! � � � � � � � � � ��

! � � � � " � � � � � � ! � � ! � � � � � � � � � � � � � + � �

� � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � + � � � � �

� � � ! � � � � ! � � � � � " � � � � � � � � � � � � " � � � � � � � � � � � � � � � � � � � � �

e � � � � � ! � �

�%�%

�%

� � � � � � + �" � � � � + �" � � � � � � � � � � � � � &

� � � � � � �

� � � � � � � � � � �

� � � � � � � � � �

� ! � � � � � � � � � � � � � � � � � � l � � � ! � � ��

� � � # � %

� ��

� ��

� ��

� ��

� ��

� �

AAA

AAA

���

���

�� �� � 1 ~ � 2 � 4 o q r � u t 2 r t u v ~ r 3 1 2 1 o � 1 o � �

�%�%�%

� � � � � � � � ! � � ! � � � � � � � � � � � � � � � � ! � � � � � ! � � ��

� # � # $ � # %d � � � � � ! � m � � � � � l � � g � " ! � � � � l � � � � � � � ! � � � � � � � �

! � � � � � � ! � � � � � ! � � � � � � � ! g� k � � � � � " � � � � � � � � � + � � � � � � � � � � � �

� � � � � � � � � l � � � � � ! � � � � � � � � " � � � + � � � � � � � � + � � � � � � � � � � � � � �

� � � � � � � � " � � � � � � l � � � � � � � ! � � � � ! � � �� d � � � ! � � � � � � � ! � � ! � �

� � � � � � � � � &

� ' / $ - W Z / / \ 6 . W / - % [ % # _ . W / - % [ % # 9

0 % [ W /W � 6 = / W 5 - V % / 0 % [ W /

/ . = 0 9

% / & % 5 � % 0 % [ W /

/ . = / \ 5 . 6 / _ a / \ # . 6 / _ a b 9

% / & 9

% / & 9

� � � � � � � � ! � � � � � � � ! � � � � � � � ! � � � � � g ! � � � � � � � � � � � � � � ! �

� � � � � � � � � � � � � � � � l � � � � � � ��� � � � � � � � ! � � � ! � � � � � � � � � � " � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � l � � � � � � ! � � � � � � � � � �

� � � �� �

�%�%�%

� � � � � � � � ! � � ! � � � � � � � � � � � � � � � � � � � � � � � ! � � ��

� � � # � %� ' / $ - W Z / / \ 6 . W / - % [ % # _ . W / - % [ % # 9

0 % [ W /W � 6 = / W 5 - V % / 0 % [ W /

/ . = 0 9

% / & % 5 � % W � \ 5 . 6 / = / W 5 _ , / & \ # . 6 / = / W 5 _ - V % / 0 % [ W / > J � @ B D

/ . = b 9

% / & % 5 � % 0 % [ W /

/ . = / \ 5 . 6 / _ a / \ # . 6 / _ 9

% / & 9

% / & 9 �

�%�%�%

� � � � � � � � ! � � ! � � � � � � � � � � � � � � ! � � � � � � ! � � � � � � �

� � � � � � " � � � � � � � � � � � � � � � � " � � � � � � � �� �� f � � � � � � ! � � � �

e � � � � � � � � � � � � � � � � ! � � � � �

" � � � + � � � �� d � � � ! � � � � � � � ! � � � � � � � � m � � � � � � � � g � + � � � � � � � �

� ! � � � � � � ! � � � g�'

�%�%�%

� � � � � � � � ! � � ! � � " � � � � ! � � � � + � � � � � � � � � � � � � � � � � � �

� � � � � � ! � � � � � � � � � � + � � � � � � ! � � � � � �

Page 62: Программирование: теоремы и задачи

� �� � 1 ~ � 2 � 4 �

� � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � ! � � � � � �

� ! � � � � � � � � � � � � � � � � � � � � � e � � � � � ! � � � � � � � � � � � � � � � ! � � � � �

�%�%�%

� � � � � � � � ! � � ! � � " � � � � ! � � � � � � � � � � � � � � � � � ! � + � � � � �

� � ! � � � � ! � � ��

� # � # $ � # %� ! � � � � ! � � � � � � � � � � � � � � g � � � � � � � � � � � � � � ! � � � � � �

� ! � � � � � � ! � � � g � � � � � � ! � + � f � � � � � � � � ! � � ! � � � � � ! � � � � + � �

� � � � � � � � � � � � � � � � � � �

" # Z $ % & ' # % " # W / - ` � ' 0 - # % % \ 6 . W / - % [ % # _ 9

0 % [ W /W � 6 = / W 5 - V % / 0 % [ W /

> ( � K A c + ( A F A J C B * D

% / & % 5 � % 0 % [ W /

U # W - % 5 / \ 6 _ 9

" # W / - ` � ' 0 - # % % \ 5 . 6 / _ 9

" # W / - ` � ' 0 - # % % \ # . 6 / _ 9

% / & 9

% / & 9

� � � � � � � ! � � ! � � � � � � � � � � � � � � � � � � � ! � � � � � � ! � � � " + � � � � � � �

! � � � � � � � � � � � " � + � � � � � � ! � � � h ! � � � ! � � � � � � � � � � � � � �

� � � � � � � � � ! � � � � � � � � � � � � � � � � � " � � � � � � + e � � � � � � � � � � � � � �

� � � � � � ! � � � � � � � � � � � ! � � �� �

��

� � � � � � � � � � � � � � � � � � � � � � � ��

� � � � �

d � � � ! � � � � � � ! � � ! � � � � � l � � � � � � � � � � � � � � � � ! � � � � � �

� � � � � � � � ! � � � � j � � � � � + � � � � � � � � � �� � ! � � � + � � � � � � � � � � � � � �

+ � � � � � � � � � � � � � � l � � � � � � �

�%�%

�%

� � � � � � � � ! � � ! � � " � � � � ! � � � � � � � � � � � � � � � � ! � + � � � �

� � � � � � � � � � � � � � � � � � � � � " � � � � � � � � � � � � + � � � � � � � � � � � � � � � � �

� � � � ! � � � � � � ��

� # � # $ � # %

� ! � � ! � � � � � � � � � ! � ! � � � � � � � � � � � � � � � � � � � �

� � � � � � ��

d � � � ! � � � � � � � ! � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � " � � � � � � l � � � � � � � � � � � � � � f � � � � � � � � � � � � � � � � � �

� � � � � � � � � � l � � � � � + � � � � � � � " � � � � � � � � � � � &

" # Z $ % & ' # % [ % / % # , - % 9

1 , # W N � . W / - % [ % # 9

�� �� � u 2 u ! 3 1 q 4 1 ~ u s t 4 q r v u 2 q x � u t � 1 ~ v u o � � 1 2 1 t u 2 � �

0 % [ W /W � - = / - V % / 0 % [ W /

� Z # W . = b - Z / & Z 0 % [ W /

U # W - % \ , . W / _ 9

% / & 9

U # W - % 5 / 9

% / & % 5 � % 0 % [ W / > - X / D

� Z # � . = b - Z < & Z 0 % [ W /

- . = - a b 9

, . - / . = � 9

[ % / % # , - % 9

- . = - ? b 9

% / & 9

% / & 9

% / & 9

! � � � � � � � � ! � � ! � � � � � � ! � � � � � � � � � + � � � � � � ! � � � ! � � &

- . = 0 9 [ % / % # , - % 9 �

� � � # � � $ � # % � � � � � � � � � � � � � � � e � � � � � � � � � � � � � � � �

� + � � � � � � � ��

�%�%�%

� � � � � � � � ! � � ! � � " � � � � ! � � � � � � � � � � � � � � � � ! � � � � � � � � �

� � � � � � � � � � � � � ! � + ��

� # � # $ � # %� ! � � ! � � � � � ! � ! � � � � � � � � � � � � � � � � � � " � � � � � ! �

� ! � � � � � � � � ! � � � � � � � � � � � � � � � �� d � � � ! � � � � � � � ! � � � � ! � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � " � � � � ! � � � � � ! � � � � � �

+ � � � � � � � � � � � l � � � � ! � � � � � � � � � � � f � � � � � � � + � � � � � ! � � � � � �

� � � � l � � � � � + � � � � � � � " � � � � � � � � ��

! � � � � � � � � ! � � ! � � � � � � � � &

� Z # W . = b - Z / & Z 0 % [ W / , . W / . = W 9 % / & 9

- . = 0 9[ % / % # , - % 9

� � � � � � � � � � � ! � � � � ! &

" # Z $ % & ' # % [ % / % # , - % 9

1 , # W N � . W / - % [ % # 9

0 % [ W /W � - = / - V % / 0 % [ W /

� Z # W . = b - Z / & Z 0 % [ W /

U # W - % \ , . W / _ 9

% / & 9

U # W - % 5 / 9

Page 63: Программирование: теоремы и задачи

� � �� � 1 ~ � 2 � 4 �

% / & % 5 � % 0 % [ W / > - X / D

� Z # � . = - a b - Z / & Z 0 % [ W /

) + I A ( H B * I A @ B C I � , . - a b / � , . � /

- . = - a b 9

[ % / % # , - % 9

- . = - ? b 9

) + I A ( H B * I A @ B C I � , . - a b / � , . � /

% / & 9

% / & 9

% / & 9 �

�%�%�%

� � � � � � � � � � � � � � � � ! � + � � � � � � � � � � � � � � � � � � � � � � � + � � � �

� � � � � � � � � " � � � ! � � � � ! � � � � � � � � � �� �

�%�%�%

� � � � � � � � � � � � � � + ! � � � � l � � � � � � � � � � � � � � � � � � � � � � �"

e � � � � � � � � � � � ! � � � � � l � � � � � � � ! � � � � � � � � � � � � � � �� � � ! � � � �

� � � � � � � � " � � � �

6

� " � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ���

� # � # $ � # %� ! � � ! � � � � � ! � ! � � � � � � � � � � � � � � � � � � � � � � � � � � �

! � � � � � � ��

� ! � � � � � � � � " � � � � � � � � � � � � � � + ! � � � � l � � � � � � � � �

� � � � � � � � � � � � � � � ! � � � � � � � � � � � + � � ! � + � � � � " ! � � � ! � � � � � � � ! � � �

� � � � � � � ! � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � + ! � � � � l � � � ! � � � � � � � �

� � � ��

� � ! � e � � � � � � � � � � � � � � � � � � � � � � � � � " � � � � � � � � � ���

" # Z $ % & ' # % [ % / % # , - % 9

1 , # W . W / - % [ % # 9

0 % [ W /W � - = < - V % / 0 % [ W /

) A K C B C B * , . b / : : , . < /

% / & % 5 � % 0 % [ W /

- . = - a b 9

� Z # W . = , . - ? b / a b - Z - ? < a / & Z 0 % [ W /

, . - / . = W 9

[ % / % # , - % 9

% / & 9

- . = - ? b 9

% / & 9

% / & 9

� � � # � � $ � # % & � � � � � � � � � � � � � � � ! � � � � � ! � � � � � � � � � � � � �

� − � * � � � � � � � ! � � � � e � � � � � � � � � � � ! � � � � " � � � � � � � � � � m � � � "

� � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � " � � � � � �

� � � � � � � � � � � ! � � � � � � � � � � � � � �� ��

�� �� � u 2 u ! 3 1 q 4 1 ~ u s t 4 q r v u 2 q x � u t � 1 ~ v u o � � 1 2 1 t u 2 � �

! � � � � � � � � ! � � ! � � � � � � ! � � � � � � � � � � &

- . = b 9

� Z # � . = b - Z b ? < a / & Z 0 % [ W /

, . b / . = � 9

[ % / % # , - % 9

% / & 9

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � m � � � � � � � e � � � � � � � � � +

+ �" � � � � � � � � � + � � � � ! � � � � � � � � � � � � � � � � � � � + � � � � ! � � � � !

� � � � � � � �� �

�%�%�%

� � ! � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � + ! � � � � l � � � � � � � � � � �

� � � � � � � � � � � � � � ��

� # � # $ � # %� ! � � ! � � � � � ! � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � ! � � � � � � � � � � � � � � � ! � � � � � � ��

� ! � � � � � � � � " � � �

� � � � � � � � � � � � � + ! � � � � l � � � � � � � � � � � � � � � � � � � � � � � � � � � � " � � �

� � � � � ! � � � � ! � � � � � � � � � " � ! � � � � ! � � � � � � � � � � � � � � � � � � � � � ! � �

� � � � � � � � � � ! � � � � � � � � � � " � ! � � � � � l � � e � � � � � � � � � � � � � � � � � � ��

� � � e � � � � � � � � � � � � � � � � � � � � � � * * � � � � � ! � � � � � � � � � � � � �

� � � � � � � � ! � � � � � � ��

" # Z $ % & ' # % [ % / % # , - % 9

1 , # W . W / - % [ % # 9

0 % [ W /W � � = / - V % / 0 % [ W /

) A K C B C B * ) + @ J A F + , C B A J * ( + @ B * , . b / : : , . - /

% / & % 5 � % 0 % [ W /

� Z # W . = b - Z # W / \ , . - / N / ? � _ & Z 0 % [ W /

- . = - a b 9

, . - / . = W 9

� . = � a W 9

[ % / % # , - % 9

� . = � ? W 9

- . = - ? b 9

% / & 9

% / & 9

% / & 9

! � � � � � � � � ! � � ! � � � ! � e � � � � � � � � � � � � � � &

- . = b 9

� Z # � . = b - Z / & Z 0 % [ W /

, . b / . = �

Page 64: Программирование: теоремы и задачи

� � �� � 1 ~ � 2 � 4 �

� . = � 9

[ % / % # , - % 9

% / & 9

� � � # � � $ � # % � � � � � � � � � � � � e � � � � � � � " � � � � � � � � ! � � � � � � � � � � � �

� � � � � � � � � � � � � � � + � � � � � " � � � � � � � � � � + � ! � � � � � � � � ! � +

� � � � � � � � + � � � � � � l � � � � � � � � � � � � � � � � � � � � + � ! � � � � � � � � � � �

+ � � � � � � l � � � � � � � � ! � � � � � � " � � � � � � m � � � � � � � e � � � � � � � � � + � "

� � � � � � ! � � � � � � � � � � � � � l � ! � � ! � � &

- . = 0 9 � . = 0 9 , . 0 / . = / 9 [ % / % # , - % 9 �

�%�%�%

� � � � � � � ! � � � ! � � � � � l � ! � � ! � � � � � � � � ! � � � � � � � � � � + � �

� � � � � � � � � � ! � � � ! � � " � � � � � � � � � � � � ! � � � � ! � � � ��

� # � # $ � # %� ! � � � � ! � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � �

� � � � � � � � ! � � � � � � + � � � � � � � � � � ! � � � � � � � � � � � � � ! � � � � " � � � � � � �

� � � ��

� � � � ! � � � ! � � � � � � � � � � � � � � &

" # Z $ % & ' # % + O E C O + B C B *` ( C F 9

0 % [ W /W � A @ B * ` @ , A E Q ;- V % / 0 % [ W /

, , A E Q ` ( C J A , + 9

+ O E C O + B C B * ` ( C F 9

U V W 5 % A @ B * ` @ ) E C , C& Z 0 % [ W /

, ) E C , + 9

+ O E C O + B C B *` ( C F 9

% / & 9

, ( � R 9

% / & % 5 � % 0 % [ W /

+ O E C O + B C B * 9

% / & 9

% / & 9 �

��

� � � � � � � � � � � � � � � � �

� � � � # � � � � � � � � � � � � % � ! � � � � � � � � � � � � � � � � � � � � � " � � � �

� � + � � � � ! � � � � � � � ! � � � � � � ! � � � � � � � � � � � �� � � � � � � � � � �

� � � � � � � e � � � � ! � � � � " � � � � l � � � � � � � � � � � � � � � � ! � � � � � � � � & � � � � � � �

� � � � � � � � � � � � � � � � � � � � � ! � � � � " � � � � ! � � � � � � � � � ! � � � � � ! � � � ! � � �

� � � � � � �� � � � � � � + � � � � � � " � � � � � � � � + � � � � � � � � � � � � � � � � � �

� � � ! � � � � � � � � � + � � � � � � � � � � � + �"

� � � + !"� � � "

� � � + � � � � � + � ��

�� � � 2 � 4 1 � 2 4 s 1 q 1 q 4 � 2 1 ~ � 2 � 4 4 � �

� ! � � � � � � � � " � � � � � � � � � � � � � � � � � " � ! � � � � � � � � � � � � � � � � � � � � " � � � + �

� � l � � � � � � + � � + � � � � � � ! � � � � � � � � � � � � � � � ! � � � ��

� + � � ! � � � � � � � � � � � � � � � � � � � � " � + � � � � � � � � � � � � ! � � � � � " � ! � �

� � � � � � � � � m � ! � � � ! � � � �� � � � � � � � � � � � � " � ! � � � � ! � � � � � � � � �

� �� � + � � � � � � � � � � � � � � � � � � � � � � � � � � + � � � � " �� � � ! � � � � � ! � � � �

� � � � ��� h � � � � � � ! � � � � � � � + � � � � � � u 2 4 1 q v 4 2 u o r q q x s 2 r z u s�� & � � � � �

� � �� h ! � � � � � � � ! � � � � � � � � � � � � ! � � � � ! � m � � � � � � � � � � � � � � � ! � � � "

� � � � � � � � � � l � � � � � ! � � � � � ! � � � � � � � � � � � � � � � � � �� � � � + � � � � � � �

+ � � � � � � v u � u � u 4 } 1 � ~ u y � u 2 v 4 2 u o ~ u y�

�%�%

�%

� � � � + � � � " � � � e � � � � � � � � � + � � � ��

� # � # $ � # %� + � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � " � � � � � � � � � ! � � �

� � " � + � � � � ! � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � �

� � ! � � � � " � � � � � � + � � � � � � � � � � � � � � � � � � � � � � � � ! � � �� � � � � �

� � � � � ! � � � � " � � � � � � � � � " � � � � + � � � � � � ! � m � � � � � � � � � �

� � � � � ! � � � � � ! � � � � � � ! � � � � � � � � � � � � � � � � � �� �

�%�%�%

� ! � � � � � � � " � � � � ! � � � � � ! � � � � � � � ! � m � � + � � � � � � � ! � � � � �

� � � � � � � � m � ! � & � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � �

� � � � � � � � � � + � � � ! � � � � " � � � � � � � � � �| | � � � � � � � � � � � � � � � � � �

� ! � � � ! � � � " � � � e � � � � ! � � � � � � � ��

k � � � � � � � � � ! � � � ! � � � � � � � � � � �

! � � " � � � � ! � � ! � � + � � � � � � � � � � � � � � � � � l � � ! � � ! � � � � � � � � � � � � �

+ � ! ·

� � * � � � � � � � � � " � � � � � � � � � ! � � � ! � ! � m � � � � ! � � � � ��

� � � # � � $ � # % � � � � ! � � � � � � � � � ! � � � � + � � � � � ! � � � � � � � � � � � � � � � �

+ � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � f � � � ! � � � � � � �

� � � � � � � � � ! � � � � ��

� # � # $ � # % � � � � ! � � ! � � � � � � � � � � � � � � � � � ! � � � ! � � �� � � �

� � � �

" # W / - % & . , # # , - . b : : / / Z � 0 Z Z 5 % , /

� � � � ! � � � � � � � � � � � � � � � " � � � � � � � ! � � � � � � � � � � � � � � � � ! �

! � � � � ! � � � � � � � � � � � ! � � � � � � � � � � � � l � � ! � � � � � # � � � � � � ! � � � "

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � ! ! � � � � � " � � � � � � � � �

� � � � � ! � � � � � � � � � � � � � � � � � � � � � � � l � � � � � � � ! � �" � � � � � � �

� e � � � � � � � � � � � � � � � � � � � " � � � � � ! � � � " � � � � � ! � � � � � � � ! � � � � � + �"

� � � � � � � � � " � � ! � � � � ��

" # Z $ % & ' # % , & & \ W . b : : /_ 9

> F C ( + . ( C ) A K C B C ( ( + A G + E E A G B ( + 9 D

> ( C F + . ( C ) A K C B C ( ( + A G + E E A G B ( + � , G J S K C A B , A E P � ( ; W D

Page 65: Программирование: теоремы и задачи

� � �� � 1 ~ � 2 � 4 �

0 % [ W /W � " # W / - % & . W / - V % / 0 % [ W / > , A E P � ( C W ; L A ( C ) A K C B C ( C D

> ( � K A c + F A J C B * ( A ( C F + D

% / & % 5 � % 0 % [ W /

> ( C ) A K C B C ( ( + A G + E E A G B ( + D

� Z # � . = b - Z / ' # . W / & Z 0 % [ W /

, & & \ , & # . W / . � / _ 9

% / & 9

> ( C ) A K C B C ( ( + A G + E E A G B ( + N , @ A , A E P � ( M N , G + B + E M A � R

W , A F ; B @ B E A J G � N ; L A ( C ) A K C B C ( M ? B C G K B + I + L ( +

) A K C B C B * W N ( A ( C E ; P C H G + E E A G B ( + @ B � D

W � / Z - " # W / - % & . W / - V % / 0 % [ W /

U # W - % 5 / \ W _ 9 " # W / - % & . W / . = � � � � 9

% / & 9

% / & 9

% / & 9

! � � � � � � � � ! � � ! � � &

� Z # W . = b - Z / & Z 0 % [ W /

" # W / - % & . W / . = � � � � � 9

% / & 9

� Z # W . = b - Z / & Z 0 % [ W /

, & & \ W _

% / & 9

� � � � � � � ! � � � � ! � � � � � � � � ! � � � ! � � � ��

�%�%�%

� ! � � � � � � � � � ! � � ! � � � � � � � � ! � � � � � � ! � � � ! � � " + � � �

� � �

W � / Z - " # W / - % & . W / - V % / 0 % [ W /

U # W - % 5 / \ W _ 9 " # W / - % & . W / . = � � � � 9

% / & 9

� �

U # W - % 5 / \ W _ 9 " # W / - % & . W / . = � � � � 9

� � � � � � � � + � � � � � � � � � � � m � � � � � � � ! � � � � !

� # � # $ � # %k � � � � m � � � � � l � � � � � � ! � � � � � � � � &

F C ( + . ( C ) A K C B C ( ( + A G + E E A G B ( +

( C F + . ( C ) A K C B C ( ( + A G + E E A G B ( + � , G J S K C A B , A E P � ( ; W 9

, @ A , ( + , * ( C ) A K C B C ( ( M A , A E P � ( M F + @ B ; ) ( M � R W : �

�� � � 2 � 4 1 � 2 4 s 1 q 1 q 4 � 2 1 ~ � 2 � 4 4 � n

�%�%�%

� � � � � � � � + � � � � � � � m � � � " � � � � ! � m � � � � � � � � � � � �

� # � # $ � # %

� � � � � � � � � � � � + � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � ! �

� � �� � � � � � � � � � � ! � � � ! � � � � � ! � � � � � � � � � � � � � � � � � � � � l

� � � � � � � � � � � � ! � � � � " � + � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � �

� � ! � � � � ! � � � " � � � e � � � � � � � � � � � � � � � � �� � + � � ! � � � � � � � � � � � � � � � �

� � ! � � � � � � � � � � �� � � � � � � � � � � � � � � ! � � � � � � � ! � � � � � � ! � � � � � � � �

� � " � � � � � � � � � � � � � � �� � ! � ! � � � � � � ! � � � � ! � � � � � � � � � ! � � � ! � � � � �

� + � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � �

� ! � � � � � � � � � � � � ! � � � � ! � � � � � k � � � � � � � + � � � � � � � � � � � � + �

� � � � � � � � � � � � � � � m � � � � ! � � � � � � � � � � ! � � � � � � � � � � � � " � � � � � ! �

� � � + � � � � � � � � � � � �"� � � � � � � � � � � � � � � � � ! � � � ! � � � � � � " � � � � � � �

� � � � � � � � � �� � � � � � � � � " � � � � + � � � � � � � � � � � � � ! � l � � � � � � � � � �

l � � � � � ! � � � � + � � � � � � � � � � � � � � � �" � + � � � � ! � � � � � � �

! � � ! �� k � � � � � � � � � � � " � � � � � � + � � � � � � � � � � ! � � � � � � � � � � � � � � � � �

! � � � ! � � � ! � � � � �� � ! � e � � � � � � + � � " � ! � � � � ! � � � � " � ! � � � l � � � �

� � � ! � � � � " � � � ! � � � ! � � � � � � ! � � � � " � ! � � � ! � � � � � � � � � � � � � � � � � � � � � �

� � � + � � � ! � � � ��� � � � � � � � � ! � � � � " � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � ! � � � !�� ! � � l � � � � � " � � � � � � � � ! � � � ! � �

� � ! � � � � � � � � � � � � � � ! � � � ! � � � l � � � � � � � � ! � � � � � �

� � � � $ � � � � � $ # $ � � � � � % � 1 u 2 4 1 q v 4 2 u o r q q x y 2 r z � � � � � ! � � �

� � � � � � ! � � � � " � � � � � � ! � � + � � � � ! � � � � � � � � � � � � � � � ! � � ! � � ��

� � ! � � � � � ! � � � � � � � ! � m � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � �

! � � � � � � � � � ! � m � " � � � � � ! � � � � � � � � � � ! � � � � � � � � � � ! � � � � ��

� o � � q u y ~ u s � u q 1 q v u y � � ! � � � � � � + � � � � � � � � � � � � � � � � � � � � �

� � ! � � � " � � � � � ! � � � � � � � � � � � � � + �" � � � � ! � � ! � � ! � m ��� � � �

� � � � � � � � ! � m � � � ! � � � � � ! � � � � � � " � � � � � � � � � � � ! � � � � � � � � � � � + � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � e � � � � � � � � � � � � � ���

�%�%�%

� � � � � � ! � � � � � ! � � � � � � � ! � m � � � � � � � � � � ! � � � � � � + � � �

� � � � � � � � � � � � � � � � � � � � ! � � � � � � � � " � � � � + � � � � � � � � � � � � � � � � � � �

� � ! � � ! � � � � �� k � � � � � � � � � � � � ! � � " � � � � ! � � � + � � � � � � � � � � � � � � �

� � � � � ! � � � � � � + � � � � � � � � � � � � � � � � � � ! � + � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � ! � � � � � � � � � ! ·

� � � � � � � � � � � � ! � � � � ! � � � !

� � � � + � � � � � � � � � � � � � �

� # � # $ � # %� ! � � ! � � � � ! � � � � � � ! � � � � � � � � + � � ! � � � � � � � � � � � �

� � ! � � � ! � � � � ! � m �� � + � � ! � � � � � � � � � � � � l � ! � m � � � � � � + � � � �

� � " � � � � � � � � � � � � " � � � � � � ! � � � � � � � � + � � ! � � � � � � � � ! � � � � � � � �

� � � � � � ! � � ! �� � ! � � � � ! � � � � � � � + � � ! � � � � � � � � � � + � � l � � � � � � � � � �

� � � � � � � � � � � � � � � � � � ' � � � � � � � � � � � � � � � � � � � � " � � � � � � ! � � � � � � � �

+ � � ! � � � � � ��

Page 66: Программирование: теоремы и задачи

� � � �� � 1 ~ � 2 � 4 �

" # Z $ % & ' # % , & & \ W . b : : / _ 9

0 % [ W /W � , A E P � ( C W R C G E C P A ( C - V % / 0 % [ W /

( � K A c + F A J C B * ( A ( C F +

% / & % 5 � % 0 % [ W /

R C G E C @ � B * W \ ( C ) A K C B C B * � ) + I A B � B * G C G R C G E C P A ( ( ; S _

F J H , @ A Q � N @ + @ A F ( � Q @ W

, & & \ � _ 9

% / & 9

% / & 9

% / & 9

� � � � � � " � � � e � � � ! � � � � ! � � � � � � � � � � ! � � � � � � � � � � ! � � � � � � � � � � "

� � � ! � � � ! � � � � � � + � � ! � � � � � l � � ! � � � � � � � � � � � � � � � " � � � � �

� � " � ! � � � � � + � � � � � � � � � � � � � + � � ! � � � � � � � � � ! � m � " � � � + � � ! � � � � �

� � � � � ��

� ! � � � ! � " � � � � � � � � � � � � � + � � ! � � � � �� � � � � � � � � � ! �

� � � � " � � � � � � � � � + � � ! � � � � � � � � � � �→ � → → �" � ! � � � � � �

� � � � � � � � � � + � � ! � � � � � � � ! � � � � � # � � ! � � � � � ! � � � � � � � � � � � � � �

� � " � � � � + � ! � � l � � � � � � � � � � ��

� + � � � � � � � � � � � � � � � � � ! � � � � �

� � � � � � � � � � � � � � ! � � � � ! � � � � ! � � � � � � � � � ! � � � � ! � � � h � � � � ! �

! � � � � � ! � � � � � ! � � � � � � � � � � � � � � � � + � � ! � � � � � � � � → → �

� � � � � � + � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � + � � � � � � � � � � � � � � + � � ! � � � � � � � � � ! � m � � � � � � � � + � � �

� � � � � � � � � � � � ! � � � � � � � � ��� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � ! � � � " + � � � � " � � � � �

� � � � � ! � � � � ! � � � ! � � � � � � � � � � + � � ! � � � � � � � � ! � � � � � � � � � � � � �

� � � � � � � ��

! � � � � � � � � � � � � � � � �� � � � � � � ! � � � � + � � ! � � � � � � � � � � � � � � � �

� � � � � ! � + � � � ! � � � ! � � + � � � � � � � � � � � � � ��

� � � � � � � � l �

� � � + � � � ! � � � � � � � � ! � + � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � + � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � ! � � � � � � "

� � � � � ! � � � � � � � � + � � ! � � � � �� � � ! � � � � � + � � � � � � � � � � � ! � � � �

� ! � � � � � � � � � � � � ! � � � ! � � � � � � + � � � � � � � � � � � � � � � � � � �� h � � �

� � ! � + � " � � � � � � � � � � � � � � � � " � � � + � � � � � � � ! � � � � � �" � � � ! � � � � � � �

� � � � � � � � � � " � � � � � � � � � � � � � � � � � � � � � � � ��

! � � l � � � � � � � � �

� ! � � � � � � � � � � � �� �

�%�%�%

d � � � � � � � � � + � � � � � � � ! � � � � � ! � � � � � � � � � ! � m � � � � � � �

� � � � � � � � � � � ! � � � " � � � � � � � � + � � � � � � � � � ! � � � � f � ! � m � � � �

� � � ! � � � � � � � � ��

� � � # � %

� � � � � � � � � � � � � � � � � � � � ! � � ! � � � � � ! � � � � � � � � �

� � � � � � � � � + � � � � � � � � � � � � � � � � ! � � � " � � � � � � � � � ! � � � � � � �

�� � � 2 � 4 1 � 2 4 s 1 q 1 q 4 � 2 1 ~ � 2 � 4 4 � � �

k � � � l � � � � ! � � � � + � � � � � � � � + � � � � � � � � � � � � � � � � � � � ! � � � � �

� ! � � � � � � � � � + � � � � � � � � � � � + � � � � � � v 1 u 2 1 s u y � � o 4 } r � �

�%�%

�%

! ! � � � � � ! � � � � � � � ! � m � � � � � � ! � � � � � � � � � � � � � � �

� � � � � � + � � � � � � � m � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � ! �

� � � � $ � % � � � � � � � � � � � � � � � � ! � m � ! � � ! � � + $ � %� k � � � � � � � �

� � � � ! � � � � � � � ! � � � � � � � � � � � ! � � � ! � � � � � � � � ! � � � � � � � � � � � � �

� � � � � � � ! � � ! � � � + � � � � � � � � � � + � � � � � � � � � � ! � � � � � � � � l � � � � � � � �

� � � � �� � ! � � � ! � e � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

" � � � + � � � �� � � � � � � + � � � � � ! � � � ! � � � � � l � ! � � � � ! � � � � � � � � l � l � � � �

� � � � � � � � � � � � � � + $ � % � � � � � � � � � � � � � � � � + � � � l � l � � � �

� � � � � � � � � � � � � � � � � � + � � � � � � � ) ��

'�

� � � � � � � � � � � � � � � � � � + � � � l � � � � � � ! � � � � ! � � � ! � � � � � �

� � � � ! � � � � ! � � ! � � � � � � � � � � � � � � � ! � � � � � ! � � � � � � � � � � � � � � � � �

� + � � � � � � � � ! � ��

� � � � � � � � � � � � � � � � � � � � � d � � � ! � � � � � � � ! � �

� � � ! � � � � � � �� � � � � � � � � � � � � ! � � ! � � � � � � � � � � � � � � � � � � � � � � � � � +

� � � � � � � � ! � � � � � � | �'� � � � � � � � � � � � � � � � � � � + � � ! � � � � � � � � � � � � � � � �

� � � � � ��" # Z $ % & ' # % � Z # - \ 5 N # . W / - % [ % # _ 9

0 % [ W /W � 5 = # - V % / 0 % [ W /

( � K A c + F A J C B * ( A ( C F + ? ; K C @ B + G ) ; @ B

% / & % 5 � % 0 % [ W /

, M O E C B * @ J ; K C ] ( + A K � @ J + � , ) + J ; � ( B A E , C J A \ 5 N #/

0 . = , . � /

) A E A @ B C , � B * � J A I A ( B M @ + E B � E ; A I + c + ; K C @ B G C B C G N K B + O M

@ ( C K C J C P J � � J A I A ( B M N I A ( * P � A 0 ? ; K C @ B + G \ 5 N 5 5/

R C B A I � J A I A ( B M N E C , ( M A 0 ? ; K C @ B + G \ 5 5 N # #/

R C B A I � J A I A ( B M N O + J * P � A 0 ? ; K C @ B + G \ # # N #/

� Z # - \ 5 N 5 5 _ 9

� Z # - \ # # N # _ 9

% / & 9

% / & 9

d � + � � � � � � e � � � � � � � � � ! � � ! � � � � � � � � � � � � � � � ! � � � � � � � ! � � � � � � �

� � � � ! � � � � � �� � � � � � � ! � � � � � ! � � � � � � � � � � � � � � � ��

� � � e � � � � � � � � �

� � � � + � � ! � � � � ! � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

! � � � ! � � � � � ! � � � � ! � � � � � � � � � � � � � � � � � ! � � ! � � � � � � � � � � � � � � � � �

� � � ! � � � � ! � � � ! � � � � � � � � � � � � � � � � � � � � � ��

����

(�

� � � � + � � � � � � � � � � � � � � � � � ! � � � � ! � � � � � � � � � � � � � � � + � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � ! � ! � � � � � e � � � � � � �

Page 67: Программирование: теоремы и задачи

� � �� � 1 ~ � 2 � 4 �

� � ! � � � � � � ! � � � � � � � � ! � �� � � � � ! � � � � � � � � � � � � ! � � + � � � � � � � �

� � ! � � ! � � � � � � � � � � ��

" � � � + � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � ! � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � ! � � � � ! � � � � � � � � �

� � � � � � � ! � � � � � � � � �� � � �

6

! � *�

� � � � − �

( � � ) � * � � � �)

� � ! � � � � � � � � � � � � � � � � � � � � ! � � � ! � � � � � � l e � � � � � � � � � � � � � � � � ! � � �

� � � � � � � � � � �� � � ! � � � � � � � e � � � ! � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � ! � � � � � � � � � � � � � � � � � � � � ! ��� k � ! � � � � � � � ! � � � � � � � � � � � � ! � �

� e � � � � � � � � � � � � � � � ! � � � � � � � � ! � � � � � � � � � � � � � � � � ) � � � � � �

� � � � � � � � � � � � � � � � � � � $ � � � � � � $ � � � � ! � � � � � � � � ) � � � � � � �

e � � � � � � � � � � � � � � � � � ! � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � �

6

!

′ � � �� � ! � e � � � � � � � � � � � � � � � ! � � � �

� � � � � � � � � � $ � $ � � � � � $ + �| | � − � � � � � � � � � � � � � � � ∫ � $ � $ $

� � � � � � � � � � ∫ � $ � $ � �� � ! � � � � � � � � � � � � � � � � !′ � � � � ! � � � ! � �

� � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � ! � � � ∫ $ � � $ � � � � � � � � � � � �

� � � � ! � � � � ��'

����

��

� � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � )�

h ! � �

� � � � � � � � � � � )� � � � � � � � � � � � � � � � � � � e � � � � � � � � � � � � � � � � � � �

� � � ! � � � � � � � � � � � ! � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � )

� ��� � � � � � � � � k � ! � � ! � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � e � �

� � ! � �� � � � � � � � � �� ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � e � � � � � �

� � � � � � � � � � � � ! � � � � � � � � ! � � � � � � � � �

)� � � � ! � � � � � � � ! � � � ) � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � ) ��" � � � � � � � �� � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � e � � � � � � � �

��

d � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � ! � � � � � e � � �

� � � � �� � � � l � ! � � � � � � � ! � � � � �#�

d � � � � � ! � � ! � � � � e � � � � � � � � � � � ! � � � � � � ! � � � � � � � � � � �

� � � � � � � � e � � � � � � �� k � � � � l ! � � � ! � � � � � � � � � � � � � � � � � � ! � � � �

� � � � � � � � � � e � � � � � e � � � � � � � � � ��

k ! � � � � e � � � e � � � � � � � � � � � e � � � � � � � � � � � � � � � � � � � � � �

� � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � e � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � ) � � � � e � � � � � � � � � � � � � � � � � � � ! � � ��

�� � � 2 � 4 1 � 2 4 s 1 q 1 q 4 � 2 1 ~ � 2 � 4 4 � � �

��

� ! � � � � ! � � � ! � � � � � � � � � � � � ! � � � � � � ! � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � e � � �

� � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � e � � � � � � � � ) � � � �

� � � � � � � � � � � � � �� � � � � � � � � � �

� � � �

6

! � * � � � � � � * � � � ! � � ! � � �|

� � �

� � � � � � � � � � � � � � � � � � j � � � � � � � � � � � � � ! � ! � � � � � � � � � � � � � � � � � � � �

� � � � � � � � ! � � � � � � � � � �|� � e � � � � � � �� � � � � � � � � � � � $ � � ! � �

� � � � � � ! � � � � � � � � ! � � ! � � � � � � � � � � � � � � � ! � � � � � � � � � $� � � � � �

� � � � � ! � � � ! � � � � e � � � � � � � � � � $ � � � � � � � � � � � � � � � � � � $�

h � � � � � � � ! � � � � � � ! � � � � � � � � � � � � � � e � � � � � � � � � � � � $�

h � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

6

! � � ! � � � l � l

! � � � � ! � e � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � * �|

� � � ��

'

Page 68: Программирование: теоремы и задачи

��

� � � � � � � � � � �

� � � � � � � � � � �

� � � � � � � � ! � � � � � � � � � � � � � � ! � � ! � � ! � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � ! � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � ! � � � � � � � ! � �

� ! � � � � � � � � � � � � � � e � � � � � � � � � � � l � ! � � ! � � � � � ! � � � ! � � �� � �

� � � � � � � � � � � � � � e � � � � � � � ! � � � � � � ! � ! � � � � � � � � ! � � � ! � � � �

� � � � � � � l � � � � � � � � � � � � � � ! � � � ! � � � � � � � � ! � � � � � � � � � � � � � ��

i � � � e � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � l � � ! � � � � � � � � � � � � � � � � � � � � � l � � � � � � ! � � � � � � � � � � � � � �

� � l � � � � � � � � � � � � � � � �� � ! � � � � � � ! � � � ! � � � ! � � � � � � � ! � � ! � �

� � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � l � � � � ! � � � ! � � � � � � � ! � � ! � �� � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � ! � � ! � � ! � � � � � � ! � �

� � ! � � � � � � � ! � � ! � � � � � ! � � � ��

� � � � � � � � � ! � � � � � � � � ! � � � ! � � �

� � � � � � � l � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � ��

��

� � � � � � � � � � � � �� � � � � � � �

� � � � � � � � � � � � �

(�

��

��

k � � � l � � ! � � � ! � � � � � � � ! � � � � ! � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � e m m � � � � � � � � �

� � � � � � � e � � � � � � � � � � � l � � ! � � � ! �

� � � � � l � ! � � ! � ��

� ' / $ - W Z / � \ / N < . W / - % [ % # _ . W / - % [ % # 9

> / Y = 0 9 0 X = < X = / D

0 % [ W /W � \ < = 0 _ Z # \ < = / _ - V % / 0 % [ W /

� . = b 9

% / & % 5 � % 0 % [ W / > 0 X < X / D

� . = � \ / ? b N < ? b _ a � \ / ? b N < _

% / & 9

% / & 9

� � � � � r t � 4 � r � q r } 1 q 4 y � 3 4 q r s 4 } 1 � ~ u 1 � 2 u 2 r s s 4 2 u o r q 4 1 � � � �

� � � � � � � � � ! � � � � � � � )� e � � � � � � � � � � � � � � � � � � � e � � � � � � � � �

� � � � � � � �� k � � � � � � � � � � ! � + ! �− � − � * ! � − � � � � � � � � � � � � � � � � m � � � � �

! � � � � � � � � ! � � e � � � � � � � e � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � )� e � � � � � � � � � � � � � � � � � � � � � � l � � l � � � � � � � � l � � l � � e � � �

e � � � � �� h � � � � � � � � � � � � � � ! �

� �

� � �

� � � �

� � � � � � � � �

� � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � e � � �

� � � � � ! � � � ! � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � �

� � � � � � �� � � � � � � � � � � � � � � � � � � � m � ! � � � �

! � +

� �

) � � � − ) � �

� � � � � � � � � � � � � � e � � � � � � � � � � � � � � � � � � � � � ! � � � � � � ! � ! � � � � �

� � � � � � � � � � ! � � � � � � ! � � � � � � ! � � � ! � � �� � � � � e � � � � � � � � � � � � � �

� � � � � � � � � � � � � m � � � � � � � � �� � � + ! � � � � � � � � � � � � � � � + �|

�|�| �

� � � � � � � � � � � � � � ! � � � l � � � � � � e � � � � � �� �

(�

����

� � � � � � � � � � � � � � � � ! � � � � ! � � � � � ! � � � ! � � � � � � � � � ! � �

� � ! � � � � � � � � ! � � � � � ! � � � � � � � � � � h � � � � � � � ! � � � � � � � ��

� � � � � � �h � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � ! � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � l � � � � ! � � � � ! � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � ! � � � � ��

! � � ! � � � � � � � � � � � � � � � �

! � � � � � � d � � � ! � � � � � � � ! � � ! � � � ! � � � � � � � � � � � � � � � � � � � � � � � � ! � �

� � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � �

� � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � ��

h � � � � � ! � � � � � ! � �

� � � � � � � � � � � e � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � ! � � � ! � � � �

� � � � � ! � � � � � � � � � � ! � � � ! � � � � � � � � � � � � � � � � � � � � � � � � ! � � � ! � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � e � � � � � � ! � � ! � � � � ! �

� � � � � � � � � � �� �

� ! � � � � � � � � � � � � ! � � � � � ! � � � � � ! � � � ! � � � � � � ! � � � ! � � � � � �

� � ! � � � � � � ! � � � ! � � � � � � � � � � � � � � � � � � � � ! � � � ! � � � � � � � ! � � � � � � � �

� � � � � � � � � � � � � � � � ! � � � � � � � � � � � ! � �� � � ! � � ! � � � � � � � � ��

� �

Page 69: Программирование: теоремы и задачи

� � � �� � r ~ u t u y v 4 � � t 1 � 2 1 ~ � 2 � 4 4

� � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � �� � � �

� � ��

� �

↙ ↘� � ��

� � � � ��

� �

↙ ↘ ↙ ↘� � ��

� � � � ��

� � � � ��

� �

i � � � � � � � � � � � � � � � � � � � � l � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � l � � e � � � � � �� � � � � � ! � � � � � � � � � � � � 3 4 q r s 4 } 1 � ~ 4 s � 2 u 2 r s s 4 w

2 u o r q 4 1 s � � � ! � � � � � � � � � � � � � � � � � � � � � � j � � ! � � � � � � � � � � � � �

� � m � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

(�

��

��

� � ! � � � � � � � � � � � � � � � � � � � � ! � � ! � ! � � � ! � � � � � � � � � ! � �

� � � � � � � � � � ! � � � ! � � � � � � � ! � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � �

� � + � � + � f � + � − � * � − � � � � � � �

(�

��

��

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � �

� � ! � � � � � � ! � � � � � � � � � �

� � � ! � � ! � � � l � � � � ! � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � − � � � � � � � � � � e � � � � � � � � � � � � � � � � � � � �

� � � � � � � k � � � � � � � l ! � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � �� � � � � � � � � � � � � l � � � � � � � � ! � � ! � � � � � �� � � � � �

� � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � ! � � � � � � � � � � � � �� � � � ! � �

� � ! � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � ! � � � � � � � � � � � � � � � � ��

� � � � � � �# � � � � � � � � � � � � � � � ! � � � � � ! � � � � ! � � � � � � � � � �

� � � � � � � � � � � � � � � ! � � � �� � � � � � )�

� � � � � ! � � � ! � � � � � ! � � � � � )�

� � ! � � � � )|

� � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � ! �

� � )�

��

� � � � � � ! � ! � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � �

� � � l � � � � � � � ! � � � � � � f � � ! � � � � )|

� � � � � � � � � � � � ! � � � ��� � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �| � �� � ! � � + ) * � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � l � � � � � ! � � � ��

q q����q��

qqA

AAAqq����q

��

��qAAAA�

) �

�� � )|

� �

� � � � � r t � 4 � r � q r } 1 q 4 y � 3 4 q r s 4 } 1 � ~ u 1 � 2 u 2 r s s 4 2 u o r q 4 1 � � � �

� � ! � � � � ) | � � � � � � � � � � � � � � � � � � ! � � ! � � � � � � � � � � � � � � � � � � � � � )|

� �

� � � � � � � � � � � � ! � � � � � � � � � �� � � � � � ! � � � ! ! � � � � � l m � ! � � � � �

� � ) | � � � � ! � � + ) * � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ) | � � + ��

� ! � � + ) * � � � � � � � � � � � � ! � � � � � � � � � � � � e � � � � � � � � � � � � � � � ) | � � + ��

� � � � � � � ) * ��

qqA

AAAqq����q

��

��q) �

��

��

��

@@

@@

@@

� � ! � )�

� � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � )|

� � � � � � � � � � � � � � � � �

� � � ! � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � ! � � � � ! � � ! � � � �� � ! � � � � � �

� � � � � � � � � � ! � � � � � e � � � � � ! � � � � � � � � � � � � � � � � � � � l � � � � � � � ! � � �

) * �| | � − �� � � � � � � � � � � � � � � � � � � ! � � ! � � � � � � � � � � � � � � � � �

� � � � � � � � � �

� {

� � � � � � � ! � )� � * � � � � � � � ! � �� � * � � ) | � * � � |� �}

� � � � � + ) * �| | � − ��

� ! � e � � � � � � � � � � � � � � � � ! � � + { * �

� � ! � { � � � � � � � ! � � � � � � ! � � � � � � � � � � � � � � � � � � � � � ! � � � � � �

� � � � � � ! � � � ! � � ! � � � � � ! � � � � � � � � �

k � � � � � � � � � � � � � � � � � � ) | � � � � � � � � � � � � � � � � ! � � � � � � ! � � � � � � �

� � � � � � � ! � � � � ! � � � � � � � − ) * � �� � � � � � � � � � ! � � ! � � � � � � � � � � � l �

� l � � � � � � � ! � � � � � � � ! � � � � ! � � � � � � � � � ! � � � � � � ! � � � � � � �

! � � � ! ! � � � � � � m � ! � � � � ! � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � �

� � � �� �

(�

����

� � � ! � � � � ! � � � ! � � × � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � �

� � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � ! � � � ! � � � ! � � × �

� � � � � � � � � � � � � � � ! � � � ! � � � ! � � × ) � � � ! � � � � � � � � � � � � � � � �

� � � � � � � � � ! � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � ! � � � ! � � � �

! � � × )�

& � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � )

� � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � ! � � � � � � ! � � �

� � � � � � � � � � � � � � � � � � � � � � � � e � � � � � � � � � � � � � � � � � � � � � � � ! � �

� � � � � � � � � � � � � � � e � � � � ! � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � ! � � �

� � � � ! � � �� � � � � � � � � � � � ! � � � � � � � � � � � � � � ! � � l � � � � � � � �

� � ! � � � � � � � � � � � � �� � � � � � � � � � � � � l � � � � � � � � � � � � � � � ! � � � �

� � � � � � � � � � � � � � � � � � � ! � � � ! � � � � � � � ! � �� � � � � � � � � � � � � � � � � �

� � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � ! � ��

Page 70: Программирование: теоремы и задачи

� � � �� � r ~ u t u y v 4 � � t 1 � 2 1 ~ � 2 � 4 4

� � � � � � � � � � ! � � � ! � � � ! � � × ��� × ��

� × � � � � � � � ! � � � � � � �

� � � � � � � � � � �� � � ! � � � � � � ! � � � � � · � · � * � · � · � + � � * � � + � ��

� � � � � ! � � � � � ! � � � � � · � · � * � · � · � + n ��

� � � � � � �� ! � � � � � � � � � � � � � � � � ! � � � � � ! � � � � � � � � � � � � � � � ! � � � �

" �|

� '� � � � ! � � � � � � � ! � � � � " � | � ' � � � � � � � � � � � � � � ! � � � � " � − �| �

'�

� � � ! � �

� � � � � � ! � � � � � " − �|

' � " |

* � ' � � l � � � � � ! � � � ! � � � � � � � � l � �

� � � � ! � � � � � � �� � � � � � � � � � � � � � ! � � " '�

h � � � � � ! � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � " � ' " � '�

� � ! � � � � |� � � � � � � � � � � � � � � � � � l � � � � � � � � � � � � � � � ! � � � � � � �

� � � � � ! � � � � � � � � � � � " | � ' � � ! � �

6

� �

6

� ��

� � � � � � � � � � � � � � ! � � � �

� � � | ���

� � � � � � � � � | * � � ! � � � � � � � l � � � ! � � � � � � � � � ! � � � � � � �

� � � � � � � � � � � �� d � � � ! ! � � � � � � m � ! � � � � � � � � � � � � �

� � |� � + � { � � |) � * � � ) | � � * " ' " ) ' " � '}

� � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � ) + * �| | � − ��

� � � � � � � � ! � � � � � � � � �

� � ! � � � � � � ! � � � � " |

) ' � � � � � � ! � � � ! � � � ! � " ' " ) '� � ! � � � � � � � � �

� � ! � � � � � � ! � � � � " )|

� ' � � � � ! � � � ! " ) ' " � '� � � � � � � � � � � � � � � � � �

� ! � � � � � � � � � ! � � � � " ' " ) ' " � '� �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � �

� � − � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � ! � � � � � �

� � � � � � � � � � ! � � � � � � � � � � � � � e � � � � � ! � ��

(�

����

� � � � � � � � � ! � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � �

� � �� � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � �� �� � � ! � � � �

� � � ! � � � � l � � � ! � � � � ! � � � � � � � � � � � � � � � � � � � � � l � � � � � � � � � ! � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � e � � � � � � � � � � � � � � ! � �

� � � � �� �

(�

��

��

i � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � ! � � � � � � � � � � �

� � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � e � � �

� � � � � � � | | � e � � � � � � � � � � � � � � � � � � e � � � � � $� � ! � � � ! � � � �

� � � � � � � � � ! � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � × × � � � � � � �

� ! � � � � � � � � � � � � � � � � � � $� � � � � � � � � ! � � � � � � � � � � � � ! � � � � � � � � � ! �

� � � � � � � � ! � � � � � � � � � � � ! � � � � � � � � � � � � � � � � e � � � � � � � � � � � ! � � �

� � � � � � � � � � � � � � � � � � �

� � � � � � �

i � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � �

� � � � � � � � � � � ! � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � ! � ! � � � � � ! � � � � � � � � � � � � � � � � �� �

� � �� � v 1 ~ u v � u ! 1 q q x � � r 3 r q 4 y � � n

� � � � � � � � � e � � � � � � ! � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � �

! � � � � ! � � � ! � � � ! � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � ��

k � � � l � � � � � � � � � � � � � � ! l � � � � � � � � � � � � � � � � � � � � � � � � � ��

(�

��

(�

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � $ � | | $ � � � � � � ��

� � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � ! � � � � � � � � � $ � |

| $ � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � ��

" � � � � � � � �� � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � �

� �� � � � � ! � � � ! � � � � � � � � � � � � � � � � � � � � ! � � � � $ � $ �� ' �

� � � � � � � � � � � � � � � � � � � � � � � � � ! � � � ! � � � � � � � ! � � ! � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � �� � ! � � ! � � � � � � � � e m m � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� ! � � ! � � ! � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � ! � � � � ! � � � � � � � � � � � � � � ! � � � ! � � � � � � � � �

� � � � � � � � � � � � � � � � � � ��

��

��

� � � � � � � � � � � � � � � �

� ! � � � � � ! � � � � � ! � � � � � � ! � � � ! � � � � ! � � � � � � ! � ! � � � � � ! � � ! �

� � � � � � � � � � � � � � � � � � � � ��

(���

��

� � � � � � � � � ! � � � ! � � � � � l � ! � � ! � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � � � ! � � � ! � � � � � l � ! � � ! � � � � � ! � � � � � � � l � l

� � � ! � � � � � � � � � � � � � � �

" # Z $ % & ' # % � Z 1 % \ W N � N / . W / - % [ % # _ 9

1 , # � . W / - % [ % # 9

0 % [ W /W � W = b - V % / 0 % [ W /

U # W - % 5 / \ � @ F A J C B * Q + F � N � N � ? Y � N / _ 9

% / & % 5 � % 0 % [ W /

� . = ? � ? / 9 > � ? B E A B � ] @ B A E L A ( * . @ ; I I C ( + I A E + , E C , ( C D

� Z 1 % \ W ? b N � N � _ 9

U # W - % 5 / \ � @ F A J C B * Q + F � N � N � ? Y � N / _ 9

� Z 1 % \ W ? b N � N / _ 9

% / & 9

% / & 9

� � � � � � � � � � � � � � ! � � � � � � � � � � ! � � � � � � � � � � � � � � � � � ! � � � � �

� � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

Page 71: Программирование: теоремы и задачи

� � � �� � r ~ u t u y v 4 � � t 1 � 2 1 ~ � 2 � 4 4

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � i � � � � � � � e � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� � � e � � � � � � � � � � � � � v 1 ~ u v � u ! 1 q q x � � r 3 r q 4 y � e � � � � � � � � � �

� � ! � � � � � � � � ! � � � � 〈 �| � | �〉� � � � � � � � � � � ! � � � � � � � � ! � ! � � � ! � � � � �

� � � � � � � � � � ! � � � � � � � � � � ! � � � � � � � � � � � � � � � � � ! � � � � � � � � � � i � � � �

� � � � � ! � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � ! � � � � � � � � � � � � � � � �

� � � � � ! � � � � � � � � ! � � � � � � � � ��

� � � � � � � � � � � l � ! � � ! � � �

" # Z $ % & ' # % � Z 1 % \ W N � N / . W / - % [ % # _ 9

0 % [ W /@ F A J C B * @ B A G R C G C R + , ) ; @ B M I

) + J + L � B * , @ B A G B E + ] G ; X W N � N / Y

> � ( , C E � C ( B . + @ B C J + @ * , M ) + J ( � B * R C G C R M , @ B A G A D

U V W 5 % @ B A G ( A ) ; @ B & Z 0 % [ W /

; F C J � B * , A E Q ( � ] � J A I A ( B N ) A E A J + L � , A c + , X � N " N Y

W � � = b - V % / 0 % [ W /

U # W - % 5 / \ � @ F A J C B * Q + F � N " N � ? Y � N _ 9

% / & % 5 � % 0 % [ W /

� . = ? " ? 9> � ? B E A B � ] @ B A E L A ( * . @ ; I I C ( + I A E + , E C , ( C D

) + J + L � B * , @ B A G B E + ] G � X � ? b N � N Y N X b N " N Y N X � ? b N " N � Y

% / & 9

% / & 9

% / & 9

� i � � � � � � � � � � ! � � � � � � � � � � � � � � � � ! � � � � � � � � � ! � l � � � � � � � � � � � �

� � � � � � � ��� k � � � � ! � � � � � � � � � � � ! � � � � � � � � � � � � � ! � � � � � � � � � � � � �

� ��� ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � " � � � � � � � � � ! � � � � � � � � � � � ! � � � � ���

(�����

� k � � � � � �

i � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ���

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � ! � � � ! � � � � � � � � � � ! � � �

��

� � � � � � � � � � � � ! � � � � � � � l � � � � ! � � � � � � � � � � � � � � ! � � �

� � ! � � � � � � � � � � � � � � � � � � � ! � � � � ! � � � � � � � � � � � � � � � � � � � � � � ! � �

� � � � � � � ! � �� � ! � � � � � ! � � � � � � � � � � � ! � � � � � ! � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � ! � � �� �

(���

��

� � � � � � � � � � � � � � � � � ! � � � ! � � � � � � � � � � � � � � � � � � � � � � � �

� ! � � � ! � � � � � � � ! � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� � � � � � �& � m ! � � � � � � l � � � � � � � � � � � � � � � � � � l � � � � � � � � � � � � �

� � � � � � � � l � � � � � � ! � � � � � � ! � � �� �

(���

��

� � � � � � � � � ! � � � ! � � � � � l � ! � � ! � � � � � � � � � l � l � � � � � ! �

� � � � � � � � � � � � � ! � � ��

� � �� � v 1 ~ u v � u ! 1 q q x � � r 3 r q 4 y � � �

� � � � � � � e � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � �

� � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � l � � ! � � � � � � � � � � � � � � � � � �

� � ! � � � � � � � ! � � � � � � � � � � ! � � � � ! � e � � � � � � � � � � � � � � � � ! � �

� � � � � � � � ! � � � � � h � � � � � ! � � � � e � � � � � � � � � � � � � � � � ! � � 〈 � � � � � � � � � �

� � � ! � � ! � � � �〉�

� � � � � e � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � e � �

� � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � ! � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � ! � � � �� �

(�����

� � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � ! � � � � � � � � �

� � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � �� � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � �

� � � � � � � � � � � � �� � ! � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � �

� � ! � � � � + � � � � � � � � � * � � � � � � � � � � � ! � � � � � � � ! � � � � � � � � ! � � � � �

� � ! � � � � � � � � � � � � � �� �

(�����

� � � � � � � � � ! � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � �

! � � � � � � � � � l � � � � � � � � � � ! � � � � � � � � � � � � e � � � � � � � � � � � � � ��

� � � � � � � � � � � � � ! � � � � � � ��

� � � � � � �

� � � � � ! � � � � � � � � ! � � � � � � � � �

� � ! � � � � � � � � � � � � ! � � � � � ! � � � � � � � ! � � � �

� � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� ! � � �� � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � ! � � � � � � � � � ! � � �

� � � � � � � � ! � � � � � � ! � � � � � ! � � � � � � � ! � � ��

e � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � ! � � � � � � � � � � � � � � �

� � � ! � � � � � � � � � � � � � � ! � �

� � � � � � � � � � � � ! � � � � g � � � � � � � � � � � � ! � � � � � � � ! � � � g

� + � � � ! � � � � � � ! � � � � ! � � � � � � � g � �

� � j � � � � � e � � � � ! � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � l � � � � � � � � � � � � ! � � � � � �

� � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

h � � � � � � � � ! � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � ! � �

� � � � � � � � � � � � � � � � � � � � ! � � �� �� � � � � � � � � � ! � � � l � ! � � ! � � � � � � � � � � � � � � ! � � � � ! � � � � � � �

� � � � ! � � � � � � � � � � � � � ! � � ! � � � � � � � � ! � � � � ! � � � � ! � � � � � � � � � � � ��

h � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � ! � � � � � � � �

� � � � ! � � � � � � � � l � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � �

� � � � � ! � � � � � � ! � � � � � � ! � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � ! � �

� � � � � � � � � ! � � � �� k � � ! � � � � � � � � � � � n�

Page 72: Программирование: теоремы и задачи

� � � �� � r ~ u t u y v 4 � � t 1 � 2 1 ~ � 2 � 4 4

(���

��

� � � � � � � � � ! � � � ! � � � � � � � � ! � � � � � ! � � ! � � � � � � ! � � � � ! �

� � ! � � � � � � � ��

� � � ��

� � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � ! � � � �

� � � � ! �� � � � � � � � � � � � � � � ! � � ! � � � � e � � � � � � �

� � � � � � � � � � � � � � � � � � � � ! � 〈 | �〉 � � � � � ! � ! � � � ! � � � � � � � � � �

� � � � � � � � � � � � � � � � � � ! � � ! � � � � � � � � � � � � � � � l � � � � � � � � � � � � � � � ��

� � e � � � � � � � � � � � � ! � � � � � l � � � � � � e � � � ! � � � ! � � � � � � � � � � � � ! � � � �

� � � � �� � � � � � � � ! � � � � � � � � � � � � � � � � � � ! � m � � � � � � � � � � � � � � �

� � �

� ! � � ! � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � l � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � ! � � � � � � � � � ! � � � � � � � � ! � � � ! � � ! � � � � � � � � � � � � � � � � � � �

� � � e � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � ! � � � � � � � � � � � � � �

� � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � ! � � � � � � �

� � ! � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � ! � e � � � � � � � � �

� � � � � � � � � � � � � � � � � � � * �� � � � � � � ! � � ! � � � � � � � � � � � � � � � � � � � �

� � � �

6

� � � � � � � � � * �|

� � � − � � �|

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � + � �� � � � � �

��

��

� � � � � � � � � � � � � � � � � � � � �

� � � � � m � � � � � � � � � � � � ! � � � � � � � ! � � � � � � � � � � � � � � � � � � � ! � �

� � � � � ! � � � ! � � � � � � � � � � � � �

� � � � + � |

� � $ � + � � $ | � � � � $ � � � � $ � � �

� � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � m � � � � � �� � ! � � � � � � � � � �

� � � � � � � � � � m � � � � � � � � � � � � � $ � � ! � � � � � � � � � ! � � � � � � � � � � � � � � � � �

� � $ � � � ! � e � � � ! � � � � � � � � � � � � � � � � � � l � � � � $ � � � � � � � � � � � � � � � � � �

$ | � � $ � | � � � � $ � � |

! � � � � � � � � � � � � � � ! � � � � � � ��

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � $ � � $ � � � � � � $ � � � � � � � � � � �

� � � � � � � ! � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � �|

� � � �|

� � � �|

(�

��

��

� � � � � � � � � ! � � � ! � � � � � l � ! � � ! � � � � � � � � � � � � � � � � � � �

� � � � � � � ��

� � � � � � �� � � � � � � � � � � � � � � $ � � � � � � � � � � � � � � � � � � � � � � � � � �

� � $ � | � � � � $ � � | � � � � � � $ � � � |

�� �� � u � 1 1 � � u ! q x 1 � � � } r 4 2 1 ~ � 2 � 4 4 � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � e � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � �� �

� � � � � � � � � � � � � � � � � � � � � � � � � � l � � � � � � � � ! � � � � � � ! � � � � �

� � � � � ! � � � � � � � � � � � � � � � � ! � � � � � � � � � � ! � � ! � � � ! � � l � � � � � ! � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � m � � � � � � � � � � � � ! � � � � � � � ! � � �

� � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � �

� � � � + � |

� � $ � + � � $ | � � � � $ � � | � � � � $ � � � � $ � � �|

� � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � m � � � � � �� � ! � � � � � �

� � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � �

m � � � � � � � � � � � ! � � � � � � � � � � � ! � � � � � � ! � � � � � � � � � � � � � � � � � � � � ��

(�

����

� � � � � � � � � ! � � � ! � � � � � l � ! � � ! � � � � � � � � � � � � ��

� � � � � � �

� � � � � � � � � � � � � � � � � � � � � � ! � � � � � ! � � � � � � � � � ! � � �

� � � ! � � � � � � � � � � $ � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � ! � � � � � � � l� i � � � � � � � � � � � � � � � � � � � � m � � � � � � � � � � �

� � � � � � � � � � ! � l� � � � � � � � � � � ! � � � � � � � � � ��� t 2 r v q u y � u � � � ~ u y � r � 4 � � � � � � � � u � v z 4 ~ � q u y � r � 4 � � � � � � ! � � � �

� � � � � � � � � l � � � � � � � � � � � � � � m � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � �

� � � � � � � � � � � � � � � � � l � � �� � � � � � � � � � � � � ! � � ! � � �

� � � � � �

� � � � � � � � � �

� � � | � � � � � � �

� � � | � � � | �� �|� � � � � � � � � �

� � � � m � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � l � v 1 ~ u o u u ~ r � � ~ � � � v u 2 r� � � � � � � � � � � � � � � ! � � � � � � � � � � � � � �

! � � � � � � � ! � � � � � � � � � � � � � ! � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � �

� � � � � l � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � m � � � � � � �

� � � � � � ��

� ! � � � � � � � � � � � � � � � � � � � � � � � e � � � � � � � � � � � � � � � � � � ��

� ! � � � � � � � � � � m � � � � � � � � � � � � l � � � � � � � � � � � � � � � � � � l � � m � � � � � �

� ! � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � ! � � � � � � � � ��

� � ! � � ! �

� � � � � � � � � � � � � � � � � � �

� � � � �

� � � � � � � m � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � l � � m � � � � � � � �

� � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � |� �

Page 73: Программирование: теоремы и задачи

� � � �� � r ~ u t u y v 4 � � t 1 � 2 1 ~ � 2 � 4 4

� � ! � � � � � � � ! � � � � � � � � � � � ��

� ! � � � � � � � � � � � � � � � � � � � � � � � �

m � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � l � � � � � � � � � � � � � � � � � � � �� � � � � ! � l � � � � � � � � ! � ! � �

� � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � ! �� � � � � ! � � � � � � � � � �

� � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � �

� � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � $ � � h � � � � � � � � � � � � � �

� � � � � � � � � � � � � $ � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � �

� � � � � ��� � ! � e � � � � � � ! � � � � � � � � l � � � � ��� � � � � � � � � � � � � � � � � � � �

� � � � � l � � � � � � � � � � � � � l � � � � � � � � � ! � � � ! � � � � � � � � �

� � � ! � � � � � ! � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� $ � � $ �

� $ � � � � � $ � �

� $ � � � � � $ � �

� $ % � � � � � � $ | % | � � �

� � � � � � �

� $ � � � $ $ � � $ � � � �

i � � � $ � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � ! � � � � ! � � � � � � � �

� � � � � � � � $ 6+ ��

� � � � � ! � � � � � � � � � � � � � � � � ! � � � � � � � �

� � $ � + � � $ | � � � � $ � � | � � � � $ � � �

� ! � � � ! � � � � � � � � � � � � � � � l � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � ��

e � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � �

� � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � ! � � � � � � � � � � � � � � � � ��

��

� � � � � � � � � � � � � � �

� � � � � �

��

� � � � � � � � � � � �

� e � � ! � � � � � ! � � � � � ! � � � l � � � ! � � � � � � � � � � ! � � � � � � � � � � � � ��

� � � � � � � � � � � � � � ! � � � � � ! � � � � ! � � � � � � � � � � � � � � � � � �� � � � � � �

� � � � � ! � � � ! � � � � � � � ! � � ��

� � � � � � � � � � � � � � � � � ! � � � � � � � � � � �

� � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � ! � � � � � � ! � � � k � � � � � � � � � � � �

! � � � � � � � � � � � � l � � � � � l � � � � � ! � � � � � � � � � � � � � � ! � � � � � ��

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � l � ! � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� ! � ! � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � ��

� � � � � �

� ! � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � �� �� � ! � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � l � � � � �

� � � � � ! � � � � � � � � � � � � ! � � � � ! � � ! � � � � ! � � � � � � � � � � � � � ! � � � � � �

� � � � � � � � � � � � � � � � ! � � ! � � � � � � � � � ! � � ��

��

��

��

� ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � ! � � � � � � �

� � � � ! � � � � � � � � � � ! � � � � � � � � �� � � � � � � � � � � � � � e � � � � � � � � � ! � �

! � � � � � � � � � � � � � � � � � � � � l � � � � � � � � � ��

� � � � � � � � � ! � ! � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � e � � �

� � � � � � � � � � � � � � � � ! � � � ! � ! � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � �� �

� � � � � � � � � � l � � � � � � � � � � ! � � � � � � � � � � � � � � � e � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � ��

��

����

� � � � � � � � � � � � � l � � � � � � � � � ! � � � � � � �� � � � � ! � � � � � � �

� � � � � � � � � � � � ! � � � � ��

� � � � � � � � � � � � � � � � � ! � � ! � " � � � � � �

� � � � � � � � � � l � � � � � � � �

� ! � � � � � � � � � � � � � � � � � � � ! � � � � � �� h � � � � � � � � � � � � � � � � �

Page 74: Программирование: теоремы и задачи

� � � � � � r � q x � r � u � � v s x q r � r z r �

� � � � � � � � � � � � � � �

! � " � � � | � | � � � � �

( ! � " � � � � � �

� �| �� � � � �

! � " � � � � ��

� � * � � � � � � � �)

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � " � � � � �� � � � �

� � � � � � � ��

� � � b �

� Z � W � � b Z � Z � [ � � . � / � � , . b / . � / � � �

� � � � � � � � � � � � . � / � � � � � � � �� ��

� � �

� � � � � ! � " � # �

� " � � � � � " � " � # �

- . � / � � � . � / �

� " � � � � � " � " � # �

� � - . � / ! � . � / $ , . � / . � / � � # �

- . � / � � � . � / $ , . � / . � / �

� �

� - . � / � � � � � � � �� �� � $ � � �

� �

� " � � � � � " � " � # � � . � / � � - . � / � � �

� � � � $ � �

� � �

� ! � � � % � � � � � � � � ! � � � � � � � � l � � � � � ! � � � � � � � � � � � � � � � ! � �

� ! � � ! � � � � � � � � � � � � � � ! � � � � � ! � � # � � � � � ��

��

��

��

� � � � � � � � � � � � � ! � � ! � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � g � � � � � � �

� � ! � � ! � � � � � � � � � � � � � � � � � � � � � g � � � � � � � � � � � � � � � � � � � � � �

� � � � � ! � � � � �

� � � � � � � � � � � ! � � � � � � � � � � � � � �

! � " � � � � �� � �

6

g � � �

6

! � " � � � � ��

� � �

� � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � ! � � � � � � � � � � � � � � � � � � l � � � � � � � � � ! � � � � � → � � �

� � � & � � ! ��

� � � � � � � � � � � � ! � � � � �� � � � � � � � � ! � � � � � � ! � � ! � � � � �

� � � � �� � ! � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � ! � � � � � � � � ��

��

��

'�

� � � � � � � � � � � � � l � � � � � � � � � ! � � � � �→ � � � � � � � ��

� � �

� ! � � � � ��

�� � � � � r v } r y p � � � � v � � � �

� � � � � � � � � � � � � � � � ! � � ( � ��

��

� � � � � � � � � � � � � � � � � � l � � � �

� � � � � � ! � ! � � � � � � � �� � � � � � � � � � � � � � � � ! � � � � � � � � ! � � ! � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � ��

h � � �

( � ��

��

� � � � � � � � � �|

( � ��

��

� � � � � ( ( � ��

��

� �|

( � ��

� ��

� � * ( � � ��

��

� � )

� � � � � ! � � � � � � � � � � � � � � � � l � � � � � � � � � � � � � � � l � � � � � � � � � � � � � l � � � � �

� � � � � � � � � � � � � � � ! � � � � � � � � � f � � � � � � � � � � � % � � � � � � � � � � � �

� � � � � � � � � � ! � � � �� �� � � � � � � � ! � � � � � � � � l � � � � � ! � � � � � � � ��

��

��

)�

� � � ! � � � ! � � � � � � � � � � � � � � � � � � � � � � � ! � m � � � � ! � � �

� � � � � � � � � � � ! � � � � � � � � � � � � � �

" � � � � � � � �� � � � � � � ! � � � � � � � � � � ! � � � � � � � � � ! � � % ! � � ! � � � � �

� � � � ( � ��

��

� �� � � � � � � � � � � � � � � � � ! � � � � � ! � � � � � � � � � � � � � �� ' �

��

��

*�

� � � � � � � � � � l � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � m � � �

! � � � � � l � � � � � � � ! � �� ����

� � ! � � � � � � � � � � � � � ! � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � l � � � e � � � � ! � � �

� � � ! � � � ! � � � � � � � � � � � � � e � �

" � � � � � � � �� � � � � � � � � � ! � m � ! � � � � � � � � � � � � � � � � � � l � � � ! � � � � � � �

� � � � ' �

��

��

��

� � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � ��

� � � � � � � � � � � � � l

� � � � � � � � � ! � � � � �→ � � � � � � � � � � � � � � ! � � � � � � �

� � � � � � � � � ! � � � � � � ! � � � � � � � � � ! � � � � � � � � � ! � � � � ! � � � � � �

� + , � � � � � + - . � � � � � � � � � � � � � � � � � ! � �� � � � � � � � � � � � � � ! � e � � �

• � � � � � � � � � � � � � � � � � � � � ! � � � � ! � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � → � f � ! � e � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � ! � � � � � � � � � � � � � � ! � � � � � � � � � � � � � ! � � f

• � � � � � � � � � � � � � � � � � � � � � � ! � � � � ! � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � → �� � � � � � ! � � � � � � � � � � � ! � � � � � � � � � � � � � � � � �

� � l � � � � � � � � � � � � � � � � � � � � ! � ��

� � � � � � � � � � � � � � � � � � � � ! � � � ! � � � � ! � � � � � � � � � � � � � � � � � � � � l �

� � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � �

� � � � ! � � � � ! � � � � � � � � � � � � � � � � � � � � � e � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � l� � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � �

Page 75: Программирование: теоремы и задачи

� � � � � � r � q x � r � u � � v s x q r � r z r �

� � � �� d � � � � � ! � � � ! � � � � � � � � � � � � � � � � ! � � � e � � � � � � � � � � �

� � � � � � � � � � � � � � � � i � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � �� �

� � � � � � � � � � ! � � � � � � � ! � � � � � � � � � � � � � � � � � � � � ! ! � � � � �

! � � � � � � � m � ! � � � l � � ! � � � � l � � � � � � � � � � � � � � � ! � � �� � ! � e � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � ! � � � � � � � � � � � � � � �

� � � � � � � � � � ! � � � � � � � e � � � � � � � � � � � � � � � � � � � � � � � � � � � � l

� � � � � � � � � ! � � � � � � � � � � � � ! � � � � � � � � � �

� ! � � � � � � � � � � ! � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � �

� � ! � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � �

� � � � � � � � ! � � � � � � ! � � � � � � �� �� � � � � � � � ! � � � � � � � � l � � � � � ! � � � � � � � � � ! ��

��

��

(�

� � � � � � � � � ! � � � � � � � � � % � � � � � ! � � � � � � � � � � � � ! � � � �

� � � � � � � � � � � � � l � � � � � ! � � � | � � � � � � � � � � � � � � � � � � � � � � � ! � � � �

� � � � ! � � � � � � � � � � � � � � � � � ! � � � � � j % � � � � � � ! � � � � � � � � � � � �

� ! � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � ! � � � � � � � � ! � � � � � �

� � � � � ! � � � � � � � � � � � � � � ��

" � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � ! � � � � � � � �

� � � � � � � � � � � ' �

� � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � l � � � � ! � ! � � � � � l

� � � ! � � � � � � ! � �� � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � �

� � � � � ! � � � � � � ! � � � �� � � � � � � � � � � � � � � � � � � � � � � � ! � � � �

� � � � � ! � � % � � � � � � � � � � � % � � � � � � � � � �� � � � � � � � � � � � � � ��

k � � � � � � � � � ! � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � ! � �

��

��

��

� � � � � � � � � � � � e � � � � ! � � � � � � � � � � � � � � � � � � � � � � � m � ! � � �

� � � ! � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � �

� � � � � � � � � � � � � � � � � �� �

��

��

� ��

� � � � � � � � � � � � � � � � � � ! � � � � � ! � � � % � � � � � ! � � � � � � � � �

� � ! � � � � � � � � � � � � � �� �

��

��

� ��

� � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � e � � � � � � � � � � � � � �

� � � � � � � l �

∞ � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � �|

� �|

� |

� � � e � � � � � � � � � � � � � � � � � � � � � ! � � � � ! � � � � � � � � � � � � ! � � � � � � � � �

� � � � � ! � � � � � � � � � � � � �� � � � � � � � � � � ! � � � � � � � � � � � � � � � � � ��

��

��

� ��

� � � � � � � � � � � � � � e � � � � � � � � � � � � ! � � � � ! � � � � � ! � � � � � � � �

� � ! � � � � � � � � � � �

� � � � � � � � � � � � m � � � ! � � � � � � � � � � � � � � � � � � ! � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � ! � � � �� � � � � � � � � �

�� � � � o � � q x � ~ u s � u q � q v x � � u � � ~ o � � t � q � � p � � � q � � � n

� � � � � ! � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � ! � � � � �� � � � �

! � � � � � � � � � �� � � � � ! � � � � � � �� � � � � � % � � ! � � � � � � � � � � � � � ! � � � �

� � � � � � � � � � � � � � � ! � � � � % � � � � � � � � e � � � � ��

��

��

� ��

� � � � � ! � � � �

� � � � � �

� � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � �

� � � � � � � ! � � � � � � � � �

� ! � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � �

� � � � � � � � � � � � � � � � � � � � � � � m � � � � � � � � ! � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � l� h � � � � � � � � � � � � � � � �

� � � � � � � � ! � �� � � � � � ! � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � e � � � � � � � ! � � � � � � � � ! � � � � � � � � ! � � ! � � � � � l � e m m � � � � � � �

� � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � �

� � � � � � m � ! � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

! � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

��

��

� '�

� � � � � � � � � � � � � � � � ! � � � � � � � � ! � � � � � � � m � � � ! � � � � �

� � � � � � � � � � � � � � ! � � � � � ! � � � � � � � � � � � � � � � ! � � � � � � � � � � � � �

! � � * � � �� � � � � � ! � � � ��

" � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � �

� � � � � ! � � � � � � � � � � � � � � � � � � � l � � � � ! ! � � � � ! � � � � � � � � � � � � � � �

� � ! � � � � � � � � � ! � � � � � � � � � � � � � ! � ! � � ��

� � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � * � �

� � � � � � �� � � � ! � � � � � � � � � � � � � � � � � � � � � e � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � ��

� � � � � � � � � � � � � � % � � � � � � � � � �� � �� ' �

��

��

� � � � � � � � � � ��

� � � � � � � � � � � �

� � � � �

� � � � � � � � � ! � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � �

� � � � ! � � � � � � � � *∞� � ! � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � �

� � � l � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � ! � � � � � �

� � � � ! � � � � � ! � � � � � � � � ! � m � � � ! � � � � � � � � � � � � � � � � � � � ! � � � � � � � � �

! � � � � � � � � � � � � ! � � � � � � � � � � � � � � ! � � � l � � � ! � � � � � � � � � � � � � � �

� � � � ��� � � e � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � ! � � � % � � � � � � ! � �

� � � � ! � � � � � � � � � ! � � � � � � � � � � � � � � �� � � � � � � � �� � � � � � � �

� � ! � � ! � � � ! � � � � � � � ! � � ! � � ! � � � � � � e � � � � � � � � � ! � � � � � � � � � � � � ��

� � � ! � � � ! � � � � � � � � � � � � � ��

k � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � �

Page 76: Программирование: теоремы и задачи

� � � � � � r � q x � r � u � � v s x q r � r z r �

� ! � � � � � � ! � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � �

� � � � � ! � � � % � � � � � ! � � �� � � � � � � � � � ! � � � � � � � � � � � � � � � � � � ! � � �

� � � � � � � � ��

� � � � � � � � � � �� � � � � ! � � � � � � � � � � � � � ! � � � � � ! � � � � � ! � � � � � � � � � ! � m � � � � � � � �

� � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � %�� h � � � � �

! � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � ! % � � ! ! � � � � � � � � *∞��

����

��

� ! � � � � � � � � � ! � � ! � � � � � � e � � � � � � � � � � � � � � � � � � � � �

� � � � � � � ! ·

� � � � � � ! % � � ! � � � � � � � � � � � � � � � ! � � � l � � � � � � � � ! � � � ��

� � � � � � � � � � � � � � � ! � � � � � ! � � � � � � � � � � � � � �� ��

� � ��

i � � � �

� ! � � � % � � ! � � � � � ! � � � � � �� � � � � � � � � � � � � � � � � � � � � � � ! % � � ! � � � �

� � � � � � � � �� � � � � � � � � �| | � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � �

! % � ! �� � � � � ! � � ! � � � � ! � � � % � � � � ! � � � � �

� � " � � � � � � � � � � � � � � � # � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � �

� , � � � � � � � � � � � � � � �� � �

� � � � � � � � � � � � � �� � �

� �� � � �� � �

� � � # � �

� # �

� � � � � � � � � ��

� � � � � � �

� � � � � � � �

� � � � � � � � � � � � ��

� �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � �

� � � � � � � � � � � � " � # �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� " � � � � � " � � . � / � " � # �

� � � " � . � / . � / �

� � � � � � � � � � � � � � � � � � # �

� � � � � � � �

� � � � � � � � � � �

� � � � � � � � � � �

� �

� �

� �

� �

�� � � � o � � q x � ~ u s � u q � q v x � � u � � ~ o � � t � q � � p � � � q � � � �

h � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � ! � � � � �� � � � � � � � � � � � � � � � � � � � ! � � l � � � ! � � � ! � � % � � � � ! �

� � � � % � � � � � � � � � ! � � ! � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � ��

� � � ! � � � � � � ! � � � � � � � � � ! � � � � � � � � � � � � � � � � � � ! % � � ! � � � ! � � � � � � �

� � � � � � � ��

� � � � � � � � ��� � � � � � � � � � ! � � � ) � � � � � � � � � � � � � � � ! � � � � ! � � � � � � � � � � � � � ! � �

� � � � � � � � � � � � � � � � � � � ! � � � � )�

� � � � � � � � � � � � � � � � � � � � � � � � �

� ) * � � � � � � � � � ! % � � ! � � � � � � � � � � ) � � \ � � � � ∪ ∪ � ) � �

� � � � � � � � � � � � � l � � � � ) � �|

�|� � � � � ! � � � � � � ! � � ! � � � � � �

� � � � � � � � � � � � � � � � � � ! � � � � � � � ! � � � � � � � � � � � � � � � � � � � �

� � � � ! � � � � � � � � � � � � � � � � � � ) � � � � � � � � � � � !

� � � � � � � � � � � � � � � � � � � � � � �| | � ) ��

� � � � ) � � � � � � � � � � � � � � � � � ! � � � � � � � � " � � � � � � � � � � � � � � � � �

� ! � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � ) ��

� � � � � � �

� ! � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � l � � � � � � � � � � � � � � � � � ! � � � � l � � � � � � � � ! � � � � � � � �

� � � � � � � � � � � ! % � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � l � � � � � � � �

� � � � � � � � � ! � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � ) * � ��

h � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � ! � �

� � � � � � � � � � � � � � � � � � � � � ) * � �� �

� � � � � � � � � � �

" � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � ! � � � � � �

! � � � � � � � � ! � m � � � � � ! � � � ! � � �� # � � � � � � � � � � � � � � � � � � � � ! � � � � � ! � �

� � � � � � � ! � m � � � � � � � � ! � � � � � � � ! � � � � � � � � � �� # � � � ! � � � � � � � � � �

� � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � ! � � � � � � � � � � �

� � � � ! � � � ! � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � �

� ! � � � � � � � � � � � ! � m �� � � � � � ! � � � � � � � � � � � � � � � � � ! � � � � � ! � m ��

� � � � ! � � � � � � � � � � � � � � ! � � � � � � � � � � � ! � m � � � � � � � � � � � � � � � �

� � � � � � � � ! � �� � � � � � � � ! � � � � � � � � � � � � � � � ! � � � � � � � � � � � ! � m �

� � � � � � � ��

" � � � � � � � � � � ! � m � � � ! � � � � � � � � � � � � � � � � ! � �

m � � � � � � � � � � � � � � � � � � � ! � � � � � � � ! � � � ! � � � � � � � � � � ! � � � �

� � ! � � � � � � � ! � � � � � ! � � � � � � � � � � � � � ! � m � � � � � � � � � � � � � � ! � ��� % � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � ! � � ! �� i � � � � � � � �

� ! � � � � � � � � � � � � � � � � � � � � ! � m � � � � � � ! � � � � � ! � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � ! � � � � � ! � m � � � ! � � � � � � ! �

� � � � � � � ! � � � � � � � � � � � ! � � � � � ! � m � � � � � � � � � � � � � ! � � � ! � � � � �

� � � � � � � � � � � � � � � % � � % ��

� � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � ! � � �

Page 77: Программирование: теоремы и задачи

� � � � � � r � q x � r � u � � v s x q r � r z r �

� � � � � � � � � � � ! � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � ! � m � �

� � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � ! � � � ��# � � � ! � � � � � � � � � � � � � � � � � � � � � � ! � � � � � ! � m � � � � � � � � �

� � � � % ! % � ! � � � � ! � � � � � � � � � � ! � � ! � � ! � � � � ! � � � � � � � h � � � � � �

� � � � � � � ! � � � � � ! � � � � � � � � � � � � � � � % ! % � ! � � � � � � � � � ! � � � � � ��# � � � � � � � � � � ! � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � ! � � � � � � � � �

! � � � � � � � � � � � � � ! % � � ! �� h � � � � � � � � � ! � � � ! � � � � � ! � � � � � � � � �

� � � � � � ��

� � � � � � � � � � � � � � � � � � � � ! � m �� � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � ! � � � � � � � � � � � � � � � � � � � � % � � � � � � ! � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � �

� ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � �

� � � % � � ! � � � � � � � � � ! � � � � � � � � � � � � � � � ! � � � � � � � l ! � � � � � � � � �

� � ! � � � � � � � � � � � � � � � ! � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

! � � � � � � � � � � � � % � � � � � � � � ! � � ! �� � � ! � � � � � � � � ! � � � � ! � � � � � � �

l � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � �

� � ! � � � � ! � m � � � � � � � � � � � � ! � � � � � � � � � � � � � � u � � ~ u s o � � t � q ��

������

� � � � � � � � � ! � � ! � � � � � � � � � � � � � � � ��

" � � � � � � � �� � � � � % � ! � � ! � � � � � � � � ! � � � � � � ! � � � → � � � � � � � �

� ! � � � → � ! � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� % � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � ! � � � � � � � � � ! � �� h � � � � � � � � � � � � � � � � � � � � � � � � � % � � � l

� � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � �

� ! � � � � � � � � � � � � � l � � ! � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � �

� � � � � � � � � ! � � � � ! � � � � � � � � � � �� ' �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � ! � � � � � � ! � � � ! � � � � � ! � � ! � � � � � � � � ! � � � � �� � � � ��

� � � � � ! � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � ! � m � � � � � �

! � � � � � � � � � � � � � � m � � � ! � � � � � � � � ��

����

��

� � � ! � � � � � ! � � � � � � � � ! � m � � � � � � � � � � � o � � u � � q x s � � � � � � � �

� � ! � � � � � � � � ! � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � l � � � � ! � � ! � �

! � � � � � � � � � � �� k � � � � � � � � � � � � ! � � � ! � � � ! � � � � � � � � � � � � � � � � � � � �

� ! � m � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � ! � � � � � � � � ! ·

� � � � � �

! % � � ! * � � � � � � � ! � � � � �

" � � � � � � � �� � � � � � � l � � � � � � l � � � � � � � � � � � � � ! � � � ! � � � � � � �

� � � � � � ��� � � � � � ! � � � � � � � � � � � � ! � � � � � � � � � � � % � � � � � � l � � � � �

� � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� ' �

�� � � � o � � q x � ~ u s � u q � q v x � � u � � ~ o � � t � q � � p � � � q � � � �

� � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � ! � � � � � � � � � � � � � � � � ! � � �

� � � � � � � � � � ��

����

'�

k � � � � � � � � � � ! � � � ! � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � ! � � �

! � � � � � ! � � � � � ! � � � � � � � � � ! � m � � � � � � � � � �� � " � � � ! � � � � � � � � � � ! � �

� � � ! � � � ��

� � ��

� � � � � � � � ! � � � � � � � � � � � � ! � m � � � � � � ! � � � � � � � � ! � � � � �

� � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % ! % � � !

� � � � ! � � � ! � � � � � � � � � � � � �| | � � � � � � � � � � � � � � � � � � � � � ! � � � � � ! % �� ! � � � � �� # � � � � � � � � � � � � � � � � � � � � ! % � ! � � � ! � � � � � � � � � � � � � � � � ! � �

� � � � � � � � � � ! � � ! � � � � � � � ! � � � � � � ! � � � � �� � � � � � � � � � � � � � � � � � �

� � ! � � � � � � � � � � � ! � � � � � � � � � � � � � � � l � � � � ! � � ! � � � � � � � � � � � � �

� � ! � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � ! � m � � � � � ! � � � � � ! � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � ! � m � � � ! � � � � �� � � � � � � ! � � ! % � ! � � � � �

� � � ! � � � � �| | �� � � � � � % � � � � � � � � � � � � � � � � � � � � � l � � � � � ! � � � � �

� � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � �� � � � ! � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � �

� � ! % � ! � � ! � m �� � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � �

� � ! � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � ! % � ! � � � � � l � � � � � � �

! � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � ! � � � � ! % � � ! �

� � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � ! � � � � � � � �

� � � � � � �

6

� � � � � � � � � � � �

� ! � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

i � � � � � � � � � � � � � � � � � � � � � � ! � � ! � � � � � � � � � � � � � � � � � � � � ! � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � l � � � � � � � � � � � � � � � �� ! � � � � � � �

� � � � � � � � � � � ! � � ! � � � � � � � � � � l � � � � � � ! � � ! � � � � � � � � � � � � � � �

� � � � � � � � � �� �� ! � � ! � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � ��� � ! � � � � � � ! � � � � � � � � � ! � � � � � � � � � � � � � � � � � ! � � � ! � � � � � ! �

� � � � � � � l � � � ! � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! �

� � � � � � � � � ! � � � � � � % � � � � ! % � ! � � ! � � � � � � �� � � � � � � � � � � � � � �

� � � � � � ! � � � � � � � � � � � �

� � ! � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � �

� � � � � � � � � � � � � � � l � � ! � � � ��

Page 78: Программирование: теоремы и задачи

� � � � � � r � q x � r � u � � v s x q r � r z r �

� � � � � � � � � � � � � � � � �

� � � � � � � . � / � � 0 � � # . � / � � � �

� � � � " � � � � � � , � � � # . � / � $ � � � � " � # �

� � � # . � / � � � . � � . � / / $ � � � # �

� � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � . � / � � � � � � � � � � � � � � �

� � � � � � � � . � / �

� � � � � � � . � / � �

� � � � � � � � # . � / � � � # . � / $ � �

� � � � # �

� � # . � / � � � . � � . � / / � � � � � � � � � � � � �

� � � � � � � �

� , � � � � � � � . � � . � / / . � # . � / / � � � � � � � � � � �

� � � , � � � � � � � � � � � � � � # �

� # . � / � � � # . � / $ � �

� � � � # �

� � � � $ � � � � . � / � � � , � � � � � # . � / � � � �

� �

� �

� �

� � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � �

� � � � �� � � � � � �� � � � � � � � � � � �

����

)�

� � � � � � � � � � � � � � � � � � ! � m � � � � � � � � � � � � � � � � � � � � � ! � �

� � � � � � � � � � � ! � � � � ��

� � � � � � � � � � � � � � � � � � � �� � � � � � � ! � � � � � � � � � � � � � � � � � � � � �

� � � � � � � ! � � � � � � � � � � � � � � � � ! � � � � � � � � � � ! � � � � � � � � � � � � l � � ��� � ! � m � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � ! � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � �

� � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � ! � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � �� �

� � � � � � � � � � ! � � � � � � � � � � � � ! � � � ! � � � � � � � � � � ! � � � � � � �

� � � � � � � � � � � � ! � � ! � � � � � ! � m �� �

����

*�

� � � � � � � � � � � � � ! � � ! � � � � � � � � � � � � � � ! � � � � � � ! � � � � � � � �

� � � � � � � � ! � � � * � � � � � ! % � � ! � � �

����

��

� � � � m � � � ! � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � l � � � � � � ! � � � � � � � � � � � � � � � � � � � � l

� � ! � � ! � � � � � � � � � � � � � � � � � �

� ��

� � � � � � � � � � � � �

� � � � � � � � �

��

� � � � � � � � � � � �

� ��

��

��

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � �

� � � � � � � � � � � � � � � � � � � � � � ! � � � � ! � � � � � � � � � � � " ��� � ! � � � �

� � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � " ���

� � � � � � � � � � � � � � ! � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� � �

� � � � � � � � � � � � � � � � � � � ! � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � ! � � � � � � � � � ! � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � �� � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � �

� � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � " � � �

� � � � � � � � ��

� � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � " � � � � � ! � �

� � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� ! � � � � � � � � � � ! � � � � � � � � � � � � � % � � � � � � �� �

� � � � � ! � � � � � � � � � ! � � � � � � � � � � � � � � ! � � � � � � � ! � � � �� � � �

� � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � ~ � � } � � � � � � � � � � � �

� � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� � ! � � � � � � � � � � � � �� � � ! � � � � � � � � � � � � � �� � � ! � � � � � � � � � � " � � � � � � � ! � �

� � � � � � � � � " � � � � ��

� � � � � � � � ! � � � � � � � � � � � � � ! � � � � � � � � � �

� � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � ! � � � � � �� � � � � � � � � � � � � � % � � � � � � � � � � �� ! � � � � � � � � � � � � � � �

� � � �� � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � ! � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � ! � � � � � � ! � � � � � � � � ! � � � � ! � � ! � � � � � � � � ! � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

Page 79: Программирование: теоремы и задачи

� � � � ��

� � � � � � � � � � � � � � � �

� � � � � � � � � � ! � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � �

� � �

� � ! � � � �

� � �

� � �

� � ! � � � � � �

� " �

� � �

� � ! � � � � " �

� � �

� � �

� � ! � � � � � �

� ! � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � �

� � � � ! � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � ! � � � ��

� � �

���

6 � � �6 � � �6 � � �6 � �

� � � �

� � � � � �

� � � � � � � � � � � � � � � � ! � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � ! � � � � � ! � � � � � � � � � � � � � � � � �

� � � � � � , � � 0 �

� � � � � � � � � � � � � � � � � � � � � � � � �� � , � � � � � � � � � � �

� � � � � � ! $ � � , � � � , ! � � " � # �

� � � , � 0 � � # �

� � � . � / � , � � # �

� , � � � �

� � � � # �

� , � � 0 �

� �

� � � � � � , � � � � # �

� � � . � / � � � � # �

� , � � � �

� � � � � � . � / � , � � # �

� ��

��

� � � � � � � � � � � � � � � � � �

� , � � � �

� � � � # �

� , � � 0 �

� �

� � � � � � , � � � � # �

� � � . � / � � � � # �

� , � � � �

� � � � � � . � / � , � � # �

� , � � � �

� � � � # �

� , � � 0 �

� �

� � � � � � , � � � � # �

� � � . � / � � � � # �

� , � � �

� � � � � � . � / � , � � # �

� , � � � �

� � � � # �

� , � � 0 �

� �

� �

� �

, � � � � � � � , � � �

� � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � ! � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � " � � � � � � � � � � � � � � � ! � �

� � � � � � � � � � � � ��� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � ! � � ���

� � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � ! � � � � � � � � � � � � � � � � �

� � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � �� � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � �

� � � � � � � � � � � � � � � � � � � � � � � � ! � � � �� � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � �

� � � � � � � � � � � ! � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � ��� � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � �� �� � � ! � � � � � � � � � � � � � � � � � � � � � � � ! � � � � ��

� � � � � � � � � � � � � �

� � � � � � � � � � � ! � � � � � ! � � � � � � � � � � � � � � � � � � � � ! � � � ! � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � " � � � � �

� � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � ! � � � � ! � � � �

$ 7→ � � � � � � � � � � � � � � � � � � � � � � � � � � � � " � � � � � � � � � � � � � � � � � � $ !

Page 80: Программирование: теоремы и задачи

� � � � ��

� � � � � � � � � � � � � � � �

��

��

� � � � � � � � � � � � � � � � � � � � � � � � � � � �

� ����

��

� � � � � � � � � ! � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � " �

� � � ! � � � � � � � � � � � � � � �

� � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � �

� � � � � ! � � � � � � � � � � � ��

� � � � � � � � � ! � � ! � � � � � � � � � � � � � � � � � �

� � � � "� � � � � � � � � � � � � � � � � � � � � � � � � � % � �� � � � � � � % � � � � � � � � � � � � ��� � � � � � � � � � � � � � � ! � � � � � � % � � � � � "� � � � � � � � � � � � �

! � � � � � ! � � � � � � � � � � � � � � % � � � � � � � � � � ! � � � � � � � � � � � � � � � � ! � � � �

� � � � " � � � � � � ! � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � �

� � � � � � � � � " � � � � � � � � � � � � � � � � � � � � � � % � � � � ! � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � " � � � � ! � � � � � � � � � � � � � � � � � �� �

� � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � " � � � ← � � � � � � � � � � �

� � � � " ← � � � � � � � ! � � � � � � � �

� � � � " ← � � � � � � � � � � � � � � �

� � � � � � ! � � � � � � � � � �

� � � � � � � ← � � � � � � � � � � �

� � � � " ← � � � � � � � ! � � � � � � � �

� � � � " ← � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � ! � � � � � � � � �

� ! � � � ! � �� � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � ��

� ������

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � �

� � � � � � �� � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � " �� � � � � � � �

� � ! � � � � � � � � ��

� ����

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � ! � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � �

� � � � � �

� � ! � � � �

� � � � � � � � � �

� � � � � � � � �

� � � � � ! � � � � � �

� � � � � � � � � � � �

� � � � � � ! � � � �

� � � � � � � � � � � � � �

� � � � � � � � � � �

� � � � � � � ! � � � � � �

� � � � � � � " � � � � � � " �

� � � � � � � � � � � � � �

� � � � � � � � ! � � � � " �

� � � � ! � � � ! � � � � � � � ! � � � � ! � � ! � � � � � � � ! � � � � � � � � � ! � � �� � � � �

� ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � ! � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � ! � � �

� � � � � � � " � � � � � � � � � � � � � % � � � � � � � � � � � �� �

� . � � � � � � � . � � � � � � � � � � � � ! � � � � � � � � ! � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � ! � �

� � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � �

� . � � � � � � � . � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � �

� � � � � � ��

� � � � � � � � � � � � � ! � � � � � � � � � � � � � � ! � � � � � � � � ! � � � � � � � � � � � � � ! � �

� ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � ! � � ! � � � � � � � ! � � � � � � � ! � � � � � � � �

� � � � � ! � � � � � � ! � � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � ! � � � � � � ! � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � ! � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � ! � � �

� � � � � � � � ! ! � � � � � ! � � � ��� " � � � � � � � � � � � � � � � � � � ! � � � � � � �

��

� � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � �

� � � � � � � � � � � � � ! � � � � ��

Page 81: Программирование: теоремы и задачи

� � � � ��

� � � � � � � � � � � � � � � �

��

��

� � � � � � � � � � � � � � � � � � �

� � � � ! � � � � � � � � � � � � � � � � � ! � � � � � ! � � � � � � � � � � � � � � � � � � ! � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � ��� � �

� � � � � � � � � � � � � � � � � � � � � � � � � ��

� # � � � � � � � � � � � � � � � � � � ��

� ! � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � " � � � � � � � �

� � � � ��� ��

��

��

� � � � � � � � � � � � � � � � � � � � � � � ��

� � � � � � ��

� � � � � � � � � � � �� � � � � � �

� � � � � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � �

� ! � � � � � � �� �

� � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� ��

����

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � �

! � � � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � ! � � � � ! � � � � � � �� �

� ��

��

��

� � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � ��

� � � � � � �� ! ! !

� � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � ��

� � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� � � � � �

� � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � � �

� � � ��

" � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � ! � � � � �

� � � � � � � ! � � � � � � � � � � � � � � ! � � � �� � � � � � � � � � � � � � � � � � � � � � ��

� � �

� � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � ! � � � � � � � �� � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � ��� � � � � � � � � ! � � � � � � � � �

� � � � � � �� �

��

��

� � � � � � � � � � � �

� � � � � � �

� � � � � �

� � � ! � � � � � � � � � ! ! � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � ! ! ! �� � �

� � ! � � � � ! � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � �

� � � � � � � � � ! � � � � � � � � � � � � � � � ! ! !� � � � � � �

� � � � � � � � � � � � � � � � � � � � ! ! ! �� � � �

� ��

� � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � ! � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � ! ! ! �� � �� � � � � ! � � � � � � � � � � � � � � � �

� � � � � � � � �

� ��

'�

��

� � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � ! � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � ! � � � � � � � � � � � �

� � � � � � � � � � � � � � � ( � � � � � � � � � � � � �

� � � � � � � � ! � � � � � � � � ! � � � � � � � � � � ( � �� � � � � � � � � � � � � �

� � � � � � � � � � � � � � ! � � � � � � � � � � � � (� � � � �� � � � � � ( � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � �

! � � � � � � � � � � � � � � (� �

� ��

'���

� � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � ! ! !� � � � �

� � � � � � � � ! � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � ! ! !� � � � � � �

� � � � � ��

� � � � � � � � � � ! � � � � � � � � � � � � � � � �� �� � � � � � � � � � � � �

� � � � � � � � � � � � � � �

� � � �

� � � � ! � � � � � � � � � � � � � � �

︸ ︷︷ ︸

︸ ︷︷ ︸

� ! � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � ! ! ! �� � � ��

� � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � �

� � � � � � � � � �� � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � �′ � ! � � � � � � � � � � � � � � �

� � � � � �� � � � � �′ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! ! ! �� � ��

� � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! ! ! �� � ��

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � ��

" � � � � � ! � � � � � � � � �

� � � � � � � � � � � ! ! ! �� � �� � � � � � � � � � � � � � � ! � � � � � � � � � � � � � �� � � � � �

� � � � ! � � � � � � � � � � � � � � � � � � � ! � � � % � � � � � � � � � � � � � � � � � � �

� � � � � � � � ! � � � � � � � � � � �� � ! � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � ��

� � � � ! � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � ! � � � � � � � � � � � �

� � � � � � � � � � ! � � � � � � � ! � � � � �� � � � � � � � � � � � � � � � � �

� � � � � � . � / � � 0 �

� � � � � � � � � . � /� �

� . � / � � � � � � � � � � � � � � � �

� � � � � ! � " � # �

� � � � . � /

Page 82: Программирование: теоремы и задачи

� � � � ��

� � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � . � / � � � . � / � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � �

� � � � � � . � $ � / ! � . � $ � / � , � � � ! 0 � � " � # �

� � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � �

� � � � . � / �

� �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � . � $ � / � � . � $ � / � " � # �

� � . � / � � � . � / � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� . � $ � / � � � $ � �

� � � � # �

� � � � � � � � � � � � � �

� . � $ � / � � 0 �

� �

� � � � $ � �

� � �

� ��

'�

��

� � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � % � � � � � � � � � � � � � � �

� � ! � � � � � � ! � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � ! � � � � � � �

� � � � � � � � � � ! � � � � � � � � � � � � � � � ! � � � � � � � � � � ! � � � � � � � � �� � � � � �

� � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � ! � � � � � � ! � � � �� � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � ! � � � � � ! �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � �

� � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � ! � � � � � � � � ! � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � �

� � � � �

6

� � � � −

� � � � � � � � � ! � � � � � � � � � � � � � * �

� � �

� � � � � � � � � ! � � � � � � � � � � � � �

6

� � � � − � � � � � * �!

� � � � % � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � �

� � � � � � � � � � � � � � � � ! � � � �� �

� ��

'�

'�

� � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � ! � � � � � � � � � �� � � � � � � � � � � �� � � ! � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � * � � � � � � � � � � � � � � � � � � � � � � � ��

� ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � ! � � � � � � � � � � � � � � � � � � ! �

� � � � � � � � � � � � � � ��

� ��

��

� � � � � � � � � � � � � �

� � � � � � � � ! � � � � � � � � � ! � � � � � � � � � � ( � ��

� ! � � � � � � � � �

� � � � � � � � � � � � � � � � � � � ! ! ! �� � � � � ! � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � ! � � � � � � � � ! � � � � � � � � � � � � � ! ! !� � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � �� �

� � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � ! � �

� � � ! � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � ! � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � �

� ��

'�

)�

� � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � ! � � � ! � � � � � � � � � � �

� � � � � � � � � � � � � � � � � ! ! ! �� � � � � � � � � � � � � � � � � � � � � ! ! ! � � � ���

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! ! !� � � � � � � ! � � � � �� � � �

� � � � � � � � � � � � ! � � ! � � �

� � � 0 � � � � 0 �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � - . � / � � - . � / �

� � � � � � ! � , � � � ! � � � " � # �

� � � � � � . � $ � / ! - . � $ � / � , � � � ! 0 � � " � # �

� � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � �

� � � � . � / �

� �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � . � $ � / � - . � $ � / � " � # �

� � . � / � � � . � / � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � $ � �

� � � � # �

� � � � � � � � � � � � � �

� � � 0 �

� �

� � � � $ � �

� �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � �� � � � � � � � � � � � � � � � �

��

��

� � � � � � � � � � � � �

� � �

� � � � � � � � ! � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � ! � � � � � � � � � � � � � � � � ! � � � �� � � � � � � � � � � � � � � � �

� ! � � � �� � � � � � � � � � ! � � ! � � � � � � � ! � � � � � � " �� � � � � � ! � � � � � � � % ! �

� � � � � � � � � � � � � � � � � � � � � ! � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

Page 83: Программирование: теоремы и задачи

� � � � ��

� � � � � � � � � � � � � � � �

� � � � � � � � � � � � � ! � � � � ! � � � � � ��

� � � � � � � � � � � � ! � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � ���

� � � ! � � � % � � � � � ! � � � � � � � ! � � � � � � � � � � � � � ! � � � � � � � � ! � � � �

� � ! � � � � ! � � � � � � � ! � � ! � � � � � � � � � � � � � � � � � �� � � � � � � � � � ! ! ! �� � � �

� � ! � � � � � � � � � ! � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � % � � � �

� ! � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � �� � ! � � � � � ! �

� � � � � ��

� � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � ! � � � � � � � � � �

� � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � ��

� ��

)�

��

� � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � " � . � / � � � � � � 0

� " � � � � � " � " � # �

� " � . � . � / / � � � �

� � �

� � ! � � � � � � � � � � � � � � � � � ! � � � � � � � � ! � � � � � � � � � � � � � ! � � �

� � � � � � � � � � � ! � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � ! � � � �� � � � � � � �

� � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� , � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � , � � � � " � # � � � � � � � � � � � � � � � � � �

� � � . / ! - . � , � / � � # � � � � � � � � � � � � � � � � � � � �

� , � � � � , � $ � � � " � . - . � , � / / � �

� � � " � . - . � , � / / � � � � � � � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � - . � , � / � � � � � � � � � � � � � �

� � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � # �

� � � � � � � � � � � � � � � � � � � � � � � � � �� �� �� � � � �

� . � / � � � . / � - . � , � � $ � / � � - . � , � / �

� � � � � � � � � � � � � � � � � � � �

� , � � � � , � $ � �

� �

� �

� � � � � � � ! � � � � � � � � � ! � � � ! � � � � � � � � � � � � ! � � � � � � � � ! � � � � � � � � � �

�� �� � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � ! � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � ! � � � �

� � ! � � � � � � � � � � � � � � � � � � � � � � �� �� � � � � � � � � � � � � ! � � � � � � ! � � � � �

� � � � � � � � � � � � � � � � � � � � � � ��

� ��

��

� � � � � � � � � � � �

� � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � ! � � �� � � � ! � � ! � � � � �

� � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � �

� � � � � � � ! � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � ! � � � ��

� ��

)���

� � � ! � � � � � � � � � � � � � � � � � � � ! � � ! � �

� � � � � � � � ! � � � � � ! � � � � � � � � � � � � � � � � � � � � � � �

� " � � � � � " � � � "� � �

� � � � � � � � ! � � � � � � � � � � � � � � � � � � ! � � ! � � � � � � � � � � � � � � � � � � � �

� � � � � �� , � � � � , � $ � � " � . - . � , � / / � �

� ! � � � % � � � � � � � � � ! � � % � � � � � � ! � � � � � � � � ! � � � � � � � ! � � � � ! �

� � � � � � � ! � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � ! � � � � � � � � ! � � � ! � � � � � � � � ! � � � � � � � � � � ! ! � � � � � ! � � �

� ��� ��

)�

��

� ! � � � � � � � ! � � ! � � � � � � � � � � � � � � ! � � � � ! � � � � � � � � � � �

� � � � � � � � � � � � � ! � � � � ! � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � ��� � � � � � � � � � � � � � ! � � � � � � � � � � � � � � ! ! ! � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � �� �� � � � � � � � � � � � � � ! � � % � � � � � � � � � ! � � � � � � ! � � � � ! � � � ! � � � � ! � �

� � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � * � � � � � � � � � � � � �� � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � ! ! � � � � � ! � � �

� ��

� ! � � � � � � � � � � � � � � � � ! � � � � � � � � � � ! � � � � � � � � � � � � � � � � �

� � � � ! � � � � � � � � �� � ! � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � �

� � ! � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � ! � � � � � � � � � � ! � � � � � ! � � � �

� � � � � � � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � % � � � � ! � � � � ! � � � � � � ! � � � � � �� � � � ! � � �

� � � � � � � � � � � � � � � � � � � � � ! � � � �� � � � � � � ! � � � � � � � � � � � � � � � � �

� � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � ! � � � � � ! � � � � � � � �

� � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � �

� � � � � � � � � � � � � � ! � � � �� � � � � � � ! � � � � � � � � � � � � % � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � * � � � � � � � � ��

��

��

� � � � � � � � � � � �

� � � � � � � � ! � � � � � � � � � � � � ! � � � � � � � �� � ! � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � ! � � � � � � � � ��

� � ! � � � � � � � � � � � ! � � � ! � �

Page 84: Программирование: теоремы и задачи

� � � � ��

� � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � ! � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � ! � � � � � � � � � � � � � � � � � � ��� � � � � � � � � � � � � � ! � � � � � � � � ! � � � � � � � � � � � ! � � � % � � � � � � � � � � � �

� � � � ��

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � �

! � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� ! � � � ! � � � � � � � � � � � � � � � � � � � �

� � � � � � � � ! � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % ! � � �

� � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � � � � ! � � � � ! � � �

� � � � � � � ! � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � % � � � � ��

� ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � ! � � � � � � � � � � � �� � � ! � � � � � � � � � � � � � � � � � � � �

� � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � ��

� ��

*�

��

� ! � � � � � � � ! � � ! � � � � � � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � ! � � � � ! � �

� � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � !� � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � ! � � � � � � �� � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � ! � � � � �

� � � � � �� � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � ! � � ��� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � �

! � � � � � � � ! � � � � � � � � � � � � � � � � �� � � � � � � ! � � � � � � � � � � � � � � � � � �

� � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

! � � � � �� �� ��

*���

� ! � � � � � � � ! � � ! � � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � � ! � � � � � � � ! � � � � � � � �� � � � � � � � � � � � ! � � � � � � � � � ! �

� � � � � � � � � � � � ! � � � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � �

! � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � �� � � � � � � � � � � � ! � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � − � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � ��

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � �

� � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � �� �−

� � � � � � � � � � � � � � � � � � �

! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� �

� � � � � � � � � � � � ! � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � ! � � � � ��

� � � � � � � � � � � � � � � ! � � � � � � � � � � � � � ! � � � � � �

� � � � � � � ! � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � ! � � �

� � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � �

� ��

��

� � � � � � ! � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � ! � � � � � � � � � ! � � � � � � � � ��

" � � � � � � � � � � � � � � � �

� � � � � � � � � � � � − � � � � � � � � � � � � � � − � � � ! � � �� � � � � � � ! � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � �� ��

� � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��� � � ! � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � ! � � � � � � � � � �

� � � � � � � � � � � ��

� ��

��

��

� � � � � � � � � � � � � � � � � � � � � � � ! � � � ! � � � � � � � � � � � � � � � � � �

� � � � � � � ! � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � ! � � � � � � � � � � � � � � � � � � ! � � � � � � � � � ! � � � � �� � � � � � � � � � � � � � � � � � � � ��

� ��

� ��

� �� � � � � � � � � � � � �

� � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � �

� � � � � � � � � ! � � � � � � � � � � � � �

� � � � � � � � � � ! � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � �

� � � � �

� � � � � � �

� � � � �

� � � � � � �

� � � � �

� � � � � � � � �

� � � � � � � � � � �

� � � � �

� � � � � � � � �

� � � � � � � � � � �

� ! � � � � � � � � � ! � � � � � � � � � � � ! � � � � � � � � � � % � � � � � � � �� � � � �

� � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� � � ! � � ! � � � ! � � � �

Page 85: Программирование: теоремы и задачи

� � � � ��

� � � � � � � � � � � � � � � �

� � � " � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � �

� � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � % � " ��

� ��

����

� � � � � � � � � � � � � � � � � � � � � � � ! � � � ! � � � � � � � � � � � � � � � � � �

� � � � � � � � � ! � � � � � � � " � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � �

� � � � � � � � � � ! � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � �

� � � �

� � �

� � � � � � � � � � � � � � � � �

� � " � � "

� � � � " � �

� � " � � � � � � �

� � " " � � "

� � " � � " � � � � � �

� � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � ��

� ��

��

��

� � � � � � � � � � � � � � � � ! ! ! �

�� � � � � � � ��

� � ! � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �

� � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � ! � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � ! � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � ! � � � � �

� � � � ! � � � � �� � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� � � � � ! � � � � � � � ! � � � � � � � ! � � ��

� � � � � � ! � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � ! � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � �

� � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � �� � ! � � ! � � � � � � � � � � �

� � � � � % � � � � � � � � � � � � � � � � � � � � � � � ��� � � � � � � � � � ! � � � � � � � � � ! ! � � � � � ! � � � �� � � % � � � � � � � � � �

� � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � �

� � � ! � � � � � � � � � � � � � �� � � � � ! � � � � � � � � � � ! � � � � � � � � � � � � � � �

� ��

��

� � � � � � ! � � � � � � � � � � � � � � � � �

� � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � �

� � �� " � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � ! � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � �� � � � ! � � ! � � � � � ! � � � ! � � � � �

{ � � �� � � �� � � � �}

� � � � � � � � � � � � � � ! � � �

r -

r -HHHHHj

��

r -�����*

r -�

� r -

r

r

r

� � ! � � � � � � � � � ! � � � � ! � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � �

� � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � ! �

� � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � ! � � � � � � � � � � � � � � � � � � �

� ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � ! � � � � � � �� � ! � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � �

� � � � ! � � � � � ! � � �� � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � ! � � � � � � � � �

� � � � � � � � � � � ��

� � � � � � � � � � ! � � � � � ! � � � � � � � � � � � ��� � � � � �

� � � � � � � � � � � � � � � � � ! � � � � � �

� ��

��

'�

� � � � � � � � � ! � � � � � ! � � �� � � � � � � � � � � � � � � � � � � � � � � � �

� � ! � � � � � � � � � � � � � � � � � � � � �� ! � � � � { �

� � ��

� � � � � � �� ! ! !}

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � �

! � � � � � � � � � � ! ! � � � � � ! � � � �� �

� � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � ��

� � � � ! � � � � ! � � � � � � � � � � � � � � � � � � � � � ! � � �

� � ��

� � � ��

� � � � � � ��� � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � ! � � � � � � � �

� � ! � � � � � � � � � � � �� � � � � ! � � � � � � � � � � ! � � � � � � � ! � � � � � � % � � � � � � �

� � � � � � � � � � � � � � � � � � � � �� � � � % � � � � � � � � � � � � � � � � � � � � � � ! � � � � � ! � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � �� � � � � � � � � � � � � � ! � � � � � �

� ! � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � �

� � � � � � ! � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � ! � � � � � ! � � �

� � � � � � � ! � � � � ! � � � � � � � � ! � � � � � � ! � � � � � � � � � � � � ��

� � � ! � � � � � �

� � � � � � � � � � % � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � �

Page 86: Программирование: теоремы и задачи

� � � � ��

� � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � ! � � � � � � � � � � � � � � � �� �

� � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � �

� � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � + & � + � � � � � . ��

� � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � �� � � � � � ! � � �

� � � � � � � � � � �� � �

��

��� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � �

! � � � � � ! � � � � � ! � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � " � �

� � � � ! � � � � � ! � � � � � � � � ! � � � � � � � � � � � � ! � � � � � �

� � � � � � � � � � � � � � � � ! � � � � � ! � � � � � ! � � � � � � �

� � � � � � � � � �� � � ! � � � � � ! � � � � � ! � � � � � � �

� � � � � � � ��

��

�� ! ! ! �

� � ! � � � � � ! � � � � � ! � � � � � � � � �

� � � �! ! !

� � � ! � � � � � ! � � � � � ! � � � � � � �

� � � � � � ��

��

�� ! ! ! �

� � ! � � � � � ! � � � � � ! � � � � � � � � �

� � � � ! ! !

� � � ! � � � � � ! � � � � � ! � � � � � � �

� � � � � � � � � ! � � � � � ! � � � � � ! � � � � � � � � � � � � ! � � � � � ! � � � � � ! � � � � � ��

� � � � ! � � � � � ! � � � � � ! � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � ! � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � �

� � � � � � � � � � � �

� � � ! � � � � � ! � � � � � ! � � � � � � � � � �! ! !

� � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � �

� � � � � � � �� � � � � � � ��� � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � ! � � � � � ! � � � � � ! � � � � � � � � � � ! ! !

� � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � ��

��

�� ! ! ! �

� �

� � � ! � � � � � ! � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � ! � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � �

� � � � ! � � � � � � � � � ! � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � ��

� � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � ��

� ��

��

� � � � � � ! � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � ! � � � � ! � � � � � � � � � � � � � � � �

� � � � � � � � � � � �� � � � � � � � � � � � � � ! � � ! � � �

� � ! � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � % � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� ��

��

)�

� � � � � � � � ! � � � � � ! � � � � � ! � � � � � � � � � � � ! � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � ! � � � � � � � � � � � � � % � � ��

� � � � � � � � � ! � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � ! � � � � � �

� � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � �� � � � � % � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � ! � � � � �

� � � � � � � � � � � | � � � �

� ! � � � � � � ! � � � � � � � � � � �

� � � � � � � � � � �

� ��

��

*�

� � � � � � � � ! � � � � � ! � � � � � ! � � � � � � � � � � � ! � � � � � % � � � � � � � � �

� � � � � � � � � � � � � � �� �� " � � � � � � ! � � � � � � � � � " � � � � � � � � � � � � � � � �

� � � � � � � � � � � " � � � � " � � � " � � �

� ��

��

��

� � � � � � � � ! � � � � � ! � � � � � ! � � � � � � � � � � � ! � � � � � % � � � � � � � � �

� � � � � � � � � � � � � � �� �� " � � � � � � ! � � � � � � � � � " � � � � � � � � � � � � � � � � � �

� � � � � � � �� � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � ! � � � � �

� � � � � � � % ! � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � ! � � �

� � � � � � � � � � � � � � � ��

��

� � ��

� � � � ! � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � ! � � ! � � � ! � � � � �

� � � � � ! � � � ! � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � ! � � � � � ! � � � � � ! � � � � � ��

� ��

��

��

� � � � � � ! � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � " � � ! � � �

� � � ! � � � � ! � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � ! � � � � � ! � � � � � ! � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � ! �

� � � � � � � � �� �� "� �� ��

� � � � � � �� � � � " � � � � � � � � " � � � � � � � " � � � � �

� � � � " � � � � � � � � � " � � � � " � � � � " � � � � �

Page 87: Программирование: теоремы и задачи

� � � � ��

� � � � � � � � � � � � � � � �

� ��

��

��

� � � � � � � � � � � � � � � � � � � � � ! � � � � � ! � � � � � � ! � � � � � � � � � �

� � � � ! � � � � � � � � � � � � � � � � � � � � � � � ! � � ! � � � � � � � % � � � � � � � � � � � � � � � �

� � � � � � ! � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� ! � � � � � � � � �� � � ! � � � � � ! � � � � � � � � ! � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � �

� � � � � � � ! � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � ! % � � ! � � � � � � � � � � � � � �

� � ! � � ! % � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � �

� ! � � � � ! � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � ! �

� � � � � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� ! � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � ! � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � ��

� � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � ! � � � � � � � � � � � � �

� � � � �

� � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � ! � � � � � ! � � � � � � � � � � � � � � � ! � � � � � � � � � � � ! � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��� � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � % � � ! � � � � � � � � � � � � � � � � � � ! � � � � � � � ! � � � � � � � ! � ���

� � � � � � � ! � � � � � � � � � � � � � � � � � � � � ! � � � � � ! � � � � � ! � � � � � �

� � � � ! � % �� � � � � � � � � � � � � ! � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � ��

� � � � � � � ! � � � � � � � � � � � � � � � � � � � �

� � � � ! � � � � � � � � � � � � � ! � � � ! � � � ! � � � � � ! � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � ��

� ��

��

� ��

� � ! � � � � � ! � � � � � ! � � � � � � � � � � ! � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � � � ! � � � � � ! � � � � � ! � � � � � � ! � � � � � �� � � �

� � � � � � � � � � � � � � � � � ! � � � � � � � � � � ! � � � �� � � � � � � � � � � ! � � � � � � �

� � � � � � � �

r -��

��

��3

QQ

QQ

QQs

r

r

��

r

� �

r��

��

��3

r - �

r

QQ

QQ

QQs

r

� ��

��

� � � � � � ! � � � � � � � � � � � � � � � � �

� � ! � � � � � � � � � ! � � � � � � � � � � � � � � � � � � ! % � � � � � � � � � � ! � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � ! � � � ��

� � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � �

r -�

r

� � � r -r

��

� r -r

� r -r

� � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � ! � � � � �

s

s -�

� �

s -

� JJ

JJ]

� s

s

� ��

��

� ��

� � � � � � � � � � �� � � � � ! � � � � � � � � � � � � � � � � � � � � ! � � � ! � � �

� � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � �

� � � �� � � � � � � � ! � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � ! � � � � � � � ! � � � �

� � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � �� �

� � � � � � � � � � � ��

��� ! � � � � �� �

� � � � � � � � � � � � � � � ! � � � � � ! � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � �

! � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � �

� � � � � � � � ! � � � � � � � � � � � � � � � �

� ��

��

� ��

� � � � � � � � � � �� � � � � ! � � � � ! � � � � � ! � � � � � ! � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � �� ! ! ! �

��

� � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � �� � � ! � � � ���� �

� � � � � � � � � � � � � � � � � � � �

� � � � � � � � � ! � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � �

� � � � � � � � � � � � � � � ! � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � �� ! ! ! � ��

� � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � ! � � � � � ! � � � � � � � � � � � � � � � � � ���� �

� ! � � � � � � ��

� ! � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � ! � � � � � � �

� ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

Page 88: Программирование: теоремы и задачи

� � � � ��

� � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � ���� � � �

� � � � � � � � � � � � � � � � � �

� � � � � ! � � � � � � � � � � � � � � � �� � � � � ! � � � �� � ! � � � � � � � � ! � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � %� � � � � � � � � � � �

� � � � ! � � � � � � � � �� ���� � � � � � ���� � |

� � �� � � � � �

�� � � � � � � � �� �� � � � �� �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � ! � � � � � ! � � � � � ! � � � � � � �� � � � � % � � � � � � � � � � � � � � � � � �

� ! � � � � � � � � � � � � � � � � �� �

� ��

��

� ��

� � � � % � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � �

� � � � � �� � � � � ! � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � �

� � � � � � � � � " � � � � � � � � � ! � � � � � � ! � � � �� �

� ��

��

� '�

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � ! � � �

� � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � ! � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � �

� � � � � ! � � � � � � � � � ! � � � � � � � � � � � � � � � ! � � � � � � � �

� � � � � � � � � � � � � � � � � � � � ! � � � � � � ! � � � � � � � � � � � � �� �

� � � � � � � � � � ! � � � � � � � � � � � � ! � � � � � ! � � � � � � � � � � � � � � � � � � � � � �

� � ��

� � � � � � � � � � � � � � � � � � � � % � � � � � � ! � � � � � � � � ! � � � � � � � � � � � � � �

� � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � �� � � ! � � �

� � � ! � � � � � ! � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � ! � � � � � � � � � � � � �

� ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � �

��

��

� � � � � � � � � � � � � � � � �

� � � � � � ! � � � � � � ! � � ! � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � ! � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � �� � � � � � � � � � � � � � � �

� � % � � � � � ! � ��

� ��

��

��

� ! � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � ! � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � ! � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � ��

� ! � � � � � � � � � � � � ! � � � � � ! � � � ! � � � � � � � � � � � � � � � � �

� � � � � � ! � � � �

� � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � ! � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � ! � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � ! � �

� � � � � � � ! � � � ! � � � � � � �� � � � � � � ! � � � � � � � � � � ! � �� � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� ����

� � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � ��� � � ! � � ! � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � ! � � ! � � � �

� � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � ! � � � � � � � � � � � � � � ! � � ! � � � � ! �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � �

� � � � � � � � � � � ! � � � � � � � ! � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� ! � � � � � � ! � � � � � � � � � � � � ! � � � � � � � � � � � � ! � � � � � � � � � � � � � � � �

� � � � � � � � � � �� � � � � � ! � � � � � � � � � � � � � � ! � � � � � ! � � �� �

� � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � ! � � � � � � � � � � ! � � � � � �

� � � � � � � � ! � � � � � � � � � � ! � � � � � � � � � � � � � � � � ! � � � � ! � � � ! � � � � � � � �

� � � � � � � ��

� � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � ! � � � � � ! � � �

� � � ! � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � ! � � � � � � � � � � � ��

� ��

����

" � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � �

� � � � � % � � � � � � � � � � � � � � � � � ! � � � ! � � � � � � � � � � � � � � � � � ��

� � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � ! � � � ! � � � �

� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � ! � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � �

� � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � �

! % � ! � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��� � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � ! � � ! � �

�� �

�� �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � % � � � � � ! � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � ��� � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � �

� � � � � � � � � � � � � � � � �� � � � � � ! � � � � � � � ! � � � � � � � � � � � � � � � � � � � � �

Page 89: Программирование: теоремы и задачи

� � � � ��

� � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � �� � � � � ! � � � � � ! � � � � � � � � � � � � � � � � � � ! � � �

� � ! � � � � ! � � � � � � � � � � � � � � � � ��

� � � � � � � � � � ! � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � �

� � � � � � � � � � � � � �� � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � �

� � ! � � � � � � ! � � � � � � � � � � � � � � � � � � � ��� � � � � � � � � � ! � � � � � � � � � � � �

� � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� �

� � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � ! % � ! � ��

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � �

� � � � � � � � � � � � � � � � � ! � � � � � � % � � � � � � � � � � � � � ��

� � � � � � � � � � � �

� � � � � � � � � � � � ! � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � �

� � � � � � � � � � ! � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � ! �

! % � � ! � � � � � � � � � � � � � � � � � � ! � � � � � � � ! � � ! � � � � � � � � � � � � � � � �

� � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � ! � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � ! � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � ��

� � � � ! � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � ! � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� ��

��

��

� � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � ! � � �

� � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � ! � �

�� � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � ! � �

� ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � �

� � ! � � �� � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � �

� � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � �

� � � � � � � � ! � � � � � % � � � � � ! � � � � ! � � ! � � � � ! � � ! � � ! � % � � � � � � �

� � � � ! � � ! � � � � �� " � � ! � � ! � � ! � � � � � � � � � � � � � � � � � � � ! � � ! � � � � � � � �

� � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � ! � � � � � � � � � %

� � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � �

� � � � � � � � � � ! � � � � � � � � � � � � � ! � � � � � � � � ! � � � ! � � � � � ! � � � � � � � � � � � �

� � � � ! � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� ! � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � �� �

� � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � �

� � � � � � � � ! � � � � � � � � � � � � � � ! � � � � � � � � � � � ! � � � � � � � � � � � � � � ! � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � ! � � � � � � � � � ! � � ! � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � ! � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � � � � ! � � � � � � ! � � � � ! � � ! � � � � � � � � ! � � �

� � � � � � � � � � � � � � ! � � � � � � � �

� ����

� � � � � � � � � � � � � � � � � � � �

� � ! � � � � � ! � � � � � ! � � � � � � � ! � � � � � ! % � ! � � � � � � ! � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � ! � � � �

� � � � ! � � � � � � � � � � � � � ��

� ! � � � � � � � � � � � � � � � � � � � � � � � � � � �• � � � � � � � � � ! � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � �

• � � � � � � � � ! % � ! � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � �

� � ! � � � � � � � � � ��

� � � � � � ! � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � �

� � � � � � � � � � � � � � � � ! � � � � � � ! � � � � ��

� � � � � � � � � � � � � � � � � � � � � � � � ��� � � � � � � � � � � � � � � � ! � � ! � � � � � � � � � � � � ! � � � � � � � � � � � � � � ! � � �

� � � � � � ! � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � ��

� � � � � � � � � ! � � � � � 6 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � ��

� � � � � � ! � � � � ! � � � � � � � � ! � � � � � ! � � � � � � � � � � � � � % � � � �

� � ! � � � � � � � � � � ! � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � ! � ! % � � ! �

� ! � � � � � � � � � � � � � � � � � ! � � ! � � � � � � � � � � � � � � ! � � � � � � � � � ! � � � � �

� � � � � � ! � � � � � � � � � � � � � � � � ! � � �� � �� � � � � � � ! � � � � � � � � � � � � � � �

� � ! � � � � � � � � � � � � � � � � � � � ! � � � � � � � �� | �

� � � | � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � ! � � � � � � � � � � � � ! � �� � � � � | �� � � | � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � ! � � � �

� � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � �

! � � �� � �� � � � � � � � � � � � � � � � � � ! � � � ! � � �� � � � � � � � � � � � � � � �� � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � ��

� � � � � � � � ! � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � ��

� ! � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � ! � �

� � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � �

� ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � �

� � � � ! � � � � % � � � � � � � � � � � � ! � � � � ! � � � ! � � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � % � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � �

! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � ! � � � �

� � � � � � � � � � � � � � � � � � � � ! � � � � ! � � � � � � � � � � � ! � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � ! � � � � � � � � � � ! � � � � � � � � � � � � � � � � ! � � � � � � � � �

! � � � � � � � � � � � � � � � � � � � � ! � ! � � ! � � � � � � � � � � � � � ! � � � � � � � � � � � � � � ! � � �

� � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � �

Page 90: Программирование: теоремы и задачи

� � � � ��

� � � � � � � � � � � � � � � �

� ! � � � ! � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � �

� � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � ! � � � � � � � � � � ! � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � ��

� ��

��

'�

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � �

� � � � � ! � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � ��

� � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � ! � � � � ! � � � ! � � � � � � � � � � � � � � � ! % � � !

� ! � � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � ! % � ! � � � � � � � ���

� � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � ! � � � � � ! � � � � � � � � � � �

� � � ! � � � � � � � � � � ! � � � � � � � � � � � ! � � ! � � � � � � � � � � � � � � � � � ! � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � ! � � � � � � � � � ! � � ! �� �

� � � � % � � � � �� " � � � � � ! � � � � � � � � � � � � ! � � � � � � � ! � � � � � � � � � � � � �

� � � � � � � � � ! � � � � � � � � � � � � � � �

• � � � � � � � ! � � � � � � � � � � � � ! � � � � � � �

• � � � � � � � � � � � � � � � � ! � � � � � � ! � � � � ! � � � � � � � �

• � � � � � � � � � � � � � � � � � � � ! � � � � � � ! � � � � ! � � � � � � � � � � � � � � � �

� � � � � � � � � � � ! � � ! � � � � � �

• � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � �

� � � � � � � � � � � � � � � � � � � � � � ! � � � � ! � � � � � � � � � � � � � � � � � � � � � �

� � � � � � ! � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � ! % � � ! � � � � � � � � �

• � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � �

� � � � � � � � � �� � � � ! � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � �

� � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

• � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � �

! � � � � � � � � � � � � � � � � � � ! � � � � � � � � �| �| � ��

� � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � ! � � � ��

� � % � � � � � ! � � � � � � � � ! � � � � � � ! � � � � � � � � � � � � � � � � � � �

� � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � ! � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��� � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � ��

� � � � � � � � � ! � � � � � � ! � � � � ! � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � �

� � � � � � � � � � ��

� � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� ����

� � � � � � � � � � � � � � � � � � � �

� � � � ! � � � � � � � � � � � � � � � � � � � � � � ! � �� � � � � � � � � � � � � � � �

� � ! � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � ! � � � � � � � � ! � � ��� � � � � � � � � � � � � � � � � � ! � � � � � � � � � � ! � � ! � � � � ! � � � � � � � � ! � � �

� � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � ! � � � � � �

��

� � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � ��

� � � � � ! � � � � � � � � � � � � � � ! � � � � � ! � � � � ! � ��

� � ! � � � � � � � � � � � � � � � � � � �� � � � � � � ′ � � � � � � � � � � � � � � � � � � � ! �

� � � � � � � � � �′ � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � % � � � � � ! � � � � � � � � ! � � � � � � � � � � � � � � � � � � � ′ � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � ��

� ��

��

)�

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � �

� � � � � � � � �

� � � � � �� � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � �

� � � � ! � � ! � � � � � � � � �� �

� ��

��

*�

� � � � � � � � � � � � � ! � � � ! � � � � � � � � � � � � � � � � � � � � ! � � �

� � � � � � � � � � ! � � � � � � � � � � � � ! � � � � � � � ! � � � � � � � � � � � � ! � ! % � � ! � �

� � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � % � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % �

� � � � �� � � � � � � ! � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � ! � � � � � � � � � � � � � �

� � � � � � � � �� � ! � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � ��

� � ! � � � � � � � � � � � ! � � � � � � � � � � � � � � �

� � � � � �′ � � �′ �� � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � �

� �′ � � � � � � � � � � � � � � � � � � � � � � �� �

� � � � � � � � � � � � � � � � � � � � � � � � ! � � ! � �

�� �

�� � �

�� � �

Page 91: Программирование: теоремы и задачи

� � � � ��

� � � � � � � � � � � � � � � �

� � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � ! � � � �� � � � �

� � � � � � � � � � � � ! � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � ! � � � � � � � � � � � � � � � ��

� � � � � � � � � � ! � � � � � � � � � � � � � � � � �

� � � � � � � � �� � � � � � � � � � � ��

� � � � � � � ! � � � � � � � �� � �� � � � � � � �

� � � � � � � � � � ��� � � � � � � � � � � � � � � ! � � � � � � � � � � ! � � � � � � � � ! � � � � �

� � � ! ! ! �

� ��

� ��

��

��

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � ! � � � � �

� � � � ! ! � � � � � � � � � � � ! � � � � � � ! � � � � � � � � � � � ! � � � � � � � � � � � � ! �

� � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� " � � � � � ! � � � � � ! � � � � � ! � � �

� � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � �

! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � ! � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � ��� � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � �

� � � � � � � ! � � � � � � � � � ! � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � ! � � �

� � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � �� � � � � � � � � � � � ! � � � � � � � � � � � � ! � � � � � � � � � � � �

� � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � ! � � � � �� �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � ! � � � � � � � � � � � �

� � � � � � � � � ! � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � ! � � �

• � � ! � � � � ! � � � � � � � � � � � � � � ! ! !

� � � � � � � � � � � � � � � � �

• � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � ! � � � � � � � � � ! � � � � � � �� � � �

• � � � � � � � � � � ! � � � � � � � ! � � � � ! � � � � � ! � � � � � � � � � � � � � � � � � � ! �

� � � � � � � � � � ! � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � ! � � � � � � � � � � ! � � � � � � � � � � � � � ! � � � � �

� � � � � �� � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � ! � � � ! � � ��� � � � � � � � � � ! � � � � � � � � � � � � � � � �

� ! � � � � � � � � � � ! � � � � � � � � � ! � � � ���

� � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � ! � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��� � � � � ! � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � �

� � � ! � � � � � � � � � � � � � � � � � � � � ! � % � � � � � � � � � � � � � � � � � � � ! � �� �

� ����

� � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � �� � � � � � �� � � � � � � � ! � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � ! � � � � ��

� � � �

� � � � � � � � � � � �!

� � ! � � � � � � ! � � � � � � � � � � � � � � � �

� � � � � � �� � ′ � � � � �!

� � � � � � � � � � � � � ′ � � � � � � � � � � � � � � � � � ! � � � � � ! � � �� � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � �

� � � � � � � ! � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � �� � � � � �

� � � � � � � � � � � � � � � � ′ � � � � � � � � � � ! � � ��

� � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � ′ � � � � � � � � � � � � � � � � �

� � � � ! � � � � � � � � � � � ′ � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � ��

� �

� � � � � � � � � � � �

� � � �

� � ! � � � � � � � � � � � � � � � � � � � � � � � � ′ � � � � � � � � � � � � ! � � ��

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � ! � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� ! � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � ! � � �

� � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � �� � � � � ! � � � � � � � � � � �

� � � � � � � � � � � � � � � � ! � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � ′� � � � � � � � � � � � � � � � � � � � � �

� � � � ��� � � �

� � � � � � � � ��

� � � � � �′� � � � � � ′ � � � � � � � � ��

� � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � ! � � � � ��� � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � ! � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � ! � � � � � � � � � � � � � � �

� � � � � � � ! � � � � � ! � � ��

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � �

� � � ! � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � ! � � � �

Page 92: Программирование: теоремы и задачи

� � � � ��

� � � � � � � � � � � � � � � �

�� �� � � � � � �

� � � �

��

� � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � ! � � � � � � � � � � � �� � ��

� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � ! � � � � ! � � �� � � � � � � � � � � � � � � � � � � � � � � �

� ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � ��

� � � � � � � � � � � � � � � � � ! � � � � ! � � � � � � � � � � � � � � � � � ! � � � � ! � �

� � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � �� � � � � � �

� � � � � � � � � � � � � � ! � � � � � �� � ��

� � � ! � � � � � � � ! � � � � � � � � � � � � �

� � ! � � � � � � � � � � �� � ′ � � � ! � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � ! � � � � ! � � � � � � � � � � � � � � � � � � � � � � �

� � ! � � � � ! � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

! � � � � ! � � � � � � � � � � � � � � � � ′� � � � � � � � � � � � � � � � � � � � � ! � � � � � �

� �� � � � � � � � � � � � � � � �� � � � � ! � � � � � � � � � � � � � � � � � � � �� � � � � � � � �

� � � � � �′� � � � � � � � � � � � � � � � � ��

� � � � % � � � � ! � � � � � � � � � � � � � � � � � ! � � �

� � � � � � ! � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � ! � � � � ��� � � � % � � � � � % � � � � � � � � � � � � � ! � � � � � � � � ! � � � � ! � � � � � � � � �

� � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � �

� � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � �

� � � � � ! � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � ��� � � � % � � � � � � � � � � � � � � �� � � � ! � � � � � � � � � � � � � � � � � � � � � �

� � � � ! � � � � � � � ! � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � �

� � � � � � � ! � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � �

� � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � ��

� � ! � � � � � � � ! � � � � � � �

� � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� ����

� � � � � � � � � � � � � � � � � � � �

! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � �� � � ! � � � �

� � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � ! � ! � � ! � � � � � � � �

! � � ! � ! � � ! � � � � � � �� �� � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � �� � � � � � � � � � � � � � � � � � �

{ � ! � � � � � � ! � � � � � � � � � � � � ! ! ! �

� �

� � �� � � � � � �

� � � � � � � � ! ! � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � ! � � � � � � � ! � � � � � ! � � � � � � � � � � � � � � � }

� � � � � � � � � � � � �� � � �

� � � � � � � � � � � � � � � � � � � � �{ � � � � � � � � � � � � }

� � � � � � ! � � � � ! � � � � � � � � � � � �

{ � � � � � � � � � ′ }

� � � � � � � �� � � ′ � � � � � � � � � � � ! � � �

� � � � � � � � � � � � � � � � � � � � � � � �� � �

� � � � � � � � � � � �

� � � � � � � � ! � � � � � �

� � � � � � � � � � � � � � � ! � � � � � �

{ � � � � � � � � � � �� � � � � � � � }� � � � � � ! � � � � ! � � � � � � � � � � � �

{ � � � � � � � � � � � ′ � � � � � }� ! � � � � � � � � � �� � � ′ � � � � ! � � � �

� � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � ! � � � � � �

{ � � � � � � � � � ′ � � � � � � � � � � � � � � � � � � �� � � }

� ! � � � � � � � � � �� � � � � � � �

� � � �

{ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � }

� � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � �� � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � �� � � �

� � � � � � � � � � � �

� � � � � � � �� � �� � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � �� � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � �

� � �

� � � �

Page 93: Программирование: теоремы и задачи

� � � � ��

� � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � ! � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � ! ! ! �

�� �� ! � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � ! � � � ! � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � ! � � � � � � ! � � � � � � �

� � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � ! � � � ! � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � ! � � � �� �

� � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � ! � � � ! � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � % ! % � � ! � � � � ! � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � ! % � � ! � � � � � � � � � � � � � � � � � � � � � � � � � �

� − � � � � � � � ! � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � �� � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � �

� � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � ! � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � �

� � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � �

� ��

� � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � ��

� � � � � � � � " � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � ! � � � � � � � � � � � � � � � � � � � � � � �

�� � � ′ � � � � � � � � � � � � � � ! � � � ! � � � � � � � � � � � � � � � ! � � � � ! � � � � � � � � � � � �

� � � � � � ! � � � � ! � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � �

� ! � � � � � � � � � � � � � � � � � � � � � ! � � � �� � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � ! � � � � � � � � �

� � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � ��

� ��

��

��

� � � � � � � ! � � � � � � � � � ! � � � � � � ! � � � � � � � � � � � � � � � �

! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � ! � � � � � � � � � �

� � � � � � � � � � �| �| � � � ! � � � � � � � � � � � � � � � � �| �| ��

� � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � �

� � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � �

� � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� �

� ����

� � � � � � � � � � � � � � � � � � � �

� ��

��

��

� � � � � � � ! � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � �� � ! � � � � � � � � � � � � � � � � � � � � � ! � � � � � � ! � � ! � � � � � � � ! � � �

� % � � ! � � � � � � � � � � � � � � � ! � � � � � � � � � ! � � � � � � � � � � � � ! � � ! � � � � � �

� � � ! � � � � � � � � � � � � � � � � ! � � � � � � �� � �

� ��

��

� ��

� � � � � � � ! � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � �

� � � � � � � � � � � � � �� � � � � � ! � � � � � � ! � � � � � � ! ! � � � � ! � � � � � � � � ! � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � ! � � � � � ! � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � ���

� ��

��

� ��

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � �

� � � � � � � � � � � � � � � ! � � �| �| � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � �

� � � � � � � � � � � � � � � � � ! � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � ! � � � � � � � � � � �

� � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � ! � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � �

� � � � � � � � � � ! % � ! � � � � �� � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � ��

� � � � � � � � � ! � � � � � ! � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � �� � � � � ��

� � � % � � � � � � � � ! � � � � � � �

� � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � ! � � � � � � ! � � � � � � � � � � � �

� � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � ! � � � � � � � � � ! � � � � % � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � � � � � ! � � ! � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � �

� � � � � � � � � � � ! � � � � ! � � � � � � � � � � � � � � � ����

�� ! ! ! � � � � � � � � � � �

� � � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� ! � � � � � � � � � � � �

� � � � � � � ! � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � ! � � � � � � � � � � � � �

� � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � �

� � � � � � � � � � � � � ! � � � � � � � � � � ! � � � � � � ! � � ! � � � � � � � � � � � � � � ! � � � � �

Page 94: Программирование: теоремы и задачи

� � � � ��

� � � � � � � � � � � � � � � �

� � ! � � � � �� � � � ! � � � � � � � � � � � � � � ! � � � � � � � � � ! � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � ! � � � � � � � � � � � � � ��

� ��

��

� ��

� ! � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � �

� � � � � � � � � � ! � � � � � � � ��

� � � � � � � �� � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � ! � % � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � ! � � � � � � ! � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! �� � � � � � � ! � � � � � � � � � � �

! � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � ! � � � � � � � � � � � � ! � � � � � � � � � � � � � ! � � � � � ! �� � �

� ��

� � � � � � � � �

� � � � � � � � �

� ��

��

��

� � � � � � ! � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � ! � �

� � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � ! � � � ! � � � � ��

� � � � � � ! � � � � � � ! � � ! � �

� � � � � � � � � ! � �

� � � � � � � � � � ! � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � �

� � � � � � � � � � � � � � � ! � � � � � ! % � � � � � � � � ! � � � � � � � � � � � � � � � � ! � � � � �

� � � � �� � � � � � � � � � � � � � ! % � ! � � � � � � � � � � � � � � � � � � � � � � � � � ! � � �

� ! � � � ! � � �� �

� ��

����

� � � � � � ! � � � � � � ! � � � � � � � � � ! � � � � � � � � � � � � � � � � � �

� � � �� � � � � � � � � ! � � � � � � � � � � � � � � % � � ! � � � � � � � � � � � � � � � � � � � ! �

� � � � ! � � � � ! � � � � � � � ! � � � � � � � � � � � � � � � � ! � � �� �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � ! � � � � � � � � � � � � � � � ! � � � � � � � � ! � � � � � � � � � � � � � � � � ! � � ��

� ��

��

��

� � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� ! � � � ! � � � � ��

� � � � � � ! � � � � � ! � � � � � � ! � � ! � � � � � � � � � � ! � �

� ��

��

'�

� � � � � � � � � ! � � � ! � � � ! � � ! � � � � � � ! � � � ��

� � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � ! � � � ! � � � � ��

� � � � � � ! � � � � � � ! � � ! � �

� � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � ! � � � � � ! � � � � � � � � � � � � � � ! � � � � � � � ! � �

� � � � � � � � � ! � � � � � � � � � � ! � � �� � � � � % � � � � � � � � � � � � � � � � � � � � � � �

� � � � ! � � � � ! � � ! � � % � � � � � � � � � � � � � ! � � � � � ! � �� � ��

� � � � � ! � � � � � � �

� � � � � � � � � � � � � � � � �� � ! � � � ! � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � ! � � � � � � � � � � � � � � � ��

� � ! � � � � � � � � � � � � � � � � � � � �

� � � � ��

� � � � �� � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � �

� � � ! � � � � � � � ! � � � � � � � � � � � � ! � � � � � � � � � � � � ��

� � � � � � � � � � � �

Page 95: Программирование: теоремы и задачи

� � � � ��

� � � � � �

���������

��� �� �

" � �� � ��

��

� � ! � � � � � � � � � ��

� � � � � � � � � � � � � � � � ��

� � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � ! � � �

� � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � �

� � � � � � � � � � � � ! � � �� � � � � � � % � � � � � � ��

� � � � � � � � � � � � � � � � � �

� � � � � ��

� � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � ! � � � ! � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � ! � � � � � � � � ! � � � � � � � � � ��

� � � � ! � � � � � � � � � � � � � � � � �

� ! � � � ! � � � � � � � � � � � ! � � � � � � � � % � � � � � � � � � ! � � � � � � � � ! � � � � � � � �

� � � � � � � � � � � � � � � ! � � � � � � � � � ! � � � ! � � � � � � � � � � � � � � � � ! � � � ��

� � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � ! � �

� � ! � � � � � � � � � � � � � � � � � � � � ! � � � � �� � � � � � � � � � ! � � � � � � � � � � �

� � ! � � � � � ! � � � � � � ! � � � � �� �

� ��

��

)�

� � � � � � � � � � � � � � � � � � ! � � � �

� � � � � � � � � � � � � � � ! � � � � � � � � � � ! � � � ! � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � ! % � � � � � � �

� � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� �

� ��

��

*�

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � ��

� �

� � � � � ! � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � ! � � � ! � � � � ��

� � � � � � ! � � � � � � ! � � ! � � � � � � � � � � ! � �

� � � � � � � ! � � � � � � � � ! � � � � � � � � ! � � � � ! � � 6 � � � � � ! � � �� �

� ��

��

��

� � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � ! � � � � � � � � ! � �

� � � � � � � � � ! � � � � ! � � ! � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � ! � � � � ��

� � � � � � ! � �

� � � � � ! � � ! � � � � � � � � � � ! � �

� � � � � � � �� � � � � � � � ! � � � � � � � � � � ! � � � � � � � �

� ��

��

��

� � � ! � � � � � � � � � � � � � � � � � � � � � � � ! ! ! � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � �

� � ! � � � � ��

� � � � � � ! � � � � � � ! � � ! � � � � � � � � � � ! � �

� ����

� � � � � � � �

� � � � � � � � � � � � � � � � � � � � � ! ! ! � � � � � � � � � � � � � � � � � � � � � � � � � �

! � � � � ! � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � ��

� � � � � � � �

� � ! � � ! � � � ! � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � ! � � � � �

� � � ! � � � � � � � � � � � � � � � � � � � � � � � ! � � �� � � � � � � � � � � � � � � � � � �

! � � ! � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � ! � � ! � � � � � � � � � � � � � � � � ! � � ! � � � � � � � � � � � � � �

� % � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � ! � � � � �

� � � � � � � � � � � � � � � � % � � � � � � ! � � � � �� � � � � � � � � � � � � � � ! � � � � � � � �

� � � � � ! � � ! � � � � � � % � � � � � � � � � � � � � � � � � � � % � � � ! � � � � � � � � � ! � � ! �

� � � � � � � � � � � � � � � ! � � � � � ! � � � � � � � ! � � ! � � � � � � � �� � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � ! � � ! � � � � � � � � � � � � � � � � � � � � � � � � � ! � �

� � � � � � � � � ! � � � � � � � � � � ! � � � ! � � � � � � � � � � � �� � � � � ! � � � � � � � � � �

� � ! � � � � � � � ! � � � ! � � � � � � � � � � � � � � � � � � � � ! � � � � ���

� ��

��

��

� ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � �

� � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � ! � � � � ��

� � � � � � ! � � � � �

� ! � � ! � � � � � � � � � � ! � �

� � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � �

� � � � � � � � � ! � � � � � � � � � ! � � � � � � � � � ! � � �� �

��

� � � � � � � �

� � � � � � ! � � � � ! � � � ! � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � ! � � � � � �

��

� � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � �

! � � � ! � � � ! � � � � � � � � � � � � � � � � � � � � % � � ! � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � % ��

��

� � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � ! � � � � ��

� ����

��

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � �

� � � � � � � � � ! � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � �

� � � ! � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � �

� � ! � � � ! � � � � ��

� � � � � � � � � � � ! � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � �� � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � ! � � � � �

� � ! � � � ! � � � � � � � � � � � � � � � � � � � � % �� � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � ! � � � � �

Page 96: Программирование: теоремы и задачи

� � � � ��

� � � � � �

� � � � ! � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � % � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � �

� � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � ! � � � � � � � � � � � � � � � � � � � � � � ! � � � ! � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � ! � � � ��� � � � � � � � � � � � � � ! � � � � � � � � � �

� � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � ! � � � �

� ! � � � � � � � � � �� �

� ������

� � ! � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � �

� � ! � � � � � � � � � � � � � � � �� �

� � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � �� ! � � � � � ! � � � � � � � � � � �

! � � � � !� � � � � � � � � � � � � � � � � � ! � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � �

� � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � ! � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � ! � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � ! � �

� � � � � � ! � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � ! � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � ! � � � � � � � � ! � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � ! ��

� ! � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � �

� � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � ! � � � � � � ! � � � � � ! � � � � � � � � � � ! � � � � � ! � � � � � � � ! � �

� � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � ! % � ! � � � ! � �

� � � � � ! � � � � � � � � � ! � � � � �� � � ! � � � � � ! � � � � � ! � � � � � � � � � � ! � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � ! � ��� � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � ! � � � � � � � � � � � � � � � � �

� � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � ! �

� � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � ! � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� ����

� � � � � � � �

� � � � ! � � � � � � � � � � � ! � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � �� � � �

� � � � � � � � � � � � ! � � � � ! � � � � � � � � � � � � � � ! � � � � � � � � � � � � ! � � �

� � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � ! � �� � � � � � � � � � ! � � ! � � � � ! � � � � � � � � � � � � � � � � ! � � � � � � ! � �

� � � � � � ��

� � � � � ! � � ! � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � ���

� � ! � � � � � � � ! � � � � � � � � � ! � � � � � � � � ! � � � � � � � ! � � � � � ! � � � ! � � � �

� � � � � � � � � � � � � � � � � � � � � � � � ± � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � �

� � � � � � � � ! � � � � � � � � − � � � � � � � � � � � � � ! � � � � � � � � � � ! � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � ! � � � � � � � � ! � � � � � � � � � � � � � � �� � � ! � � � � � � � � � � � � � � � � �

� � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � ! � � � � � � � � � � � � � � � � � � ! � � � % � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � ! � � � ��� � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � ! � � � � � � � � � � � � � �

� % ! � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � ! � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � ! � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � �� �� � � � � � � � ! � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � ! � �

� � ! � � � % � � � � � � � ! � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � ! � � � � � � ! � � � � � � � � � � ! ��

� ����

��

� � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � �

� � ! � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � ! � � �

� � � � � � � � � � � � � � � � � � � ! � � � � � � � �� � � ! � � � � ! � � � � � � � � � � � ! � � � �

� � � � � � � � � � � � � � � ! � � � � � �

� � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � ! � � � � ! � � � � � � � � ! � � � ! � � � �

� � � � � � � � � � � � � � � � ! � � � � � ��

� � � � � � � � � � � � � � � � � � � � ! � ��

� � � � � � � � � � � � � � � ! � � � ! � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � ! � � � � � � � � � ! � � � � ! � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � ! � � � � � � ! � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � �

! � � � � ! � � � � � � � � ! � � � ! � � � � � � � ��

� � � � � ! � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � ! � � � � � � � � � � � � �

� � ! � � � � � � � ! � � � � � �� � � � � ! � � ! � � � � � � � � � � � � � � � � � � � � � � � � � �

� � ! � � � � � � � � � � � � ! � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � �

! � � � � ! � � � � � � � � � ��

Page 97: Программирование: теоремы и задачи

� � � � ��

� � � � � �

� � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � ! � ��

" � � � � � ! �

� � ! � �� � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � ! � ��

� ! � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � ! � �

� � � � � � � � � ! ��� � � � � � � � � � � � � � � � � � � � � ! � ��

� � � � � � � � � �

� � �� � � � � � � ! � � � � � � ! � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � ��� � � � � � � � � � � ! � � � � � ! � � � � � � � � � ! � � � ! � � � � � � � � ! � � � � � � � � � � � �

� � ! � � � � � � � � � � � � ! � � � % � � � � � � � � ! � � � � � � ! � � �

• � � � � ! � � � � � � � � ! � � � � � � � � � � � ! � � � ! � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

• � � � � � � �{ � � �′ �} � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � ! % � �

� � � � � � � � � ! � � � � �′ � � � � � � ! � � � � � � � � � � � � � � � � � � � �

� ! � � � � � � � ! � �

• � � � � � � { � � �′ �} � � � � � � � � ! � � � � � � � � � � � � � � � � � � � ! % � � �

� � � � � � � ! � � � � �′ � � � � � � ! � � � � � � � � � � � � � � � � � � � � ! � � � �

� � � � ! ��

� � � � �

� � � � � ! � � � � � � � � � ! ! � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � ! � � � � � � � � � ! � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � �

� � � ! � � � � � � � � � � � � � � ! � � � � � � � � �

� � � � � � � � � � � � � � � � ��

� � � � � % � � � � � ! � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � ! � � � ��

� � � � � � � � � � � � ! � �

� � � � � � � � � � � � ! � � � � � � � � � � � � � ! � � � � � � � � � � ! � � � � � � � � � � � � � � � � �

� � ! � � �� � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � ! � � � % � � � � � � � � ! � � � � � ! � � � � � � � � � � � � � � � � � � � � � � ! � � � � �

! � � � ! � � � � � � � � ! � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � �

� � � � � � � � � ! � � � � �′ � � � ! � � � � � ! � � � � �′ � � � � � ! � � � � � � ! � � � � ��

� � � � � � ! � � � ! � � � � � � � � ! � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � ! �

� � � � � ! � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � ! � � � � ��

� � � � � � � � � ��� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � ���

" � � �

� � � ! � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � ! � � �

� � � � � � � � � � � � � ! � � � � �′ � � � � � � � ! � � � � � � � � � � � � �′ � � � � � � � � � �

� � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � �

� � ! � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � ! � � � � � � � ! � � � � � � �� � � � � � � � � ! � � � � �

� ����

� � � � � � � �

� � � � � � � ! � � � � ! � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � �

� � � � � � � � � � � � � � � � ! � � � � �′ � � � � � � � � � � � � � � � � � � � �′ � � ! � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � ! � � � � � � � � � �

� ! � � � ! � � � � � � � � � � � � � � � � �

� � � ! � � � � ! � � � � � � � � � �� �

� ����

'�

� � ! � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � ! � � � � � � � � � � � � ! � � � � � � � � ! � � � � � � � � � ! � � � � � � � � � � � � � � � � %

� � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � �� � � � � ! � � � � � � � � � � � � � � ! � �

� � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � ! � � � � � � � � � ! � � � � � � � � � � �

� � � � � � � � �� � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � �

� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � ! � � � � � � � � � � � � � ! � � � � � � � � � ! � � �

� � ! � � � � � � � � � � � � � � � � � � ! � ��

� � � � � � � � � � � � � � � � � � ! � � � � ! � � � � � � ! � � � � � � � � � � � � � � � � � � � �

� � � � � � ! � � � � � � � � � ! � � � � ! � � � � � � � � � � ! � �� � � � � � � � � � � � ! � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � %

� � ! � � � � �� � � � � � � � ! � � � � � � � � � � � � � � � ! � � � � � � � � � � � � ! � �

� � � � � � � � � � � ! � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � �

� � � � � � � � � � � � � � � � � � ! � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � ! � � � � � � � � � � � � � � � � � % � � ! � � � � � � �

� � � � � � � � ! � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � �

� � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � ! � � � � ��� � � � � � � � � � � � � � � ! � � � � � � � � ! � � � � � � � � � � � � � � ! � � � � � � � � � ! � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � ! � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � ! � � � � ��

! � � � � � � � � � ! � � � � � � � � � � � � � � ! � � � � � � ! � � � � � � � � � � � ! � � � � �

� � � � ! � � � � � � � ! � � � � � � � � � ! � � � � � ! � � � � ! � � � � � � � � � � � � � � �

� � � � � � � � � � ��

� � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � �

� � � � � ! � � � � � � � � � � � � ! � � � � � � � � � � � � � � � ! � � � � � � � ! � � � � � � � � � � � �

� � � � � � � � ! � � � � � � � � ! � � � ! � � � ! � � � � ! � � � � � � � � � � � � � � ! � � � � � �

! � � � �� �

� ����

)�

� � � � � � � � � � � � � � � � � � � � � � � ! � ! � � � � � � � � ! � � � � � � � � � �

� � � � � � � � � � � � � � � � ! � � �� �

� ����

*�

� � � � � � � � � � � � � � � ! � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � ! � � � � � � � � � � � � � � � ! � � � � ! � � � � � � � � ! � � � � �

� � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � ! � �

� � ! � � � � � � � � � � � � � � � � � ! � � � � � �

Page 98: Программирование: теоремы и задачи

� � � � ��

� � � � � �

� ����

��

� � � � � � � � � � � � � � ! � � � � ! � ��

� � � � ! � � � � � � � � � � � � � �

� � � � � � � � � ! � � � � � � � � � � � � � � � � ! � � � � � � � � � ! � � � � � � � � � � � � � � �

! � � � � � � � � � � � � � � ! � � � ��" � � � � � ! � � � � � � � � � � � ! � � � � � � � � � �

� � ��

� � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � ! � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � �

! � � � � �� � � � � � � � � � � � � � � � � � ! � � �� �

� � � � � % � � � � ! � � ! � � ! � � � � � � � ! � � � � ! � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � ! � � � � � � � � � � ! � � � � � � ! � � ��

� � � � � ! � � � � � � � � � � � � � � � ��

� � ! � � ! � � � � � � � � � � � � � � � � � � � � � � �

�� �� � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � �� �� � ! �

� � � � � � � ! � � � � � � � � � � � � ! � � � � � �� " � � � � � ! � � ! � � � � � � � � � � � � � �

� � � � � � � � ! � � � � � � � � � � � � � � � � � � ! � � � � � � � ! � �� � ��� � � � � ! � � � �

" � �� � ����

� � ! � � � � � ��

� � � � � � � � � � � � � � � � � � ! � � � � � ! � � � � � � � � � ��

� � ! � � � � � ! � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � �� � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � ! � � � � ! � �

� � � � � � � � � �� � � � � � � � � ! � � � � � ! � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � ! � � � � � � � � � � ! � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � �

� � � � � � � % ! � � � � � ! � � � � � � � ! � � � � � � � � ! � � � � � � � � � � � ! � � � � � � � ! � � �

� ! � � � � � � � � � � � � � � � � � � ��

� ����

��

� � � � � � � � � � � � ! � � � � ! � � � � � � � � � � � � � � � � � ! � � � � � ! � �

� � � � � � � � ! � � � � � � � ! � � � � � ��

� � � � � � � �� � � ! � � � � � � ! � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � ! � � � � � ! � � � � ! � �� � ��

� � � � � � � � � � � � � � � � � ! � �

� � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � �

� � ! � � � � � � � � ! � � � � � ! � � � � � � � ! � � � � � � � ! � � � � � � � � � � � � � � � � ! � � � �

� � � � � � � � � � � � � � ! � � � � ! � � � � � � � � � � ! � � � � � ! � � � � � � � � � � % � � � � �

� ����

� � � � � � � �

� � ! � � � � �� � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � �

� � � � ! � � � � � � � � ! � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � ! � � � � � � �

� � � � ! � � � � � � � � ! � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � �

� � � ! � � � � � � � � � � � � � � � ! � � � � �� � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � ! � � � ���

" � �� � ��

��

� � ! � � � � � � � � � � � ! � � � � � � � ! � � � ��

� � � � � � � � � ! � � ! � � � � � � � � � � � � � � � � � � � � � ! � � � � � ! � � � � �

� ! � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � �

! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � ! � � � � � � � � � � � � � � � � �

� � � ! � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � ! � � � � � ! � � � � � � � � � � � � � �

! � � � � � � ! � � � � � � � � � � � � � � ! � � � � � � � � � � � ! � � � � � � � � � ! � � � � � � � ! � �

� ! � � � � � � � � � � � � � � � � ! � � � � � � � � � � � ! � � � � � � � � � � � � � � � ! � � � � � � � � ��

� � ! � � � � � ! � � � � � � � � � � � � � ! � � � � � � � � � � � � � ! � � � � � � � � � � � � � � ! � �

� � ! � � ! � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � ! � � � � � � ! � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � ! � � � �

� � � ! � � � � � ! � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � ! � � � � � � �

� � � � � � ! � � � � � � ! � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � ! � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � ��

� � � � � � � � ! � � ! � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � ! � � �

� � �� � � � � � � � ! � � � � � � � � � � � � � � � ! � � � � � � � ! � � � � � � � � � � � � � � � �

� � ��

� ! � � � � � � � � � � � � � � � � � � � � � % �� � � � � � � � � � � � � � � � � � � � � ! � � � �

� � ! � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � ! � � � � ��� � � � � � � � � �

� � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � ! � � � �

� � ! � � � � � � � � � � � � ! � �� � ��

� � � � � � � � � � � ! � � � � � � � � � � ! % � � ! � � � ! � � �

� � ! � � �� " � � � � � � � � � � � � � � ! % � ! � � � � ! � � � � � ! � � � � � � � ! � � � � � � � � �

� � � � � � � � � � � � � � � � � ! � � � � � �� � � ! � � � � � ! � � � ! � � � � � � � � � � � � � ! � �

� � ! � � � � � � � � � ! � � ! � � � � � � � � � � � � � ! � � � � � � � � � ! � � � ! � � % � � � � � �

Page 99: Программирование: теоремы и задачи

� � � � ��

� � � � � �

! % � � ! � � ! � � � � ! � � � � � � % � � ! � � � � � � � � � ! � � ! � � ! � � � � � � � ! � � � ! � � � �

� � � ! � � � � � � � � � � � � � � � % ! � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

" � �� � ��

��

� � ! � � � � � � � � � � � ! � � � � � � � ! � � � � � �

� � � � � ! � � � � � � � � � � � � � � � � � � ! � � � � � � � � ! � � � � � ! � � � � � % � � � � ! � �

� � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � ! � � � ��

� ����

��

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � �

� � � � � � � � � � ! � � � � � ! � � � � � � � ��

�� � � � � � � � � � � � � � � ! � � � � � � � ! � � ! �

� � � � � � ! ��

� � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � ! � � � � � �

� � � � � � � � � � � � � � � � � � � � � ! � � � � � ! � �� � ��

� ��

� � � � � � � ! � � � � � � � � � �

� � � � � � � � � � � � � ! � � � � � � � � � � � � ! % � ! � � � � � � � � � � � ! � � � � � � � ��

� ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � ! � � � � � � � ! � � � � � � �

� � � � � � ! � ! % � � !� � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � �

" � �� � ��

��

� � ! � � � � � � � � � � � � � � � � � ! � � � � � � � ! � � � � � ��

� ��

��

� � � � � � � � � � � � � � � � � � � � � � � � � �

� � � ! � � � � � � � ! � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � ! � � � � ! � � � � � � �

� � � � � ! � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � ! �

� � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � �

� � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � !�

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � ! � � ! � � � � � � � ! � � � � � � � � � � � � � ! � � � � � � � � ! � �

� � � � � ! � � � � �� � � � � � � ! � � ! � � � � � � � � � � � � � � � � � � ! � � � � � � �

� � � � ! � � � % � � � � � ! � � � � � � � � � � � ! � � � ! � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � �� � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � �

� � � � � � � � � � � ! � � � � � � � � � � � !� " � � � � ! � � � � � � � ! % � � � � � � � ! � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � ! � � � � � � �

� � � � � � � � � � � � � � � � ! � � � � � ! � � � � � � � � ! � � � � � � � � � � � �

{ � � ! � � � � � � � � � � � ! � � � ! � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � �}�

� � � � � � � � � � � � � � � � � � � � ! � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � ! � � � � � � � � � � � � � � � � � � � �

� � � % � � � � ! � � � � � � � � � � ! % � ! � � � � � � � ! �� �

� ����

� ��

� � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � �

� � � � � � ! � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � �

� � � ! � � � � � � � � � � ! � � � � � � � � � � � � � � ! � � � � � � � ! � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � ��

� � � � � � � �� � � � � � � � � � � � � � � � ! � � � � � � � ! � � ! � � � � � ! � � � � � � � �

� � � � � � � � � ! � ! � � � � � � � � � � � ! � � � � � � � � � � � � � ! � � � � � � � � � � � ! � � �

� � � � � � � � � � ! ���

� ����

� ��

� � � � � � � � � � � � � � � ! � � � � � ! � � � � � � � � � � � � � � � � � � � � ! �

� � � � � � ! � �� � ��� � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � ! � � � � � �

� � � � � � � � � � � � � � � � � � ! � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � �

� � � ! � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � �� �

��

� � � � � � � � � � � � � � � � � � � ��� � � � �

� � � � � � � � � � � � � � � � � � � � � � � � ! � � � � ! � � � � � � � � � � � � � � �

� � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! �

� � �� � � � � ! � � � � � � ! � � � � � ! � � � � � � � � � � � � � � � � � � � � � ! � � � ! � �

� � � � � ! � � � � � � � � � � � � � ! � ! % � ! � � ! � � � � � � � � � � � � � � � � � � � � � � �

Page 100: Программирование: теоремы и задачи

� � � � ��

� � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � ! � � ! � � � � � � �

� ! � � � � � � � � � � ��� � � � � � � � � � � ! � � � � � ! � � � � � " � � � � � � � � � � ! � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � ! � � � � ! � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � ! �� " � � � � � � � � � � � � ! � � � � � � � � ! � � � ! � � �

� ! � � � � � � � ! � � � � � � � ! � � � ! � � � � � � � � � � � � � � � � � � � � � ! � � � " � � � � � " � � � �

� � � � � % � � ! � � � � � � � � � � � ! � � � � � � � � � ! � � � ! � � � � � � � � � � � � � � � �

� � � � � ! � � � � � � � ! � � � � � � � � � � ! � � � � � � � � � � � ! % � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � ! � � � � � � � � � ! � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � ��� ��

��

��

� � � � � � � � � ! � � ! � � � � � � � ! � � � � ! � � � � � � " � � � � � � � � � � �

� � � � � � � � � � � ! ��

� � � � � � � � � � � � � ! � � � ! � � � � � � � ! � � � � ! � � � � � � ! � � � � � � � � � � � � � �

� � ! � � � � ! � � � � � � � � � � � � � � ! � � � � � � � � � ! � � � � � � � � � ! � � � � � " � � � � �

� � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � �

� � " � � � � � � �� � " � � � � � � � � # � �

� � � � � # � �

� # �

� � � � � � � � � � # �

� � � � � � � � � � �

� � � � � � � � � � � # �

� � � � � � � � � � � � �

� � �� � " � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � " � # �

� � � � � �

� � �� � " � � � �

� � � � � � �� � �

� �

� � � � � � � � � � � � � � � � � � � � � �

� � � �

� � � � # � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � �� �

� �

� �

� � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� �

� ��

����

� � � � � � � � � � ! � � � ! � � � � � � � ! � � ! � � � � � � � � � � � � � � � � � �

� � ! � � � � � � � � � � ! � � � � � � � � � � ! � � � � � � " � � � � � �

� ��

��

� � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � ! � � � ! � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � ��� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � ! � � � �

� � � � � � � � ! � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � ! � � � � � � � ! � � �� � � � � � � � � � � � � � � ! � � �

� � � � � � � � � � ! � � � � � � � � � � � ! � � � � ! � � � � � � � � � � � � � � � � ! � � � � � � � �

� � � � � � � � ! � � � � � � � � ��

� � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � ! � � � � ! � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � �� � � ! � � ! � �

� � � � � � � � � � � � � � � � � � � � ! � � � � � ! � � � � � � � � � ! � � � � � � � � � � � � � � � � �

� � � � � ! � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � �

� � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � �

� � ! � � � � � � � ! � � ! � � ��

� � � � � � � � � � � � � � ! � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � �

� � � � � � ! � � ! � � � � � � � � � � � � � � % � � ! � � � � � � "� � � � � � � � � � � � � � �

� � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � �

� � � � � � � ! � � � ��

� � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � �

� � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � " � � � � �

� � !� ��

� � � �

• { � �� � � � � � � � � � � � �} � � � � � � � � � � { � � �} �

� � � ! � � � � � � " � � � � � � � � � � � � � � � � � � � � � � � � � � �

• { � �� � � � � � � � � � �} � � � � � � � � � � � { � �} �

� � ! � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � �

� � � � � � � � � � � � � � � � � � −∞

� �∞ � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � ! � � � � −∞ � � � � � � ! � � � � �∞ �

• { � � � � � � � � � � �� � �} � � � � � � { � �} �

� � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � ! � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � ! � �

� � � � � " �

• { � � � � � � � � � � � � � � � � � � � ! � � � � � �} � � ! � { � � �} �

� � � ! � � � � � � " � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � ! � � � � � � � � � � � % � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � �

� � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � �

� � � � � � � � � � � � � � � ! � � � � � � � � ! � � � � � � � � � ! � � � � � � � � � � � � � � �

� � � � � � � ! � � � � � � " � � � � ! � � � � � � � � � � � � � � � � � � � � ! � � � � � � ! � � ! � �

� � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � " � � � � ! � � � � � � � � � � ! � �

Page 101: Программирование: теоремы и задачи

� � � � ��

� � � � � �

��

� � � � � �� � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � ! � � � � � %

� ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � �

� � � � � � � � � ! � � ! � � � ! � � � � � � � � � � � � � � � � � ! � � � � ! � � � ! � � � � �

� � � � ! � � � � � � � � � � � � ! � � � � ! � � � � � � � ! � � � � � � � � � � � � � � ! � � � � � � � � �

� � � � � � � � � � � � � ! � � � �

� � � � � � � � � � � � ! � � � � � ! � � � � � � � � � � � � � �

� � �

� � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � ! � � � � � � � � � � � � � � �

� � � � � � � ! � � � � � � � � � � � � � ! � � � � � � � � ! � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � ! � � ! � � ! � �

� � � � ! � � � � � � � � � � � � � � � � � � ! � � � �

�� � ! � � % �

>

� � � � � � � �

� � ! � � � � � ! � � � � � � � � � � � ! � � � � � � � � � � � � � ! � � � � ! � � � ! � � � � � �

� � �� ��

" � � � �

� � � � � � � � � � � � � � � � � � ! � �

>

� � � � � � � � � � � � � � ! � � � � � � � ! � � � � � � � � � � � �

� � ! � � � � � � � � ! � � � �

� � ! � � � � ! � � � % � � ! � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � ! � � � �

� � � � � � � � � � � � ! � �

6

� � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � �

� � � � � � � � � � ! � � � � � � � ! � � � � � � � � � � � ! � � � � � � � � � � � � � ! � � � � � �

� ! � � � � ! � � � � � � � � � � � �

� ! � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� ! � � � � � � � � � � � � � � � � � � � � ! � � � � ! � � � � � � � � � � � � � � � � ! � � � ! � �

� � � � � � ��

� �� � � ! � � � � � � � � � ! � � � � � � � � � � ! � � � � � � � � � � � � � � � ! �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � �

� ! � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � ! � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � ! � � � � � � � � � % � � � � � � � � � � � � �� � � �� � � � � � � � � � � ��� � �

� ��

��

� � � � � � � � � � � � � � � �

� ! � � � � � � � �

� �� � � �

�� � � � � �

6

� �

�� � � � � �

6

6

� �

�� � � � � �

6

� ! � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � ! � � � � � � � �� � �� � � �

� � ! � � � � � � � � � � � � � � � ! � � � % � � � � �� � � � � � � � � � � � � � � � � ! � � � � �

� � � � � � � � � � � � � � � � � ! � � � � ! � � � ! � � � � � � � ! � � � � � � � �

� � � � � � � � ! � � � �

� � � ! � � � % � � � � � �� �∞ � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � �� � � ! � � � � � � � � � � � � � � � � ! � � � � ! � � � ! � � � � � � � � � ! � � � % � � � � � � � � � � �

� � � � � � � � � ! � � � � � � � � � � � � � � � � � ! � � � � � � � � � ! � � � � � � � � � � �

± � ! � � � � � % � ! � � � % � � � � � � � ! � � � � − ��

� � � � � � � � � � � � � � � � � � ! � � � � � �

� � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � �

� � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � ! � � �

� � � � � ! � ! � � � ! � � � � � � � ! � � � � � � � � � ! � � � % � � � � � � ! � � � � � � � ��� �

� � � � � � ! � � � � � � � � � � � ! � � � � �

� � " � � � � � � �� � � � � �� � " � � ��� � � # � � � � � � � � # � �

� � � � � # � �

� # �

� � � � � � � � � � # �

� � � � � � � � � � � � � � � � � � � � � � �� � �

� � � � � � � � � � � # �

� � � � � � � � � � � � �

� � �� � � � � �� � " � � ���� � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � �� � �

� � � � � � � � � � � � � � � � � � � � " � # �

� � � � � �

� � �� � � � � �� � " � � ��

�� � �

� � � �

� �

� � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � �� � � �

� � � �

� � � � # � � � � � � �

� � � � � � � � � � � �

� �

� �

� � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � ! � � � ! � � � ! � � � � � ! � � ! � � � � � � � � � ! � � ! � � � � �

� � ! � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � ! � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

Page 102: Программирование: теоремы и задачи

� � � � ��

� � � � � �

! � � � � � � � � �

� � ! � � � � � � � � � � ! � � � � � � � � � � � � � � � � ! � � � � � �

� � � � � ! � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � �

� � �� � � � � � � � � � � � � � � � ! � � � � � � ��� � � ! � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � ! � � � � � � � � � � �

� � � � ! � � � � � � � � � � � � � � �

� ��

'�

��

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � ! � � � � � � � � � � � ! � � � ! � � � � ! � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � ! � � � � � � � � � � � � � � � � � ! � � � ! � � � �

� � � � ! � � � � �

� � � � ! � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � �

� � � � � � � % � � � � � � � � � � � � � � � ! � � � � � � � � � � ! � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � ! � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � �

� � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � �

� � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � �

� � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � ! � � � � � � � � ! � � � � � � � � ! � � � � � � � � � � � �

� � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � �

� � � � � � � � � � ! � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � ! � � � � � � ! � � � � �

� � � � � � � � � � � � � �

� ��

'���

� ! � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � �

� � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � ! �

� � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � �

� � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � ! � � � � � � � � �

� � � � � � � �

� ��

'�

��

� � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � ��� � � ! � � �

� � � ! � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � % � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � �

� ��

��

� � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � ! �

� � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � ! � � � � � � � � � � � � � � ! � � � � �

� % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � �

� � ! � � � � � � � � � � � � � �� � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � �

� � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � �

� � � � �

� � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � ! � � � � � � ! � � � � � � � �

� � � � � � � � � � � � � � ! � � � % � � � � � � � � � � � � � � � � � � �

� � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � ! � � � � � � � � � �

� � � � �

� � � � � � � � � � � � � � � � � � ! � � � � � � � � � � " � � " � � � � ! � � � � � � �

� � � � � � � � � ! � � � � � � ! � � � � � � � � � ��� � � ! � � � � � � � � � ! � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � ! � � � � ! � � � � ! �

� � � � % � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � ! � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� ��

'�

'�

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � ! � � � � � � � � � � � ! � � � � ! � � � � ! � � � � � � � � � � � � � ��� �

� ! � � � � � � � � � � � � � � � � � ! � � � � � � � � � ! � � � � � � � � � �

� � � � � � � " � � � � � ! � � � � � � � � � � � � � � � � �� � � � � � � � ! � � � � � � � � �

� � � � � � � � � � � � � ! � � � � � � � � � � � � � �� � � ! � � � � � � � � � � � � ! � � � � � � � � �

� � �� � � � � � � ! � � � ! � � � � ��

� � � � ��

� � � ! � � � � � � � � � � � � � � � � � � �

� ! � � � � ! � � � � � � � ! � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � ! � � � � � � � � ! � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � �

� � � � � � � � � � � � � � � � � � ! � � � � � � � � ! � � � � � � � � � � �� � � ! � � � � � � � � �

� � �� � � � � � � � � � � � � ! � � � � � � � ! � � � � � ! � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � ! � � � � ! � � � � � � � � �

� � � � ! � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � �

� � � � ! � � � � � � � � � � � ! � �

� � � � � � � � � � ! � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � �

� ! � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � �� � � � � � � � � ! � � � � � � � � � �

" � � � � � ! � � ! � � � � � � � � � � � � � � � � � �

� � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � ! � � � � �

� � ! � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � ! % � � � ! � � � � � � � � � � � � �

� � ! � � � � � ! � � �

� � � ! � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � ! � ! � � � � � ! � � � � � � � � � � � ! � � � � � � � � � � � � � � ! � � � � � � � � � � �

� � ! � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � ! � � � � � � � � ! � � �

� � � � � � � ! � � � � ! � � � � � � � � � � � � � � � � � � ! � � � ! � � � � � ! � � � � � � � � �

� � � � � � � ! % � � � ! � � � � � � � � � � �

Page 103: Программирование: теоремы и задачи

� � � � ��

� � � � � �

� ��

'�

)�

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � ! � � � � �

� � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � √ � � � � � � � �

� � � � � � � � � � � � � � � � √ � � � � �

��

� � � � � � � � � � � � � �� � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � ��� � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � �

� � � � � � � � � � � � � � � � � � �

� � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � �

� � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��� � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � �

� � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � %

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � �

� ��

)�

��

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � �

� � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� ��

��

� � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � − � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � − �� � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � �� � � � � � � − �� � � �

� � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� �

) �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � �

! " # � "

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � $ � � � � � � � � � � � % � � � � � � �

� � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % �

� � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � �

� � � � � � � � � � � � % � � � � � � � � � � � � � � � � % % � � % � � � � � � � � � � � � � � � � �

� � � � � � � � % � � � � � � � � � � � � � � � � � � & �

� � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � % �

� � � � � � � � � � � � � � � � � � � � � � � � � � � �

Page 104: Программирование: теоремы и задачи

� ��

� � � � � � � � � � �

� � � � � � � � � �

��

�� � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � �

� � � � � � � � � �

� � % � � � � � � � � � � � � � � � � � ��

��

��

� � � � � � � � % � � � � � � � % � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� ��

� � � � �

� � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � % � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � �

� � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � �

� � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � % � � � % � � � � � � � � � � � � � � � � � � � � � � �

% � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � �

� � �

� � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � % � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � % � �

� � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � �

� � � � � � � � � % � � � � % � � � � � � � � � � % � � � � � � � � � � � � � � % � � � � � � � �

� � � � � � � � � � � � � � � � % � � � � � � � � � � � % � � � � � � � � � 6 � � �′ � � � � � 6 � �′

� �

" � � � � � � � � % � � � � � � � � � � � � � � { �� �� �} � � � � � � � � ��

� � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � % � � � � �

� ����

� � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� �

� � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � �

� � � � � � �

� � � � � � � � � � � � � � � � � � % � � � � �

� � � � � � � � � � �

� � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � �

� � � � � � � � � � � � � � � �� � % � � � � � � � � � � � % � � � � � � � � � � � � � �

� � % � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � �

� % � � % � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � ! � � � � � � � � � � � � � ! � % � � � � � �

� �

"

� � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � �

� � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � �

� � ! � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � ! � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � �

� � � � � � � �

� �

#

� � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � �

$ � % � � �

� � � � � � � � � �� � � � � � �� � � � � � �

� � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � �

�� �

)

� � � � � � � � � � � � � % � � � � � � � % �

% � � � � � � � � � � � � � � � %

� � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � % � � � � � � � � �

��

��

& � ' � � � � � � � � � ' � � � � �

( � ) * � + + � � �

� � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � % �

� � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � �

� � � % � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � �

�� � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � �

Page 105: Программирование: теоремы и задачи

� � � � ��

� � � � � � � � � � � � � � � � �

� � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � �

� � ∑ �� � � | � � � |

� � � � � � � � � � ∈ �� � � � | � � � | � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % �

� � � � � | � � � | � � � � � � � � � � � � � � � � � � � � � � ∑ � � � � � | � � � | � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � �

� � � � � � � � % � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � �

� � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � %

� � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � �

� � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � �

% � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! ! ! � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

�− � � � �− � � �! ! !

� �− � �

6

��

� % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � % � � � � � � � � � � �

� �

� � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � �

� � � � � � % � � � % � � ��

� � � � � � � % � � � � � � � � � � � � � � � �

� % � � %

� � � � � � � � � � � � �

� � � � � � % � � � � % � � � % � � � � � � � � � � % � � �

� % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � % � � �

� � � � � � � � � � � � � % � � � � � �−| � | � � � � | �| � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � �

% � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � % � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � �

" � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � �

% � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � �

� ����

� � � � � � � � � � � � � � � � � � � � � � �

� �

"

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! ! ! � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � % � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � ��� � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � ! ! ! � � �

� � � � � � � % � � � � � � � � % � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � % � � �� � �� � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � �

� � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � �

� � � � � � � � � � % � � � � � � � � � � % � � � � % � � � � � � � � � �− � �� � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � % � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � % � � � � � � � � � � � � % � � � % � � � � � � � % � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � �

� �

#

� � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � �� � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � % � � � � � � � �− � �� � � � � % � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � �

$ � % � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � % � � � � � � � � � � �

� � � � � � � � � � � � � � � � % � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � �

� � % � � � � � % � � � � � � � � � � � �

� �

)

� � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � % � � � � � � � � �

� � � � � � � � � � � � % � � � � � � � � � � � � � � % �

� � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � %

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � �

� � � � � � � �

� � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � ! ! ! �

� � � � � � � � � �

% � � � � � � � � � � � � � � � � | � � | � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � �

� � � � � � � � � � � � � � % � � � � � % � � � � � � � � � � � � � � � � � � � � � � � �

Page 106: Программирование: теоремы и задачи

� � � � ��

� � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � �! ! !

� � �

� � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � % � � � � � � � � � � � % � � � � �

� � � � � � � � � � � � � � ��

� � % � � % � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � �

� � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � % � � � % � � � � � � � � � � � � � � � �

� � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � �

� � � � �� �! ! !

� � � � � � � � � � � � � � � � � � !

� � � � � � � � � � � � � � � � � � � � ��

� ��

� � � � � � � � � � � � � � � � % � � � � � � � � �

� � �� � �� � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �!

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

% � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � �

� �− � � � �− � � �! ! !

� �− � � � �

� � � � � � � � � � � % � � � � � � � � � % � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � % � � � � � � � � � � � �− ��

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � �

� � � � � � � � � � � � � �

� �− � � � �− � � �! ! !

� �− � � � � � � � � � � ��

� � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � �

� � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� ��

��

� � � � � � � � � �

��

��

�� � � � � * � � �

� � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! ! ! � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! ! ! � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � �

� � �� � � �

� � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � ! ! ! � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � �

� � � � % � � � � � � � % � � � � � � � � � � � � �

� � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � �

� � �

� � � � � � � � � � � � � � � ! ! ! � � � � � � � � � � � � � � � � � � � � � � � �

� �

"

� � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � %

� � � � � � � � � � � � � �� � � % � � � � � � � � � � � � � � � � �

� � � � � � � � � � � % � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � �

% � � � � �

� �

"

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � �! ! !

� � �

� � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � 6

� � 6 ! ! ! 6

� �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � �

� �

"

"

% � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � % � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

��

� � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � % � � � � � � � � � � � � � � � � � �

� � � �

Page 107: Программирование: теоремы и задачи

� � � � ��

� � � � � � � � � � � � � � � � �

� �

"

#

� � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � �

�� � >

� � > ! ! ! >

� ��

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � 6

� � 6 ! ! ! 6

� �

� � � % � � � � � � � �

� � � �� � � �

>

� �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � , � � � � � � - � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � % � � � � � � � �

� � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � �

� � % � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � �

� �

"

)

� � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � �

� � >

� � > ! ! ! >

� �

� >

� �

� >

� �

� % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � − � � � � � �

� � � � �� ! ! ! � � �

�� � �

� � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� �

� � � �

� � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � �

�− � � � �− � � �! ! !

� �− � �

� � �− � �

� � �− � � �

� �− � � � �− � � �! ! !

� �− � �

� � �− �

6

��

� � � � � � � � � � � � � �

� � � � � � � �

� � � � � � % � � � � � � � � � ! ! ! � � �

� � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � − �

� � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � �

� � � � �! ! !

� � �

� � �

� � � �

� � �

� � � � � � �

� � � � � �! ! !

� � �

� � �

� � � � �

� � � ��� � � �

� � � ��

!

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � ! ! ! � � �

�� � � � � � � � � � � � � � �� ! ! !

! ! ! � � �

�� � �

� � � �

� � � � � � � � � � � � � � � � � � � � � � �−

� � � � � � � �� � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � % � � � �

� ��

��

� � � � � � � � � � � �

� � � � � � % � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � % � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � % % � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � % � � � � � � �

� � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � % � � � � �

� �

� � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � �

� � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � �

� � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� �

"

*

� � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � ! ! ! � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � %

� � � � � � � � � � � � � � � � � � � � � � � � � � � �

$ � % � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � % � � � � � � % � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � % � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� % � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � % � % � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � �

% � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

% � � � � � � �

��

��

�� � � � � � � � � �

� � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

% � � � � � �

� � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � % � � � � � % � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� �

#

� � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! ! ! � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

Page 108: Программирование: теоремы и задачи

� � � ��

� � � � � � � � � � � � � � � � �

� � � � � ! ! ! � � � � � �� � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � �

� � � � � ! ! ! � ��

� � � ��− � � � � � �� �

! ! !

� � � �− � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � �

� � � − � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �− � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � % � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � ! ! ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! ! ! � � ��

� � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � �

� % � � � � � � � � � � � � � � � � � � � � � � � �− � �

6

� �

� � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � − � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � % � � � � � � � % � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � % � � � � � � � % � � � � �

� �

� � � � � � � � � � � � � � � −b � � � � �c � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � �− � �

6

� �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � %

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � ��

� � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! ! ! � � ��

� �

#

� � � � % � � � � � � � � � � � � � � � � � � % � � � � � % � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � ! ! ! � � ��

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � % � � � � � � � � � � � � �

�− � � �! ! !

�− � �

6

��

� �

� � � � �! ! !

� � � � � >

� � � � � ! ! ! � � � � !

� � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � $ � � � � � � � � � � � � � � � �

� � � � � � � � ! � � �− � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � ! ! ! � � � � ! � � ! ! ! � ! � � � � � � � � � � � � � � � � � � � � � � � � � � �

� ��

��

� � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � �

� ��

− � � � ! � � �! ! !

� � � �− � � � ! � � > � ��

− � � � � � � �! ! !

� � � �− � � � � � � !

� � � � � � � � � � � � � � � � � � � �

� ��

− � � � ! � � �! ! !

� � ��

− � � � ! � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � ! � � ! ! ! � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � ! � � � �

% � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � % ! � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � �

− � �� ! � � ! ! ! � − ��

� !�

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � �� ! � � � � �

� � � � � � � � � ∑ � � � ∑ ! � � �� � � � � � � % � � � � � � � �

� � � ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � �

� � �

(∑

� � �

)

>∑

� � � � � � !

% � � % � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � ∑ � � � � � � ! � � � � � � � � � � � � % � % � � � � � � � � �

� � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � % �

� � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � % � � � � � � � � � � � � � � � �

� % � � � � % � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� � � ��

� � � � � � � � � � � � % � � � � % � � � � � � � ��

� � � � � �

� � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � �

� � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

% � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

Page 109: Программирование: теоремы и задачи

� � � � ��

� � � � � � � � � � � � � � � � �

� � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � �� �� � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � % �

� �

#

"

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � % � � � � � � � �

� � � � �

��

� � � � � � � � � � �

� � � � � � ��

� � � � � � � � � �

��

� � � ' � � � � � � � � � ) ' � � �� � � ' � � � � � � �

� � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � �

� � � �

� � � � � � � � % � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � �

� � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � % � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � �

� � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � % � � � � �

� � " � � # $ % & � � '

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � (�

� � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � ) � � � � � � � � � � � � �

� � � * � � � � � � +� � � � � � � � � � � � � � % � � � � � � � � � ( � � � � � � � � � �

% � � � � � ,- - - ) . /

0 � � � � � � � � � � � � � � � � � � � * � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � % � � � � � % � � �

% � � � � �

� � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � ! � � * � � � � � � � � � � � % � � � �

� � " 1 2 � & 3 4 � � 5� � � � � � � � � � � � � � � � �

2 � & 3 4 � � 5 � % 6 6 7 % & # $ 3 �

� � � � � � � � � � �

8 � 9 : � ; ; � < � => > ? @ A � B C D E

F G H I : � ; ; � < � => > ? @ A � B C J B B 9 H � ? E

� � � % � � � � � � � � � � � � � ) . / � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

Page 110: Программирование: теоремы и задачи

� � � � ��

� � � � 6 7 % $ � � & � � ' & # � � 6 7 $ � � " � � # $ % & � �

� � � � � � � � � � � � � � � % � � � � � � �

� � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � �

� � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � % � � � � �

� � � � % � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � + � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � �

� � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � % � � � � � � � � � � � � �

� � � � � % � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ) � � � � � � � � � � � �

� � � % � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � * � � � � � � � � � � � � � � � � � � � � � � % � � � � � � 0 � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � + � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� "

� � � � � � � � � � � % � � � � � � � � � � �

� � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � �

� � % � �

� � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � �

� � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � �

� � � � � �

� � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� "

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � �

� � � � � � C F ? � � � B ? � � � � � � � � ! � : D " : J B B 9 H � ? E

8 � ; � : � ? � H # H ; EJ H # � ?� : $ % ! � " E

& % � 9 H F G H I � � � � ? I ! 8 � 9 � � � ' ( � " I B J H # � ?

� : $ ! � ) A " * B I ? E

H ? I E + ? B � F G H I � � � B ; ! 8 � 9 � � � $ � " ,

� ��

��

� " � � # $ % & � � 6 # 7 3 � - 7 # 5 % � � � 6 % 4 � � 5 � � .

� � � � � � � � : $ F G H I � � � � ? I ! 8 � 9 � � � $ � " E

H ? I E/ ; B � H I F ; H � 0 1 � 2 3 4 ! � : D " E

8 5 ; � : � ? � H # H ; EJ H # � ?� : $ % ! � " E

& % � 9 H F G H I � � � 5 ? I ! 8 5 9 � � � ' ( � " I B J H # � ?

� : $ ! � ) A " * B I ? E

H ? I E + ? B � F G H I � � � B ; ! 8 5 9 � � � $ � " ,

� C ? B � F G H I � � � � % H ? J H # � ?

F G H I � � � : $ � ; F H E

8 5 9 � � � : $ � E

H ? I EH ? I E

/ ; B � H I F ; H 3 6 7 � 8 9 3 4 ! � : D " E

8 5 ; �: # 5 / : � ? � H # H ; EJ H # � ?

� : $ % ! � " E

& % � 9 H F G H I � � � 5 ? I ! 8 5 9 � � � ' ( � " I B J H # � ?

� : $ ! � ) A " * B I ? E

H ? I E + ? B � F G H I � � � B ; ! 8 5 9 � � � $ � " ,

� C F G H I � � � 5 ? I ! 8 5 9 � � � $ � " � % H ? J H # � ?

F G H I � � � : $ C 5 9 G H E

# 5 / : $ � E

� : $ ! � ) A " * B I ? E

+ # 5 / @ � ; � �: 7 0 0 � � < = 0 � � > � 7 � ; 4 6 < 0 � � 3 = 3 > �:

� ) A : > > > ,

& % � 9 H F G H I � � � I B J H # � ?

� C � $ % ! 8 5 9 � � � " � % H ? J H # � ?

+ � � 6 2 0 � = = � 6 �: � 3 9 � ? 0 � � � � � � 4 ,

H ? I H 9 G H � C I � G � ! % ! 8 5 9 � � � " : � " ' I � G � ! # 5 /: � " � % H ? J H # � ?

+ # 5 /> > > % ! 8 5 9 � � � " > > > �: � 3 9 � ? 0 � � � � � � 4 ,

H ? I H 9 G H J H # � ?

F G H I � # 5 / � : $ � ; F H E

8 5 9 � # 5 / � : $ 8 5 9 � � � E

F G H I � � � : $ C 5 9 G H E

# 5 / : $ � E

H ? I E� : $ ! � ) A " * B I ? E

H ? I EH ? I E

H ? I E

@ � � � � � � � A BC � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

Page 111: Программирование: теоремы и задачи

� � � � ��

� � � � 6 7 % $ � � & � � ' & # � � 6 7 $ � � " � � # $ % & � �

� � � � � � � � � � A � � C� � � � � � �

� � � A BC � � C . A � ) � � )-

� � � � � � � � � � ) � � � � � * � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � ��

� "

"

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � % � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � % � � � �� � � � � � �

� � � � � � � � � � � � � � � � � � � � � ��� � �� - - - � � � � � � � � � � � � � � � � � � � � � � � � - - - �

� � � � � � � � � � � � � � � � � � � � � � � { �� - - - � � − �} � � � � �

� � � � � � � � � �

� � � � � � � � � � � � � �

! " # � "

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � * � � � � � � � � � � � � � �

� � � � � � � � � � � � % � � � � � � � � � � � � � � � % � � � � � � � � ��

� "

#

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � * � � � * � � � � � � � � % � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � A � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � ' � � � � � � � � � � � � ) � * �

� � � � � * � � � � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � * � � � � � � � % � �

� � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � � � � % � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � % � � � � � � � � � * � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � �

� � � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � % � � � � � � � � � � * � � � � � � � �! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � % � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � �� � % � � � � � � � � � � � � � � % � � � � � � � � � � � �

� "

� � � � � � � � � * � � � � � � � � � � � � % � � � � � / - - - ��

� � � � � �

� � % � � � � � � � � � * � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� ��

��

� " � � # $ % & � � 6 # 6 � � 6 3 % ' � � � �

� � � � � � � � % � � � � � � � � � � * � � � � � � �

0 � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � �

� 0 � � � � � � 3 � : 5 ; ; 5 < � A > > ? � B C D E

� � � � � 8 � 3 : 5 ; ; 5 < � A > > ? � B C A > > ? E

! � � 2 � 2 0 1 : A > > ? E

" � � # 3 � � : 5 ; ; 5 < � A > > $ � B C A > > ? E

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � $ � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � % & ' ( ) * + � � � � , � �

� � , � � � / - - - �� - � � % � � � � , , � � � � � - � � � � � � � � � . � � � � � � � � � . � � � �

� � � � - � / & ' 0 1 0 2 3 � / - 1 4 & 5 6 7 8 ) 9 � � � � � � / : � � � / - - - ) . /� ; < ; � ,

1 4 & 5 6 7 8 ) 9 � ) � � ,�

C F ? � � � B ? � � 3 � � � � � � 3 ! � : D " : J B B 9 H 5 ? E

8 5 ; � : � ? � H # H ; EJ H # � ?� : $ " � � # 3 � � � % ! � " � E

+ 0 6 � � 0 6 4 3 6 7 � 4 2 6 � 3 6 7 � : � � 9 3 � � < 6 � ,

& % � 9 H ! � ' ( = " 5 ? I ! � 0 � � � � � � 3 � � � � ' ( � " I B J H # � ?

� : $ � � � � � 8 � 3 � � � E

H ? I E + ! � $ = " B ; ! � 0 � � � � � � 3 � � � � $ � " ,

� � 3 � � � � � � 3 : $ ! � ' ( = " 5 ? I ! � 0 � � � � � � 3 � � � � $ � " E

H ? I E/ ; B � H I F ; H � 0 1 � 2 3 4 ! � : D " E

8 5 ; � : � ? � H # H ; EJ H # � ?� C ? B � � � 3 � � � � � � 3 ! � " � % H ? J H # � ?

� : $ ! � � 2 � 2 0 1 E

+ ! � � 2 � 2 0 1 ' ( = @ 6 9 3 � � =: 9 0 � � � � � � � 0 � � < � 6 < ,

! � � 2 � 2 0 1 : $ � � � � � 8 � 3 � ! � � 2 � 2 0 1 �

� 0 � � � � � � 3 � � � � : $ � E

� � � � � 8 � 3 � � � : $ " � � # 3 � � � % ! � " � E

" � � # 3 � � � % ! � " � : $ � E

H ? I EH ? I E

/ ; B � H I F ; H 3 6 7 � 8 9 3 4 ! � : D " E

8 5 ; �: / ; H I : � ? � H # H ; EJ H # � ?

� : $ " � � # 3 � � � % ! � " � E / ; H I : $ = E

Page 112: Программирование: теоремы и задачи

� � � � ��

� � � � 6 7 % $ � � & � � ' & # � � 6 7 $ � � " � � # $ % & � �

+ 0 6 � � 0 6 4 3 6 7 � 4 2 6 � 3 6 7 � : � � 9 3 � � < 6 � E / ; H I @

� � � � ; � � � 3 : � 6 � 3 0 � � 6 4: 3 =: � 6 � 3 � � ,

& % � 9 H ! � ' ( = " 5 ? I ! � 0 � � � � � � 3 � � � � ' ( � " I B J H # � ?

/ ; H I : $ � E � : $ � � � � � 8 � 3 � � � E

H ? I E + ! � $ = " B ; ! � 0 � � � � � � 3 � � � � $ � " ,

� C � ' ( = � % H ? J H # � ?

+ � 0 � � � � � � 3 � � � � $ � : � � � = � � � 6 4: � � � 0 � � � � 3 4 ,

� C / ; H I $ = � % H ? J H # � ?

+ � � � = � � 0 7 � > � � 6 < � � � 2 ; = 2 6 � 3 6 7 � ,

" � � # 3 � � � % ! � " � : $ � � � � � 8 � 3 � � � E

H ? I H 9 G H J H # � ?

� � � � � 8 � 3 � / ; H I � : $ � � � � � 8 � 3 � � �

H ? I E+ 0 6 � � 0 6 4 2 � � � � 4 � 2 6 � 3 6 0 7 6 2 0 1 0 � � ; � ,

� � � � � 8 � 3 � � � : $ ! � � 2 � 2 0 1 E

! � � 2 � 2 0 1 : $ � E

H ? I EH ? I E �

� "

� � % . ; � � � � � � < , � � - , � � , � � < . � � < , � �� � � � < � � , � � * � . � �

� - � � % . ; � , . - � - - � � � � � % � , < � : � � � ; . , . - � . � , � < � ; � � � � < � � � �

� : ; . . � � � � � , . < � � � , � , . < � � �

� � � ; % ; < � � � < � � ; < , � ; < - � , � � � , � - : ; �

. - , � - � � ; : , � � < � - � � � � , : � : � � , � % ; : ; � , . , � � , � � � � � : - < � � � � � � � �

, � � - : � � ; � � , � � � � : ; � , � � � � - � � � � � � � � � , � , . < � � � � � ; . � � � � ; � . � �

� � - � � � � � , % . ; � , . - � � � - � , � < � ; � . � , � , � � < . � � < - � ; � . � , � � � ��

� � � � � � � � � - � � � � : � - . ; � � - � � ; � � � � < � , < � < � � � � , � � � , � � % . ; � , �

. - � � � < � � - � � � � � , � ; � - � . , � � , � � � � � : - < � � � � � � � � � � � � � , : . � � � �

� � � ; , � - � � � � � � � < � , < � < ; � � ; � ∑ �� � � � � � �

! < ; � � , . � ; � � . � � ; . ; . ; � � , : � � � � , . - - � � ; � . � � � , � � < . � � < � �

� : �

. ; � � � � � . � � , < . � � � - < � ; � - - � � � � � , < � � < � � � � � , � . , < ; � � - % . ; � , . -

� , � � * � . � � - - � � � � < � � � � � - � ; < � � � : � � ; : � � � � . � � , < . � � � , � � * � . � �

� - - , � < � � . , � : ; � . � , � - < � ; � - - � � � � : ; � - � � � : , � � < � - � � � ; % � � ; , < �

� � � � � - � �

� � - � � � . ; % � � ; , � � � � & � $ � � 6 % � � & - ' � � " � � # $ % & � � ' � � � % � � � �

, < � � � � < - � < � � � � � � , � � �

� : , � � � < � - < � < � � � � < � � , � � < � � , � , � � < � �

� , � � * � . � � - � � � � - � � � � � � ; � - < � ; � - � � ; % � � ; , < � . , � : ; � . � � � - � � : �

. , � � � � � � � � ; � < - � < � � � � , � , � � < � ; �

� � � < � � � � , � , � � < � � * � . � � - � � � ; : ; - % � � < � � � � � < � � � ; ; , <

� . � , � < � � � � . � , < � � - % � � � , � , . < � � . ; � � - � , � ��- - - � − � ��

� � � �

� < � � < � � � � & � $ � � 6 % � � & # � 6 � ' � 5 6 7 $ # � � " 1 2 � & 3 4 � 5� , � - : � � � � � �

: � � � � ; % � - � . � � % . ; � , . - � � - � - % � . � , < � ; � , � � < . � < � � � � < -

� � � � � � � : � � � � ; � . � � * � . � � - - � - % , � , � < � ; � � ; � . ; � � ��

� � �

� ��

��

� " � � # $ % & � � 6 # 6 � � 6 3 % ' � � � �

� - � - � � � ; � - � < , * � . � � - - - % �� : � � � < � � � � � � � � � � �� � < ; � � � <

� � � � � � ; < � � , � * � . � � - � � ��

� � � � � � � � 0 � � , , - � � . � , < � , � � � ; . - , � , � , � < � � � � � � � � � � �

< � < � � < � � � � < � � � : � � � � � � : � � � � ; % � - � . � � � � , � , . < � � � - � � . �

, < � ; % . ; � , . - � � � - � � � � � � ; � . � � * � . � � - - � � � � < . , % ;

� - - � � � - � � � ; � . � � - � , � - � - . ; � - � � ; � . � � , � . � � ; � � , : , � � . . � � - . ;

�- - - � − ��

� "

"

� � < � � � � - - - ��� � � � � - % � � � � . ; � � � , : � � ; < , � � . � < � � ; % � - �

. � � � � , � , . < � � � . � , < � ; �

� ; � � < � - � � � � - � , < � � : , � < � - � � � � � -

� � : � - � � � - � � � , � , . - - � � , � , . < � � � � � - - - ��� � � . � , < � � � � , � - � � , � � ,

� � � � � � � * � . � � - - � - % � . - � , � ; � � . � � � , � , � < � ; ��

� � � ; % ; < � � � < �

� , : . , , � � � - � , < � � : , � < � - � � � , : . , . - , � � � � , � � - % � � . , � � , � �

� � : - < � � � � � � � � �

� � � � � � � � � % . ; � - � � , � , % � � � � � - � , < � � � � , � , . < � � � � � , : � � ; < , � �

. � < - � : � � � < � � � � � , � * � . � � - � ; � . ; � � - � ; � � � - - - � � �

� % ; � -

< � � � . , � . � � � < � � � � � ; � , � * � . � � - - � � � � � - � , < � � : , � < � - � � � � < �

� � , � � � � < - � � � , . - < � � < � � . � < � � : � � � < � . . � � � � . � - < , � � ; �

. � � � � � � �� � - - - � � � �−

� � � � - � � - , � � � � ; : ; � < � � : . � � , � � ,� � � �

< � : � � < � � � < � , � � , < � � - � , � . � � � � � - - - � � : , � < � - � �� ! < � ,

� � � � � � ; : � ; < � � � � . � % ; � - ; < � � ; � � - � � � ; � 〈 �� !〉 � : � � � < � � � �

� ��� � � ����

� � � , : . , , � ; � , . < � � � , � - , � � � ; � ; < � - � ; < � � ; � � � �

< - , � � - * - � - � � � ; . . � � � - ! � - � , , < � , � � < . � < � � � � � � - � 6 � ! � � � � < � � �

� , : . , , % . ; � , . - , � � < � , < < � � � � , � � � � , . ; � � � � � ; � . � � � �� ; : � � , �

� � � � � � � � � ; , � � � , . � � � � � : � ; � � � � � ; < � � . , , � � � � � � � , � - � � , < �

� � , . � � � ! � �! < ; % ; : ; � ; � � � ; % � � ; , < � � < � . ; � ; : � � : � � ; � � , � � � � � , � , . < � � -

� � : - < � � , : . , � � � � � � � � � � , � ; � - � �� � < � � � � , . � , : � � � � � � � - � , , <

� � � � � � � * * - � - , . < ; % ; � � � . , . - � , � < ; � � - � � �

� "

#

� � � ; % ; < � ; . ; � � � - � . � , � < � , � : , . - , : � � � � - % � � � � . � � � � � ,

: � � ; < , � � . � < - � � , � ; � - � : � � ; � � , . - � � � - � ; - � : ; � , . - ; . , < � � � � � : �

: � � ; � � , . - � � ; � � � � , : � : � � , � % ; : ; � , � �

� $ � ; % ; . - , �

0 � : , � � � , : < ; � � < � , � , � � < � � � � : , � � - � ; � : � � ; � � ,

. - - � : ; � , . - � � , � , . < � � � < ; � � - � ; , < � � � - � � � � - � � � � � , � < , �

, � , � % . ; � , . - , � � � � � ; . , . ; � : � < � � , � � : � � � . - � ; - � - . , : � � : � < : �

� � . � ; � - � ; �0 � : , � . ; % � � ; < � � < � � � . � � , . - , � < � � � . � � , . - , � � �! �

� . � � - � � � � � - % � � : � < � � - � � . , < � � % ; � - - � � < - � < ��

� � � � , , � -

� � : , � < � - � � � - � , � . � � ; � . � � - � � � , � � � � - � , : � - � < � � � . � � , . - �

� � � � - � � � � , � , . < � � �� � - � � 6 � �! � , � � < . � < � � < � � � . � � , . - . , � � ,

� � � � : - < � � ��

� < ; � � � � � � � , : - < � % ; < � � � . � � , . - � - � , : � � ; � . � � -

Page 113: Программирование: теоремы и задачи

� � � ��

� � � � 6 7 % $ � � & � � ' & # � � 6 7 $ � � " � � # $ % & � �

� � , � , . < ; � - �

� - � - � � , � . , � � < � � � , % . ; � , . - , � - % � . � , < � ; - � � � �

< � - � . ; � % ; . . � , . - � � � , � ; � - - �

� . - - : � < � � � - � � � � : � � ; � � , . - , �

� � � � , � � - � � : ; � , . - , � : � � ; � � , . - , � � � � � , � � - � � : ; � , . - , � � � �� < � � � . � � , . - � � � - � � : < � , : � : � � ; � � , � � � � � , � , . < � � - � , : � � � -

� - % ; . - � � � � � , � � ; � - : � � : ; � , . - � � � � � - < , � � . � � � � � � < � � � � � � , , - �

� - � � . , � � , � � � � : - < � - � ; � � , � , . < � � � � ; � . � � � ��

� , � , � � � � - � , : � � � � - � , � � � . - � , � ; � � . � � , � , � < � �

� � , � - : . � � : �

� � � � � � � . , � . � � � . � , < � � - � , � , � < � � � , � * � . � � - � � � < � � � ; ;

� � - � � � �� � � , < � . - � , � ; � � . � � �

� : . ; � � � < � < � � - � , � � � ; � < -

� , � � � < � � � - % � , . - � , � � � , % , . � : � % ; � � � - . ; . - � � � ; � . � � * � . � � - -

- % � < � � � , � , � < � ; . � , . � ; - � � � - � � � � , � , . < � � � � � < � � � � � ; � . � � -

� � � � , � , . < � � � � . � , < � , ��

� , � - � � � � , � , � , � � % � � � - < � < ; � � �

� ; - � � < � . - � ; � � � � � , � - � � � ; . - . , < � , � � , < & �0 � � , , � � ; � < - � . � , � � - � , � � � . - � , � ; � � . � � , � , � < � � � � � < � � < �

� � < � � , . � � � � � � � � . , � � . � � ; � � , � � ; - � , � - � � � . < � � � � - � �

,

� , %

Z� � � � � � % . ; � ; , � � . � , < � � � � � , < � � � � � � � < � � � � � : � � � � � < � , �

{ ��

�� - - - � � − �} � ; � - * � , < - � , � - , � � , � ; � - - � � < � � � . � , < � , � � � � �

. � < � � � � : � � � � �

$ . - � , � ; � � . � , , � , � < � � � � � ; % � � < � , � - . , �

. � , * � . � � - � . ; � � . ;

Z

�� � % . ; � , . - � - �

Z� �

0 � � , , � � : � � � . � � � � < �

� � � - - - � � � � � � � - % � � � � . � , � � , � , . < �

Z� � � ; � � < � - � � < � � � ; , . - ,

� � 〈 � � - - - � � 〉 7→ � � � � � - - - � � � � � -

� � � � � � � ; , � , � , � < � � - % �� � < � � � ; , . - �

Z

�� → Z� � ; � ; � , < � - % � � ; .

. � , . ; � � � ; � - 〈 � � - - - � � 〉 �

� "

� � � ; % ; < � � � < � � < � , � , � < � � � � , < � . - � , � ; � � . � � �

� $ � ; % ; . - , �� � < � � - � � ; % � - � . � , < � � � - � � � < � ; . < � ;

Z

�� �

� ; � � � ;

� , � � < . � < � < � � � � � < � � � � ; � . � � * � . � � - � . ; � � � - . - � ; , < . ; . - � � : -

. ; � � � � , % . ; � , . - � � � � - � - � � � ; � - � � ; � � � ; � , � � < . � < � < � � � � � < � � .

� ; � , . . � � � . ; - � � ; % . � < - � − � < � , < : ; � < < ; � - � � < � , � : , . - , � �

� � < � � � . , . � � , � � � � , � < � � � < � � : ; � , % . ; � , . - � � � ; � . � � � * � . � � - �

. ; � ; . ; . � � � ; � . � � , � � < . � ��

� � , : � � � , � % ; : ; � , � . � , < � �

B

� { ��

�} � ; � ; < � - � ; , < � ; � � . �

, < � � � � � , < � � � � � � : � � � ��

� "

� , � , � < � � � , � � - . , � . � � � < � � � ; , . - � - %

B

� �B

� � � , <

� . - � , � ; � � . � � � �

� � : < � , . . � , � , � - � � � ; . - � - : , - . , � - : ; . . � � � ; % � � ; � < � � � , % . �

� - � � , : � � � , � - < � ; � - - � ; � ; % ; � ��

� ; � ; . � * � , � � �

� � < � � � � �

� ��

��

� " � � # $ % & � � 6 # 6 � � 6 3 % ' � � � �

< - � . ; � - ; < � � � � � � ; � � � � � � < � � ; � � . ; � � - � ; � ; � � � � � - . < � � � � � , � ;

< � � � < , � < , � . � . , � � < - � � � ; . - < � � - � � � , � � � ; � - � � . � � � � � � * � � � �

� � , : � ; � ; , < � � < � � - < � < ; � � � � � � ; < � . , � � < � � � , � - . ; � � � * � . � � - �

� � - - - �

� � � < � � � ; ; � � - � � � � - , � � � ; � �� - - - � ��

� � ; - � , - % � � -

< � � � � � � - � � , � - < � � ; � . � � - . � � � � � � � � , < , � � � � < � � � , � � � < % . ;

� , . - , � � ; � � � < � * � . � � - - . ; � � � ; . ; � ; � � � < � � � ; � - � � . � � � � � � * � � � , �

� , � , � � � � - � � - � . . � � < , < . ; � � ; � - � � . � < � � � � � * � � � � < ; � � � � � � � � ,

� - < � � � < � % . ; � , . - � , � * � . � � - � . ; � � � ; . ; � < � � � � � � * � � � , � � � ; : ; � <

. ; � , < ; � % ; . < � , , : - . - � ; � - � ! < � < < , < � � , < . , % ; � , < - < � . , � � < �

� � � � � - � � � � . � � , � � ; � - � � . � , � � � � * � � � � � � : � < � : � � � , . � ��

Page 114: Программирование: теоремы и задачи

��

� � � � � � � � � � �

� � � � � � ��

� � � � � ��

� � � � � � � � � � � � �

� � � � � �

� ' � � � � � � + � � � � * � � � � � � �

� � � * � � � � ' � � � �

� � � � � � � � � � � � � � � � � � � � � � � �

� ; � - � , � < � � � � �� % . , � � � � � , : � � : � , < � , � � - � � , � � � � , � � - � � � ; � �

� � , � � � � : � , : � � � - , < � � � - �

� % � ; : � � - % � < - � < � � , � � � � � , : � � � � : � ,

< � , � � - - < ; � : ; � , , �

� � � � � , . . � � � ; � < - . � � � � � � � � , � � : , < � � −

< � � , � � . ; % � � ; � < � � � . � � : � � - � . � � � � � � $ # ' � � - . � � < � � � � . ; % � � ; � <

3 # � & � ' � $ � ; : � � � , � � - . � , < � : � ; 6 - & % : � , � , � � - . � � � � � < � � � ,

- : � < < � , � � - � � � � $ - 5 - � � % $ - 5�

$ � � � � � , � � - . � � � � � � , � � � . �

, < � , : - . < � , . . � � # 7 � 4 �� � < � � � � � ; . � . , � � < � � � , � � . , � . � , � . � , < � � � , � � - . � � � . � � � : � �

- � . � � � : , � , � ; � � : , � ; � , , � � , < , � ; : � � � , � � - . � � - � , � , � � � , :

� � � �� � < � . ; � ; : � � � , � � - . , � < � � � � . � , < � ; . ; � - ; . � % . ; � , . - , * - �

- � � � ; . . � � � < - � ; < � , < � % ; : ; . � � < � � � ; , . - , � . � , < � ; � , � � - .

� � . � , < � � % . ; � , . - � < - � ; ��

� � � � < � � � � � � - < � � � : , � . ; % � � ; < � : ,

� , � � � �

� . � , < � � � , � : , � , � � , � � � � % . ; � - � � � � � ��

� , � � � - � . � , � � � , : , � , . - , �� � � , . , � � < � , : , � , � � � ; % � - � ; , < . ;

< � - � ; < - � � � � , . � . , � � - � � � � , < � � - % � � � , � � , - � � ; � � , � � : : , � , � �

� � < � � � , � � � � < � � < � � � < � � - � �

! < � � ; % � - , . - , � < ; . ; � � - � ; , < � % ; - �

. � � : . � % . ; � . � , � � < � , < < � - , � , : � � . � , < � � � . , � � < � � : , � , � � , �

- � � � - % � , : , . - , � × � � � � � × � � � � ��

� � � % . ; � - � � , � , % � � � � � � < � ,

: , � , � � � � � . � . ; � - ; < �

� � � � � � { ! � � � } � × � � � � � × � � � � �-

" #�

"�

� � � � 6 7 % $ � � & � � ' & # � � 6 7 $ 6 � # ' # $ � � � � � � $ � � $ � � %

� � � � � � � � � � & ' ( ) *

� - � - � � , � . , � � < � � � , : , � , � � �

� � � ; : � � , � � � , � � - . � � � � � ,

: , � , . � , � � � $ # � � # � � � � � $ # � , � � � � . � , � � - . � � - � , , � � � � < � � � - � �

� � % $ # � � # � � � � � $ # � � ; � � � � . � , � � - . � � - � , , � � � � < � � � - � - � # � � � 1

� � $ # 6 3 # � & � ' $ � � , � � - . ; � - � , , � � � < � � � - � �q@

@@

@

BB

BB

��

��

����

� � � , . �

� , � � , � � ; � � ,

+ , � � , - � � ; � � , � � : : , � , � � � , � � - . � � � � � � < � � < � � � < � � - � ; � � : : ,

� , � � � � � . , � � � � , � : ; . , � � < � � : , � - < � � � � ; � . , � � , � , � � �

, - 1

6 # 7 # 5 � � : : , � , � ; � � : , � � - < ; < � � ; � - � ; � � . � � : � - . � � , � - � - - - � , � �

� , � � - . � � � � < � � � � � - � � � . � : � � , � � � � � � < ; : , � , � ; - . � : . � � �

� � � . � ; � . ; , : - . - � , � � � � < ; � � < � � � : , � , � ; � . � � � � �

/ 0 � � � � � � � ' � � � � � � � � �

� � < � . ; � . � , < � , . . ; � , . - � < - � ; * - � - � � � ; . � � � : � � �� ; . � � � �

: , � , � � 1 � # � 2 � # 3 � & & - ' � , � - � � � � � . , . � < ; � � , � � � < � � � : � � � � � �

� , � � - . � � � , � � � , < � - � , � � , � � � � � : : , � , � , � , . � � , � � � , < � - � � � ; � ,

� � � , < � - � , � � � ; � � � � � : : , � , � , � � � � � , � � � , < � - � � �

r@@

@@

@

BB

BBB

��

��

�����

� � � �

� #4 � 4 � 4 � � � ; . ; < � � � < � � � � � � : � � , . . � � : , � , � , � , � � � , < � - � ; . � - �

. � �� $ � ; . ; . - , � � . : � � � - � � � � � < , : , � , � ; ��

� � � � ( ) * � � � � � � 5 � 6 � ( ) � ( 0 5 7 � 8 � � � � � � � �

9 ; : � , : , � , � � � � : , � � - < ; < � � � , : < ; � � , . - , � � . � , < � ; � , � � �

� , < � � . ; , � � � , � � - . ; � �

� � - � < � � � : . � - < � , � . � , < � � � � , < - � , < �

� ; . � - � . � , � � , : < ; � � , . - �

Page 115: Программирование: теоремы и задачи

� � � " #�

� � � � $ � 2 � � � % � % & 6 � � # $ % & & - � � � � � $ � 2

0 � ; � � : ; � � � � � : � � , . . � < - � ; : � � � � , � , . < � � , < � , � � � � . ; � < -

� � � � , < � � : , � , � , � � � - : � � ; � � � < � � , � � - . � - � ; � . - � , � < , � �

� < � < ; � . ; � � : - < � � � , � , . < � , � ; , < � - : < - , � � . ; � , � � - � - . ; � � ; � � �

6

@@I

���

� � � �

� ; � ; � � � � . - : � - � ; � � � � < � � � � � ; � - � � � � . � - � � � � . ; � � - < � � < �

< ; � � � � � , � , . < � , , < � � � - � � . ; � : � < � , < � � � � � < � � � � � . : � � , . � � < � �

� � : � : ; � , , � � � � , : � � � ; � ; , � � � < � . ; . . ; � , . - � < - � ; . ; : ; . � �

� : � � � - � ; � ; < � - � ; , � < � � � � � � � � � : � � , . . � , : , � , � � �

� � * � � � � � � � � � � � � � � 0 � � � * 5 5 �

� � . � � � � � � � � � � < ; � - < � � , � � - . � � � � . � � � : � � - � . � � � : , � , � ;

� - � ; � - ��

��

�� - - - � - < ; � � < � � , � � � � . � , < � � � � � � ; � � � � . � , < �

� � � � � - � � ; . - < � � � � , < � - � � ; - � , � A � � /� � �

��

� : . ; � � � < � < � � � �

. , � � � . � � , . � � � � � � � � � < � ; < - < � , < � . ; � � ; . , . - , � � < � � � ; � ; . - �

� � � � . � � : � � - � . � � : , � , � , �0 � � , , � � � . � � , . < ; � � � � � � � �� � , : � � < � - � ; - � ;

8 5 9 : 5 ; ; 5 < � A > > ? � B C D E9 H C � : ; � # % � : 5 ; ; 5 < � A > > ? � B C => > ? E

) � � ; � - � ; � � . � , � � . � � . � , � - � � � , � � - . : , � , � ; � - � , � , � , . . � �

� � � � � ,� � ) � 9 ; : ; � , � � - . ; � � ; . - � � � � : , � , � ; � � : , < - � , < � . � � , � �

� - � � � < / : � ) � � ; . . � , � , � � - . � � � : � < - � , < � � ; . . � , . � � , � ; �� � � , < � ;

� � , � � - . , . � � , � � � � � ; � . ; � A � � � � �

9 � � , . � - � , , < . � � , � � � � � �

� - � , � � - . ; . � � , � � � � - � , , < � . � � , � � < � - � . � � , � ; � ; � . � � � � � �

- � � � + � � � � � � < � < < � � � � - � � . � � � � � � < � , < < � � , < � - � � ,�

� . ; � �

� - � . � � � � � ; . � � . . ; � , . - , � � � � � , � � < � , < < � � , < � � < � � � : , � , � � �

� � � � ; . , . - : , � , � ; - � � � � . � , < � - � � � ; < � � ; - � ; � : � < , � � �

� � < � � � , � � � � : . � . , � � � < . � � , � ; � - � , � � - . � � . . ; � , . - � A � � � �

� , . � ; . � - � . � � � ; � � � : , < � : � � . � � � < � � � � , � � � � : . � , � - � ; � � � - � �

. ; . � � � - � � � � , � � � , � � ; . - < � � , � - ; � � . � � � , � , � , . . � � � � ,� � ) �

; � , : � � � , , . ; � � � � � : . � , � - � � � � ; . - < � � � � � � � < ; � � < � � � � � :

. � � - � ;

� B

� � � � � B

� � � � � � � � � B � � �

" #�

"�

� � � � 6 7 % $ � � & � � ' & # � � 6 7 $ 6 � # ' # $ � � � � � � $ � � $ � � .

� � � � � , : . , � � � � � � : . � � � � - � ; � . . ; � , . - , � � � � � � ; � . � ,�

� ; � , .

< � � � � , � . . ; � ; , < � � < � � � � � : . � � � - , � � � � � � , . , < �

� � � � � � � � 4 � � - � � � � . � � ; � - : � � . � � ; . - � � � � : . � � � , � � - . � ;

- � � � � . � � � � . , � . � � < , � , � � , � � � � � . � � � � � - � � � � . � � ; < � � ;

- � � � � + ��

� � , < � . . ; � , . - , � � � . . ; � ; � � , � � � < � < < � - , � , � � - . � � � � . � � �

� � � � � � � � � � . � � ; < � � � � � � : � � � - � � - � � � � . , /� � ) � < � � � � � : � , � �

. � < � � < � � � � : , � � � , < � , - � � � � . � � ; < � � � . < ; . < � ) � � � � , �

� #4 � 4 � 4 � � < ; � - < � � � � � � ; � � � � � � � , : , � � � � � � � : , � - < � - � � ,

� , . < � � ( � � � � � : � � , . . � � : , � , � , � � ; . - � � � < ; � � � ; � < � � � � � � < � � � -

; . � � �� � � � � � 4

� C ; B B � $ ? F 9 9 � % H ? J H # � ?

> > � � � � 3 � � � � � � 3

H ? I H 9 G H J H # � ?

: $ ; B B � E

+ 3 � 2 � � 3 � � : 0 6 � � 6 < � � 0 2 � � 3 4 � � � 3 9 3 � � 2 � � � � 6 0 =

� 0 � � � � � 2 � 6 7 0 � � � = ,

& % � 9 H ! ! � ' 8 5 9 � � " 5 ? I ! 9 H C � � � ' ( ? F 9 9 " " B ;

! ! � ( 8 5 9 � � " 5 ? I ! ; � # % � � � ' ( ? F 9 9 " " I B J H # � ?

� C � ' 8 5 9 � � � % H ? J H # � ? + 9 H C � � � ' ( ? F 9 9 ,

: $ 9 H C � � � E

H ? I H 9 G H J H # � ? + � ( 8 5 9 � � : ; � # % � � � ' ( ? F 9 9 ,

: $ ; � # % � � � E

H ? I EH ? I E

+ � 3 1 0 � $ 8 5 9 � � : � 3 1 0 � 0 6 � 6 2 � � 2 � � � � 2 � ,

> > 0 2 � $ ! � $ 8 5 9 � � "

H ? I E �

� #4 � 4 " 4 $ � � � < - < � � , � , . - , � - � � � � . � � , : � � � - � < � � � � � ; � - � - �

� � � ; < � � � � , : , � , . - � ; - � ; � A � � : � � ; � - � � , � � � . � � , � � � ) � � � - � �

� � - � � A � � ) � � � � � � �

� � � � � � 48 5 9 � ? F 9 9 � : $ � E

: $ ; B B � E

& % � 9 H � ' ( 8 5 9 � � I B J H # � ?

� C � ' 8 5 9 � � � % H ? J H # � ?

: $ 9 H C � � � E

H ? I H 9 G H J H # � ?

Page 116: Программирование: теоремы и задачи

� � � " #�

� � � � $ � 2 � � � % � % & 6 � � # $ % & & - � � � � � $ � 2

: $ ; � # % � � � E

H ? I EH ? I E

> > 0 2 � : ! ' ( ? F 9 9 " > �

� #4 � 4 #4 � � < ; � - < � � � � � � ; � � � : � � ; � � , . - � � , � , . < ; � � � . � , < � � �

� � , : < ; � � , . . � , � � � � : � � , . . � � : , � , � � � , � - � � , � , . < � � , , < � � . -

� , � � : , � ; < � . , . ; : � � �

� � � � � � 4 � � � , : , � - � � � � � , : � � � � � � � � A � � � � ) � � � � : ; �

� � � � � � � : . � , . , � � � � , , . � � , � � � � � - � � � - � � < � , < < � � � � - �

� � � ; . � � � � � � , � < - � � � � � � � - � � � � � � : . � � � - , � �

/ ; B � H I F ; H # H �� C ; H H ! 8 5 ; � : � ? � H # H ; " EJ H # � ?+ C ; H H ' ( ? F 9 9 ,

� : $ C ; H H EC ; H H : $ 9 H C � � C ; H H � E

H ? I E

� , � - � � � � . � � ; . - , � � � � � � ; � � ; � � - � � � , < ; , < < ; � � � � - : �

� C ; B B � $ ? F 9 9 � % H ? J H # � ?

# H � � C ; H H ! ; B B � " E9 H C � � ; B B � � : $ ? F 9 9 E ; � # % � � ; B B � � : $ ? F 9 9 E

8 5 9 � ; B B � � : $ � E

H ? I H 9 G H J H # � ?

: $ ; B B � E

+ 3 � 2 � � 3 � � : 0 6 � � 0 6 4 � 0 1 � 2 3 4 � 7 � � � � 6 0 = � � 0 � � � � � 2 � 6

7 0 � � � = 2 ,

& % � 9 H ! ! � ' 8 5 9 � � " 5 ? I ! 9 H C � � � ' ( ? F 9 9 " " B ;

! ! � ( 8 5 9 � � " 5 ? I ! ; � # % � � � ' ( ? F 9 9 " " I B J H # � ?

� C � ' 8 5 9 � � � % H ? J H # � ?

: $ 9 H C � � � E

H ? I H 9 G H J H # � ? + � ( 8 5 9 � � ,

: $ ; � # % � � � E

H ? I EH ? I E

� C � ' ( 8 5 9 � � � % H ? J H # � ? + � � � 2 � � � � 2 � ,

# H �� C ; H H ! � " E9 H C � � � � : $ ? F 9 9 E ; � # % � � � � : $ ? F 9 9 E

8 5 9 � � � : $ � E

� C � ' 8 5 9 � � � % H ? J H # � ?9 H C � � � : $ � E

H ? I H 9 G H J H # � ? + � ( 8 5 9 � � ,

" #�

"�

� � � � 6 7 % $ � � & � � ' & # � � 6 7 $ 6 � # ' # $ � � � � � � $ � � $ � � �

; � # % � � � : $ � E

H ? I EH ? I E

H ? I E �

� #4 � 4 � 4 � � < ; � - < � � � � � � ; � � � � : ; � , . - � � , � , . < ; � - . � . � , < � ; �

� � , : < ; � � , . . � � � � � � � : � � , . . � � : , � , � � � , � - , � � < ; � . , < � . - � , � � : ,

� ; < � . , . ; : � � �

� � � � � � 4� C ; B B � $ ? F 9 9 � % H ? J H # � ?

+ � � � � 2 0 � � 6 0: � 3 9 � ? 0 � � � � 4 � � � � � 0 ,

H ? I H 9 G H J H # � ?

: $ ; B B � E

+ 0 6 � � 0 6 4 � � � � 3 4 � 3 > � 0 � � � � � 2 � 6 7 0 � � � = 2 E � 0 6 7 0 � 4 7 �

� 0 = 0 � � � 0 � � 1 0 2 � 4 3 > = � � � � 3 2 0 � � : 2 2 � � � =

� � � � = � � � ; � C 5 � % H ; : A > > ? 3 I � ; H � � � B ? : ! 9: ; " E

� 0 � � � � � 3 2 � � = � 7 0 3 � 2 � � 3 � � : � 6 � 3 � � 7 0 � � � 4: 0 C 5 � % H ;

@ � ? 0 0 � �: � I � ; H � � � B ? � � 2 � 0 9 3 � 3 ; 2 > � 2 3 6 3 = 0 6 3 0

0 ? 0: � � 2 ; = 3 � 3 � � � 2 ; = 6 ; � 0 = < 2 � < � 6 < ,

& % � 9 H ! ! � ' 8 5 9 � � " 5 ? I ! 9 H C � � � ' ( ? F 9 9 " " B ;

! ! � ( 8 5 9 � � " 5 ? I ! ; � # % � � � ' ( ? F 9 9 " " I B J H # � ?

� C � ' 8 5 9 � � � % H ? J H # � ?C 5 � % H ; : $ E I � ; H � � � B ? : $ 9 E

: $ 9 H C � � � E

H ? I H 9 G H J H # � ? + � ( 8 5 9 � � ,

C 5 � % H ; : $ E I � ; H � � � B ? : $ ; E

: $ ; � # % � � � E

H ? I EH ? I E

+ � $ 8 5 9 � � 3 � 3 � � � 2 � � � � 2 � ,

� C � $ 8 5 9 � � � % H ? J H # � ?

> > � � � � � � 3 � 2 � � # 3 � ; 6 0 � 0 = C 5 � % H ; 3

� � � � � 2 � � � 3 � = I � ; H � � � B ?

H ? I EH ? I E

$ : ; � , . - , � , � � - . � - � � � � . � , < � � � � , : � � �

/ ; B � H I F ; H * 5 $ H� C ; H H ! � : � ? � H # H ; " EJ H # � ?9 H C � � � � : $ C ; H H EC ; H H : $ � E

H ? I E

Page 117: Программирование: теоремы и задачи

� � � " #�

� � � � $ � 2 � � � % � % & 6 � � # $ % & & - � � � � � $ � 2

� . ; � � � � � ; , < � - � � � � � - � � � � � � : . � � � � � - � : ; � , . - - � ; . � - � ; � <

� � � ; � . ; � - - � � < - � < . ; � - � - - � - � < � < < � - � . � � , � � � : ; � , � � �

� , � � - . � �� C ! 9 H C � � � $ ? F 9 9 " 5 ? I ! ; � # % � � � $ ? F 9 9 " � % H ? J H # � ?

+ @ � 3 6 : > �> � � 3 = � � 6 ; � 0 2 � ,

* 5 $ H� C ; H H ! " E

� C $ ; B B � � % H ? J H # � ?

; B B � : $ ? F 9 9 E

H ? I H 9 G H � C I � ; H � � � B ? $ 9 � % H ? J H # � ?9 H C � � C 5 � % H ; � : $ ? F 9 9 E

H ? I H 9 G H J H # � ? + I � ; H � � � B ? $ ; ,

; � # % � � C 5 � % H ; � : $ ? F 9 9 E

H ? I EH ? I H 9 G H � C ! 9 H C � � � $ ? F 9 9 " 5 ? I ! ; � # % � � � ' ( ? F 9 9 " � % H ? J H # � ?

+ � � � � < � 6 <: � ; � # % � � � > � � 3 = � � = � 6 0 ,

* 5 $ H� C ; H H ! " E

� C $ ; B B � � % H ? J H # � ?

; B B � : $ ; � # % � � � E

H ? I H 9 G H � C I � ; H � � � B ? $ 9 � % H ? J H # � ?9 H C � � C 5 � % H ; � : $ ; � # % � � � E

H ? I H 9 G H J H # � ? + I � ; H � � � B ? $ ; ,

; � # % � � C 5 � % H ; � : $ ; � # % � � � E

H ? I EH ? I H 9 G H � C ! 9 H C � � � ' ( ? F 9 9 " 5 ? I ! ; � # % � � � $ ? F 9 9 " � % H ? J H # � ?

> > 6 3 = = � � 3 9 � 0

H ? I H 9 G H J H # � ? + 9 H C � � � ' ( ? F 9 9: ; � # % � � � ' ( ? F 9 9 ,

> > � � � � 3 4 2 � � # 3 � � 6 � 2 � = < 6 ; � 0 2 4 < = 3

H ? I E

$ : ; � , . - , � , � � - . � : � � � � . � � � � - . , � � . : , � ; < � � � � < � < ; � � . � , �

� � . � � � , : � ; � - < , � � . � � � � , . < � � , � � - . � � � � � � , < � ; . ; � � < � � � � � �

, < . , � � � , : < � , . . � � , : � � � - � � � � � : � , � � . � ; < ; . - � � � , � , . < � � . ;

� � � , < � � � . ; � �

< : $ ; � # % � � � EC 5 � % H ; : $ E I � ; H � � � B ? : $ ; E

+ � � � � 4 C 5 � % H ; 3 I � ; H � � � B ? 0 � 0 6 < 6 < 7 2 � � # 3 � � < ,

& % � 9 H 9 H C � � < � ' ( ? F 9 9 I B J H # � ?C 5 � % H ; : $ < E I � ; H � � � B ? : $ 9 E

< : $ 9 H C � � < � E

H ? I E+ 8 5 9 � < � @ = 3 � 3 = � � 4 � � < 3 > � 0 = � 0 7:

1 0 � 4 # 3 � 8 5 9 � � :< � � 3 = � � � � 2 0 ? 0 6 ; � � ,

" #�

"�

� � � � 6 7 % $ � � & � � ' & # � � 6 7 $ 6 � # ' # $ � � � � � � $ � � $ � � �

8 5 9 � � : $ 8 5 9 � < � E

> > � � � � 3 4 2 � � # 3 � � < ! 7 � 7 � � � � < 4 2 � � # 3 � �: � 7 0 0 � 0 � �

� � 2 0 ? 0 6 ; � �: = ; � � � > � � � = " �

� #4 � 4 � 4 $ � � � < - < � � � � � � ; � � � � : ; � , . - � . ; � , < - � � � < � . , � � < � � � ,

� � � ; - . ; � � - � , � � � , � � � , : � ; - . � , < � � � � � � � . � � � � , : - . - < � � �

� #4 � 4 �4 � � � � � . � � ; < � � � � � : � � , . . � , : , � , � � : � � � , : < ; � � , . -

* � . � � - � � � � � ; < � � � � , : , � , . - � � < � � � � � � � . , � . � , � . � , < � ; . . ; � ,

. - � < - � ; ( � ; . . ; � , . - - � , � < . , � � < � � � � < - � ��

� � , � ; � - - � � � � - � , . - ,

. . ; � , . - . ; : ; . . � � ; � � � � , . < , � - . � , . , . - , . . ; � , . - . ; : ; . . � � ; � � � � , .

< , � : � � � � , : , � , . - , * � . � � - - . ; : ; . . � � ; � � � � , . < , � - � � � � , . - , � � , � , . < ;

- . � � � ; < - � � � , : , � , . - * � . � � - - �

� � � � � � 4 � , � ; , � � ; � � ; . � � , � : � � ; � - � , � � � : - . � ; - �

C F ? � �� 8 5 9 : 5 ; ; 5 < � A > > ? � B C � E

, � - � A � � � � � � � � ) � � A � � � � � � � < � . . ; � , . - , � � ; . - � � � * � . � � - - . ; �

� ; � . � � � �

� #4 � 4 � 4 � � , : � � � � - � � � < � . , � � � � : - � � � � , < � < ; � , � < � � - � ; < �

� � � � � , � , . < � . � , < � ; � � � � : � , � � . � ; < ; . - � � � � - � � � � � � - � , < � �

: , � < � - � : � � . � � � < � . , � � � , , � · � � � < ; : , � , � ; � �

9 ; � � � : � � � � . -

< , � � . � � - . * � � � ; � - � . ; : � � � ; . - < � � � , � � - . ; � : , � , � ;

� � � � � � 4 � � ; : � � � , � � - . , � � : , � � � ; . - < � � - � � � , � , � � � < � � � � � �

� � � ; � � , . - , - - � � � � , . - , � , � � - . � < � , � � , < � � � � , � � - - � - � � . ; � � < -

� < � � � . � � < � � � , � � - . , �

� � � � � , , � � - � ; � �� � , � � - . � � � : : , �

- � ; , < < ; � � � - . � ; � - ; . < � - � � � ; � , � � - . ; � � , < � � � , � � - . � �

� � : : , � , � ; � � � . , � � � . : , � - � � � , � , � , . . � , � � �� � � � * � � � � ( ) � * � � ( ) � �

� � � ; : � � - . � � , � ; � - � � � � � , � � - � : � � ; � � , . - - - � � � � , . - � � � � -

� , < � � : , � < � - � . , � � , � � � � : - < � · � � � < ; : , � , � ; � �

� � � � � � . � � � :

< � - , . . � � � : , � , � ; � � � : ; � , � - < � . ; � : . � � � � � < , � � � � < ; � �

� � � : � � � , � - � - . � � ; � . ; � � � ; � - * � � � - � ; � , � � - . �

� : . ; � � : � � � - � �

� � � � � � : , � , � ; � � � � , < � � < � � � � ; . : � � � , � � . ; - � � : � , � � � � ; , � ,

� , � � - . � � � � ; . � � < � , � � - � � � < ; � ; � . ; � - � � � , � � - . �

� ; � � � � - <

� , � - � � , � , . < � � . � , < � ; : � � ; � � � < � � � . � ; < ; � � , � - � - � � �

� ; � � , � � � � : � , �

� � . � : � � ; . ; < � � � : . ; � � � � < � � � - : � � ; � � , . - - � � ,

� , . < � � � � � � � ; � . � � � � � : � , � , : . � � � < ; : , � , � ; � � : , < . , � � � � � ,� � � � � - � � � , � � - . � �

� � - � < � � � � , . � - � � � , : . , � � ; � � � . , � � � � : - � �

: � � � � . - < , � � . � , : , � < � - � � � � : : , � ; . - � � � ; � ; . - � � � ; . . � < - : , � ,

� ; �� � � < � � � � < � - � � , : � � � , � � � . � < , �

Page 118: Программирование: теоремы и задачи

� � " #�

� � � � $ � 2 � � � % � % & 6 � � # $ % & & - � � � � � $ � 2

� � � + � � � � ' � � � � � � � � � ' � � �

� , � , � � . ; . � � ; , < 6 � % � % & 6 � � # $ % & & - ' - � - � � + : , � , � � � � � , < �

- . � � � , < ; < , � , � � < � � � � , < � : ; ��

��

� : , � � � . ; � , � � � � � � - ��

��

+ ; . : -

; � � , � - : � � � � � � , � � � , � � - . � � � � < � � , � � � � - � � ; � � � � � � : : , � , � � , �

� < � � � , � � - . � � < � - � ; � < . , � � � , , � , � . ; ��

� � ; < . � < - � � � � : ; � :

. � � � - . � . � � , � . , < � : � � � � � � , � - � . , < � � � � . ; . � � < � � - < � � ��

� #4 � 4 � 4 � ; � < - � - . - � ; � � . � , - � ; � - � ; � � . � , � � . � � . � , � � � - � , < � �

� , � � - . � � ; � ; . - � � � ; . . � � : , � , � , � � � < � � �

� � � � � � 4 � ; � - � ; � � . � , � - � � � , � � - . � ; � . � � � − ��

� � - � � � � -

. - � ; � � . � , � - � � � , � � - . � < � � � ; � � , � � � � - : , < � � � � - � � � � � � � � � - � �

� < � � : ; � � � �� - � − � �� � � � , � - � � � - � � . ; � � - � � � � � � � � � � �

�� - � � �� � �� - � � � �

� #4 � 4 � 4 � � � ; . ; < � � � < � � ; � ; . - � � � ; . . � , : , � , � � � � , � � - . ; � - - � ,

, < � � � < � . , � � � � � , � � � � � : � . , � � < � � � � � � . < ; . < � � � . , . ; � - � , �

� < � �� � � � � � 4 � . : � � � - , � � � � � , � � � : � � ; . ; < � � � < � �� - � > �� � � : ,

� � � � � � � - � � � � , . � � � ; : � ; < . � � � � � ; � . , . - � � � � � � � < � , < � � �

√ � � � � � ��

� < ; � < � � � � � � . � � ; < � � � , : � : � � , � . ; : ; � , � � �

& � * 7 � � � �

� � � � < - � � � < ; . ; � � - � ; < � � ; � ; . - � � � ; . . � < � : , � , � ; � � � , � � � �

� , . - - � : ; � , . - � � , � , . < � � �� � � < � � � . , � � � � : - � � � ; � - , < � � � , � � � ;

. � � ; . - : , � , � ; � . , � , . � � - , � . � , < � ; � � � , < � � . ; , � � � , � � - . ; � - . ,

. ; � � � ; � � - , � � � � : � � , . . � < - � . � � � � � < � � � � - , � � � � , � � ; � ; . -

� � � ; . . � < - �

� � - � , � . , � � � � � � < ; � - � � � , � � � ; . � � ; . - � �

t����

t

tHHHH

t�

@@

@@

@@

BB

BB

BB�

@@

@@

@@

BBB

BBB

��

��

��

������ �

→�

��

���

������

@@

@@

@@

BB

BB

BB�

��

��

��

������ �

� � < � � , � � - . ; � - � , , < � � ; � � � � � . ; � �

� � � . . ; � - � � , � , . � � , � � , � � :

: , � , � � � , � � - . � � � � , � , . � - � � � , � � , - � � ; � � , � � : : , � , � � � , � � - . � � �

$ � � � : � � , . . � < � : , � , � ; < � , � � , < � � < � � � � � � � � � � � � < � � . , , � ,

" #�

��

� � % � % & 6 � � # $ % & & - � � � � � $ � 2 � � �

: � � ; � � � � � ; . ; < � � � � � ; � � � , < � ; . ; � � , . � � , � � � , < � - . ; � � � � � � , < � ;

. ; � � , . � � , � � � � � � � � , < � - . ; � - < � : � � . � � � � � . � � � - � , � , � < � � �

. , : , � ; < � � �� � � . � < � � � , < � , � � , < � � � � : � � , . . � < � : , � , � ; � � � . , � � �

, � � � , � � � � . � � � � � � � < � � � � � - � � � , � � , - � � ; � � , � � : : , � , � � � �

� � � � ; � � , � � : : , � , � � � �

� � � < � � � � , � � � , : , � , � � � � . � � � , � � � ; . �

� ; < � � � � < � � � , � . , . ; � � � ; � � � � : � � , . . � < - �

� ; � � , � � , � � � ; . � � ; . - ,

. ; . � � � � ' % � - ' � � % $ - ' $ � % $ � & � � ' � � ; � � � � � � � � � � � � � � , < � � , <

- � � , < � - � . � , � � , � � , � � ; � � � � � � � � � � � � , < � - � � � � � � , � � � < � � � , � �

, � � ; � � - � , � � �� � < � � � � � ; � � � � . � � � � � , � � � � . � � � � � , � � , � � : : , � , � � � �

� - � � � , � � , - � � ; � � , � � : : , � , � � � � � � � � ; � � , � � : : , � , � � � �

� � � : ;

� � � � � � � � � � � � ��

s

ss

s

s s

��

��HHH

��

��

QQ

QQ

� �

@@

@@

BB

BB

� @@

@@

BB

BB

��

��

����

��

��

����

→@

@@

@

BBBB

JJ

JJ

����

��

��

� ; � � � , � � � : � � � � < � , < < � � , < : , � , � � � � � . , � � � - � , � � - � � , � � � �

� . ; � - � � ; � � � � � . ; � � : � � � < � � � � � � - � � � � : : , � , � � � , � � - . � � �

; � - � � � � : : , � , � � � , � � - . � � �

� � � < � , < < � � � � , , � � , � � � ; . � � ; . - ,

� � : , � . ; . � � ; < � � # � � " � ' � � % $ - ' $ � % $ � & � � ' � � . ; � � � - � . � � � � , : , � , <

- � � , < � - � . � , , � � � # � � " # � � � $ # � $ � % $ � & � � � �

� #4 � 4 " 4 � ; . � : , � , � � � � ; � ; . - � � � ; . . � , � � : � � � � � � , � � � . � � � �

< � � � � � ; . . - � ; � � � < � ; � . ; � < � , � � , � � , - � � ; � � , � � : : , � , � � � � � .

� ; � ; . - � � � ; . � - - � � � � < � � < � - � ; � < . ; � ��

� � � ; . ; < � � � < � � . � � �

, < � � < � � � , � � ; � , . � � � ; � ; . - � � � ; . . � , � : . - � - . � , < � � � � � � - ; . . � �

� � , � � � ; . � � ; . - � � � � - � � � � � � < ; , � � � < ; . , < � � , . , � - � - � � , . � � - <

. ; ��

� � � � � � 4 � � < � � � � , , . - . � - � � � , < � . ; � � - � , � � � , � � , � � : : , � , � � �

- , � � � � � < ; � ; � . ; ��

� � � : ; � � � < ; � � ; � � � � � � : : , � , � ; � ; � . ; � � ��

� � � . . ; � - � � � � , . � � , � , . � � ; , � � � � ; � � � � � . ; � . � � . ; < , � � . � , < � �

� , � , . � �� ; � � < � - � � , � � , - � � ; � � , � � : : , � , � � � , � � - . � � �

� : . � - . . - �

� � . ; < , � � . � - � , , < � � � < � � � � � ; : � � � � , � � , < - � , < � � � � < � � - � -

� � � � , . � � , � � � < � . , � � , < � < ; � � ; � � � : : , � , � � � ; � ; . - � � � ; . � � �

Page 119: Программирование: теоремы и задачи

� � � " #�

� � � � $ � 2 � � � % � % & 6 � � # $ % & & - � � � � � $ � 2

� � - � � � < ; � , � � � � � � : : , � , � ; � ; � . ; � � � � ; � � ; � � � � � � � < � � � < � , � �

, < � � � � � � , � � ; � � , � � ; � , . - , � � � < ; � � . � � � � � ; � � � � � � ; , < � ; � � , �

� � < � ; � � � � � : < < � - � � � ; � ; � ; . - � � � � - : , � , � ; �

�� @@�

s

s

s

s

@@@

AAA @@@

AAA

����

→ @@@

AAA

���

���

��

��

�� @@s

s

s

s

@@@

AAA J

JJ

J

BB

BB

����

→ @@@

AAA

����

��

��

��@@

@@ ���

s

s

s

s

s s

@@

@@

JJ

JJ

��

��

@@@

AAA

���

���

@@

@@

AA

AA

��

��

����

→@

@@

@

JJ

JJ

AAA

��

��

���

AA

AA

����

� #4 � 4 #4 � � ; � ; . - � � � ; . . � , : , � , � � : � � ; � - � - - � - - . . , � � � : ; � - � -

� - < �

� � � ; . ; < � � � < � � � . � � � < ; . � � - < � � ; � ; . - � � � ; . . � < � � � � �

� � � . , � � � � � - � � � ; � , . - � � � � - � � � - � � - � � . , � � � � � , � � � < � : , � , � ; �

� � � � � � 4 0 � : , � : � � ; . � � ; < � � � � , , � � � - � * ; � < �

� � � � �4 � � - � � ; � ; . - � � � ; . . � � : , � , � , � � : . � - . , � � � � : : , � ,

� � , � � . ; � , . - � - . ; � ; � ; . - � � � ; . . � , : , � , � � � � � � - � � � � � � < ; � � <

� - � ; , < � < � � � < � � . , � � � , , � , � . ; � � < � � � � � � , . . � , < ; � � � � � � - � - �

� � � : , � , � � � � . � � � , � � ; < - < � � � ; � ; . - � � � ; . . � , � � ; � , . - � - � � -

� � � � � � - � , < � � � � ; � , . - � . , � � , � � � � : - < � � � < � � . ; � � < � � � � : , � ; , <

� � - � - � � ; � �

; < . � � � � � ; , � � � - � - � � - � � , < . ; � , . ; � � < � � � � � : : , � , � ; . ;

� - < - � - . ; � � � � � < � < ; � � < � : � < ; < � � . � : � � ; . ; < � � < � � , � � � �

� � � ; . ; < , � � < � � � , � � � � � . : � � � - � � � � � < , � . ; � � < � � � � : , � ; , <

� � - � - � � ; �

� � - � . ; � � � - � � : - < � � � � . , . ; � , . , < � � : , � , � � � , � -

" #�

��

� � % � % & 6 � � # $ % & & - � � � � � $ � 2 � � %

� � � � � < � � � � � , � - : . � � � � - � � � � ; � ; . - � � � ; . � � � � � � - � � � � � < � . ;

� , . , < . , � � < � � � , � � : : , � , � � � . ; � � - � , � � � , � � , � � : : , � , � � . , � � < � � � �

� , � � - . � � �

� � . � � . � : � ; � � � ; �

� � � � � , � � - � - � � - � ; � ; . - � � � ; . . � < � � � , � � - . , � . , . ; � � � - � ; �

� � < � � � . � � . � � . ; � � � - � ; � � ; � ; . - � � � ; . . � < � � � � , : � ; � � �

� � � < ; � � : : , � , � ; � � � . , � � � � � � � ; - . � , . - < � � �

� � � : ; � � . �

� � ; < � . ; � � , : � � � � , . - , - . : � � � - - � � - < ; � � < � � � � � - � - � ; � -

� , � - � � � � � : : , � , � � � � � . , � � � �

� � � � ; � ; . - � � � ; . . � < � � � . ; � � � - � ; � �

� � - � < � � � ; . . - � ; � � � <

� ; � . ; � � � � � � , � . ; � � < � . , � � , < � < ; � � ; � � � � < ; � � < � - � ; , <

� < � � � < � � . , � � � , , � , � . ; � ��

� ; . � , � � � : � ; � ; � - ; . < ; �

s������

��

��

��

@@@

AAA

@@

@@

AA

AA

�� �

s���

���

AA

AA

AA ��

; � � �

; � � � � , � � ; � � , . , . ; � , . � � , , � � � : : , � , � � � , � � - . � � �

� � < �

� � � < ; � , � � � � < � , �

� � � ; � . ; � � � � ; � � � � � � � ��

� � � < ; < ;

� � � � � , � � � � � � : : , � , � ; � , � � - . � � < � , �

� � � � � ; � ; � . ; � � ��� � : : , � , � � � � � . , � � - � , � � � - � � : . � � : , � , � , � � � < � � � � �

- � < ; � � � < ; . , - . � , . - � ; � � � � , � � - � - � � - �

� � � � , : � : � � , � . ; : ; � , � � ; � , . - , � � , � � � ; . � , < � � : : , � , � �

� � � . , � � � � � ; � ; . - � � � ; . . � , � � : : , � , � � � � � < � � � � - � -

� � ��

� � , < � � � � < ; � � : : , � , � ; � � � . , � � � � � ; � . , . - -

, � � � � , . , � � � � < � � � . , - . � , . - � ; � - � - � � , . � � - � ; �

. ; � � - � � � � , � � � � � � � . � � ; < � � � , : � � � � , . - , � - . : � �

� - - �

� � � � � , � , � � , � � : : , � , � � � , � � - . � � �

� � < � � � � < ; � , � � � �

< � , �

� � � ; � . ; � � � � � � ; � � � � � ��

� � � < ; < ; � � � � � , � � � �

� � : : , � , � ; < � , �

� � � � � ; � ; � . ; � � ��

� � : : , � , � � � � � . , � �

� - � � : . � � : , � , � , � - � , � � � � � < � � � � � � � � , � � - � - � � - � . ;

< ; � ; � ; � . ; � � ��

� � � , � � : � � : � , � � � � ; � , . - � � � � , : �

: � � � � . ; : ; � � � � � : : , � , � � � � � . , � � � < ; . , < � ; � ; . - � �

� ; . . � � � , � � � � � < ; � � : , < � ; � . ; � � � - � - � � � � < ; � � < �

- . � , . , . - , � � � < � � � � ; � . , . - � � � � < � � � � : : , � , � ; � � �

. , � � � : , � , � , � . , � � , � � � � : - < � - � � . � � � ; < � . ;

� � , : � � � � , . - , - . : � � � - - � �

Page 120: Программирование: теоремы и задачи

� � � " #�

� � � � $ � 2 � � � % � % & 6 � � # $ % & & - � � � � � $ � 2

� #4 � 4 � 4 � � < ; � - < � � � � � � ; � � � : � � ; � � , . - - � : ; � , . - � � , � , . < � � � �

� � ; . � � - , � ; � ; . - � � � ; . . � < � �

- � � : , � < � - � . , : � � . � � � , � � � �

: - < � � · � � � < ; : , � , � ; � �� ; . � , � ; , < � � ; . - < � � � , � � - . ; � : , � , � ; : �

� � � . - < , � � . � � - . * � � � ; � - � � . , � � � � : - � � � � � - � ; � ; . - � � � � , �

� � � � � � 4 0 � : , � � � ; . - < � : � � ; : � � � , � � - . � � ; . . - � � � , : � � �

� < � � , � � � ; � � � � - � , � � � � � � : : , � , � � , � �

� � � � � � � � < ; � � ; � � � � � � : : , � , � ; � , � � - . � � � −− � � � < ; � , � � � � � � : : , � , � ; � , � � - . � � ��

� ; � � � < � , � � � < � , < � � , � � � � , : � � � � � � < � , < < � � � � - , � � � � � - � - � ;

� � � � � ; � � � - � , � � � � � ; � , . - � - �

� � � . ; � ; � , : � ; . ; � , � ; . - � � � � ; �

. � . � � � < � � � � � - � � ; � , . - - � � : : , � , � ; . � � , � , � � � � � . . , � , . � �

� � � � < - � . � � � � � ; , � � < � , � � � ; � � � � � � � � � , � < - � � � ; < � - . * � � � ; � - �

� � < � , � � � . � � < � . , , � ; < , � � . � �� ! < � � � � � . � : � < - � � � < ; � � ; � . �

� , � ; � , � � - . : , � , � ; � � . � � � � - � ; < � . , . ; � - - � � � < - � . . ; � , . - � � � ;

� ; � < - . � ; � . � � , � � � ; . ; . � � � � � < � , � � - . � � ; . . ; � , . - , � � . � < � - ��

����� @@@@I@@

@

� @@

@

���

� →@

@@

���

���

��

� �

�����

@@@@I

HHHY���*

@@

@

@@

@

���

���

→ @@

@

���

@@

@

���

� �

� � � � � , � � , � � � ; . � � ; . - � � � : � � . � < ; � , - . � , . - < � � � < � , < < � , . . �

. . ; � , . - � � ; - � , � �

� � � < � � � : � < ; < � � . � . . ; < � � � � < � : , � , � � , �

��

�� - - - < � � . � < � � : � � � . < ; . < � � � � � < � � � � � . � � � , : � � � ; � ; < � � � < �

� : . ; - . � � � < � ; � . ; . � � � �

" #�

��

� � % � % & 6 � � # $ % & & - � � � � � $ � 2 � � .

� � < � � � � , : � � � � � ; � , . - � �

/ ; B � H I F ; H � � ! 5 : � ? � H # H ; " E + = � � 0 � � � � 2 0 � 2 � � � � � 3 � 6 7 0 � � � = 5 ,

8 5 ; J : A > > ? E 8 5 9� 5: 8 5 9� J : D E %�

�:

%�

�:

%�

� : � ? � H # H ; EJ H # � ?J : $ ; � # % � � 5 � E + J ' ( ? F 9 9 ,

8 5 9� 5 : $ 8 5 9 � 5 � E 8 5 9� J : $ 8 5 9 � J � E

%�

� : $ = E %�

� : $ I � C C � J � E %�

� : $ ! * 5 ! %��:

%�

� " ) A " @ I � C C � 5 � E

8 5 9 � 5 � : $ 8 5 9� J E 8 5 9 � J � : $ 8 5 9� 5 E

; � # % � � 5 � : $ ; � # % � � J � + � 0 � � � � � 2 0 � ,

; � # % � � J � : $ 9 H C � � J � + � 0 � � � � � 2 0 � ,

9 H C � � J � : $ 9 H C � � 5 � + � 0 � � � � � 2 0 � ,

9 H C � � 5 � : $ J EI � C C � J � : $ %�

� @ %�

� EI � C C � 5 � : $ %�

� @ ! * 5 ! %�

�:

%�

� " ) A " E

H ? I E/ ; B � H I F ; H � � ! 5 : � ? � H # H ; " E + 1 0 � 4 # 0 � � � � 2 0 � 2 � � � � � 3 � 6 7 0 � � � = 5 ,

8 5 ; J: � : A > > ? E 8 5 9� 5: 8 5 9� J: 8 5 9� � : D E

%�

�:

%�

�:

%�

�:

%�

� : � ? � H # H ; EJ H # � ?J : $ ; � # % � � 5 � E � : $ 9 H C � � J � E +: � ' ( ? F 9 9 ,

8 5 9� 5 : $ 8 5 9 � 5 � E 8 5 9� J : $ 8 5 9 � J � E 8 5 9� � : $ 8 5 9 � � � E

%�

� : $ = E %�

� : $ I � C C � � � E %�

� : $ ! * 5 ! %��:

%�

� " ) A " ) I � C C � J � E

%�

� : $ A ) * 5 ! %�

�:

%�

� @ I � C C � J � " @ I � C C � 5 � E

8 5 9 � 5 � : $ 8 5 9� � E 8 5 9 � � � : $ 8 5 9� 5 E9 H C � � J � : $ ; � # % � � � � + � 0 � � � � � 2 0 � ,

; � # % � � � � : $ 9 H C � � � � + � 0 � � � � � 2 0 � ,

9 H C � � � � : $ 9 H C � � 5 � + � 0 � � � � � 2 0 � ,

9 H C � � 5 � : $ � EI � C C � J � : $ %�

� @ %�

� EI � C C � � � : $ %�

� @ %�

� EI � C C � 5 � : $ * 5 ! %�

�:

%�

� " @ * 5 ! %�

�:

%�

� " E

H ? I E

+ , � � , � � ; � , . - � � � � � � , - � ; � � , � . ; � - � � ; � < - � � , < � - � . � �

� � � � , : � � � : � � ; � � , . - - � : ; � , . - � � , � , . < � � � - � � < � ; � � ; . � � , �

. � < � � � � � : � � ; � � , . - , - � : ; � , . - , : � � . � � � � � � � : ; < � � � � � , � � - , �

� ; - � ; � - � � < ; . � � � , . - , � � ; � ; . - � � � ; . . � < - �� � - � < � � - � � � � . � , < � � � � , : � � ; < ; � - � - � � � < � ; � - �

� � � , � � , - � � ; � � , � � : : , � , � � � , � � - . � . � � , � � � A

� ; � ; . - � � � ; . � � � ; � � � � , � � - . , � ; . . - � ; � � � < . , � � � �

� , � � � � � : : , � , � , � � � . , � A � ; - � � . ; � � � . , . � � ;

� - � � . � �

Page 121: Программирование: теоремы и задачи

� � " #�

� � � � $ � 2 � � � % � % & 6 � � # $ % & & - � � � � � $ � 2

� � � � : : , � , � � � � � . , � A � ; � ; . - � � � ; . � - � ; - � �

� � < � , < < � , . . � - . � , . � . � � - . � , . , . - , , � � � � � < � � ; � . � ,

- � - . / � � � � < ; � � . � � � ; < - � � � < ; � � � � ; � � � � � � � � ; <

. � < - � . . ; � , . - �

/ ; B � H I F ; H J 5 9 5 ? � H ! 5 : � ? � H # H ; E 8 5 ; I : � ? � H # H ; " EJ H # � ? + @ � ' $ I � C C � 5 � ' $ � ,

� C I � C C � 5 � $ � � % H ? J H # � ?J : $ ; � # % � � 5 � E

� C I � C C � J � $ @ A � % H ? J H # � ?

� � ! 5 " E I : $ @ A E

H ? I H 9 G H � C I � C C � J � $ = � % H ? J H # � ?

� � ! 5 " E I : $ = E

H ? I H 9 G H J H # � ? + I � C C � J � $ A ,

� � ! 5 " E I : $ @ A E

H ? I EH ? I H 9 G H � C I � C C � 5 � $ @ � � % H ? J H # � ?J : $ 9 H C � � 5 � E

� C I � C C � J � $ A � % H ? J H # � ?

� � ! 5 " E I : $ @ A E

H ? I H 9 G H � C I � C C � J � $ = � % H ? J H # � ?

� � ! 5 " E I : $ = E

H ? I H 9 G H J H # � ? + I � C C � J � $ @ A ,

� � ! 5 " E I : $ @ A E

H ? I EH ? I H 9 G H J H # � ? + @ � ' I � C C � 5 � ' �: � 3 9 � ? 0 � � � � 4 � � � � � 0 ,I : $ = E

H ? I EH ? I E

� � < ; . � � � , . - , � ; � ; . - � � � ; . . � < - < � , � � , < : � - , . - � < � - < � , �

� � � � . � � � � � < � � � � � : , � � � ; . - < � � < , � , � � < � � < � � � . � � ; � ; < � - � ; ,

� � � � : ; . . � � � � � , . < � , � � - . , �! � , � , . < ; � - < , � ; � � : � < � ; � � 〈 � , � � - . ; �

. ; � � ; � � , . - , : � - , . - - . . , �〉 � < � , � . . ; � , . - < - � ;

; H � B ; I8 H ; � : A > > ? E + 2 � � # 3 � � ,I � ; H � � � B ? : ! 9

: ; " E + 9 @ � � 2 0 �: ; @ � � � 2 0 � ,

H ? I E

� � � � � ; � � ; : � � ; � � , . - � � , � , . < ; � < , � , � � � � � � : - < < ; � �

� C ; B B � $ ? F 9 9 � % H ? J H # � ?

# H � � C ; H H ! ; B B � " E9 H C � � ; B B � � : $ ? F 9 9 E ; � # % � � ; B B � � : $ ? F 9 9 E I � C C � ; B B � � : $ = E

" #�

��

� � % � % & 6 � � # $ % & & - � � � � � $ � 2 � �

8 5 9 � ; B B � � : $ � E

H ? I H 9 G H J H # � ?

: $ ; B B � E > > 6 � � � � 4 6 � 7 � � 6 ; =

+ 3 � 2 � � 3 � � : 0 6 � � 0 6 4 � 0 1 � 2 3 4 � 7 � � � � 6 0 = � � 0 � � � � � 2 � 6

7 0 � � � = 2 E 6 � 7 6 0 � � � � 3 � � 4 7 ,

& % � 9 H ! ! � ' 8 5 9 � � " 5 ? I ! 9 H C � � � ' ( ? F 9 9 " " B ;

! ! � ( 8 5 9 � � " 5 ? I ! ; � # % � � � ' ( ? F 9 9 " " I B J H # � ?

� C � ' 8 5 9 � � � % H ? J H # � ?

> > � 0 1 � 2 3 4 2 6 � 7 � � � � ' :

9 (

: $ 9 H C � � � E

H ? I H 9 G H J H # � ? + � ( 8 5 9 � � ,

> > � 0 1 � 2 3 4 2 6 � 7 � � � � ' : ; (

: $ ; � # % � � � E

H ? I EH ? I E

� C � ' ( 8 5 9 � � � % H ? J H # � ? + � � � 2 � � � � 2 � ,

# H �� C ; H H ! � " E 8 5 9 � � � : $ � E9 H C � � � � : $ ? F 9 9 E ; � # % � � � � : $ ? F 9 9 E I � C C � � � : $ = E

� C � ' 8 5 9 � � � % H ? J H # � ?

> > � 0 1 � 2 3 4 2 6 � 7 � � � � ' :

9 (

9 H C � � � : $ � E

H ? I H 9 G H J H # � ? + � ( 8 5 9 � � ,

> > � 0 1 � 2 3 4 2 6 � 7 � � � � ' : ; (

; � # % � � � : $ � E

H ? I EI : $ A E

+ 3 � 2 � � 3 � � : 6 � 7 6 0 � � � � 3 � � 4 7 3 > = � � 3 2 # � = � 6 <

� 0 � � � � � 2 �: 2 ; 6 0 � 7 0 0 � 0 ? 0 � 2 � � 3 9 3 � � 6 4 � 0

6 � � 2 � � � 3 8 6 2 ; 6 0 0 2 3 6 � 0 � � 0 = � � � � 2 �

� � I ! $ = 3 � 3 A " E � 0 � 0 � � � � � 2 0 6 1 � � � � 6 3 � 0 2 � � 0 E

> � � 9 � � 3 < I � C C � � < � ? 0 2 � � # 3 � � � � 2 3 � 4 � ; E 2

0 6 � � 4 � 0 = � � � � 2 � 2 6 � 0 6 � � 0 6 4 7 � 7 1 ; � 0 @

2 9 � 6 � 0 6 3: > � � 9 � � 3 < I � C C ,

& % � 9 H ! I ' ( = " 5 ? I > > 6 � 7 � � � � 6 I B J H # � ? + I $ A ,

> > 2 > < 4 3 > 6 � 7 � � � � � 2 ' 8: I � ; H � � (

� C I � ; H � � $ 9 � % H ? J H # � ?

� C I � C C � 8 � $ A � % H ? J H # � ?

� : $ = E

H ? I H 9 G H J H # � ?

� : $ A E

H ? I EI � C C � 8 � : $ I � C C � 8 � @ A E

H ? I H 9 G H J H # � ?

+ I � ; H � � $ ; ,

Page 122: Программирование: теоремы и задачи

� � " #�

� � � � $ � 2 � � � % � % & 6 � � # $ % & & - � � � � � $ � 2

� C I � C C � 8 � $ @ A � % H ? J H # � ?

� : $ = E

H ? I H 9 G H J H # � ?

� : $ A E

H ? I EI � C C � 8 � : $ I � C C � 8 � ) A E

H ? I E+ � $ 3 > = � � � � 3 � 2 ; 6 0 ; � 0 � � � � � 2 � 6 7 0 � � � = 2 8 � 0

6 � � 2 � � � 3 8 6 3 6 � 0 � � ; = � � � � 2 0 = E = � 6 6 3 2 I � C C

6 0 � � � � 3 � � � 2 3 � 4 � ; � > � � 9 � � 3 < � � < � 0 ? 0 � 0 � � � � � 2 � E

2 0 > = 0 � � 0 � � � � # � � 3 � 6 1 � � � � 6 3 � 0 2 � � � 0 6 3 2 8 ,J 5 9 5 ? � H ! 8: I A " E I : $ � ) I A E

H ? I EH ? I E

H ? I E

+ , � � � � � � � , � - < � � � < � . . ; � , . - , � � , < � � < � � ; � . � < � � � � � , - � - / . �

. , . / � � , � - � � , � < � � � � � � , - � ; � ; . - � � � � ; . , � � � - . � � : - < �� � � � � ; � � ; � : ; � , . - < � � - < ; . ; � � � - � . � �

� � � . � � . � � * � ; � � , . <

< ; � � � �+ 3 � 2 � � 3 � � : 6 � 7 6 0 � � � � 3 � � 4 7 3 > = � � 3 2 # � = � 6 < � 0 � � � � � 2 �:

2 ; 6 0 � 7 0 0 � 0 ? 0 3 > = � � 3 � � 6 4 � 0 6 � � 2 � � � 3 8 6 2 ; 6 0 0 2

3 6 � 0 � � 0 = � � � � 2 � � � I ! $ = 3 � 3 @ A " E � 0 � 0 � � � � � 2 0

6 1 � � � � 6 3 � 0 2 � � 0 E > � � 9 � � 3 < I � C C � � < � ? 0 2 � � # 3 � � � � 2 3 � 4 � ; E

2 0 6 � � 4 � 0 = � � � � 2 � 2 6 � 0 6 � � 0 6 4 7 � 7 1 ; � 0 @

2 9 � 6 � 0 6 3: > � � 9 � � 3 < I � C C ,

& % � 9 H ! I ' ( = " 5 ? I > > 6 � 7 � � � � 6 I B J H # � ?

+ I $ @ A ,

> > 2 > < 4 3 > 6 � 7 � � � � � 2 ' 8: I � ; H � � (

� C I � ; H � � $ 9 � % H ? J H # � ?

� C I � C C � 8 � $ @ A � % H ? J H # � ?

� : $ @ A E

H ? I H 9 G H J H # � ?

� : $ = E

H ? I EI � C C � 8 � : $ I � C C � 8 � ) A E

H ? I H 9 G H J H # � ? + I � ; H � � $ ; ,

� C I � C C � 8 � $ A � % H ? J H # � ?

� : $ @ A E

H ? I H 9 G H J H # � ?

� : $ = E

H ? I EI � C C � 8 � : $ I � C C � 8 � @ A E

H ? I E

" #�

��

� � % � % & 6 � � # $ % & & - � � � � � $ � 2 � �

+ � $ 3 > = � � � � 3 � 2 ; 6 0 ; � 0 � � � � � 2 � 6 7 0 � � � = 2 8 � 0

6 � � 2 � � � 3 8 6 3 6 � 0 � � ; = � � � � 2 0 = E = � 6 6 3 2 I � C C 6 0 � � � � 3

� � � 2 3 � 4 � ; � > � � 9 � � 3 < � � < � 0 ? 0 � 0 � � � � � 2 � E

2 0 > = 0 � � 0 � � � � # � � 3 � 6 1 � � � � 6 3 � 0 2 � � � 0 6 3 2 8 ,J 5 9 5 ? � H ! 8: I A " EI : $ � ) I A E

H ? I E

+ , � � � � � � � , � - < � � � < � . . ; � , . - , � � , < � � < � � ; � . � < � � � � � , - � - . / . �

. , . � � � , � - � � . / � < � � � � � � , - � ; � ; . - � � � � ; . , � � � - . � � : - < �

� < � , < - � < ; � , � � < � . ; � - � - , < , � ; : , � ; , < - . � - � . - � - � , � , � , . . � ,

A � + � - � � � � � � ) - � � � � � < , � , � � - � � ; , < � , � � - . ; < , � ; � � �

� #4 � 4 � 4 � � � ; . ; < � � � < � � � - : � � ; � � , . - - � � , � , . < ;

; � � < � � � � - . < � � � � � � ; , � � ; � ; . - � � � � - � � � - � . � � . ; �� � � �

. , � � . � � , . �

� � � � � . ; � ; � ; . - � � � � ; < � , � � , < . , � � � , , � : . � � � � � ; � , . - � � � ,

� , � � � � : , � , � � < ; . � � - < � ; � ; . - � � � ; . . � � � � � < � � � , � � ; � � � -

� : ; � , . - - � � , � , . < ; � � , < � � . ; : � � - < � � . � � � � � ; � , . - � � �

� � � � � � � � 4 � � < ; � ; � - � . ; � - ; < � � � � � � ; � � � : � � ; � � , . - - � : ;

� , . - < ; � � � < � � � � . - � � � - � ; � � � . � � � � , , � � � � - � - : � � � . ; : � � � ; �

� � � � � . � � , � - * - � � � ; : � � - . . - � � � � . � � . � � � , � � � � < - < � �

� � � , < � � � < - : � � � - , � � � � � � � , : < ; � � , . - � . � , < � � � ; � ; . < -

� � � � - , � - � � : , � < � - � � � � : � ; � � � � . ; � ; : � � � � , � ; � - � � � � - � , �

� : - . - . . - � . ; . � � ; , � � � � 1 � � � � $ � 2 ' � � �

� � - � � � � � ; : ; � , � � - . ; � : , � ; � ; � : - . � � , � , . < � � ; . - � � � � � . �

, < � ; �! < � < � � , � , . < � � - � � � ; . - � , � � , : � � , � � � - � � ; � � � � � : : ,

� , � � � �

0 � : , � < , � , � � � � ; . - < � � � , � � - . , �

>

� � � , � , . < � � � . � , < � ;

� - � � � � � , < � , . < � � < � , � � - . � � � , � � - . , � ; < ; � , � � - : � � ; � � ,

. - - - � : ; � , . - - . � � � � � � , � , . < � � � � � : ; � , , � �! < - � � � , � , . < � � � � ; <

� ; . : , � - < , � � - : � � � � � � : : , � , � ; �� � < � * - � - � � � ; . � . , � � < � � � , � -

� � �

>

��

0 � : , � � ; � ; < � - � ; < � : , � , � � � � � � ; : ; � � - , < ; � - � - � � � <

� ; � - �� � 9 ; : ; � , � � - . ; � : , � - < � < � : � � � � � , � , . < � � . ; - � � � � , . -

, � � � � . � � � < � � � � � � , < � : , � ; < � � � � � , � - � � � � , � , . < � � � < �

: � � � ��

� � � , � � - . ; � � � , � , . < ; � - � - � � - � , , < � � � � . ; � � - � � . , - � , , <

� . � � , � � � � � � , � � , < � � 6 7 # ' � �

� � � , � - < � . ; � � : < . ; � : . � � - < � � , � � � < , �

Page 123: Программирование: теоремы и задачи

� " #�

� � � � $ � 2 � � � % � % & 6 � � # $ % & & - � � � � � $ � 2

� � � � � � � � � � � � � � � � � � � � � - � � : - < < ; � �

� � - � - < � � � � < � � � � � . � �

� ; : ; , < � . , � � � � . < � , � � : , � - < � , . , , � � � � , � , . < � � � � < � . , < � � � � � , � �

� � - � . � � � � . � < � � � � � � � , � , . < � , � � , � , . < � � - < ; - . � � � � � � , � , . < �

� ; . � - � ; , � . ; : � ; � - < ; � � � � � , � , . < � � - � ; . : , � � � - � - � , � , : - . . � �

� � , � , . < �! < � < , � , : - . . � � � � , � , . < . ; : � : � � ; � - < � � � , � � - . � � � , : � : �

� , � � � � � � . �! < � � � . � � . � � , � - � . , � � , . , , � � � � , � , . < � � �� � - - � . ;

� � � . ; � < � , � � ; . � - � ; � < . ; : � , � � � : , � � < , � , : - . . � � � � , � , . < - < � : ��

� - � � � . � , � � . � � � � � . ; � � < - � : � � ; � - < � � � , � , . < � � � � , . � � ; � . � � ; , <

� � � . � � � < � � � � , . � � ; � , � � , < . ; : � , � , � � - . � � ; � � � < ; : , � , � ;

� � , � - � - � ; , < . ; ��

� � � � � � � � � � � � � � � . ; � � : � , � � . , � � - < , � � � : - < � � : ; � , . - �

. , � � � , : < � , . . � � , : � � � , � � . ; . - � � � � < � � � � . ; � � : - < � � - < , �

� � � < � � � : � < ; < � � . � . ; � � - < � � : ; � < � � � , � , . < - . � - < ; �� � - � - < � � -

� < � � < ; . � � - < � - � � � � � ; � , . � � - � � < � , � � � � . � � � � � � . - < � . ; � � <

� , : . , � � � - < ; � , � - < � � � � � - � . . , - � , , < � - . - � ; � � . � � � . � � . � �

� ; . � , � ��

� � - , � � ; � - < ; - � , � < � ; . � , � � � < � . ; . - � � � , < , � � � � ,

� , . < � � � � � , < , � ; . : , � - < , � , � � � � � ��

� � � , � : ; � , . - � : . � � � � � ,

� , . < ; � < ; � < � � � � , � , . < � � � � ; � � ; . . ; � : - . � - < �

� � - � � - � < � �

� , � � - . ; � � , : � : � � , � � � � � � . < ; . � � - < � , . � � , . � � � � � � � � � , � �

� < � � , < - < � : �

� #4 � 4 �4 � , ; � - . � � ; < � � � - ; . . � � � , � � � � ; . , . - � . � , < � � � � , : - �

� - � � � < � � . ; < ; � , � � . � � � , < � � � � < - � � � � � � : , � < � - � : � � � , � ; � - �

� � � � � , . - � - � � � � , . - - � � � � , � � - � � - . ; : � , . � < - � �

� #4 � 4 � 4 � � . � � � � , : , � < � � ; � ; . - � � � ; . . � < � : , � , � ; - . ; � , � < � , � �

� ; < � � � < � � � : � � ; : � � � , � � - . � , � � , � � , - � � ; � � , � � : : , � , � � - � , � -

. , � - � � � � - � � . � � < � - � ; � � - , � � � - � , < � ; � , � � - . � � � , - � � � , < � �

< ; � � � � � � � , : , � , . - � < � - < � < � � � � < � � � - � � ; � , . - � . , . ; � � � ; , <

� ; � ; . - � � � ; . . � < � � � , � � - . ; � � . ; � � : � - � . - , < � � � - � � ; � , . - � �

� , ; � - . � � ; < � . ; � . � � , � < � � - : , - � � � � � � ; . , . - � . � , < � � � ; � ; . < -

� � � � - � � � , . � � � � � � � � : , � < � - � : � � � � � � , . - � � : ; � , . - - � � � � , � � -

� � - . ; : � , . � < - �

� $ � ; . ; . - , � � . < ; � , - � � � � . � , < � � � � � - , - � ; � � , � � ; � , . - � � �

: � � � . � < - � � � � . - � , � , � . � � � � : ; � � - � , � � , � � < ; - � , � � 9 � � � - . ; < � � . � ,

; � � � � - < � � � � �

��

� � � � � � � � �

� � � � � � � �

� � � � � � � � �

� � � � � + � � ' � � * ' � � � � ' �

< � � � � � � , : , � - < � < � � � < � . ; . � � ; � < 3 # & 7 � 3 6 7 & # 1 6 $ # � # � & # 5 � � % ' 1

' % 7 � 3 # 5 9 � � � ; � � ; < - � � � � � . ; : � �

• � � ; . ; < � � � . , � . � , � . � , < � � � � . ; . � � ; , � � , % � 2 % $ � 7 # ' � , � � � � ,

� , . < � . ; . � � ; � < 6 � ' $ # � % ' � � � � . , � . � , � � � , : � � ; < , � � . � < - - �

� � � � � . ; . � � ; � < 6 � # $ % ' � � : ; . . � � ; � * ; � - < , � �

• � ; . : , � - < � � , - � � � � � ; � * ; � - < ; � . ; : � , � � � � � � � 7 � � ' � & % � � & - �

� � � � . � ; < , � � . � , � - & � 7 � � ' � & % � � & - � � � � � � , � < � � . � , � �

• � � � � ; < � � , : - . , < , � � - . ; � � . � � - � � � � � � � : - . � . ; . � � ; , � � � & % 1

3 % � � & - ' �

• � � ; . ; < � � � . , � . � , � - � � � � % $ � � � � ; � � ; < - � - � � ; : � , - . � � < � � � �

: � � . � - � , < � � - : �→ � � � : , � � . , � � < � � � � . , < , � � - . ; � � . � �

- � � � � � ; � � � � � � � . , � � � � � � < � � � : - < � - < , � � - . ; � � . � , � - . ,

< , � � - . ; � � . � , - � � � � � � �

� � < � * - � - � � � ; . ; 9 � � � ; � � ; < - � ; � � � ; < � � � : , � � � � � ; < � � � , * - � � 9 � � < ; � � ; � : � � � - � � � ; � � ; < - � � . ; . , � � : , < � �

, - $ # � # ' � � < � �

� � ; � � ; < - � , . ; . � � ; , < � � � , : � � ; < , � � . � < � � � � � � �

� � � - - - �

�� � � � �

< � � � � � � � < � - < - . � : . � � � - � � � � ; � - � < � < - � � � � � . ; � ; � � . � � �

; � � - � � � � � � ; , < - . � � . ; � , . � � . , � � < � � � � � . , < , � � - . ; � � . � � � - �

� � � ; � . ; � � � � � � � � : . � � � - . � � ; � - � � � ; � � ; < - � - �

� � � � � � � < ;

� � , . . � , - . < , � � - . ; � � . � � - � � � � � � � . ; . � � ; , < $ - $ # � � ' - ' � , � - � � ,

< � � , < � � � � : � � � < � � � � - � � � . � ; , < �

� . � , < � � � , � � � � � : - � � � � � �

- . < , � � - . ; � � . � � - � � � � � � � . ; . � � ; , < 2 � - 3 # ' � � # � # � � % � ' - ' � % & & # 5

� � % ' ' % 7 � 3 # 5�

Page 124: Программирование: теоремы и задачи

� � " ��

� # & 7 � 3 6 7 & # 1 6 $ # � # � & - � � � % ' ' % 7 � 3 �

� � < � � - � , : � � � , � � � ; � , . ; � � : , < - . < , � , � � ; < � < ; � � � � � � � � �

: ; . ; 9 � � � ; � � ; < - � ; � � � < � � - < � ; � � � � - < � � � � < � � � � � � � � � � � � � � � �

� � � � , � , < � � � � � : - � � � - � . � � � < � � � � ; � � ; < - � , �

� � � � � � � 4 � � * ; � - < �

� � � �

� , < � � , < , � � - . ; � � . � � - � � � � ; - � : - . . , < , � � - . ; � � . � � - � � � � � ��

� ;

� ; � � . � � - � � � � � ��

� � ; � - � ; �

�→ � �

�→ � � �

�→ � �

�→

� � � � , : . , � � � ; � - � , � � ; � ; < � - < � � < � , � � � � � �� � - � , � � � � � � : - � � � � � � �

� � < � , � � � � �

� � �

� � � � � �

� � � � � � � �

� � - � , � � . , � � � � : - � � � � � � �

� �

� �� � �

! < ; � � ; � � ; < - � ; � < � , � ; � ; � � � ; . : , � , ��

� � : , � � � � : - � � < � � . , � � � �

� , � � ; � � � � � � � � < , � ; � �

� � � � � � � 4 � � � � ; � � ; � � ; < - � ; � � � � � : ; � � ; < � < , . � � �

� � * ; � - < � � � � ( �

� � ; � - � ; �

�→

�→ ( �

(→ � �

(→ � � �

" ��

"�

� � $ � 5 % � � # � � 7 ' � % � � # � % � %

� ; � ; � � . � � - � � � � � � � � � , � � � - � � : - � � � : ; � , , � � - � , � ; � � � : , � � -

< ; < � - � � � � � < � � - � � � , � � � � ; < - � , � � � � � � � ; � - � ; � : ; . . � � � � � ; ,

� < � - � � � � � � � . , � � � � ; � - � ; � < � � � � � � � �

� � � ; : � � � . , < , � � - . ; � � . � � � - � � � � ; � � . � � ; � � < � , < � � . � ,

< � � � , � � � � - . < , � � - . ; � � . � � - � � � � � � � � � < � � � , - . . , � � � � � � : <

; . ; � � � - � . � < � � � � � ; � � < � : , � ; . � : � . ; � ; � � . � � � - � � � � ; � � � � , : , � ,

. - - � � � � : - � � < - � � � ; � � ; < - � , � �

9 ; : � , � � ; � - � � � � ; � � ; < - � - � � . �

� ; � ; < � - � ; < � � ; � � � � < � � � < - � � . � , < � �

� � � ; , � � < � . ; � � - � , � ,

< � � � � � � < � � � - � , : � . . � � � � ; � � ; < - � - �

� � < � - � � � . � , < � ; � � �

- . � � � � � � � � � � � : - � � � - . . , < , � � - . ; � � � ( - � � � < � , < < � , . . � �

� � � : ;

� � ; � - � ; � � � ; � � ; < - � - � � < � , < < � � � < < ; � - , � � � < � ; �

�→ � � : , � - < � � < � , � � � �

�→ ( � , � - � � � � � � � - . ; : � , - < �

; � � � � � � � - . ; : � , - < � � < �

� � � � � � � � - . ; : � , - < �

(→ � � � , � - � � � - . ; : � , - < � � < � � � � �

� � � � � - . ; : � , - <

(→ � � , � - � � � - . ; : � , - < � � < � � � � �

� � � � - . ; : � , - <

� * � � � � � - � � � ; . . � , � � � < � ; � . � , < � � � . , � � � , : , � � < � < - � . �

, < � ; � : . � . . ; � . � . ; � � - � , � � � . - � < ; � < � , � . � � - � , � - � � ; � ,

< � , � - � . < � � . � , < � � � , � � � � � �

� : . ; � � � � . � : � � ; . ; < � � � < �

� . � , < � ; � . ; : ; � ; , � � , � � ; � � ; < - � � � � � � � < � - . - � ; � � . � � - � , : -

� : � � � , < � � � � � - � � < - � � � � � - � �

� � 4 � 4 � 4 � * � � � � � - � � � ; < � < � � . � - : � � ; . ; < � � < � � < � , � : , . - , : �

� � � - . � � � � . � � � � . < , � < . � � � � � : . �� � � ; � � ; < - � - � �

� � 4 � 4 � 4 � � < � � - < � � � ; � � ; < - � � � � � � < � � � � � � � � : - � � � � � ;

; � , ,� �

, , / /� �

/ / � - � � . � � , � � ; � . � � - � � , : - . - � � �

� � , ,� �

, , / /� �

/ / � - � � . � � , � � : � � , � � � � � , � - � ; , : - . - � � �

� � , ,� �

, , / /� �

/ / � - � � . � � , � � � � � � , � - � ; , : - . - � � �

- < � � � � � � . - � � �

� � 4 � 4 " 4 � � � ; . ; < � � � < � . , � � , < � � , < 9 � � � ; � � ; < - � - � � � � < � � � � � �

� - � � � � � � : - � � � � � ; � - : ; , ,� �

, , / /� �

/ / � �� �

� � � � � � < � � � � � - � ; . �

� , � � , : - . - � - : � � , � � ; � . � � - < � � � � � � . - �

� $ � ; . ; . - , � � � � ; . ; < � � , : � � � � � � , � � � � � � � - . � � � � . � � 9 � � � ; � � ;

< - � , � : � � � � � � � : � < ; < � � . � : � - . . � � � � � � ; � � � � � � : - � � � � � � < � �

� � ; � � ; < - � , � � � , < � � , < < ; � � , , � � � � , : < ; � � , . - , � � - : , � � � � � � � < �

Page 125: Программирование: теоремы и задачи

� � " ��

� # & 7 � 3 6 7 & # 1 6 $ # � # � & - � � � % ' ' % 7 � 3 �

� � � � , � � � � � - : ; � � - - - � � � - - - � � � � : , � - � � � � < � � , . � � : - . ; � � � � ,

� - � � � ; . � < ; � , � � � � : - � � � � < � � � � ; � � ; < - � , � ! < � � � . � � < ; . �

� - < � � . ; � : . , < , � � - . ; � � . � � - � � � � � � � ; . � � ; � � - � � � - � � � < � , .

. � � � . ; � , : . - � � � � � � � � , , � � � � : ; �� �

� , < , � � - . ; � � . � � - � � � � � � . � � ; � ; < � - � ; < � � ; � � � � : � � � , - �

: � � � � � : - � � � - . . , � � � � � �

� � , : � � � , � � � - � , � , : � . ; � � : . � < -

� � ; � , < � , . , < , � � - . ; � � . � � - � � � � � � - � � � � . � � ; . � * � ; � � , . < � � �

� - � � � � � . ; � � � � � . . � , � � � � � � � , � � � � - � � < � � � - . � , . - � � ; � � ; < - � -

� ; : � � < ; � � � * � ; � � , . < � � : - . - � � � � � �

� � � � � � " 4 � � * ; � - < �

< , � � - . ; � � � � � � �

. , < , � � - . ; � � � 〈 � � �〉 〈 � < � � �〉 〈 � ; �〉 〈 � < � ; �〉 〈 � . � 〉

� � ; � - � ; �

〈 � � �〉 → 〈 � ; �〉 〈 � < � � �〉〈 � < � � �〉 → � 〈 � � �〉〈 � < � � �〉 →〈 � ; �〉 → 〈 � . � 〉 〈 � < � ; �〉

〈 � < � ; �〉 → � 〈 � ; �〉〈 � < � ; �〉 →〈 � . � 〉 → �

〈 � . � 〉 → � 〈 � � �〉

� � � � ; . � � < � � � � ; � � ; < - � , � � � � ; , . - , 〈 � � �〉 � � < � � � � , : � � ; < , � �

. � < � � ; � ; , � � � 〈 � ; �〉 � � ; . : , � � . . � � � � � ; � - � � ; � ; , � � , � � < � � � � ,

: � � ; < , � � . � < � � . � - < , � , � 〈 � . � 〉 � � ; . : , � � . . � � . � � . : � � � ; � - . . ; � ; � -

� � . � , . - � � ; � . � - < , � � � � < � � - � � � � � � ; � � � - � � � � � ; , . - , � � � �

� ; � �� � 4 � 4 #4 � � - � , < - � � - � , � : � � � � � � � ; � � ; < - � - � . ; : ; � � , � < � < , . � � �

� � � � � 4 � � < � : - . - . � ; � - ; . < � � �

〈 � � �〉 → 〈 � � �〉 � 〈 � � �〉〈 � � �〉 → 〈 � � �〉 � 〈 � � �〉〈 � � �〉 → �

〈 � � �〉 → � 〈 � � �〉

! < ; � � ; � � ; < - � ; � � < � - � � � � , � . � � . , � � < � � � � � < . � � , . - � � � , �

� � � � � � , � � � � : , � � � � � � - < � �

" ��

"�

� � $ � 5 % � � # � � 7 ' � % � � # � % � .

� � 4 � 4 � 4 � ; . ; � � � - . � � � � . ; 9 � � � ; � � ; < - � ; �

� � < � � - < � ; � � � � - < �

� � � � , � � - � � - . ; : � , . � < - . ; : ; � ; , � � � � , � . � � � � � ; � � < ; � � - � � � � -

. � � - ; � � . � , � � , � < � , � � - � � : , � < � - � . , � � , � � � � : - < � � � - . � � ; � <

: � - . � � � � � , � , � � � � � � � ; � � � � - . � � � � , < . ; � - , < � � < � � ; � � ; < - � - � �

� � � � � � 4 @ ; � , < - � � � < � < � , � � � ; . - , � � � - . � � - ; � � . � < - - � � � � ; , <

� � . � � . � < � � , � , . - � � . � � ; . . � � . ; � , � , � � � , � , � � � . � � . � � � �

� � : � � �

� , � . , � , . , , � � � - . � � - ; � � . � � ; � � � � - < � � � , < � � , < �

� � � � � �

� � � � ; � < - � , � � � � . . ; � , . - � . . , - � , , < - � � � � . � , � � , . ; � � ; � < - � ,

9 � � � ; � � ; < - � - � � � ; : ; � < : � � � � . - < , � � . � � - � � � < � ; � - � � � . � � � � � -

� - < � � - < � � � � , , � * * , � < - � . � , ; � � � � - < � � � � � � - . � � - � � - � � � � � � �

� , � � � , � , . - �

� � � � < � � � � ; � � ; < - � , , < � . , < , � � - . ; � � � � � - - - � �� �

� � < � � - � . �

� � � � � ; � � ; < - � � . , < , � � - . ; � ; � - �′ � � - - - �

�′� < ; � � � < � � � � � � � � . � � �

< ; � � , � � � < � � � - . �′� � � � � : < � . � � � � � � ; � � ; < - � , � < , , � � � ; � � < �

- . � � � < ; � � � � . ; - � � � � , . - , � � � < � � � � � � ; � � � < � � � , . , � � � � : - < �

< � � � � � � � � . - < � < ; � � , � � , � � � ; . � � ; . - , � � ; � � ; < - � - � . ; : � � �

. - < � � - . � ; � - � . , < , � � - . ; � � � - � � : . � � � � ; � � ; < - � - � � � � : - < � � < � ,

� � � � � ; . ; < , � � ; : � , � � ; � - � � . ; � , . - < � . ; � � � � � � . � < � � � ; � - � � � � � �

� ; � � - � � , � - � � � ; � � � � ; < - � � � < - < � � ; � - , � - � � - . . , < , � � - . ; � � � �

- . � � < � � � � � � � � : - < � � < � , � � � � � ; � � < ; � � . � � � � < ; � - < � � < � - � - �

� ; � � - � , � � , � - � - � � : . � � � � ; � � ; < - � , � � � � � � ; � - � �

� → � � �B

� � - � � � - . � - � � � � � : - < � � < � , � � � � � ; - . � . , < � < � � < � � � ; � - � �

. ; : � . ; � , . - < � . ; � � ; � - � ;

�′→ �′ �′ �′

�′→ �′ �′

�′→ �′ �′

�′→ �′

� � � < ; � � � � � , � - : , � � � � � ; � � ; < - � , � � : , . - - . � : . � � � . , < , � � - . ; � ;

. , � � � � : - < � � < � , � � � � �� , � , � � � < � ; . - � � � - � � � � - : ;

�→ �

�→ �

�→ �

�→ �

� � � ; � � � � ; < - � ; : � � � � � ; � - � ; � : - . - � � � � � - � < - - � � � � � � � � ;

. � � < � - � � � � � - . � � � � . � � : � - . � � � � < � � , � � � : , � ; < � � � < � : , < � - � � ,

� � � : � - , � � - � � . , < , � � - . ; � � �

Page 126: Программирование: теоремы и задачи

� � � " ��

� # & 7 � 3 6 7 & # 1 6 $ # � # � & - � � � % ' ' % 7 � 3 �

� � � , � , � � � � � � , � � ; � � - . ; : � , . � < - � ; � � � � � - � � � � � ; . � � � � � �

� � : � . . � � � � � ; � � ; < - � � � � � � , < � � � � � . < � < ; � � : � � ; : � � � � � :

� � � ; � � � � , � , � � � � � � � ; - : � � ; : � � � . , < , � � - . ; � ; � � . , � � � � � �

: ; , < � - � < � � � : � � � � � < - � . , < , � � - . ; � � � �� � - � < � � � � : � � � ; � � � � ,

� � < � � � � : � , � � . � ; < ; . - : � - . � ; . , < , � � - . ; � � � � < ; � � � � � � : � , �

� < � � � � � - . ; � - � - - � � ; � - � ; �→ � . , < , � � - . ; � � � � � � , � � � ; . � � ,

. , < , � � - . ; � ; ��

! < � � � . � � . � � - � � � < � < < � - � - � � � � �� � � . - �

� < � < � � � � , . ; � � - � , � , �� � < � � � � ; � � ; < - � , , < � � � ; � - � ;

�→ �

�→ � � �

- : � � � - � � � ; � - � � � : , � ; � - � � � � , � � � � ; < - � . , < �

� � � � < - � � . . ; < � �

� � � � : - < � - : ; . . � , � � � � � - . . , < , � � - . ; � ; ��

! < � � � : , < < ; � � � : . � �

- . � � � ; , � �

• , � - � � � � � : - < - . � �

• , � - � � � . � � ; . � - < � . ; . , � � < � , � � � ; � � � � � � : � � � < � � � �

� � � � � : - < - . � � � � � � � : - < - . � � ; � � � � � : - < - . ��

� � < ; - . * � � � ; � - � , , < � � � � ; � � � � � � � � � � , � � ; � � ; � � < � , .

: � � ��+ , � � � � - : , < � � � < � � - � � : , � < � - � � < � � � ; � � � � - < � ; � � � - . � � - ; � � . � �� < , � , . � � � � - . � � ; . ; � - - < � < � - � ; . , < , � � - . ; � � � � � � ; � � � � ; < � � � ;

� - � - � � , < � � < � � � . - , . ; � , � - � � ; � � ; < - � � � � , � � � ; . � � ; < � � * � � � , �

� � � < � � � � � � ; � ; � ; < � � ; : � � � � � ; � - � ; . , � � � , , � . , < , � � - . ; � � � � < �

� , � � � : , � ; < � � � � � : . � � � , . , < , � � - . ; � � � . ; � � - � , � � � � ; � - � � � → � � �

� � . � . ; � , . - < � . ; � → � � - � → � � � � : , � � . � � � � . , < , � � - . ; � � � �

� � 4 � 4 � 4 � ; � � < � - � � � ; � � ; < - � � , : - . < � , . . � � . , < , � � - . ; � � � � �

. , < , � � - . ; � ; � - /B

�B

� - � � ; � - � ; � -

�→ ,�→ / �

�→ � � �

�→ � � � �

9 ; � � � � � , � - < � � � � � : - � � < � � � � ; � � < � � � � ; � � ; < - � , � � - < ; � � � � � ,

� ; . ; � � ; � � - � � : , � < � - � � � - � � � � < , . - - � : . � � � � � � � : � � . � � � < �

� � � ; . - � , . � ��

" ��

��

� � 7 # � � � 3 1 � 6 � $ & # � # 6 � 1 6 3 % � � �

� � � � � � 4 � � ; . - < � , � ; � , � , � , . . ; ) � - . � ; � - ; . < � � � � � � � � � : -

� � ⇔ . , � � � � - < ; . . ; � ; < � � � , : < ; � � , < � � � � � � . � ; < , . ; � - � � , : -

. , . - , � ) � � � � : - � � � � � � � �

� � 4 � 4 �4 � � < , � � � � � : � � � ; � � ; < - � -

�→ ,�→ � /

�→ � � �

�→ � � � �

( � � � � ' � ) � ' � � � � � � � � � � � ) �

� � < � - � - , � < ; � � � � - < � ; � � , : � : � � , � � � ; . : , � ; � � , : < ; � � � � ,

� � � - < � < , � � , < - � , � - � - . < , � , � � ; � � � � - < � � . ; � . � � , � , � � � - � . � � �

� � � ; � ; < � - � � � � . � � < . ; � � ; � < - � , �! < � < � , < � : � � - � , . - � � � : . ; � � �

: ; � , � � . , � � � , � � � ; � � ; < - � ; � �

� � � � � : - � . , � � � � : - � � , � � � ; . - � , . -

� � . : . , , �� : , � , < � : ; � , � � � - � . � � � � � � ; < ; � � � ; �� � � ; : � � � . , < , � � - . ;

� ; � � � < � � - � � � � � , : � � � � A � � � � < � � ; � � � � - � , . , . - - � � � � � � �

� � � : . � � � � � � � � � : , � ; , < : � , � , � - �

• . ; � � : - < . ; - � � � � � , , . ; � ; � � � � � � ; � � � � < � � � , � � , < � � < � . ; � ;

� � � � � � � : - � � � � - . � � � � ; �

• � � � � ; , < � � � , < � - . ; � : , . . � , � � � � � � � � � : - � � � - . ��

� � , : , � , � � � - � � ; < � � < � < � , < � : � � � , , � � : � � � . � � : � � � � � � - �

� < � � � � ; � � � � � , : � � � � � � � � ; � < � , : , . - � � � � : . � � � � � , - � ; � � � �

� ; � < � � , . � � � < ; < ; � � � , � � ; � � < � � � � � � , : � � � ; � ; , � � � < � � � � � � � � � :

. � � � � � � ; � � < � � ; � < � . - � � � � : . � � � < � , � - � , , < � � ; . - � ; � � < : , �

� � ; � � � � � - < ; . . � � � ; < � � < � . , � � � � - < ; . . � � � 0 � : , � � - < ; < � � � < �

, < � * � . � � - � , . � ; � ; � , < � � � �� � � � � � � C � �

: ; � � ; � , � � � � . , � � � � - < ; . . � � - � � � � �� � . . ; � , . - � - � � � � < � � < � < , �

� - . ; � � . � , - � � � � � � ; < ; � , � , � - ; � � . � � - � � � � � � � � � � � � � � �

� � . , � � � � : ; � � � . . ; � ; � � - � � � < � � � � � � � � , � � � � - < ; . � �� � . � � � < � �

* � . � � - - � � ; . � � , , < � . , : � - � ; , < � � ; . - � � � , : � � � � � - < ; . . � � - . ,

� � � � - < ; . . � � � ; < � � � : � � < � � � , < � � � � � , : � � ; � � � � � � < � � ; : � -

� ; , < � � ; . - � � . ; � : - . - � � � � � � . ; � � - � , . - � ; � , � - � � � � � � � � �� � � < � �

. ; � � . , � � - � , , < � � � , � � ; � , � , � , . . ; C�

Page 127: Программирование: теоремы и задачи

� � � " ��

� # & 7 � 3 6 7 & # 1 6 $ # � # � & - � � � % ' ' % 7 � 3 �

� , � , � � � � � � , � * � � � � � - � � � ; < � . ; � - < � , � � � ; . - � � � � � , : � � ,

� A � �

� . - � < � < � � , : � � � , � �

� A �

• � � � � - < � � ; , < - . � < ; � � , � � ; < - � � � ; � ; � - � ; � � . � , . ; � ; � � � �

� � � � , , . ; � ; � � � . , � � < � � � � � � � � ; � � � � � : - � � � � - . � �

• . . ; � , . - , C < ; . � � - < - < - . . � � - � - � � . � � � . ; � - - � � < - � <

< � � � � � � , < � - � � � � � : - � � � - . � - � - � - � � . , � � � � : - � � � . ;

� ; � � � � � � � : - � � � � - . � � � � � ; �

� � � : � � < � ; � � , : � � < ; � � � < , � � - . � � � � - � � � � � � : - � � , - . � � � � � � �

: , � . ; . � � ; < � � � � � � � � ; � � � � , . ; � ; � � � � � � � � � � � � : - � � � � - . � � � � ; �

� . ; � ; � � � �� � ; * � � � � � - � � � ; . . � � < � , � � � ; . - � � , < , � � : , � � � � ; ; < �

� � � ; � - � � A � � � � � , � < . ; : � � �

� ; � . � � � � - � , � ; �� � < � � � ; � - � �

� → � �

� � , < , : - . < � , . . � � � � ; � - � � � � � ; � � ; < - � - � � : , � ; � - � � � � , � � �

� ; < - � � � < � � � � � . , < , � � - . ; � � - � A � � � A � � � � � � , � < . � , : �

. - � � � � � � , : � � � �

� ; � � < � - � < ; � � � � � � � , : � � � �

/ ; B � H I F ; H � H 5 I � EJ H # � ?� H 5 I � E

� C J � % H ? J H # � ?

� H 5 I � E

H ? I EH ? I E

� � 4 � 4 � 4 � � - � , < - � � - � , � � � � � : ; � < ; � � � � , : � � ; � � : , < . , � � � � , � < . � �

: � ��

� � � � � 4 � � < � - . � � � � � : - < � � � � , � � � � � - : ; , ,� �

, , � ; - . � � � � �

: - < � - � � � � � � , /�

� � � : ; - . � � � � � : - < � � � � , , , , / � . � � � � � , : � � ;

� A � � < � � � . , . ; � , < - < � �

$ � ; , � : � < ; < � � . � , � � � � - � � � � , � < . � < - � � � � , : � � � � A � �

� �

� < � � � . ; � � � . ; : � � < . , � � < � � � , � � � . . ; � , . - � � � < � * - � - � � � ; . �

9 � � � ; � � ; < - � ; - . , � � < � � � � . , < , � � - . ; � � � < � � � � ; � � ; < - � - �� ; � �

< � - � � � � � �

� � � � < � � � , - � , , < � � < � , . . � , . ; � ; � � � � < ; � , � � �

" ��

��

� � 7 # � � � 3 1 � 6 � $ & # � # 6 � 1 6 3 % � � �

� , , � � � � � � , � - < ; � - , , < � � �

� � � � � � � � ; � � < ; � - � � � � � - �

� ; � � < � - � < , � � - . ; � � . � � - � � � � � - : � � - � � � . , � � � , : < � , . . � . ; ��

� . � , < � � � , � < ; � - � < , � � - . ; � � � � � � . . ; � - � � � � � ��

� � - . - � ; � � ,

� � � � � . , � � , < � � < � , . . � � . ; � ; � � � : � � � � � � � � � � ; � < � � . � ,

< � � � � � � � � � < � ��

� � 4 � 4 � 4 $ � ; . ; < � ; � � � � � � : � � � - � , � ; � �� � � � � � � � � � � � -

� � � ( � : � � � - � , � ; � �� � � � � � � � � � 〈 � ; �〉

� - � � � 〈 � . � 〉

� : �

� � - � , � ; � �� � � � �

� � � � � 4 ; � � � � � � � { ��

�} � � � � � � � � � { ��

�} � � � � ( � � � < � . -

� ; � � , ( � � � � . , � � , < . ; � ; � � � : � � � � � � � � � � � � � 〈 � ; �〉

� � { �} �� � � 〈 � . � 〉

� � � < � � �

9 � � � , < � � � � : � � ; : � � � . , < , � � - . ; � ; � � � � . . ; � - � � , � , . � ; � � �

� . � , < � � � , � < , � � - . ; � � � � � � � � - � � , � � � � - � � � � ; � - . , � � < � �

� � � � �

! < � � � � . . ; � , . - , � � � , < , � � , : � : � � - � � � � . � � � - < : ; < �

: � < ; < � � . � , � � � � - , � � � � , � < . � < - � � � � , : � � � � A � � � � - ; . . � � � � � ,

- < � ; � - - �

� � 4 � 4 " 4 � � � ; . ; < � � � < � , � - � � � � � . , � , � , , � ; , < � ; � � � - � . �

, < � � � , � � � � � . , � � < � � < � � A � � � � � , � < . ; �

� � � � � � 4 � ; � � < � - � : � ; � � � ; �

� � � � < � � � � , � A � . . ; � , . - , � , � , � , . . � � C � � . � �

� � < � � � �

� ; , � A � � - < ; , < � � � � : ; � ; � - � ; � � . � , � . ; � ; � �

� � . , � � � � , ,

� � � � � � �

� . � � � , < � . ; � ; � � � . : , � � ; . � � � < � � . � , < � � � � � �

. , � � < � ���

0 � : , < � - � . � � ; � - � ; � � . � � � . ; � ; � � � � , : - . ; � ; � � � � : ;

� � - . , < � < � � � � , < . ; � ; � � � � � � ; � � � � : , � , < � � � � � � � � , < �

� . ; � ; � � - � � � � � � , , : � - . . � , . ; � ; � � � � � : ; � � , � ��

� � - � : � - . . , , � �

< � � � . , � ; � - � ; � � . � , . ; � ; � � � � � : ; � � � � � , , � . ; � ; � � � � � < � � � �

< - � � � , � - < � � � � , � < . � < - � A � �

� � - � � � � < � � � � � � � � � � � � � � �

; � < � . , < ; � �@ . ; � - < � � � � � � � , � � � . , � � < � - � , � � � � - � � � � � � � ; �

� , : � , < � � . ; � � � , : . - � - � � � � � � � � � ; � � < � , �

� � � � � � , � , , � ; , <

� ; � � � �

� � � < - � � � , � - , �

� < ; � � � � ; � - � ; � � . � �

� . � ; . ; . . � � � � , : �

, < < ; � , � � < � � . , � � , < � � � � � � �

9 � � � , � < . � < � � � � � , : � � � � A �

� � < � � � � � ; , � � � � , � , . ; �

� � � � < � � � � , � A � . . ; � , . - , � , � , � , . . � � C - < - . . � �� � � : ; � � � � -

< ; . . � , � � � � , : � � � � � A � . ; � ; � � � � � : ; - � , , < � - : � � � � : , � , < � � � �

� � � ; � , < � � . ; � ; � � �� , � ; � � � � � , < � � . ; � ; � � �� � � � , � - � , � � � ; �

- � ; � � . � < � �

� � < � � , < � � � � � � , , � . ; � ; � � �� � � : ; � - � � � , < � � . ;

� ; � � � < � . , � � . � � . � � < ; � � ; � � � � � � � ; � - � ; � � . � � � . ; � ; � � � � � � - � �� � �

′ �′ � � : , �

′ � � � � � � � �′ � � . ; � ; � � �� � - �

′ � � � � � , � � < � �′ . ,

Page 128: Программирование: теоремы и задачи

� � � " ��

� # & 7 � 3 6 7 & # 1 6 $ # � # � & - � � � % ' ' % 7 � 3 �

� � < � - . ; � - . ; , < - � � � � ; � � � - . ; : � , ; � , � � - � ; � � � � - � � � � � �

� < � . , � � . � � . � �� � - �

′ : � - . . , , � � < � � � . , � ; � - � ; � � . � , � . ; � ; � � �

� < ; � � �

′ � ��

� � � � < � � � � � ; , � ′ , < � � � � : � � , . - , � � � < � � � � < - � �

� , � - < � � � � , � < . � < - � A � �

� < ; � � � � � � ; � - � ; � � . � , � . ; � ; � � �� < ; � < � � � � , � - < � � � ; � - � � . � < � � � : ; � ; , � � � � � � � � , : � � � � � A � . . ; � , . -

� , � , � , . . � � C�

� � - � . � - < - . . � � < � � < � � � , � - : . � �� � - � . � � � . � � < �

� . , , < � � � � � � � - . ; : � � � � � , � - < � � � < � � � � . , � � � � � �

� ; � � �

: , � , � , � - � � � � � � � . � � � � � � �

′ �′ � � : , �

′ � � � � � � � �′ � � � � � � �

< � �

′ . , � � , < � � < � : � - . . , , � � A � � - < ; , < � ; � - � ; � � . � , � � � � � �

′ . , � � , < � � < � � ; � . � � < � � : ; �′ � ; � . � � - . , � � , < � � � � � � �

- �

′ . , � � , < � � < � � � � � � , � < � � : ; � , � � � � - � � � � � ′ � � - . ; : � , - <

- � ; � � � � - � � � � � � � @ ; : ; � ; � , � , . ; � �

� , � , � : � � < , � , � � � : � � � � � � � ; < . � � � � � � ; � � � � < � � 9 � � � ; � � ;

< - � , , < � � � ; � - � ;

�→ �

�→ �

�→ �

- : � � � - � � � ; � - � � , � � � � ; < � � � . , < �

� � 4 � 4 �4 � � - < ; � � < � � A � � � A � - � A � � � � � , � < . � : � � � � - � �

- � < � � . � , < � ; � ; � � � � � ; � � � - � ; � � � . , � , � , , � ; � < � . ; � - ; < �

� � � � , : � � � � � � � � , � < . � � : � ��

� � � � � � � 4 � � , � ; � � � � , : � � � < ; � � � ; �

/ ; B � H I F ; H � H 5 I � EJ H # � ?� C ! � H � � � 3 � � � � � � 3 � � 9 ! � " " � % H ? J H # � ?

� H 5 I � E

H ? I H 9 G H � C ! � H � � � 3 � � � � � � 3 � � 9 ! � " " � % H ? J H # � ?

� H 5 I � E

H ? I H 9 G H � C ! � H � � � 3 � � � � � � 3 � � 9 ! � " " � % H ? J H # � ?

� H 5 I � E

H ? I H 9 G H J H # � ?J : $ � ; F H 3 � 3 C 5 9 G H 2 > � 2 3 6 3 = 0 6 3 0 0 ? 0:

2 ; 2 0 � 3 = 0 � 3 � � 6 0 � 6 � 0 2 0 3 > � 3 � 3 � �

H ? I EH ? I E

� � � ; , � � � < � � A � � � � � , � < . � � , ; � - . � , < ��

� � - � � � . , � � - . ; : � , - <

. - � : . � � � - . � . � , < � � ; � � � � � ; � � � � � ; � � � � < � � � < � , � � � � � � , <

" ��

��

� � 7 # � � � 3 1 � 6 � $ & # � # 6 � 1 6 3 % � � �

. ; - � � � � � - � . ; � ; � � � � � � : ; � � � � � - � � . ; � ; � � � �� � - � � � � � - . ; :

� , - < � : . � � � - � � , : � � ; < , � � . � � < � � � � � � : . � � � � - . � < - � � . � , < � � < �

� ; � - � ; � � . � , . ; � ; � � � � � : ; � � � � � , , � . ; � ; � � � � . , � � < � - � - < ; , <

� � < � , < < � � � � , � � � � � , : � � � � � �

� � 4 � 4 � 4 � � � � � . � � ; . ; . . � , � � < ; � - < � � � � � , : � � � � ; � � . . ; � ; . -

� � � ; , . - � : � � � ; � � ; < - � - � � - � , � � � �� � � � �

〈 � � �〉 → 〈 � ; �〉 〈 � < � � �〉〈 � < � � �〉 → � 〈 � � �〉〈 � < � � �〉 →〈 � ; �〉 → 〈 � . � 〉 〈 � < � ; �〉

〈 � < � ; �〉 → � 〈 � ; �〉〈 � < � ; �〉 →〈 � . � 〉 → �

〈 � . � 〉 → � 〈 � � �〉

� � � � � � � 4 ! < ; � � ; � � ; < - � ; . , � � � . � < � � � � : � ; : ; , < � � : � ; � � < � , .

. � , � ; < . � , � � � ; - � � � � ; � � � � ; < � , < � � � � � - . ; � - - < , � � - . ; � � � - . ,

< , � � - . ; � � �

� 〈 � � �〉

- � � � � � � - . < � � � - � � � � � �

� 〈 � � �〉

� � � ; � � ; < - � , , < � < ; � , . , � � � � � � � � ; � - � � : . � � � , � � � � ; < � � -

� � ; � � � - � ; < � - � ; . . � � � � � : ; � . ; � � - � , �

〈 � < � � �〉 → � 〈 � � �〉〈 � < � � �〉 →

! < - � � � ; . - � , . - . , � � � < � � - . � - � - ; � � . � � - �

� ; � � � � ; � - � � < - � ;

�→ � � � � � . � � � � � � � . ; � , . - < � . ; : � ; � � ; � - � ; �→ � - → � � �

< , � � - . ; � � . � , - � � � � � � � � ; � � � � ; < - � . ; . , < , � � - . ; � � , : - .

< � , . . � � � � ; � - � � � . ; � , . � . ; � � < � , < < � � � � - , < , � � - . ; � � � � � , � � � �

� � � � ; � - � � : . � � � , � � � � ; < � � - � ; . . � � � : . � � - � � ; � � � - < ; � , � � . �

Page 129: Программирование: теоремы и задачи

� � � " ��

� # & 7 � 3 6 7 & # 1 6 $ # � # � & - � � � % ' ' % 7 � 3 �

� , < - � � , � ; . � � � ; . . � � � � � � ; � � . ; � � - � , � �

�→ � � �

�→ �

�→

� � . � . ; � , . - < � . ; � � ; � - � ;

�→ � �

�→ � �

�→ � �

� � → � � �

� � → �

� � →

� � � � . , � � : , � � < � � � : , � ; < � � ; � ; . � , . ; � - � , � < � � � < � � � � � � - < �

, � - � � : < ; � - < � � � - ; . - � � � � , : � � : � . � � � � < , � � - . ; � � . � � - � � � � � �

� � , < ; - � - � � � � . � � ; . - �� ; � � - � , � � : � � � ; � - � ;

�→ � � �

� < � : ; � < � � � � , : � � �

/ ; B � H I F ; H � H 5 I � EJ H # � ?� H 5 I � E

� C J � % H ? J H # � ?

� H 5 I � E

H ? I E� C J � % H ? J H # � ?

� H 5 I � E

H ? I EH ? I E

� � , � � � � � , � < . � < - . ; : � � � < � � � � � � � � . , � , � , , � ; � � � � ; � � � �

� � < � � � , � ; � . � � ; � � � � , � - - . � . , � � � � : - < � � < � , � � � � � - � ; � . �

� � � , : - . , . - � � ; � � � - � ; � � � � , � - � � � � : - < � � ; < ; � , � < � � � � � � � �

. , � , � , , � ; � � � � ; � � � �

� . ; � � � - � . � � � � � ; . � � � � ; � - � ;

�→ � � �

�→ �

�→

" ��

��

� � 7 # � � � 3 1 � 6 � $ & # � # 6 � 1 6 3 % � � %

� � - � � : < � � � � � , : � � ,

/ ; B � H I F ; H � H 5 I � EJ H # � ?� C ! � H � � � 3 � � � � � � 3 � � 9 ! � � � " " � % H ? J H # � ?

� H 5 I � E� C J � % H ? J H # � ? � H 5 I � E H ? I E

� C J � % H ? J H # � ? � H 5 I � E H ? I E

H ? I H 9 G H � C ! � H � � � 3 � � � � � � 3 � � 9 ! � � " " � % H ? J H # � ?

� H 5 I � E� C J � % H ? J H # � ? � H 5 I � E H ? I E

H ? I H 9 G H J H # � ?J : $ � ; F H E

H ? I EH ? I E

� � � � , � < . � < � � � < � � � � < � , � � , < � � < � � � � ; � � � � � . , � , � , , � ; � � �

� ; � � � �

- < ; � � - � , : � . . � � : ; � , , � � � � � ; � � � � � � � , . . � - � , < � � � - : � � � <

� , < < � - , � , : � � � � - � - - ; . � � - � � - � - � � � ; � - �

� � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � ! � " � # $ $ � % � & � % � ' (

� � � � � � � � � ! � " � ) � � � � � � � # $ $ � % � & � % � ' (

* + � , � � - . / # 0 � � '

/ ; B � H I F ; H � H 5 I � < * J ! � : � < * J B 9 " EJ : $ ! � H � $ � " E

� C J � % H ? J H # � ?

� B 8 H E

H ? I EH ? I E

/ ; B � H I F ; H � H 5 I 1 / ; E

� H 5 I 2 I I E

� C J � % H ? J H # � ? � H 5 I � H G � 1 / ; E H ? I E

H ? I E/ ; B � H I F ; H � H 5 I � H G � 1 / ; E

� C � H � $ 3 ) 3 � % H ? J H # � ?

� H 5 I � < * J ! 3 ) 3 " E

� C J � % H ? J H # � ? � H 5 I 1 / ; E H ? I E

H ? I H 9 G H J H # � ?

Page 130: Программирование: теоремы и задачи

� � � " ��

� # & 7 � 3 6 7 & # 1 6 $ # � # � & - � � � % ' ' % 7 � 3 �

J : $ � ; F H E

H ? I EH ? I E

/ ; B � H I F ; H � H 5 I 2 I I E

� H 5 I � 5 � � E

� C J � % H ? J H # � ? � H 5 I � H G � 2 I I E H ? I E

H ? I E/ ; B � H I F ; H � H 5 I � H G � 2 I I E

� C � H � $ 3 � 3 � % H ? J H # � ?

� H 5 I � < * J ! 3 � 3 " E

� C J � % H ? J H # � ? � H 5 I 2 I I E H ? I E

H ? I H 9 G H J H # � ?J : $ � ; F H E

H ? I EH ? I E

/ ; B � H I F ; H � H 5 I � 5 � � E

� C � H � $ 3 3 � % H ? J H # � ?

� H 5 I � < * J ! 3 3 " E

H ? I H 9 G H � C � H � $ 3 ! 3 � % H ? J H # � ?

� H 5 I � < * J ! 3 ! 3 " E

� C J � % H ? J H # � ? � H 5 I 1 / ; E H ? I E

� C J � % H ? J H # � ? � H 5 I � < * J ! 3 " 3 " E H ? I E

H ? I H 9 G H J H # � ?J : $ C 5 9 G H E

H ? I EH ? I E

� < ; � � � � � � : - < � � � � � � , � � � � � . ; . . � , � . ; - � . � � � , � � � - � . � < � �

� < - � � � � � , : � � � � : . ; - � � � � . � , < : � � � � � - . ; � � � � � < �

� � ; � ; � , � < �

: � � � � ; , < � � < � � � � � < � , � � , < � : ; < � � � , : � ; � - < , � � . � , � � - ; . - , � � � � ,

: � � � � � � � � � � 9 ; � � , � : ; : � � � , � � � - � . � � � � � � , : � � � � � � - � � : � � ; . ;

< , � � < � ; < � � � � � < � � ; : ; � � � � � , : � � ; � ; � � < ; , < � � ; � - � � . � � � � , : � � � �

, . - - � � < � - � � � � . � , � � , � . , � � � . � � � � � � � , : � � � ; � � < ; � < � � ; � - � � . � �

. ; : � : � � ; . ; < � � < : , � � . � � � < � � ; � � < ; . ; � , � � ; , < � �� ! < � . , � � , � - : . � �

, � - � � � ; � � ; < - � , , < � � � ; � - � � �→ � � � < � - . � . - � , � � . , � � � � : - < � �� � � � � - � ; � � � � � < � � . � . ; � - ; . . ; � � � . ; � - � � ; . � . ; � � � � � , : � � ;

/ ; B � H I F ; H � H 5 I � EJ H # � ?� H 5 I � E

� C J � % H ? J H # � ?

" ��

��

� � 7 # � � � 3 1 � 6 � $ & # � # 6 � 1 6 3 % � � .

� H 5 I � E

H ? I EH ? I E

. , . ; � ; . � - � ; , < � ; � � < � �

� : ; . . � � � � � ; , � � � � , : � � � � A � � � � � � � � A � � � � � A � A � �

� - � � . ; � , � � ; � < � � � - � � � � , . � � ; � < : � - . � . , � � � � - < ; . . � � � ; < - � � � : ; �� � � � � � � � � � � � � � - � � � � . � � � � � � � � � ; , < � : . � - . . - � � < � . ; � - � � - � ; . - ,

. , � � . � � . � � �

� � 4 � 4 � 4 � � < � � � � ; � � ; < - � , - � , � < � : � ; � � ; � - � ; . , < , � � - . ; � � � �

� � , � � � � ; < - � - � , � � - � � - :

�→ � �

�→

� � � � < � � � � � � � � � � � , : < ; � � � , < � � � � � � . , � . � � � � � , : � � ; < , � � . � < �

� � � � � � � - � � � � . � , < � ; � � � � � - � ; � � � � � : ; . . � � � � � ; , � ; �

. � , � ; � � � . , � , � , , � ; � < � � � � � � � . � � � � � � , � < . � � : � � � � � � � , : � � �

� A � � . ; � - ; < � � � � � , � < . � � : � � � � � � � , : � � � � A � � . , - � � � � . � � � ,

� � � - - �

� � � � � � � 4 � � . ; � - � � � ; � - � ; � � , : � � ; � � � � . ; � - ; < �

/ ; B � H I F ; H � H 5 I � EJ H # � ?� C ! � H � � � 3 � � � � � � 3 � � 9 ! � " " � % H ? J H # � ?

� H 5 I � E� C J � % H ? J H # � ? � H 5 I � E H ? I E

H ? I H 9 G H J H # � ?J : $ � ; F H E

H ? I EH ? I E

. ; � , � � , . - , � ; � � < � � ; � ; . < - � � , < � < , � � � < � � , � , : � , � � � - � . � � � � . �

� � � : � - . ; . , � � � � - < ; . . � � � ; < - � � , . � � ; , < � �! < ; � , � � � - � . ; � � � � � , : � � ; � � � - � ; � , . < . ; . , � , � � � - � . � � �

/ ; B � H I F ; H � H 5 I � EJ H # � ?J : $ � ; F H E

& % � 9 H J 5 ? I ! � H � � � 3 � � � � � � 3 � � 9 ! � " " I B J H # � ?

� H 5 I � E

H ? I EH ? I E

Page 131: Программирование: теоремы и задачи

� � � " ��

� # & 7 � 3 6 7 & # 1 6 $ # � # � & - � � � % ' ' % 7 � 3 �

� � � � ; � � . � � � . � � � � � , � - < � � < � � � � - � ; � , . < . � < � < ; � �@ ; � , � � ; , � � < �

� � � � - � � � � ; � � � . ; �� � < ; < � � . � � � � � , � - < � � � � < � � � � � < � < , � � � , � � �

- � . � � � � � � , : � � � � � � - � ; � , . < . � . , � , � � � - � . � � � � � , : � � � � , . - - � � < �

, � � , � � � - � . � � � � . � � � � � - � ; � , . < , . � � . � � � . , � , � � � - � . � � � � � � , : � � � �� � : < ; � - � �� C ! � H � � � 3 � � � � � � 3 � � 9 ! � " " � % H ? J H # � ?

� H 5 I � E� C J � % H ? J H # � ?J : $ � ; F H E

& % � 9 H J 5 ? I ! � H � � � 3 � � � � � � 3 � � 9 ! � " " I B J H # � ?

� H 5 I � E

H ? I EH ? I E

H ? I H 9 G H J H # � ?J : $ � ; F H E

H ? I E

� , � � � � � � � ; . : � C � � � � � � � . � � � � - . � < � � � � < � � � , < , - < ; � C -

< - . . � �� < � � � � � � � ; . : � � � . � � , � , . , < - � . ; � ; � � �

J : $ � ; F H E

� C ! � H � � � 3 � � � � � � 3 � � 9 ! � " � % H ? J H # � ?

� H 5 I � E� C J � % H ? J H # � ?

& % � 9 H J 5 ? I ! � H � � � 3 � � � � � � 3 � � 9 ! � " " I B J H # � ?

� H 5 I � E

H ? I EH ? I E

H ? I E

� , � , � � � . � < � , . . - � � � � . � � � � - . � < � � , � - C � � . � � � - � � � + � � � �

� ; � . � . , � � � � � . � , < � - : � � ; � - < � � � � � � - , � . , � . , � � � � � � � - , C

� � � < � � � , � � � ; � . � - < - . . � �

J : $ � ; F H E

� C J 5 ? I ! � H � � � 3 � � � � � � 3 � � 9 ! � " " � % H ? J H # � ?

� H 5 I � E& % � 9 H J 5 ? I ! � H � � � 3 � � � � � � 3 � � 9 ! � " " I B J H # � ?

� H 5 I � E

H ? I EH ? I E

� < � � � � - � ; � , . < . � � � - � , : � . . � � � � � , . , � , � � � - � . � � � � � � , : � � , � - . � �

< � � � � � � . , , . ; � , � � ; � - < , � ; � - � � - � � ; � �

" ��

��

� � 7 # � � � 3 1 � 6 � $ & # � # 6 � 1 6 3 % � � �

� � 4 � 4 �4 � � � ; . ; < � � � � � , � < . � < � � � - � , : � . . � � � � � , . , � , � � � - � . � �

� � � � � ; � � � . , � � � , : < � , . . � � � , . � � � � . ; � , � � � - � . � � �

� � � � � � � 4 � ; � � < � - � . ; - � � � � � , , . ; � ; � � � � � : ; � � � � � � � , , � � . ;

� ; � � � �

� . � � � , : < ; � � � , < � � � - : , � � . � ; < , . ; � - - � � � � , : � � ; < , � � . � � �

� � - � - � � ; . - � . , � � � � � - � . , � � < � � � � � � - � � � . � � . � � � : . � � � . , � �

< � � � � . ; � ; � ; � . , � � � � � � , � � � � � � � � � �� . � ; � - ; . < � - � � ; � � � � � - < ; . �

. , � � � � � � - . . - � � C ⇔

� � � � , : . , , � � � � - < ; . . � , � � � � , < � � � � � � � �� � � � ; . , . - , - . � ; � - ; . < ; � , � - � < ; � � � � � � , : . , , � � � � � � < � � � , � - :

. � � , � - � < ; � � � . , � � � � � � � < � . ; � , � � � � � � � � � � � - . � - � ; � < ; � � - �

� - : � < - � � � � - . � ; � � � � - � � < � � � � < � � � � � � � ; � - � ; � � . � , . ; � ; � �

� � � : ; � � � � � � � , , � � . ; � ; � � � � �

� ; � � ; � < - � , � � - . ; � - - � � ; � � ; < - � - - � � � � . � � < � � � ; � , . - � � � � -

� � ; � - � ; : � � � ; � � � � < � . , < , � � - . ; � ; � - � , � < � - :

�→ � �

�→

� < � , �� � � � ; � � < � � � � , : � � ; < , � � . � < - � � � � � < � � < - � � � ; � - � . , � -

� � < � ; � � , < � � � - � � < � � * - � � � . � � � � � � ; � �� , � � � � � � � � ; � - � � : . � �

� , � � � � ; < � � - � ; . . � � - � � ; � � � - . ; � - � � ; � < � ; � � : . � � � ; � - � � � � ; .

: , � � � ; � � < , � . ; < - � . � , � � ; � � , � ; < - � , � < - � ; � � . � � � , � < � � �

� ; � � - � , � � � ; � � < � , . . ; � � � � , � � ; � � ; < - � ; : � � 〈 � � �〉 � � , < � � < �

. ; � - ; . ; < ; � �

〈 � � �〉 → 〈 � ; �〉 { � 〈 � ; �〉 }〈 � ; �〉 → 〈 � . � 〉 { � 〈 � . � 〉 }〈 � . � 〉 → � |

〈 � � �〉

� � 4 � 4 � 4 � ; � - ; < � � � � � , : � � � � � � � � , � < . � � : � � 〈 � � �〉 � � , : � � � < � �

� � ; � � ; < - � , - - � � � � . � � � - � � � � , < � � , � � � - - � � : , � � . � �

� � � � � � � 4/ ; B � H I F ; H � H 5 I � < * J ! � : � < * J B 9 " EJ : $ ! � H � $ � " E

� C J � % H ? J H # � ? � B 8 H E H ? I E

H ? I E/ ; B � H I F ; H � H 5 I 1 / ; EJ H # � ?

� H 5 I 2 I I E

Page 132: Программирование: теоремы и задачи

� � � " ��

� # & 7 � 3 6 7 & # 1 6 $ # � # � & - � � � % ' ' % 7 � 3 �

& % � 9 H J 5 ? I ! � H � $ 3 ) 3 " I B J H # � ?

� B 8 H E � H 5 I 2 I I E

H ? I EH ? I E

/ ; B � H I F ; H � H 5 I 2 I I EJ H # � ?� H 5 I � 5 � � E

& % � 9 H J 5 ? I ! � H � $ 3 � 3 " I B J H # � ?

� B 8 H E � H 5 I � 5 � � E

H ? I EH ? I E

/ ; B � H I F ; H � H 5 I � 5 � � EJ H # � ?� C � H � $ 3 3 I B J H # � ?

� B 8 H E J : $ � ; F H E

H ? I H 9 G H � C � H � $ 3 ! 3 � % H ? J H # � ?

� B 8 H E � H 5 I 1 / ; E

� C J � % H ? J H # � ? � H 5 I � < * J ! 3 " 3 " E H ? I E

H ? I H 9 G H J H # � ?J : $ C 5 9 G H E

H ? I EH ? I E �

� � 4 � 4 � 4 � � � � , : . , � � � � � , : � � , � � � ; . : � C � � � � � � � . � � � � < - < � �� � � , � �

� � � � � � � 4 � � . � � � , : � � � ; � ; < � � � < � � , � � � � , : � � � � � . � � ; � < � � � -

C � � � � � �

� �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � < � � � ; . : , � , � � � ; � � < � - � , � � � : - . � , < � : � � � � , � � - � � � � : - � �

< - � 9 � � � ; � � ; < - � , � . ; . � � ; , � � � � � < � ; : - � - - � � � � � ; . � � � � � �

� � <

, � � - : , � � � : . � � * � ; . , � � � . � � - < ; < � � � < � � � � � � , , � � � � : ; � �

� , � : ; . ; � , . � , � ; � � � � , � � � . , < , � � - . ; � - . � . � � - � � � � � � ; < � � : . �

- . � � ; � - � � , � - . ; � � � � , . � < � � ; � � ; < - � � � � < � � � � � ; < � � � ; � - � � � �

. � � � � � : � . ; � , � � � � - � � � � � � � � : - � � � � - . � < � � � . , < , � � - . ; � ; � � � ; �

� � � � � � � � , , * � � � ; � � . � � : ; : - � < ; � � ,

� � � � � � � � � � � 4 � � $ - ' $ - $ # � # ' � � � � ; � � � ; � � ; < - � , . ; . � � ; , < � � �

� � : � � � � < � � � � . ; � ; : � � � ; � , . ; � , . , � � : � , � � ; , < � ; � � � � , � � � - .

. , < , � � - . ; � � � �

" ��

��

� � � # � � 7 ' � % � � # � % � � 2 � � � " � 1 � � % ' ' % 7 � 3 � � �

� � 4 � 4 � 4 � � � � ; : � � � � � � � : - � � � � � � � ; � - . < , � � - . ; � � � � � , < � � , <

, � � � , � � � � � � � : �

� � � � � � � 4 � ; . � - � . � , . , < , � � - . ; � � . ; � , . � � < � . , . ; � - - � � � , � - �

� � � � , , � � � � : ; � � � � - � � � � � � � - - - �- - - �- - - � � : , � � � � . , < , � � - . ;

� � � < � . ; � , . � � - � � � . � � � � - . � � : - < � � � � � � � � � � � : � , �

� � � < � � �

� � . � � , � , < � � - < � � � � � : < ; � � � < � � � < � � � - � � , � , , . , < , � � - . ; � . ; � ,

. � � � � ; . � � , �� � � � � ; � � . � � � � � � � � . ; : � : � � ; . � � ; < � - . : � � � - , � � � : � -

. , � � � � : ; < ; � � � * ; � < � , � - - . . , � � < � � � � � . , < , � � - . ; � ; � � � � � : - < �

. , � � < � � � , � � � � � � < � � � , < � � , < � , � � � � � � � : � - . ��

� � 4 � 4 � 4 � � � ; � � ; < - � , � � � ; � - � ; � -

� � �→

� � �→ ( �

� � (→ � �

� � (→ � � �

. ; � < - � , � � � � � � � : � � � ; � � � � � � � � - : � � ; . ; < � � � < � � . , : - . < � , . �

� � � � � � � 4 � ; � , � � � � � ; � , � � . � � � - � , . - < � < � � � � � � � ; � - � � � � �

�→ ( �

< � � � : , < : ; � � � , ( � ; � � ; � � � � � � . ; � - . ; , < � . ; � � < � � � , <

� � - � , . - < � � < � � � � � � � ; � - � � � � �

�→ ( �→ � � � �

� , � � � , � : � � . � . ; � , . - < � � . ; ( � � - . ; � , � < � � � � - � � � � � � � � � ; � �

� � � � ; � �

�→ ( �→ � � � �→ � ( � � �

- ( : � � . � . ; � , . � < � � � � � � �

�→ ( �→ � � � �→ � ( � � �→ � � � � � �

� ; � , , � , � � � , � : � � . � . ; � , . - < � � . ; � � < � , � � � � � - . ; � , < � , < � , � � � �

� � � � � � ; � � : , < � - � - � � < � � � � � . ; � < - - � � � � � � � , < . ; � - . ; < � �

� � � � � � � � � : - � � , - . ( �

�→ ( �→ � � � �→ � ( � � �→ � � � � � �→ � � � � �

- : ; � , ,

- - - → � � ( � � �→ � � � � � � �→ � � � ( � � � �→ � � � � � � � � � �→→ � � � � � � � � �→ � � � � � � � �→ � � � � � � �→ � � � � � �

Page 133: Программирование: теоремы и задачи

� � � " ��

� # & 7 � 3 6 7 & # 1 6 $ # � # � & - � � � % ' ' % 7 � 3 �

< � < � , � � , < � � < � � ; � � ; < - � - � � < � � � < ; � � � � , < � : � � - � ; � , � � � �

� � � � : ; � � � � � - � , . - � � � < � � . ; � � - � , � � . ; � � , � , : . � � � ; � , ; � � �

� , � � � . , < , � � - . ; � � � � � ; . ; � � . , < , � � - . ; � � � < � , � � � - � , , � � � � � � -

: ; � � � � � : , � � � � � � - . < , � � - . ; � � � � ; � � � � � � - . < , � � - . ; � � �

- . , < , � � - . ; � � � �� � < � � � � ; � � ; < - � , , < � � � ; � - � ;

�→ � � �

�→ � �

�→ �

� ; � . ; : � � � � � ; < � � : . � - . . - � �

� � � � : , � � � < ; < � � : , � ; < � � < � < � � � � � �

� � � : � . ; � , � � � � - � � � � < � � � ; < - � � � : . � � � � � � ; � � � < � � ; � � � � � : - < �

- . � ��

� ; � � < � - � � . � , < � � � ; � � � � � < , � < , � � - . ; � � � � � � < � � � � . ;

� - . ; � < � . , � � < � , � � � ; � � � � � : - � � , - . � � ��

� ! < � � . � , < � � � ; � . �

� ; � � � � � � � , : - . � . . � � � � ; � � � � , � - - . � � � � � : - < � � � < � , � � � � �

; < ; � , � ; � � � � , � - - . � - - . � � � � � : - < � � � < � , � � � � � < �

� � � � - ; . . � � � , < � : � � � � � - � , . - � � . ; : � � � < � � � � ; � � � � � � � ; � � � �

- � ; � � � . , � , � , , � ; � - � �

� � � < � � � � ; � � �

� , : � � � , < � � < � < ; � � . ;

� � - � , � � � < � - . � � � � � : , < � � � , : , . � � � < � , � � � � � ; - . � � � ; � � � : , <

� � � , : , . � � � � � � . ; � - . ; � � , , � . ; � � � � � - . � ; � � � � �

� � , : � � � - , � � � ,

: , � , . - � � � - < � � ; � < � < � � � � � � , � � �

� ; � � � . - � � � < � � � � , : , � , . - , � � � � : - � � < - � 9 � � � ; � � ; < - � , � � � � : ;

. � < � � � � � : � � � � � ; - . < , � � - . ; � � � �� . � � � , � - : . � � � � � ; . � � � � � � � ; , < �

. ; � � � ; � � � � - . < , � � - . ; � � � - . , < , � � - . ; � � � �

� � . � < ; � , � � � � � - < �

� � � � � : - � � < - � : . � � � � � � ; � � : , � ; � , � � < , � � - . ; � � - . , < , � � - . ; � �

- . : � � � � � � �� � � - � � � � � - < � � � � � � : - � � < - � � � ; � , . � � ; . ; . - � < � � � �

� < � � : ; � . � � � � � : - < � � < � � , � : ; � � : � ; . � � , � ; , < � � � � � : - � � < � � � � ; �

� ; < - � , � < � , � � � � � : - � � < � - . . ; � ; � � . � � � . , < , � � - . ; � ; �

� � � � ; : � � � � � � ; � - . < , � � - . ; � � � - . , < , � � - . ; � � � � , � , . � ; � � �

� � � . . ; � ; , � � . � , < � � � , � < , � � - . ; � � � � � � < � � � � . ; � - . ; � < � . , � �

< � , � � � ; - . < , � � - . ; � � � � � � � � : - � � , - . ��

� � � � � ; , � , � - - . � � � � � �

. , < , � � - . ; � ; � � � � : - < � � � < � � : . � � � � � - . < , � � - . ; � � � � . , - � � ; , < � � � - �

� ; � ; < � - � ; , � � - � � � � - � � � , : , � , . - - � ; � � � � � � ; < � � � � � - . < , � � -

. ; � � � - � - � � � � , � � � ; �

� � � � : , � � � , : � � � ; � ; < � : ; � , , � � < � � < � � � � � - ,

� � � � � . , . � �

� � � � ; : � � � . , < , � � - . ; � ; � � , � , . � � � , : � � � � � . . ; � - � � . � , < � �

< , � � - . ; � � � � � � < � � � , � < � , � ; � < � � � � � � : - � � � � � � � ; � � ; < - � , � � � ; �

� ; . � , . ; ��

� � , � , � - � ; < � � � � � � � � , : � : � � , � � � ; . : , � ; � 9 � � � ,

< � � � � � � � � , : � � � � � � � ; , < � - � � � � � � � � , � - � � , < � � , < � � � � : - � � ,

� � � � � � � ; . � - � ; � � , , � . ; ��

" ��

��

� � � # � � 7 ' � % � � # � % � � 2 � � � " � 1 � � % ' ' % 7 � 3 � � �

� � � � ; : � � � � � ; � - � ;

�→ �

� � : , � � . , < , � � - . ; � � � � � � � � � � : , � ; � , , < , � � - . ; � � - . , < , � � -

. ; � � � � � , : , � - � � . � , < � � & % � � % $ � 2 � $ � � 7 � � ' � & % � # $ � � � � . . ; � ; , � � ,

� ; � � � �→ � �

� � � � � , : , � , . - � � . � � ; � . � � ; � � � � � � � < � � � � � : � � ; � � ,

. � � � � , : � � � , � - - . � � � � � : - < � � � < � , � � � � �

� � � � � � � � � � � 4 � ; � � ; < - � ; . ; . � � ; , < � � � � " � 1 � � % ' ' % 7 � 3 # 5 � , � - : � �

� � � � � � � ; � - � �→ � - �→ � � : - . ; � � � � � - � , � � � - � ; < � � - � . �

, < � ; � ; � � � �→ � - � ; � � � �→ � . , � , � , , � ; � < � �

� � 4 � 4 � 4 � � � , < � � - � � ; � � ; < - � ;

�→ �

�→

� � � � � : - � � � - � � � ; � - � � � � � < � � � � , : � � ; < , � � . � < - : - , . � � � � � �

� � ; � � ; < - � � �

� � � � � � � 4 � , < � - � � � � � � - . ; : � , - < � . � , < � ; � . ; � � ; � � � � � - �

- � � � � � � : � � � � � - � � � ; � - � � : � � � < � � � � � � � � � � � � � � � � - . ; : � , - <

� � � , : � � � �

� � 4 � 4 � 4 � ; � - ; < � � � � � � � ; � � ; < - � � : � � < � � � , � . � � ; �

� � � � � � � 4

�→ �

�→ �

9 ; � � � � � � � < � � � , � � � , � � � - � . � , � � ; � - � � . ; � , . , . � . ; � � � ; � � � , � � �

- � . � , �� � , : � � � ; � . ; : ; � ; � � � ; . � � ; , < � � < � : � � � � � � � � ; � � ; < - � - � � ,

< � � , < . , � � � , , � : . � � � � � . � � . � � � � � � : � � , . - � � , � � � � � � � � : ; �

� � 4 � 4 � 4 � � < � : ; . � � � � � : - � � , � � � � � � � ; � � ; < - � , � � � � � � � � �

< � � � � � � : , � , . ; � � � � , � � � . , < , � � - . ; � � � � � � � � � � : , � � � � � �

- . < , � � - . ; � � � � � � � � � � - . < , � � - . ; � � � - . , < , � � - . ; � � � �� � < � � � ,

< � � � < : � ; � ; . � - � . � � � � ; � - � ; � � ; � � ; < - � - . , < , � � - . ; � � � � � � , � � �

� ; < - � - � � � � - � , . - � - - � � � � : , � , . . � � � � � . , < , � � - . ; � � � � . ; < , �

� � � : � � - � - � � � � : - � � � . � , � � . � � � � � � � � - � - : � ; � � � ; - . < , � � - . ; � � � �

. ; � - . ; � � - � � . ; ��

� � � ; . ; < � � � < � � � < - � � � � ; � . ; . ; � ; � � � � - : � <

� ; . . � , � � � � � � � @ : , � � � - � � � � � � � � � < . � - � � � ��

Page 134: Программирование: теоремы и задачи

� � � " ��

� # & 7 � 3 6 7 & # 1 6 $ # � # � & - � � � % ' ' % 7 � 3 �

� � � � � � � 4 ! < - � � � � � � � - . ; : � , ; < . ; � � ; � � � � � - � � . � , < � ; � � ; .

� - � . � � � � ; � - � � �

� � 4 � 4 � 4 � � � ; . ; < � � � < � , � - � � � � � � � � : - � � � � � � � � � ; � � ; < - � , � < �

, � � � , � � � � � � � : , : - . < � , . �

� � � � � � � 4 � � , : � : � � ; � . ; : ; � ; � � � ; . � � ; , < � � < � . ; � ; : � � � ; � , � ,

� � � � � � � : � � � : � � ; , < � � : . � . . ; � . � � �

� � 4 � 4 �4 � ; � � ; < - � ; . ; . � � ; , < � � � $ # � � 3 1 � 6 � $ & # 5 � , � - - . . , � � < � � �

� � . , < , � � - . ; � ; � � � � � : - < � � � � � � . ; � - . ; � � , , � � � . � . , � � � ; : ;

� � , , . - � �� � � ; . ; < � � � < � � , � � � , � � � - � . ; � � � ; � � ; < - � ; � � � � < � � � � - .

� ; : � � � . , < , � � - . ; � ; � � � � : - < � � � < � � � � : . � . , � � < � , � � � � - . < , �

� - . ; � � � - : � � � ; : � � � . , < , � � - . ; � ; � � , < � � , < � � � � : � . ; � - . ; � � - �

� . ; � ; � � . � � � . , < , � � - . ; � ; � � � � < � � � � � . � < � , � ; , < � � . , � � � � , < �

� � � � � � ; � � ; < - � � � �

� � � � � � � 4 � � < � - . � � � � � : - < � � � � � : , � � . , < , � � - . ; � � ; � �

. , � � < � , � � � � �

� � . � � - < ; < � � � < � � < � � , � � � � � � � : � : � � � - , . , < , � � -

. ; � � � � . � . , . ; � , . � < � �� ; � � < � - � � � � � :

� �

� � �

- - -

� . . ; �

� � � . . ; � ; , < . , � � � � � � � ; � � � � � � � : ; - � , � � � � � � � : �

� �

� : , � � . , � � < � , � � � � - . < , � � - . ; � � � �

� ; � ; � � � < � � ; � , � < � � � � � �

� � : � < � � � . � , < � � < � , � � � � � � ; � , : � < , � � � � � � - � � � < � � � � , < � � < �

� � � � � , . � � � � � � . ; � - . ; � � , , � . ; � � � � , � � � � � � � ; , � < � � � . � � . � � < ; �

� ; � � � � ; . � , < � . , < , � � - . ; � � � � � < � � � � � � , < � � � � , : < � - - � � < � . ;

� , . � . . ; � �

! < � � � � < - � � � , � - < � � . � � . � < - � : . � . . ; � . � � � � � � , : , � ,

. - � � � ; � - � ; � � � - � , . � , � � � � . ; � � , � , : . � � � ; � , � � - � ; � , � � � � � � � � : ; �

� � : . � . . ; � . � < � � � � � � . � , < � : � � � � � � : � � - . . ; � ; � � . � � � . , < , � � - . ;

� ; � - . ; : � � � � � � � . � � ; < � � < , � � � < � � � � � � , : � � � � , . - � � < � , � ; , < �

� < ; � � � � � � � : , � �

� ; � - � � � � ; . � � � � � , � � � , � � � - � . � � � � ; � � ; < - � ; � � � � � � , < � - � - ; � �

. � � � � � ; , � � � � � � , < � : . , � � - � , . - � �

� � � � - � � : - < � � � , � � � ; . � � �

� ; < � � � � � - � ; � , . < . � � � � � � � � ; � � ; < - � ; � � - � - � � � � . � � ; < � � : � � � -

� - � , < � : ; � - � ; � � . . ; � ; . - � �

� � 4 � 4 � 4 � � � � � . � � � ; . ; . . � , � � � < � � - < � ; � � � � - < � � � � � , � � - � � � �

: - � � < - � � � ; - . < , � � - . ; � � � � � � � � � � ; � � ; < - � , �

� � � � � � � 4 � � � , : � , � � � - ; . . � � � � � � , � , < � : � � � - � ; � , � � � � � �

� � : ; � � � ; . � � - � � � ; < � � � � ; � . ; � � : � � � � � � � ; � , , � , � � � � - < ; . . � �

" ��

��

� � � # � � 7 ' � % � � # � % � � 2 � � � " � 1 � � % ' ' % 7 � 3 � � %

� ; < - � � � : . � � � � � � ; �� � � � - � - � � � ; � - � � � � � ; . - � � � � � � - . < , � � -

. ; � � � - . , < , � � - . ; � � � � � � � ; : ; � � , , < ; � - � - � � � < � ; � - � � � � � - < ; . . � �

� ; < � � � � : ; � � � . . ; � ; , � � , � , . � �

� � � � � � � � � � � : - � � � � � ; � � ; < - � , �

� � � � � � � , � � � � � � � : � � � : . � � � � � � ; � � � � � : - < � , � , . < ; : - � � �

! < - � � � < � ; � � , < , � � : , � � � � . . ; � ; < � � � � �

� . ; � ; � , � � � < � � ; � � < � - < - . , : - . < � , . . � � � - � � � � ; � . ; � ; � �

. � � � . , < , � � - . ; � ; �

� � - � . , � � < � � � � � � � , . < � . ; � - . ; , < � . ; < , � � - . ; � � - � � � � � �

< � � � . � � � � � � . - < � � � � ; . : � � � � - � : ; � - < � - � � � � � � � � � � � � - � �

. ; � ; � � . � � � � � � � � � � � � � � � - � < � � � � . , � , . � , < � �

� � - � . ; � - . ; , < � . ; < , � � - . ; � � - � 6 � � � � � < � � � � : . � , � � � � . ,

� � � � : - � � � - � � � � � � � � - � � � � � � , � � � � � � : : � � , . � � � � � : - < � � , � , .

� ��

� ! < � , � � ; � , : � - � � - � � � � ; , � � � � � � � �

� � - � � � < � � < � - . � � � � - � � � � , : � , < � � < � � � � : . � , � � � � � � � � : - � �

< � � : ; - < � � � � � < � � : ; � � � � : ; � � � � � � � �

� < ; � < � � � � ; � � � � � : ; � . ; � - . ; , < � . , � � < � � � � � . , < , � � - . ; � ; ��� � : � � ; . ; . . � � � � � � , � , � , � � , � � � � : � - . � � � � � . ; � - . ; � � - � � . ;

- � � � � � � � � . ; � - . ; � < � � � - � , . , . - � � � � : . � � � - < � � � , � � ; � - � ; �

< � � � � : � � � � < � � � � � � � � � � - . ; : � , - < . ; � � ; � � � � � , � � � . � , < � � �� � -

< ; � - � � � ; � - � . , < � < � � � � : . � , � � � � . , � � � � : - � � �

� � - < ; � � , � � ; � - � �

, < � � < � . � . � � � - � , . - < � , � � � � , � � � � � - � � � � � � � � ; � � � � - � < � �

� � � < � � � � . , . ; � � � - < � �� � - � � : - � � < ; � � � � ; � � � � - < � � �

G : $ � � 6 0 � 6 � 0 2 0 E

H ; ; B ; : $ C 5 9 G H E

+ H ; ; B ; $ ( 2 � 0 � � 0 � 6 � 0 2 0 � � 2 ; 2 0 � 3 = 0 E ,

+ ? B � H ; ; B ; $ ( ! � " ,

& % � 9 H ! ? B � H ; ; B ; " 5 ? I ? B � ! ! � H � $ 1 � � " 5 ? I ! � � � 6 0 " "I B J H # � ?

� C ! � � � 9 3 � � � 6 < � � � � = 3 � � �: � � 2 � ; � H � " � % H ? J H # � ?

� B 8 H E � � � � 3 4 3 > � � � � 2 ; 6 3 = 2 0 � E

H ? I H 9 G H � C ! � � � 9 3 � � � 6 < � � � � = 3 � � �: � � � � 2 � ; � H � "

� % H ? J H # � ?

H ; ; B ; : $ � ; F H E

H ? I H 9 G H � C ! � � � 6 0 " 5 ? I ! � H � ' ( 1 � � " � % H ? J H # � ?

H ; ; B ; : $ � ; F H E

H ? I H 9 G H � C ! � � � 9 3 � � � 6 < � � � � � � = 3 � � � 3 � H � 2 � 0 � 3 2

� � � � � 2 � < 8 � � � = � 0 � � 6 2 0 0 � � 0 ? 0 3 > � � � 2 3 � � � < � 0 ? 0

� � � � = 3 � � � � " � % H ? J H # � ?

� � 3 = � � 3 4 � 0 � � � 2 3 � 0

Page 135: Программирование: теоремы и задачи

� � � " ��

� # & 7 � 3 6 7 & # 1 6 $ # � # � & - � � � % ' ' % 7 � 3 �

H ? I H 9 G H � C ! � � � 9 3 � � � 6 < � � � � � � = 3 � � � 3 � H � � � 2 � 0 � 3

2 � � � � � 2 � < 8 � � � = � 0 � � 6 2 0 � 3 0 � � 0 ? 0 3 > � � � 2 3 � � � < � 0 ? 0

� � � � = 3 � � � � " � % H ? J H # � ?

H ; ; B ; : $ � ; F H E

H ? I H 9 G H J H # � ?

+ � 7 � � 1 ; 2 � � ,

H ? I EH ? I E

+ 2 � 0 � � 0 � 6 � 0 2 0 2 ; 2 0 � 3 = 0 ' $ ( ? B � H ; ; B ; ,

� � � � � - < � . ; � ; . � - � ; , < � ; � � < � � � � � � � � � � � � - � � � � � , . - - < , � � - . ; � ;

� . ; � ; � , � � � ; � � � � - � � : - < � < , . - , � � � � : ; - � - � < ; . � � � ; � ; � , � � . , �

. � � � - � � � , . � � � - � : � � � : � � � ; . , < , � � - . ; � � � � . ; � ; � , � � . . ; � ; � � � �

� < � � � ; � � ; < - � ; � , � � � , � � � - � . ; �� � � � � � , � � � , : � � � ; � ; < � � � � � ; . �

� � , : � : � � , � . ; : ; � , � � < � � < � . , < ; � � . , < , � � - . ; � � � . , � < � , � ; � � - , �

� � � � � : ; � � ; < ; � , . , < , � � - . ; � � � - . � � < � � � � . , � � � � : - < � . , � � < � � �

� � � ; � . , � � . � � : ; � - < � - . � � ; � � ; < - � - �

� � � � � � � � �4

• � � - � , : � . . � � ; � � � � - < � - � � � � . � , < � � ; � < , � � � , : , � < � - � � � �

- . � � : � < � � , � � � � � � . � ; �

• � , � < � - � : � � � � � � , : . - � � ; � - ; . < � � � . � < � - � - � � ; . , � � - � � : � <

� � < , . - � � � , � , : . � � � - � � � � ; � � � � : ; � � � � < � � � - � � � . � . ; � ;

. , , � � , : � � � - � - < � : � � � ; : � � � . , < , � � - . ; � ; - � ; : � � � - � � � � ;

� � � �

� � � , � < � � � . ; � ; : � � � ; � , � - � � ; � � : , < � - < ; < � � � � , � , :

. � � - � � � � � � � : ; �

• � � - � � ; � < - � , � � � � , ; � - . ; � - - � : � � . � � < ; � - < � < ; � � - � � � � � �

< � � � � . ; � - ; . � � ; � - ; . < � : , � < � - � � . ; � - - � � < - � < � � � : . � � �

- � � � � ; - � , � � � � � - � � � � ; � � - . , � � � � � � � � � � � � ; � � � � � � � � � . �

� � � � : , � < � - � � � � < � , < < � - - � < � � < ; � � - � , � �

� � 4 � 4 � 4 � � - � � � � , � � , < � � � � � < . � - < � � - : ; . . ; � � � ; � � ; < - � ; � < - � �

� � � � � . , � � � � : - � � � � � - � - < � � � � , : � - � ; � � : � � � , � . , < , � � - . ;

� � � �

9 ; � � < � : , � ; < �

� � � � � � � 4 � � < � � . ; � � - � , � � � � � ; � � ; < - � , , < � � � ; � - � � �→ � � ��

� � � : ;� ; � � � ⊂ � ; � � � �

� ; � � � ⊂ � ; � � � � , � - - . � � � � � : - � � � � < � , � � � � �

� ; � � � ⊂ � ; � � � � , � - - . � - � � � � � : - � � � � < � , � � � � �

" ��

��

� � � # � � 7 ' � % � � # � % � � 2 � � � " � 1 � � % ' ' % 7 � 3 � � .

� � � , : � � ⊂ � � � , : � � �

� � � , : � � ⊂ � � � , : � � � , � - - . � � � � � : - � � � � < � , � � � � �

� � � , : � � ⊂ � � � , : � � � , � - - . � - � � � � � : - � � � � < � , � � � � �

� ; � � � ⊂ � � � , : � � �

� ; � � � ⊂ � � � , : � � �

� ; � � � ⊂ � � � , : � � � , � - - . � � � � � : - � � � � < � , � � � � �

� � : � � . � , � � ; � - � ; � � . � � � � � < � ; � . ; � ; � � � � � � � : ; < � � . � , < � ;

� ; � � � ; . ; < , � - � � � , : � � : � � � , � < , � � - . ; � � � - . , < , � � - . ; � � � �� � - � < � � . ; � ; � � � � � - <

� � � ∈ � � � , : � �

: � � . ; � ; � � . � � � . , < , � � - . ; � ; � -� ∈ � ; � � �

: � � � � � � � � < , � � - . ; � ; � �

� � � � : , . - , . ; � ; . � - � ; , < � � � � � : ; � � - � ,

. , . - , � � ; � - � � , � , < ; � < : ; � ; < � . � � � , � � , � , . < � � . � , < � � ; � �

- � � � , : � � �

Page 136: Программирование: теоремы и задачи

� ��

� � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � �

� , � � ; � � � ; � � < � - � , � � � : - . � , < � : - . < ; � - � , � � � � � ; . � � � ; �

. ; . � � ; , � � � � � � � � ; . � � � � � � ; < ; � , . , � � < � � � , � � � � � � . . � , , � � � ;

� - ; . < � �� �

� � � � � � � � � �

� � ; � < � - � - � � � � � � ; . � � � ; � < � � � � � ; . � � � ; � � � � , � � � � � < � � - < �

. , � , � � � � � � � : � ; � � ; � � � � � � � < � � � � � � . < � � - < � . , . ; � ; � ; � ;

� � . � ; �� � � � � : � 9 � � � ; � � ; < - � , . ; . � � ; , < � � � % $ - ' � , � - . ; � ; : � �

� ; � , . ; � , . , � � : � , � � ; , < � ; � � � � � ; � � � . , < , � � - . ; � �

� � 4 � 4 � 4 � � � ; . ; < � � � < � , � - � � � � � � < � � � , , - . < , � � - . ; � � � � � � � �

: - � � � < � � . � - � , , < � � ; � � � � � � � : � �

� ; � � � : , < � : � � . � � � < � , < � . ; � � ; � � � � � � � : � . ; : � � . ; � , � � : � � � � ,

: , � - � � � . � < - , � � 1 � � # 4 � 6 6 % & % � 6 � # $ # ' ��

� � < � � � � � � , , � � � � - � � � �

� � : , < � � ; < � � � ; < � - : � � � � , � � � � � � � � < � � � , � � , < � : , � ; < � � ; � < , �

� - . ; � � � < ; � - . , < , � � - . ; � � � � . ; � ; � , � � � � � � � < � �� � � : , � � � � � � , ;

� ; . � , � , . � : � ; � - : ; : , � < � - � �

� � � � � . � � , � , . , < - � , � � � � - � � � � � � � ; � � , � � . ; . � � ; � < # 3 � � � � & - '

- � � � � � � - � � � . . ; � ; � < � � � � � � . , � � � � ; � � � : ; � - � , � � - . �

� � < � : , � < � - , . ; . � � ; � < 6 � $ � � # ' �

� � � , � - � � ; � ; � � ; < � � : . � � � - . � � ; � - � � � ; � � ; < - � - � � ; . ; � ; � � � . � � �

� � � ; � � < � � ; . � , � ; , < � . ; � , . - < � , � . ; . , < , � � - . ; � � < � � � - � � � ,

� � � � ; < - � < � � � � � ; � - � ; � � � - � < � � � � � � � . , � , . � , < � �� ! < � : , �

< � - , . ; . � � ; � < 6 $ � � 7 3 # 5 � - � - � � � $ � � � & � � ' �

� < � , < - � � � < � � � � � � � , . , � � � � , < � : , < , � � - . - � � � ; . . � � � � � : . � �

- < � � , - < � ; � - - � � � � < � � < � � ; . � , � , . � � ; . . � , : , � < � - � �

" ��

"�

� � 1 � � # 4 � 6 6 - � % �

� � � � � < � � < � � � � � � � , . ; � � � , � 1 6 � � " & # � % $ � � " % � 7 6 2 � , � - � �

� � � < ; . � � - < � � � < � � � ; � � � � , � � < ; � < � , : - . < � , . . � � . , < , � � -

. ; � � . ; � ; � � . � � . , < , � � - . ; � � � ; � � ; < - � - �

� � 4 � 4 � 4 � � � ; . ; < � � � < � : � � � � � � � � � � � ; � � - . < , � � - . ; � � � � � , �

. � . ; � , � � ; � � - � � � � � � � � , � � , < � � , < < � � : ; - < � � � � � < � � : ; � � � � : ;

� � � � � � � � � : - � � � � � ; � � ; < - � , �� � � : , : � � ; . ; < , � � < � ; � < ; . � � - < � � . ;

- � . � � : . � . . ; � . � , � � < � , < < � - , � , : � � � ; � � � - � � � � : ; � - - � � , � . �

. ; � , � � ; � � - � - � � � � � � � , ; � - �

� � � � � � � 4 � � - : � - � , � � � � � � . , � , . � , < � � � � - � � � < � , � � � � � �

� � : � , � � ; , < � � � , � � � ; . � � ; . - � � � � � ; < . � � � � ; � � � � � � : ; � ! < � < � � � � :

� � : , < � � ; � � � � < ; � � ; � � � � ; � - � ; , < � � � . , � � � ; � � � , - � � � � � �

< , � � - . ; � � . � , � � ; � - � � � � ; . � � � � ; : � � � � � � � � � , � � � < � , < < � � , <

� � ; � � � � � � � : � � � � ; < . � , � � < � , < < � - , � � � < � : ; . � � ; � � � � � � � : � � � , :

< ; � - � , � , � � < � . ; � � � , : . - � . , < , � � - . ; � � � � � � � , < � - < � , � , � � � � :

� ; �

� � - � , . - � � � < � � � . , < , � � - . ; � � � � ; � - � � � � ; � � ; < - � - � � � : � � . �

: � - . � < � � , � , � � � � : � � � � , � � � , � - � � ; � ; � � ; < � � � ; � - � ; � � . � ; , < � . ;

< , � � - . ; � �� ; . � - � ; � � < � < : � - � . ; � < : , � � . � , � ; � - � � � � � � - � � � � � , �

� < � � . � < - � � � ; < . � � � � � � � � , � � �

� � � � � � � � � � � : , � � � � � � , ; � , - . � , . , . - � � � � � , � � � � - � � : � <

� � ; � � � � � � . � ; � � � � � � . ; . � � ; � < 6 7 � 3 # ' � � � � � � , ; �

@ ; : ; � ; � � < � � , . - � � � ; � � � � � � � � : ; : � � : ; . . � � � � � � ; � � : - < � � < ;

� - � � � � ; . � � � � � � ; � - � � . � � � � � � � � � � � , � , : . � � � � ; � ; � � � � � � , ; �

� ; � . � . � � , � - < � � � � : , � � - � � : , � ; < � : � - � - � - � � � < � � � - , � -

� � � < � � � < � � � � ; � � � � � � ; � - � � � � , : � � � : � � : � � - � � � ; � - � � � , < � � < �

. , � � � � � � �

� � � � � ; � � � � - < � , � < � � , � , . - , � � - . - � ; , < � . ; � . � � , �

- � , � � � � � - � � � � ; � � � ; � � , � - - � � � � . � , < � < � � � � � � � < � � � � � � � <

� � � � � ; � � � � - < � , �� � � � . � , � � � , : , � , . - � � � < � - . - , �

� � < � * - � - � � � ; . ; � � ; � � ; < - � ; � � � � < � � � � - . � � � � � � . , < , � � - . ; � ;

� � . � � � � , < - � ; � � , � - � � � � � � - . < , � � - . ; � � � �� ! < � � � � ; . - � , . - , � �

� � : , < � , � : ; � � , : � � � ; � ; < � � � � � � . , . . � � �

� � < � �→ � � � : . � - . � � ; � - � � � ; � � ; < - � - � � � . , < , � � - . ; � � � �

� � � � - . < , � � - . ; � � � - . , < , � � - . ; � � � �

� � � , : , � - � � . � , < � � � � � � - .

< , � � - . ; � � � - . , < , � � - . ; � � � � . ; . � � ; , � � , � � $ - ' 3 # & 7 � 3 6 7 # ' � � ; � - � ;

�→ ��

� � � � . . ; � , . - , � + , � 9 � . < � �→ � �

� � � � � , : , � , . - � � . , � � � � � : � <

� , � � � ; � � � < � � � , � � � � � < � � : , � - � � � < , � ; . , � � � , : < � , . . � � , � , :

� � � < � � � � � � � � � : , . , � � < � � � � � � � , � . � . ; � , � � ; � � , � � � � � � � �

� , ; �� � 4 � 4 � 4 � , � , * � � � � � - � � � ; < � � < � � � � , : , � , . - , . ; � . � � , � � ; � � � � �

� � : � � �

Page 137: Программирование: теоремы и задачи

� % � " ��

� � & 7 % 3 6 � 3 � 6 3 � 5 � % � � # � 6 � � $ % & % � � % $ # � � � �

� � � � � � � 4 � ; � � < � - � � , � � ; � � , � � � � : � � - : ;

〈 . ; � ; � � . � � . , < , � � - . ; �〉 � � �→ � � ��

� : , � � � � � � - . < , � � - . ; � � � � � � � � � � - . < , � � - . ; � � � - . , < , � � - . ; � � � �

� , � � . . - � ; � � - , � � - � < � � � � � ; � � - � � � ; . � � < � , � � � � � . < , � < � � ; � -

� ; �→ ��

< � � � � � , : - < � � � � < � � � � , : � , < � � � � . - < � � � < � � � � � , : � �

� ; � ; , � � � < � - . � � � � � � . , < , � � - . ; � ; � � . � � � � , < - � ; � � , < � � � � � - .

< , � � - . ; � � � � < ; � � < � � � ; � � � � � � � : � � � ; � � � � � , < � � < � � � � : � � , .

: � � � ; � � � � � � � � : ; � ; � � � � < � � � � ; - . < , � � - . ; � � � � �

� � 4 � 4 �4 � , � � � ; - . + , � 9 � . < � �→ � � � . � ; � < � � � � , � - : . � � . ; ��

� �

� ; . ; < � � � < � , � - � � , � . - � � < � < � � . , � � � < � � � - < � � < � � � � � � , . . � ,

� . � , < � � � � � . , . ; � - - < � < < � � � � � ; � � , - . � � ; � - � : � � . , < , � � - . ; � ; �

� � � � ; . � �� ! < � � . � , < � � � � � . . ; � ; , < � + , � � � �

� � � � � � � 4 � . � � , : � : � � , � . ; : ; � - � . � � � < � + , � � � � � < � � � � � < �

� � , < � � � � - < � � � � � ; � � � � � � � : ; � � , � , , ; � � � � � � ; � � � � . , < , � � - . ;

� ; �� �

� � 4 � 4 � 4 � � � ; . ; < � � � < � � � � , : � : � � , � * � ; . , � � . � � < � � � - < � � � � ;

� ; � � � � � � ; � � � � � + , � � � � � < � � � < � � � < � � � , < � � � � � � < � � � � � ; � � �

� � � � : ; � � , � , , � � � � � � � � � : , . - � . , < , � � - . ; � ; ��

� � � � � � � 4 � � � : � � - � � � < � � , . - , � � ; � � � � � � � � : ; � � , . , < , � � - . ; � �

� � ; � ; � < � � � . � . ; � , . - < � . ; < , � � - . ; � � � ; � , � ; � < � � � - � < � � . - � , � �

. , - . � , . - < � � �

� � 4 � 4 � 4 � � < � � - < � � � ; � � ; < - � � � � : , � ; � � � : � � � ; : � � � . , < , �

� - . ; � ; � - � � : . � � � � ; � � ; < - � - . , < , � � - . ; � 〈 + , � �〉 � � � - � � � � , : � �

� , , � � � < � � : � � . � � � � � � . � < � � : � � � � � � � � . , < , � � - . ; � ; � - � � : . � �

� � ; � � ; < - � - � � . � � � � � � ; � � ; < - � , - . 〈 + , � �〉 � � � � : - � � � , � � , � , . < �

+ , � � � - < � � � � � � . - �� � � - � < � � < , � � - . ; � � - . , < , � � - . ; � � - � � : . � �

� � ; � � ; < - � - � � � � � < � < , � � - . ; � ; � - . � � � � �

� � � � � � � 4 � � < � � � . ; � ; � � . � � . , < , � � - . ; � � � ; � � ; < - � - �

� � � : ; �

. � � � � � � ; � � ; < - � , � � : , < � � ; � - � �

〈 + , � �〉 →

� � � < � , � � � �

� � � � ; : � � � � � ; � - � ; - � � : . � � � � ; � � ; < - � - � . ; � � - � , � � � � ; � - � ;

�→ � � � � � � � � � � � . , < , � � - . ; � � � � � < , � � - . ; � �

" ��

"�

� � 1 � � # 4 � 6 6 - � % �

� . � � � � � � ; � � ; < - � � � � : � � ; � - � � � ; � - � ;

〈 + , � �〉 → 〈 + , � �〉〈 + , � �〉 → 〈 + , � �〉 � �

〈 + , � �〉 → 〈 + , � �〉 � � �

- ; . ; � � � - � . � � � < � � - � : � � � - � - � � ; � - � ; � - �

� � � � . � � � � � � ; � - �

< ; � � � � � � < � , � � � � � � , < � � � � - < � � � , � ; � < � � , � - � � � � � � � , <

� � � � - < � � � , � ; � < � � < � � � � , < � � � � - < � � � , � ; � < � � � � � � � , < � �

� � - < � � � , � ; � < � � � � � � � � , � ; � < ��

� . : � � � - , � � � : � - . , � � ; � � � �

� � � � : ; � , � � � � � � � , � - < � � � < � � � � � < � � � , < � � � � - < � � � , � ; � < � ; � �

� � < � . , < , � � - . ; � ; � � � � � � � , < � � � � < � , < < � - - � < - � - � � ; � - � ; � - � �

� � 4 � 4 �4 � � � , � � � � � , : � : � � , � . ; : ; � , � ; . � � � < � � � � ; � ; < � - � ; , �

< � � � � � � � ; � � , � � � � : �

� � � � � 4 � � � � < - � . � � � � � ; , � , : � � ; � � � � � � - < � � ; < � � � , � � � ; . � � ;

. - � � � � � - � � : � � - , � . � < � - � � � ; � < � � � , � � � , � ; � < �� �

� � 4 � 4 � 4 � � � : ; . . � � � � ; � � ; < - � - � � < � � - < � ; � � � � - < � � � � < � � � � � �

� � � � � � � � � � � � � . � , < � � ; � - � - . � . � , < � � & 0 � � � . � � � - . ; : � , - < �

� @ ; � , � ; . - , : � � . . ; < � � � � �� � � , < � � � ; . - , < ; � � � � ; � � � � - < � ; � - : ;

, � � . , � . � � � ; � < � � ; < ; � < � , < � - . : � � < - � . � � � � ; � - � , . - � � � . , � . � �

� - � � � . . ; � , . - � � � � � ; . : , � ��

� � � � � < , � ; , < - . � � , : � : � � , � . ; : ; � - �

< ; � � ; � � � < � � , . . ; � � . , � � � ; � � ; < - � ; - � , , < � , � - ; � � . � � � - : � � � � ;

� � � � ; < � � � , � � � : - . . , < , � � - . ; � � � � - � � � � . < � - < � � , � � � � � � ; � �� , �

. , � , . , , � � � � - � , : � � � � . � , � � < � � , . - , �

� � � � � � � 4 0 � : , � . ; . � � ; < � 6 � 7 1 % 4 � � 5 : ; . . � � � � ; � � ; < - � - � : . � - . , �

� � ; � - � � � � � ; � � � � ; < - � � < � � � � � � < � , � , . ; � : . ; - . � � . - � - � � : � � , � � � �

� � � � � � � , : � � , � � � � - � < � � � � � � � � � � � - - - � � � � , � � � , : . , � � � � � � �

� ; � � - � , � � � � ; � - � �

� → � � � �

� � � � � � � � � . , < , � � - . ; � � � � � < , � � - . ; � � � � � : ; , < � � < � - < � ; � - �

�→ � � � � �→ � � � � �→ � � � � �→ � � � � �→ � � � �

� � � . - � - � � � ; . � � ; , < � . . ; � � � � � : � � � � - � ; . - � �0 � : , � � � � � � - < � � � < � � � � � � 6 # � � % 6 # $ % & # - < � ; � - , � �→ � � � , � -

� � � . � ; , < � . ; � � < � , < � � � ( � � � - . , � � < � � � � ( � - � � � � � , < � � � � ( � � -

. ; : � , - < + , � � � � � � � � � � < � � � � � � , : , � , . - � � � - � , � . � < ; � � � � � < , � , �

Page 138: Программирование: теоремы и задачи

� % � " ��

� � & 7 % 3 6 � 3 � 6 3 � 5 � % � � # � 6 � � $ % & % � � % $ # � � � �

� � : � � < � � � , . ; � ; < � � : � � � � : � � , � � � � < � - � � � ��

� � < - � < , � � - . ; �

+ , � 9 � . < � �→ � � � < � � . � , < � � � , � � � � � � � � ; � � ; . . � � - < � ; � - , �

�→ � � ; + , � � � � � < � � . � , < � � � , � � � � � � � � ; � � ; . . � � - < � ; � - , �

�→ � � � : , �→ � � � � � � , � � ; � - � � : � � . , < , � � - . ; � ; � �! � � - � ; � , . < . � , � � � , : , � , . - , � < , � � - . ; � � � � � � � , ; � � � � � ; � � ; . �

- < � ; � - , � �→ � � � , � - � � , < � � , < � � , � . � � � � � � � � , � � � � < � � � �

� � � < - � � ; . � - � ; � < � < ; � �

• � � � : , � � � � , ; � < , � , � � � � � � , < � � � � � � � - � . � � � ; . � - � ; , < �

. ; � �

• . , � � < � � � , � � , � � � . , . ; < � ; � - � ; , < � � ; � � ; � ; � < . , � � � � � � � � , <

� � �

• � � � � � ; � - � ; , < � � � �

• � � � � , � � � : � � ; , < � - � � , � . � . ; � , � � ; , < � �

� � 4 � 4 � 4 � � � ; . ; < � � � � - � ; � , . < . � < � � < - � � � � , : , � , . - � �

� � � ; . ; . - , � � � - � � ( � - ( � � - . ; : � , - < + , � � � � < � � � . � � � � � � - < �

� < , � , . ; � ; � ; ( � � � < � � � � � � < � � � � � � < � � � , � . � < � � � � � - . ; < , �

� � , � . � . ; � , � � - < � � � � � , �� � � - � � � � . � , � . , � � � � � � � ; . < � < * ; � < �

� < � - . � � � � � � . , < , � � - . ; � ; � < � < � : ; � � � � : - < � � � � ; � � : ; � � � < � � � � �

� � , � : � � ; � - < � � < , � � � � � , � � � � � �

� ; � ; � , � � � � � < � � , . - , ; � � � � - < � ; � � ; � � . . ; � � , � � � � - . ; : � ,

. � < � � � � - . � � � � . � � � � � � ; � + , � � � � � ; � � < � - � * � . � � - � � � � � < ;

� � � � � � � � ; : � � � � � � � � � - . < , � � - . ; � � � - . , < , � � - . ; � � � � . � ,

< � � � , � � � � ; � � ; . . � � . - � - < � ; � - � �

! < � � . � , < � � . ; . � � � � 6 # 1

6 7 # 2 & � � ' � 6 # # 7 $ � 7 6 7 $ 1 � $ � ' 6 � # $ 1 ��

0 � : , � � � � . . ; � ; < � , � � � � < � � �

� � < ; < � � . � � � � ; . ; < � � � < � * � . � � - � � � < � � - . : � � < - � . ; � < � , < � � < �

. . ; � , . - , � � < � � � � � : , � � < , � � - . ; � - � - . , < , � � - . ; � � � � , < � � < � � �

� - � , . � � , � - - . � , < . � � � < � � - - � � � � ��

� � � � - : , � - � ; . , , � � ; � � � -

. ; : � , . � < � � + , � � � � � � ; ; , < � � < , � � - . ; � � < � � * � . � � - - � @ . ; � , . - ,� � < � � � � � � - � � , < � � � < ; � - � � � ; � - � ; � �

� � � � - � � � � � � � � ; � � ; . � - < � ; � - , � �→ � � � � � - � � �

� � � � � . ; � - . ; , < � . ; � � � � � � � < � , < � � � � � � < � � � � � � ;

� � ; . � - < � ; � - , � �→ � � ��

" ��

"�

� � 1 � � # 4 � 6 6 - � % �

! < � � � ; � - � � � � � . � < � � � � � , : , � � , < � , - < � ; � - - . , � � < � � � , � � �

� � � � � - . � � � < � , < � . , . ; � - . ; � � - , � � � : � � � � - � ; . - � � � � � ; � � ; . . � ,

� ��

� < ; � � � � � � , : , � - < � � : � � � ; � - � . , < , � � - . ; � � � � � � � � � � � � - . ; :

� , - < + , � � � � ! < � : , � ; , < � � � : � � � � � ; � - � ; � �

� � � � - - < � ; � - � �→ � � � � � ; � � ; . ; � � � � � � ; . � � � ; � -

� � � � � ; � . ; � - . ; , < � . ; . , < , � � - . ; � � � < � � � � � - . ; : � , - <

+ , � � � �

� � � � - � � � � � : - < � + , � � � : � � . , � � < � � � � � � � � � - � � � �→→ � � � � ; � - � � � � ; � � ; < - � - - � . ; � - . ; , < � . ; . , < , � � - . ; � � �

< � � � � � - . ; : � , - < + , � � � �

@ ; � , < - � � � < � � � ; � - � � � � � � . � � ; � ; < � - � ; < � � ; � ; . ; � � � � � ; � - � ;

� � � � � � ; . ; . . � � � � � � � , : � � � � , . - � � - < � ; � - � � → � � � � ; � � ; . ;

� � � ; � . ; � - . ; , < � . ; . , < , � � - . ; � ��9 � � � , � < . � < � � < - � � � ; � - � � � � � , � < � � � , � - : . ; � , � - � � � � � , . �

� � � � : � � ; < � �� : - . < � , . . � , � � < � < � , � � , < . , � � < � � � � � � � . , . - � � � < �

< � � � � � , � � � � � � � � � � � ; � - � � � - � � � � . ; � � ; < � � � � < , � � - . ; � � � �

: � � � � < � � � � � � � � - . ; : � , - < + , � � � � � � � � < ; , � � � < � � � � � . - < � �� ;

� � < � - � � � ; � � � � � � � : � � � � < � � � � � � < � - < � , � ; � < ��

� < � � : ; � � �

� . � < � � � . � � . , < , � � - . ; � � � � - � � ; � - � � � � � < � � � , , � � � � � � : - � � � � �

� � : - � � < ; � , - � � . , � � � � ; � � � < � � � - . ; : � , . � < � � � � + , � � � � � : , <

� � . ; � � , . ; � � � � ; � - � � � � �

� � - , � � � � � � , � � � � � � � � � � � � � ; � � � � �

: � . . � � � � ; � - � < � : � � � - � . , < , � � - . ; � � � � � < � � � � ; � � : ; � � � � ; � - � �

� � � : � < ; < � � . � � < ; . � � - < � � � - . ; : � , . � < � � � � + , � � � � � < ; � � �

� � - � , . - < � < , , � ; � : , . - � � � - < ; � : ; � , , �

� < , � � - . ; � � � � � � � , ; < � , ; � � , � � . � � ; . ; < � < ; � �

� . ; � ; � ;

. , < , � � - . ; � � � � , < � � ; < � � � ; < � � . , � � � � � - � � � � < � ; � � . , . ; < � ; � -

� ; � � - � � � � � . - � � < � , < < � � � < � � - � , . , . - � � � ; � - � ; � � � . � . ; < , � � .

� � � . ; . � � : � , � � . � < � � � � � < � , � . ; < � ; � - � ; � � , � � � � � < � � � < � , < < � � , <

� � - � , . , . - � � � ; � - � ; � � �

� < ; � � � � � � . - < � � � ; � - , - < � ; � - - � � � ; � � ; . � � � < � � � � � � � �

< � , < � : � � � ; � - � . , < , � � - . ; � � � � � � < � , � � � � � � - . ; : � , - < + , � � � �! < � � � � , : , � � , < � � � � , : � � � - � � � ; � - � ; � �

� � . ; � ; � � . � � . , < , � � - . ; � < ; � � � �

� � , � - � < ; � � � - � → � � � � ; � - � � � � ; � � ; < - � - � � � - � � �

� � � � � . ; � - . ; , < � . , < , � � - . ; � ; � � < � - � < ; � � � � �

Page 139: Программирование: теоремы и задачи

� % � " ��

� � & 7 % 3 6 � 3 � 6 3 � 5 � % � � # � 6 � � $ % & % � � % $ # � � � �

� � 4 � 4 � � 4 � � � : , � ; < � � � - ; . . � � ; . ; � - . : � � � � ; � � ; < - � -

�→ � � (

�→ ((→ ( � �

(→ �

�→ �

�→ � �

� . ; : ; � � , � < � < , � . � � � � < � - � � ; � � ; < - � ; � � - � , � ; � � �� � � �

� � � � � � � 4 � . � , < � ; � � < � � : � � � ; . � - � . � � � � � - � , : , . � � < ; � � - � ,

. ; �� % %�

@ . ; � � ; � , . < � ; � . . ; � ; , < � � < � � . � , < � ; - < � ; � - � � � � � � � � -

, � . . ; � , . - � � - * � . � � - - � � < � � . ; � � � ; � � < � � � - � � , � ; - � � ; � ; � <

. . ; � ; � ; � , . < � ; � � : - . ; � � � � �� � ; � - � � � � � , : , � , . - � � � < � � � � , � - - . � , < . � � � < � � - � � . : , � � �

� � � � - . < , � � - . ; � � � - . , < , � � - . ; � � � � � � < , � � - . ; � - � - . , < , � � - . ; � �

< ; � � � � �. ; : � . ; � < - � � < � � � � � ; � � � � � � � . � , � � . � < � � � < � , < < � � �

� , , , � � � � � � ( � � , � � � � � � � . � , � � � - � - ; < � � . , � � � - � . � < �

� . � , < � � � < � � � , , . ; � � � < - � � � � ; ( � � , � - � � � � ( � � < ;

� � - � , � < � < < � � , < � < � � � < � � � � � < � � �

� �

� � � � � � � � � � � � � �

� ; � � � . - � � � < � . ; � ; � . � � . ; � � , � � � � < � � � - � � � � � : ; . ; : ; . . � � �

� � � ; � - � - � : � � � - � - � � � ; � - � � � - � � � , � . � � � � � � � � � , ; . ; : . - � �� �

� , � � ; � ; < � - � ; , � � � . ; � - � � ; � � ; < - � ; � � � , � . � � � � � � � � , � . ; :

: ; . . � � � � � � � , : - . < � , . �� � ; < � � < � < , : - . < � , . . � � � � , � . � � � � �

� , � � � � : , � � � < , � , . . � � � � ; : � � � � � , . < � � � � < � - � � � ; � � � � ; �

� � . � � , . � , : � � � - � �� � � � < � � � . ; � � ; � � ; < - � � . ; : � . ; � � - < � : � � � �

. - < , � � . � , < � , � � � ; . - � � - , � � ; � � � ; � � < � - � � � � < , � � - � � � � ; �

< ; � . ; . � � ; , � � � � � � � � � ; � � ; < - � �

� � � , . . ; , � �

� � � � � , � . � � � � � � � � , , � � . � � . ; � � � < � ; � � � � ; � - � �

�→ � � � - � : , � - � � � < , � ; � < � � : ; - < � � � � � < � � : ; � � � � : ;

� � � - . ; : � , - < + , � 9 � . < � �→ � - � - � : � � � - � - � � � ; � - � � � � : ;

� � � � � � � � ; � � ; . � - < � ; � - , � �→ ��

" ��

��

� � � � � 1 � � % ' ' % 7 � 3 � � % %

� � � � � � � � < � �

� � < � , �→ � � ( �→ ( (→ ( � �

(→ � �→ � �→ � �

� �→ � � (

( �→ ( (→ ( � �

� (→ �

� �→ �

� �→ � � �→ � � ( �→ (

(→ ( � � (→ � �→ � �→ � �

� � �→ � � ( (→ ( � � (→ �

�→ � �→ � �

( � (→ ( � � �→ � �→ � �

� � �→ � � �→ � � (

� ( � (

� � � �

� � � �

� � � �

� � ( �→ � � ( (→ ( � �

� � � � �

� � � � �

� � � � �

( � � (→ ( � �

( � � � �

( � � � �

� � �→ � �

� � � � � �

� � ( � � ( �9 . ; : ; � , � ��

��

� �

Page 140: Программирование: теоремы и задачи

� % � " ��

� � & 7 % 3 6 � 3 � 6 3 � 5 � % � � # � 6 � � $ % & % � � % $ # � � � �

� . ; � � � - � . � , � < � , � : , . - , � � � : � - � � � ; - < �

� � � � � , � . � � � � � � � � , , � � - � : , � - � � � < , � ; � � � .

� � , . : � - � � � , � , : . � � - � � � � � � A < � � : ; - < � � � � � < � � : ; �

� � � : ; � � � � ; � � ; . � . , � � < � � � � - < � ; � - , � �→ � A � �

� � 4 � 4 � 4 � � � ; . ; < � � < � �

� � � ; . ; . - , � � � < � � � � - . � � � � : � - � - � < , � � � : � � ; � - � ; � � � � � ; A �

� ; � � < � - < , � , � � � � � � � < � � � . ; < � ; � - � ; � � � � � < � � � � � � ��

� ; � � < � - � . , � � < � � � � � � ; � � ; < - � � - � � � - . � � � � . � , � � � � � - . < , �

� - . ; � � � - . , < , � � - . ; � � � �� � - � . � , < � � � � < � � � : , � - < - < � ; � - � �

� � � < � � � � � � ; � ; � < � � : � � � � - � ; . - � < � - < < , � � - . ; � � < � � � � � � � < � � < �

� � 2 6 � # $ % � $ # � ' # � � & 6 � $ � � � � � - � � � < � � , < � - < � ; � - � � � � � < � � � �

� � ; � ; � < � � : � � � � - � ; . - � . - � , � � . , < � < � � � � � � � < � � < � � � 2 6 � # $ % � $ # � 1

' # � & % 6 $ � � 7 3 % � � � � � < � , < < � � � � , � � � � ; � - � � � � � � � � < � � < � : � � � �

� ; � � � . . - � ; , < 3 # & 2 � � 3 7 7 � � % 6 � $ � � � 6 $ � � 7 3 % � , � - � � . � � . � - : � - � �

- � � � < � ; � � � � � � < � � < � : � � � � � ; � � � . . - � ; , < 3 # & 2 � � 3 7 7 � � % 6 $ � � 7 1

3 % � 6 $ � � 7 3 % � , � - , < � . , � � � � � � � � ; � - � � � � � � < � � � � � � . � � . ; � � � < � ; �

� ; � � ; < - � ; . ; . � � ; , < � � � � � � 1 � � % ' ' % 7 � 3 # 5 � , � - � . , � . , < � � . * � - � < � � < - � ; : � - � � � � � < � ; - � � � < � ; � � � � < � ; . - : � � � : . � � � � � � ; ��

� � 4 � 4 � 4 � � � , < � � - � � - � , : � . . ; � � � � , � � ; � � ; < - � ; � � � � � � ; � � ;

< - � � � � � � � � � � 4 � , < � . , � � � � , < � �

� � � � � � ( - � � ( - � , � < � � � . * � - � < �

< - � ; : � - � � � � � < � ; � �

� � 4 � 4 � 4 � � � � < � � - � � � � � � ; � � ; < - � ; � - < ; � - , �

� ; (→ ,(→ ( /

(→ ( ( �

(→ ( ( ( �

� � (→ ,(→ / (

(→ � ( (

(→ � ( ( (

� � � � � � � 4 � � � � < � � � � < ; � � - � � . ; �� % .�� � � � � � . * � - � < � � . , < � �

! < ; . ; : ; � ; � � � ; . � � ; , < � � < � � � � � � � ; � � ; < - � - � � � � < � � < � � ; � � ,

� � � , � � � - � . � � - � < ; � - � � ; � � � , � � � - � . � � - �

� � 4 � 4 �4 � � < � : ; . ; � � � � � � ; � � ; < - � ; �� � � ; . ; < � � � < � � � � � � � � � �

� ; � � , < � � , < . , � � � , , � : . � � � � � ; � � � � � � � � : ; � � � < � � - < � ; � � � � - < �

� � � � , � � - � � � � : - � � < - � � � � � � � ; � � ; < - � , �

" ��

��

� � � � � 1 � � % ' ' % 7 � 3 � � % .

� � � � � � � � < � �

� � < � , �→ , (→ ( / (→ ( ( � (→ ( ( ( �

, �→ ,

� �→ � / (→ ( ( � (→ ( ( ( �

�→ , (→ ( / (→ ( ( � (→ ( ( ( �

( / (→ ( /

( ( (→ ( ( � (→ ( ( ( �

(→ ( / (→ ( ( � (→ ( ( ( �

(→ , (→ ( / (→ ( ( � (→ ( ( ( �

( ( � (→ ( ( �

( ( ( (→ ( ( ( � (→ ( ( � (→ ( ( ( �

(→ ( / (→ ( ( � (→ ( ( ( �

�→ , (→ ( / (→ ( ( � (→ ( ( ( �

( ( , � ,

( ( ( � (→ ( ( ( �

( ( ( � � ( ( �

( ( ( ( � ( ( (

( ( ( , � ,

� ;

� � � � � � � � < � �

� � < � , (→ , (→ / � (→ � � � (→ � � � �

, �→ ,

/ �→ / (

(→ , (→ / � (→ � � � (→ � � � �

� (→ � ( (

(→ , (→ / � (→ � � � (→ � � � �

� (→ � ( ( (

(→ , (→ / � (→ � � � (→ � � � �

� � � . ; � ; � �

9 . ; : ; � , � ��

��

��

Page 141: Программирование: теоремы и задачи

� � � " ��

� � & 7 % 3 6 � 3 � 6 3 � 5 � % � � # � 6 � � $ % & % � � % $ # � � � �

� � � � � � � � < � �

/ ( (→ / (

/ , � ,

/ / � /

/ � � �

/ � � �

� ( (→ � ( (

(→ , (→ / � (→ � � � (→ � � � �

� , � ,

� / � /

� � � �

� � � �

� ( (→ � ( ( (

(→ , (→ / � (→ � � � (→ � � � �

� , � ,

� / � /

� � � �

� � � �

� ( ( (→ � ( (

� ( , � ,

� ( / � /

� ( � � �

� ( � � �

� ( ( (→ � ( ( (

(→ , (→ / � (→ � � � (→ � � � �

� ( , � ,

� ( / � /

� ( � � �

� ( � � �

� ( ( ( (→ � ( ( (

� ( ( , � ,

� ( ( / � /

� ( ( � � �

� ( ( � � �

� � � � � � . � ; . - ,

" ��

��

� � � � � 1 � � % ' ' % 7 � 3 � � � �

� � � � � � � 4 � � < � : ; . � � � � - . � � � � . � , � � � � �

0 � : , � < � � - < � � � � � �

� , . ; : . - � � � � ; � ; � �

� � < � < , � � � , , � < � � . - , < , � ; � � � � � � , ;

� ; � . � ��

� ; � . ; : � � , � - < � � : , � ; < � : � - � - � - � � � < � � � - , � - � � � <

� � � < � � � � ; � � � � � � ; � - � � �

� � � � ; . � � � � , : , � , . - � � � � � � � ; � � ; < -

� - � � . ; � , � � < � � . - - � � � . � � , . � - � � < � � � � � : � - � � � - � � < � � �

� � � � � < � ; � � � - � � � � - � � � � � : . � � � � � ; � - � � �

� ; � - � � � � ; . � � � � � - �

� � . � � . � � � � � : � � , . - � � � � � � � , ; � � � - � � : - < : , < , � � - . - � � � ; . . �

� . ; � ; : � � � ; � , � � . � � � � , : , � - < � � � ; � � , : , � < � - , < � � � � � - � � .

� � . � � �

� � 4 � 4 � 4 < � � � � - . � � : � < � , � - ; . ; � - . - � � , � � , � � � � . , - � , , < � � � � : ;

� : ; . . � � � � ; � � ; < - � ,

� � � � � 4 + - � � . ; . , � � < � � � � � ; � , . , � � : , < � � . � � , . . - : � - � � . -

� � � < � ; � � - � � � , � � . � � . � , : � - � - � � : , < - � , < � . , � � : � � : � � - � � � ,

� , : . � � - � � � � � �

� � � � � � � � �4 ��

� � - � , ; � - . ; � - - � < � � � ; � � � � - < � ; . , < . , � � � � : - � � < -

� ; : � � � ; . . ; . � � � � � � - � � < � � . � , < � � � � < � � : � � < , � � � , � � . . ; � ,

. - � ��

! < - � . � , < � ; � � . � < ; � , � � ; . - < � � < , � , � � � ; : � � � � � , . <

� � ; . � < � � . � , < � ; � � < � ( : � � � , � . ; � ; � ( < , � � � , � � � � � ; � �

��

� ; ; � � � : , � , ; � � � � � � � � � . � . , � � ; . - < � � : � < ; < � � . �

� � ; . - < � � . � , < � ; - < � ; � - � � � < � ( : � � � , � , � � . ; � ; � ( � � � � � � ; �

; � � � �

� ; � � � � - < � , � � � � , � � - � � � � : - � � < - � � � � � � � ; � � ; < - � , � � -

� � � � . � , � . , � � - . * � � � ; � - � � � � < � � � � � � � � - � � � � � < � � ; � � � � - <

� , : � � � ; : � � � � < � � . - � - . � , < . � . ; � ; . , , � � < � � . � � � � . � � , . < �

� � � � : � - � - � - < � � � � � � � � < � ; � � � - � � � � � � � , : . , � � � � ; , - . � , <

. � � � � � ; � � � � � � ; � - � � �

0 � � , , - . � � � � . . � � ; � � � � - < � � � � � � � � -

. - � ; < � � , � , . - , � � � � � � , � , : � : � - � � � - � � � < � � � � � � � � < � , �

. ; � � , � , : . � � - � � � � � � � � �

� � : � . ; � < � � . - , � � � . � � ; . ; < � �

� � - � ; � - � . . ; � , . - � � � � � � � . � � , . : � - � � � < � < , < , � � - . ; � � � � �

< � � � , � - < � ; � - � � � < � � � � < � � . - � < � � < . , � � � , : < � , . . � . ; � � :

� � � � - � ; . - , � �

� � � . , , � � � � � � . � � ; < � � - . * � � � ; � - , � � - � � � � , � � �

: � � � , � , . - � � � � � � ; � < � � � � � . � � . ; � - � � � < � ; �

� � � � < � � � , < �

� � � � � � . . � � � , < � : � � � ; � � ; < - � - � � � � < � � � � � . � � - � , . - � � . ; . � � ;

� < � � � � � � � ; � � ; < - � ; � - � � � � ; � , . - , � < � � � � �� � � � � � - � � � . � �

� , < � : � � � � , , � � . � � � . � - � � � � . � � � - � � � � � . � � . � � - . * � � � ;

� - � � � � ; � � ; < - � - � � � � < � � � � � . � � - � , . - � � . ; . � � ; � < � � � � � � ; �

� ; < - � ; � - �

� < � - � � � � , � < � � . � � � � ; � � ; � � ; < - � � . ; . � � ; , � � �

� � � � � � �

Page 142: Программирование: теоремы и задачи

� � � " ��

� � � 7 � 3 6 � 3 � 6 3 � 5 � � � � � � 6 � � � � � � � � � � � � � � �

� �

� � � � � � � � � � � � � � �

� ; � � � . - � � � < � : � � � � � � � � . , < , � � - . ; � ; � � � � � � , : , � � � - � �� � �

� . � , < � � � � � , : � � < , � < , � � - . ; � � � � � � < � � � , � � � � < < � � < � . , � � � , :

< � , . . � . ; � � � � � � : - � � � � - . . ; � ; � � . � � � . , < , � � - . ; � ; � � � , � � � < � � . �

, < � � : � � ; � � � � < < ; � , - � � � � � � � � , � - . , < , � � - . ; � � � � , < < � � < �

� � � . � , � � � � : - � � � � � � � ; �

� � 4 � 4 � 4 � � � ; . ; < � � � < � , � - � : ; . . � � � � � , . < � � � � � � , ; � � � , :

. - � - � � � � < , � ; � � ; � , . � � � � - � � � � � � � , � < � < � � , < � : ; � � . , � � , �

� � , � . � . ; � , � � - < � � � < � � � � � � - . ; : � , - < � � � , : � � �

� � � � � � � 4 � < � < � ; � < � � � � , < � . , � � � , : < � , . . � � � , : < � - , � � � � ,

: , � , . - � � � � � � . - � � � < � , < < � - , � , : � � � ; � � � - � � � � : ; � - - � � � � �

� , ; � - � �

� ; � � < � - � . , � � < � � � � � � ; � � ; < - � � � � � � - . � � � � . � , � � � � � - . < , � � -

. ; � � � - . , < , � � - . ; � � � - < , � � - . ; � � �� � - � . � , < � � � � < � � � : , � - <

- < � ; � - � � � � � < � � � � � � ; � ; � < � � : � � � � - � ; . - � < � - < < , � � - . ; � � � < �

� � � � � � < � � < � � � 2 � � � - 〈 �� �〉 � � � � � � � � � � � � � �

� � - � � � < � � , < � -

< � ; � - � �→ � � � � - � � � � � � - . ; : � , - < � � � , : � � � < � � � � � � � < � � < � � � 2

� � � - 〈 �� �〉 � � � " � � � � � � � � � � � � � � � � � � � � � ; � - � � �→ � �

� � � � � < � � < �

: � � � ; � � 〈 �� �〉 � � . . - � ; , < � � � " � � � � � � � � � � � � � � � � � � � � � � � � � � , � -

� � . � � . � - : � - � � - � � � < � ; � � � � � � < � � < � : � � � ; � � 〈 �� �〉 � � . . - � ; , <

� � � " � � � � � � � � � � � � � � � � � � � � � � � � � � � � , � - , < � . , � � � � � � � � ; � - � �

� � � � < � � � � � � . � � . ; � � � < � ; �

� ; � � ; < - � ; . ; . � � ; , < � � � � " � � � � � � � � � � � � � , � - � . , � . , <

� � � � � � � . � � - � < � � < - � ; : � - � � � � � < � ; - � � � < � ; � � � � < � ; . - : � � � :

. � � � ; � � 〈 �� �〉 �

� � 4 � 4 � 4 � � < � : ; . ; � � � � � � � ; � � ; < - � ; �� � � ; . ; < � � � < � � � � � � � � � �

� ; � � , < � � , < . , � � � , , � : . � � � � � ; � � � � � � � � : ; � � � < � � - < � ; � � � � - < �

� � � � , � � - � � � � : - � � < - � � � � � � � � ; � � ; < - � , �

� � � � � � � 4 � . ; � � � - � . � � � � ; � � � � � � � ; � � ; < - � � < � � � � � � � - � � � � � ,

� , : � : � - � � � - � � � < � � � � � - < � � ; , < � � � , � , : . � � - � � � � � � � � � �

� � 4 � 4 � 4 � � � � , � - < � � � � � � , < � � - � � - � , : � . . ; � � � � , . ; �� % � � � ; � � ;

< - � ; � . , < , � � - . ; � ; � - � � � - � � � � � � � � ; � � ; < - � � � �

� � � � � � � 4 � ; � � � � � , < � � < ; � � ; � � � ; � � . � � - � < ; � � , � ; � � - , , � � � < �

� � � � � � ; � � ; < - � � � � � ; . � , � ; � < � � � � < � � � � , � , : . � � � - � � � � ; � - : � �

� � � ; � � - : � � � � � ; � � � : � - � � � . � � , . < � � � � � � � - � � � � � � ; - � � � � �

. , � � - . ; : � , - < . - � � � , : � � � { � � ��

��

} � . - � � � , : � � � { � � ��

�� ��

} �

- � � � < � � � � � - � � � � � � � � < � ; . , � � . � � . ; � �

" ��

#�

� � � " � � � � � � � � � � � �� � � � � " � � � � � � � � � � � � � �

� �

� � � � � � � � � � � � � ��

� � � � � � � � � � � � � � � �

� � - ; . . � � � � � , � � � � � � � : � � : - � � � � . � � < . , � � � � . � � . � � - .

� � � � ; � - � � � - � � � . , . - - < � � � � � � . � � . ; � - � � � < � ; �� � , . . � � � . � <

: , � � . � � � � � , � � , < � � � . � � . ; � - � � � < � ; � � - : ; . . � � � < � � . - - < , � ; �

- � < : , � � . � � � � . � � . ; � - � � � < � ; � � : ; . . � � � � � ; � - � � � � - : ; . . � �

- � � � � , � � � �

� , : � < , � � < - � � � � , � � - . , � � � � � < � . , . ; � - - � � � - �

� � , � � � � < : ; < � � � � � - < , � � . � � � < � , < � . � < , � . , � , . , , � � � < � ; � � -

< , � , � - � � , � , : . � � - � � � � , � � � . , � � . � � . ; �

� � � � � � � : � � : , � < � <

. , : � < ; < � � � < � ; . � , < � �

� � � � � � : � � : � < � - < � � < � � � � � � , . ; � - � � � , : , � , . - � - � < � , � : ,

. - � � � : - � - � - � � � < � < ; � � � < � � � � � , < � � � ; � � � - � � � � < � - < � � ; � ;

� < � ; . � � � ; � - � ; , � � � � . , < , � � - . ; � ; � : � � � - � - � � � ; � - � � , � � � ; � , . � � �

� � - � � � < � , �� � < � �→ � � � : . � - . � � ; � - � � � ; � � ; < - � - � ; � � . , � � < � � � � < , �

� - . ; � - � - � , � - � � � � � � � � � � < � � � � � � : � � � � - � ; , � � � � . � , � � � :

. � � � � � � ; �

� � � , : , � - � � . � , < � � + , � 9 � . < � � → ��� � ; � � . � , < � �

� , � � � � � � � < � � � , � � � � � < � � : , � - � � � < , � ; . , � � � , : < � , . . � � ,

� , : � � � < � � � � � � � � � : , � � , � . � � � � � � � � � , ; � � � - � � � � - - � � � � �

� � � � � , . < � � � < � - �

� � - � < � � � - < � � � , � � � � - . + , � 9 � . < � �→ � - � � � . , � � � < � � � � �

� - < � � . � , < � � � , � � � � � � � < � � � , � � � � < � � � � - < � � � � � ; � � � � � � � : ; �

� , � , : . , < , � � - . ; � � � � � . ; � � < � � � � < � - < - � � � � ��

� < � � . � , < � � � . ,

. ; � - � � , , � < < � � � � � ; � � , - . � � ; � - � �→ � : � � . , < , � � - . ; � ; � � � � � ; . �

� � � � : , � � � � . . ; � ; < � + , � � �� � �

� � 4 � 4 � 4 � ; � - ; < � � � ; � � ; < - � � : � � � � � � : , . - � � . � , < � + , � � �� � �

� � � � � � � 4 � � . , < , � � - . ; � ; � - � � : � < - � � � � � 〈 � & 0 � �〉 : � � � ; : � � �

. , < , � � - . ; � ; � - : � � � ; : � � � < , � � - . ; � ; � � ; < ; � , : � � � � � � � �

� �

� � ; � - � ; < ; � � � � � � � < � � � . ; � ; � � . � � . , < , � � - . ; � - � � : . � � � � ; � � ;

< - � - �

� � � : ; � . � � � � � � ; � � ; < - � , � � : , < � � ; � - � �

〈 � & 0 � � � �〉 →

� � � < � , � � � � �

9 ; : � , � � ; � - � � - � � : . � � � � ; � � ; < - � - � � � � : ; , < . , � � � � � � � � ; � - � . �

� � � �� ; � � - � , � � : � � � � ; � - � ;

�→ � � � �

� � � � � � � . , < , � � - . ; � � � � � < , � � - . ; � � . � � � � � � ; � � ; < - � � � � : � � ;

� - � � � ; � - � ;

〈 � & 0 � �〉 → 〈 � & 0 � �〉

Page 143: Программирование: теоремы и задачи

� � � " ��

� � � � � � � � 3 � � � � � � � � � � � � � � � � � � � � � � � � � � �

� : � � � , � < , � � - . ; � � � � �

〈 � & 0 � �〉 → 〈 � & 0 � �〉 � �

� : � � � , � � � � � < � � � , � � � � < . ; � - . ; < � � � � ; � � � � � : - � � , - . � � - : � �

� , � � � ; < ; � , : � � � , � � ; � � � � � , � - - . � � � � � : - � � � � < � , � � � � �

〈 � & 0 � �〉 → 〈 � & 0 � �〉 � � �

� : � � � , � < , � � - . ; � � � � � �

� � 4 � 4 � 4 9 ; � � , . � , < � � � � , : , � , . - , - < � ; � - -

� � � � � � � 4 � - < � ; � - , � . ; . � � ; , < � � ; � ;

� - < � ; � - � � < ; � � � � � � ,� < , � � - . ; � - � - � � � �

� � 4 � 4 � 4 9 ; � - . � , . - < � � � � , : , � , . - , � � � ; � � ; . . � < -

� � � � � � � 4 � � � � � � - . < , � � - . ; � � � - . , < , � � - . ; � � � � � � ; � � ; . � -

< � ; � - , � � �→ � ��

� � � . : , � � � < , � � - . ; � - � - � � � � , � - � � � . � ; , < �

. ; � � < � , < � � � � � � - � � � � � , < � � � � � � � - . ; : � , - < + , � � �� � � �

� � 4 � 4 �4 9 ; � � � � � � ; � - � ; : � � - . : � � < - � . � � � � � � - � , . - � � . � , < � ;

� � < � � - < � ; � - � � � � � ; � � ; . . � � : ; . . � � � � � � � �

� � � � � 4

� � � � - � � � � � � � � ; � � ; . � - < � ; � - , � � � → � ��

� � � � � - � � �

� � � � � . ; � - . ; , < � . ; � � � � � � � < � , < � � � � � � < � � � � � � �

� � � ; � � ; . � - < � ; � - , � � �→ � � ��

� ��

� < � � � ; � - � � � � � . � < � � � � � , : , � � , < � , - < � ; � - - . , � � < � � � , � � �

� � � � � - . � � � < � , < � . , . ; � - . ; � � - , � � � : � � � � - � ; . - � � � � � ; � � ; . . � ,

� ��

� < ; � � � � � � , : , � - < � � : � � � ; � - � . , < , � � - . ; � � � � - < , � � - . ; � � � �

� � � � � � � � - . ; : � , - < + , � � �� � �

� < � : , � ; , < � � � : � � � � � ; � - � ; � �

� � � � - - < � ; � - � � �→ � ��� � � � � ; � � ; . ; � � � � � � ; . � � � ; � -

� � � � � ; � . ; � - . ; , < � . ; . , < , � � - . ; � � � < � � � � � - . ; : � , - <

+ , � � �� � : � � � , � < , � � - . ; � � � � � � � < � � � , � � � � < . ; � - . ; < �

� � � ; � � � � � : - � � , - . � � � ; � \ � � � � � � � � , . � , � � � � � � � � � � �

; < ; � , : � � � � � � , � - - . � \ � � � � � : - < � � � < � , � � � � �

� � � � - � � � � � : - < � + , � � �� � : � � . , � � < � � � � � - � � � � - � � �

� → � � � � ; � - � � � � ; � � ; < - � - - � . ; � - . ; , < � . ; . , < , � � -

. ; � � � < � � � � � - . ; : � , - < + , � � �� � : � � � , � < , � � - . ; � � � � �

� � < � � � , � � � � < . ; � - . ; < � � � � ; � � � � � : - � � , - . � \ � � ; < ; � ,

: � � � � � � , � - - . � \ � � � � � : - < � � � < � , � � � � � �

" ��

#�

� � � " � � � � � � � � � � � �� � � � � " � � � � � � � � � � � � � �

� � 4 � 4 � 4 � ; < � � � � , : , � , . - � � � � � � � . � � - � < � � : � - � � � � � < � ; - � � � <

� ; � � � � < � ; � � ; . ; � � � - - : ; . . � � - � � � , �

� � � � � � � 4 � � < � : ; . ; . , � � < � � ; � � � ; � � ; < - � ; �� � < � � � � � � - . � � � �

. � , � � � � - . < , � � - . ; � � � - . , < , � � - . ; � � � �� � - � . � , < � � � � < � � �

: , � - < - < � ; � - � � � � � < � � � � � � ; � ; � < � � : � � � � - � ; . - � < � - < < , � � -

. ; � � � < � � � � � � � < � � < � : � � � ; � � 〈 ���〉 � � . � � , . : � - � �� � < � � � � , : , � , . - ,

. , - . � , . - � � � � � � ; � . , . - � � � � � � � � � ; , � � � < � � � , � � � � � . , . < �

� ; � - . � � < � � . , � � - < � � ; � < � �

� � - � � � < � � , < � - < � ; � - � � � � � < � � � � � � ; � ; � < � � : � � � � - � ; . - �

. - � , � � . , < � ; � < � � � � � � , . � � � ; � � � � � � , < � < , � � - . ; � � � < � � � � � � � < �

� < � : � � � ; � � 〈 ���〉 � � � " � � � � � � � � � � � � � � � � � � � � � < � , < < � � � � , � �

� � ; � - � � � � � � � � < � � < � : � � � ; � � 〈 ���〉 � � . . - � ; , < � � � " � � � � � � � � � � � �

� � � � � � � � � � � � � � , � - � � . � � . � - : � - � � - � � � < � ; � � � � � � < � � < � : � �

� ; � � 〈 ���〉 � � . . - � ; , < � � � " � � � � � � � � � � � � � � � � � � � � � � � � � � � � , � -

, < � . , � � � � � � � � ; � - � � � � � � < � � � � � � . � � . ; � � � < � ; � �

� ; � � ; < - � ; . ; . � � ; , < � � � � " � � � � � � � � � � � � � , � - � . , � . , < � � � �

� � . � � - � < � � < - � ; : � - � � � � � < � ; - � � � < � ; � � � � < � ; . - : � � � : . � � � ; � �

〈 ���〉 �

� � 4 � 4 � 4 � � < � � - < � ; � � � � - < � � � � � , � � - � � � � : - � � < - � � � ; � � � � �

� � ; � � ; < - � , �

� � � � � � � 4 9 ; � - � ; . � � , � . ; � ; : � � � ; � , � � � � � � , ; � � . � � : . �

. . ; � . � � � � , : , � - < � � � ; � � � � ; � < � � � � � - � � , < � � < � � , : � � � - � � �

� � � , . . � � � � ; < . � < - � : � � � � � � � � � ; . � � � ; � � � < � - . - , � � . � < � �

� ; � � � . ; . � � � . � < - � � � � � - � � � � � � � ; � � ; . . � < - �

� � 4 � 4 �4 � � � � � � � - � � � ; < � - : � � ; . ; < � � � < � , < < � � � � , , � < � , � : , . - , �

� � � � � 4 � � < � � - � - � � � ; . ; . , � � < � � ; � � � ; � � ; < - � ; �� � � � � � - . < , �

� - . ; � � � - . , < , � � - . ; � � � � � � � , < � � � � � � � � ; � � ; . . � � - < � ; � - , �

� → � � < � � : ; - < � � � � � < � � : ; � � � � : ; � . � � � � � � � � ; � � ; . � � ; � � �

� �→ � ��� � : � � . , � � < � � � � � < , � � - . ; � ; � � - � - : � � � � � � � �� � , ; � � ,

: � � � - � - � � � ; � - � + , � � � , < � � � � , : - . , . - , + , � � �� � � � � , � ��

� � �

� , : . , � � � � � , � < � � � , � � . � � �

� � � � � � � � � 4 � ; � - � � � � ; . � � � � � . � � - � � � < � � � � � � � � � � , � � � �

, < � � ; � - � , . - , � � � . � � - - � � < � � � � � � � � � � , � � � < � � � � �� � � � � �

� ; , < � - . � � < � � � �� � � , � - � � � , � � ; � ; � � � � � � - < � � < � � � , � � , . � �

� , � , � � � � � � , � : ; < � � � � , : , � , . - , � � � � � � � � ; � � ; < - � - �

� � < �

� - � - � � � ; . ; . , � � < � � ; � � � ; � � ; < - � ; � � � � � � � - . . , < , � � - . ; � � � - < , �

� - . ; � � � � � � . , � � < � � � � < , � � - . ; � � - � - � � � �

� : , � � � � � � - < � � � < � : � �

Page 144: Программирование: теоремы и задачи

� � � " ��

� � � � � � � � 3 � � � � � � � � � � � � � � � � � � � � � � � � � � �

� ; � � 〈 ���〉 � � � � � � � � . � � . ; � � � < � ; � � . , � � < � � � � � � � ; � - � � � , � - �

� , < � � , < : � � � � , � � � � � � � < � � � � �� � � � � � < � � � � �� � � � � - � � � : � �

� ; � � 〈 � ��〉 � � � � � � . � � . ; � � � < � ; � � � ; � ; < � - � ; , � � � � � � ; � - � � �

� ; � , , � � � , : , � � � < � � � . � � - � < � � , < , < � , . . � � � � � ; . � � � - � � ; � � ; < -

� ; . ; . � � ; , < � � � � � � � � � ; � � ; < - � � � � , � - � � . � � - � < � � . , < �

� � 4 � 4 � 4 � � � ; . ; < � � � < � � � � ; � � � � � � � � ; � � ; < - � ; � � � � , < � � � � � � �

� � ; � � ; < - � � � � ; � � � ; � � � � � � � � � ; � � ; < - � ; � � � � , < � � � � � � � ; � � ; < -

� � � �� � � ; . ; . - , � � < � � � � � < � , � , : < � - , � � � , : , � , . - � ��

� � 4 � 4 � 4 � � < � � - < � ; � � � � - < � � � � � , � � - � � � � : - � � < - � � � � � � �

� � ; � � ; < - � , � � � < � � � � � � ; . - < � < , � , � , . � � , - . � � � � ; � - - � � , � � � <

� , < < � � � � - � � � � � ; � � � � - < � �

� � � ; . ; . - , � � � < ; < � � . � � � ; . - < � � < , � , � . � , < � ; � � < � � � � �� � �

� � � � � � � � � � � ; . � � � � , : , � , . - � � � � � � � � � . � � . � < � � � � < � - - � -

� � � , : , � � , < � �� � ; � � < � ; � ; � � � � - < � . - � , � . , � < � - � ; , < � � < � � � � �

� � � ; � � � � � � , < ; � � - � � � � . � � . � � � � � < � � � �

� � 4 � 4 � � 4 � � - � , < - � � - � , � � � � � � � � � ; � � ; < - � - � � � < � � ; � . , � � � �

, < � � � � � � � � ; � � ; < - � � � � �

� � 4 � 4 � � 4 � � - � , < - � � - � , � � � � � � � ; � � ; < - � - � � � < � � ; � . , � � � � , < �

� � � � � � � � ; � � ; < - � � � � �

� �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � - � , . , . - , � < - � � , < � : � � . ; � � ; � < - � , - � , , < � � - � - < � � < - - < � .

� � < - � � � < � � � � � � . , � ; ; � - � �� � ; � � - � , � � < ; � � - � � � , : � , < � � ; . - < �

� � � � . � � . � < - � � � . � � . � � ; < � � � ; . � � ; , < � < ; � , � � < � : � � . , � �

< � � � � � � � � : . � � � � . � � ; . ; - � � � , , , < , < � , . . ; � � � ; � � ; < - � ; . , � � � � , < �

� � � � � � ; � � ; < - � � � � . � � � � � , < � � � � � � � ; � � ; < - � � � � ; < ; � , � � , <

� � < � . ; � , . , . ; . ; � � � � � � ; � � ; < - � � � , . - . � , . , . - � � . � � ; � 9 ; � � � - .

� < - � � ; � - ; . < � � � � � � ; < � � . , � , � : ; � . � �� - � , < ; . < � - � � � , < � , � - � �

; � - � � � , � < - � � , < , � � � : . � � � . � � � < � . , � , : � , < � � � , . : � - � ; < � � - � � �

< � , � � � < � � : . - - < , , - � � � � � : � � � ; . . � � � , � , � � - < � � : ; � � � � . � . ,

� � . � . ; � - ; < � � � � � � � ; � � ; < - � � - � - � , � � � - � . � � ; . ; � - . ; < � � �� � -

, � � � : . � � � . � � . ; : ; . . ; � ; . , , � � � � � � � � � � � � � ; � � ; < - � - � . , � � � �

� � , � � � � � � � � ; � � ; < - � � � � < � � � � � , , � . , < � � � ; < � � ; � ; . � - � ; < � � ; �

, < � �

� � - � < � � � � � � < � � ; . ; < � � � � � , . . � , � , : < � ; ; � < � � ; < - � , � � � �

" ��

��

� � $ � � � � � � 3 � � � 2 � � � � � � � � � � � � � � � � � � � � � � � %

� � � � : , . - � ; . ; � - . ; < � � � � � . ; - � � � , , - . � , < . � � - - . � � < � � � � � � � � � < �

� � � � � � � � - � � � � � � �

� � � � � , � � � - � , < � � � � � , . . � � - � � � � � � - . � � , . . � � - . � � � � ; � - -

� < , � � - - - � � ; � < - � , - . < ; � - � , � � � � � ; . � � � ; - � , , < � � � . - � , � � � �

� , < - - � � � � ; . ; � � � � - � � � . - � : � � � < , . - � �

Page 145: Программирование: теоремы и задачи

�� � � � � � � � � � � � �

��

� � � �

� � � � � � � �

� � � � � � 9 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � �

� � � � � � � � � � � �

��

� � � � � �

� � � � � � � � � � �

� � � � � �

� � � � � � � � � � � � � � � � . � � � � � � �

� � � � � ! � � � � � � � � � � � �

�� � � "� # $ #

%�

& � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ' � � � � �

�� � � "� # $ $

%�

& � � � � � � � � � � � ! ( � � � � � � � � ! ' � � � ! � � � � � � � � � � !

� �

� � � "

� # � )

*�

+ � � � , � � � � � � � " ' � � � � � � � � � � � � ' � � � � � � � � � � � � � � � � � � � � � � � � - �

� � � � � � � ' � � � � � � "� � � .

*�

+ � � � � � � � � � � � � � � � � � � � � � � � � �

�� � � "� # � �

��

+ / � � "*�

* � � � � � � � ! � � � � � � � � � � ! � � � � � � ! � � � � ' � � � � � � � � ! � 0 �

' � � � �

�� � � "� # � �

1�

* � � � � � � � 2 � � � � � � � � � � � � � � � � � � � � � � � � � �

�� � � "� # $ �

3�

� � � � � � "4�

� � � 5 � � � � � "�

� � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � � 0

� �

� 6 � � �"

7 � � �

��

+�

� 8 9 � � � � � � � "+�

&�

� � : � � � � - � � � � � � � � � � � � � � � ' � � � � � � � � � � � � �

� �

� � � � � "; # � �

&�

� � � � � � � < � � � � � � � � � � � � ' � � � � � � � � � � � � � � � �

�� � � "; # � �

1�

� � � = � , � � "

>�

% � � � � = � , � � "%�

* � � < � � � � � � � � � � ! � � � � � � � � � ! ? �

� � � � � � � � � � � � � �

�� � � "; # � �

* � �

� � � � � � � "�

� � � � � � � "* � �

� , � � � � � � � ' � � � � � � � � � � � � � � � � �

� � � " � 0 ! � � � � � ! � � � � � � � � �

�� � � � � � � "7 � � 7

�@ � � A � � B C

D

E F G H G � � � I

� � J K � ' � � � � � 7 � 7

� � L M N � � L � � O ) #

P N � � � Q R

� � � � � ; � � � � � � � � � � 7 � �

� � � ; � � � � � � � � � � 7 � )

� � � ; � � 0 � � � 7 � 7

� � � � � � � � � � � � � � 7 $ �

� � � ; � � � � � � � � � � 7 � )

� � � � � � � � � 7 $ �

� S � � � � � � � � #

� � � � ; � � � � � � � � � � 7 � 7

T U N S � � � � R"

7 #

� � V ' � � � � � 7 . R

� � � � � � � � � � � � � ! � � #"

; ) )"

; � $

W W � � ' � � � � � � � � � � � � � � ! � ; $ 7

� 0 � � � � � � � 0 � 7 � $

� � � � � � � � � � � < � � � � � ; � �

� � � � � � � ; ) $"

7 � �

� � � � � � � � � � � � � � ' � � � 7 � �"

7 � ;

' � � � � � 7 R .

� � � � � ! � � � � � � � � ! � " � � � � � ) $

� � � � � � � � � � ! � � � � � � � � � � � � ; . )

� � � � � � � � � � � � � . ;

� � � � � �

� � � � � � � � � � � � ; � .

� � � ' � � " � � � � ; # R

� � � � � 7 � �

W � � � � � � � 7 � $

� ! � � � � � � � � � � � � � 7 �

� � � � � � � � � � � � ; � )"

; 7 #

� � � � � � � � � � � � � � � � � ' ) #

� � � � � � � � � � � � � " � � � � � � 7 . )

W � � � � � "� � � � � � � 7 . )

� ! � � ' � � � � � � � � � � � 7 R )

W � � � ! � 7 � 7

W � � � � ! � 7 $ �

� ! � � ' � � � � � � � < � � � � � � � � � � � "

� � � � � � � � � � � � ! � � � � � � � � � 7 R #

� ! � � � ! � � ! � � � 0 � � � � ; � #

� ! � � � � � � � � � � � � � � � �"

; � #

� ! � � � � � � 7 R �

W � � � � � � � � � � ; $ �

� ! � � � � 7 7 $

� � � � � � � ! � � � � � ; $

� � � � � � � � ; # R

� � � � � � ' � � � � � � � � . �

� � � � � � � � � � � 7 �

� � � � � � � � � � � � � � � ; 7 � �

W � � � ; 7 � )

W � � � � 7 $ �

W � � � ; 7 � )

W � � � � ; 7 � 7

W � ! � � � � � � 7 R �

W � � � � � � � � � � � � � � � ' � � � 7 R )

W � � � � � � � � � � � � � � � 7 � �

Page 146: Программирование: теоремы и задачи

7 # � � � � � � � � � � � 8 � � 5 � � � , �

� � � � " � � � � � � � ; � )"

; 7 #

W ' � � ' � � � � ! � ; ) 7

W " � � � � � � � � � � � � � � ; R )

W � � � � � � � � � � � � � � � ! � ; 7 #

W � � � � � � � � � � � � � ! � ; � )

W " � � � � � ; 7 #

W " � � � 0 � � � � � � � � � � � � � ; ; 7"

; 7 #"

; R #

W � � � 0 � ! � ; � )

� � � � � ' ! R �

' � � � � � � � � � � � � � ) �

' � � � � � ! � � � � � � . .

' � � ' � � � � ! � � � � � ; ) 7

2 � � � � � � ! � � � � � � � � � � � � � � � � � � �

� � � � ; R $"

; R #

' � � " � � � � � 0 � � � � � � � � � � � � ; � R

W " � � ! � � � � � � � � � � � 0 � � � � ; � #

' � � � � � � � � � � � � � � � ; �

W W � ! � � � � � 7 7

' � � � � � � � J K � 7 � 7

W � ' � � � � � 7 R .

W " � � � � � � � ; 7 �

W " � ! � � � � ; 7 ;

W ' � � � � � � � $ R

W " � � � � � � ; 7 �

W " � � � � ' � ;"

; 7 7"

; 7 �"

; R ;

W W � � � � � � � � � � � ! � ; R �

W � � ' � � � � ; $ R

W � � 0 � � � � ) #

W W " � � � � � 0 � � � � � �

W � � � � � � ' � � � � � � � 7 7 �

W " � � � � � � � � � � � � � � � � � � � � ; 7 ;

W � � � � � � � � � � � � � � � � 7 . R

W � � � � � � � � � � � � � � � ; $ �

W " � � ! � � � � � � � � � � � 0 � � � � ; 7 �"

7 7 �

W � � � � � � � � � � ; $ R

W � � � � � ' � � � � � � � 7 7 $

W " � � � � � � � � � � � ; 7 ;"

; R ;

W W � � � � � � � ; 7 ;

' � � � � � � � � � ' � � � � " � � � � � ' 7 �

' � � � � � � � � � 0 � � � � � " � � � � � � ; $"

7 �"

; R �

W W W � � � � � � � � � � � ; ; �

W W " � � � � � � # ;

' � � � � � � � � 0 � � � � � � � � � � � � �

� � � � � � � � ; $ .

' � � � � � � � � � � � � � � � � � � � � � � � � � � �

; . R"

; . �

W W " � � � � � � � � � � � � � � ; R )

' � � � � � � � � � � � � � � � � � � ; 7"

; R

' � � � � � � � � � � � � � � � � � � � �

� � � � � � � � ; $ R

� � � � � ' � � � � � � � � � ; 7"

; .

W W ' � � � � � ! � ; .

� � � � � � � � � � � � � . ;

0 � ' � � � � S � � � � � � � #

W � � � � 0 � � � � #"

; . #

� � � � � � � � ; # R

W � � � � � � � � � � � � � � ; # .

W � � � � � � � � ' � � � ! � � � � � � � �

7 � )

W � � � ; � �

W " � � � � � � � � � � � � � ! � � � � � � 0 7 � R

W � � � � � � � � � � � � � � ; # �

W " � � � � ; � #

� � � ! � � � � � � � � � � � � � � � � � � ; � $

� � ' � � � � � � � � � � � � � � � . $

� � ' � � � � � � � � � � � � � � � � � � . �

� � � � � � � � ; $ 7

� � � � � � � � � � � � � � � � � ! � ; R .

< � � � � � � � � � � � � ) R"

) �"

; . �

< � � � � �

� � � � � � � �

- � � � � �

� � � � � � � � ; ) #

� � ' 7 � �

W � ' � � 0 � � � � ! � 7 � �

W � � � � � � � � ! � 7 � $

W " � � � ' � � � ' � � � � 7 � �

� � � � � � � � � � 8 � � 5 � � � , � 7 # ;

� � ' � � � � � � � � 7 ; ;�

7 ; .

W � � � � � � � �

� � � � 7 ; .�

7 ; )

� � ' � � � � � � � � � 7 � �

� � ' ! � � � R �

� � � � � � � � � � � � 0 � � � � ! � 7 R"

$ #

� � � � � � � � � � � � � � � � � ! � # �

W " � ' � � � � � � # �

� � � � � � � � � � ; ) $

� � � � � � ! � � � � � � � � � #"

; ) )"

; � $

W W � � ' � � � � � � � � � � � � � � ! � ; $ 7

� � � � � � � � � � � � � � � ' � � �

� � � � � � � � � � 7 R )

� � � � � � � � � � � � � � � � � � � � � � � 7 $ �"

7 � 7"

7 � )

W � ' � � � � � � � � � � � 7 $ �"

7 � 7"

7 � )

� � � � � � � � � � � � � � � � � � � � � ! � ; . )

< � � � � � �

� � � � � � � � � �

� � � � � � � � � � � 7 � $�

7 ; �

� � � � � � � � � � � � � � " � � � � ; # .

< � � � � � � � � � � � 7 R )

V � � � � 7 $ 7

V � � � �" �

7 � .

V � � < � � � � �→ � 7 $ ;

V � � < � � � � �→ �" �

7 � .

� � � ! � � � � � � � � � � � � � � � � 7 $ ;

� � � < � � � � � � � � � � � � � ; $ �"

; � �

� � � � � � 7 .

W � � � � � � � � � � ! � � � � � � � � � � ; ; R

W � � � � � � � � � ; � )

� � � � � � � � � � 0 � � ' � � � � ; R �

� � � � � � � � � � ; R �

� � � � � � ! " � � � � ' � � � � � � � � � � ; . $

� � ' � � � � " � � � � � � �"

; . 7

� � � � � � � " � � � � � � 7

� � � � � � � � � � � � � � � � � � � � � � � � � � ) �

� � � � � � � � � " 0 � � � � � � � 7 �

W " � � � � 0 � � ' � � � 7 $

W " � � � � � � � � 7 $"

7 �

� � � � � � � � " � � � ' � � � � � � � � � 7 ; $"

7 7 �

W W ' � � � � � � 7 7 �

W " � � � � � 0 � � � � � � � � � � � � � � � � � � �

; ; �

W W � � � � � � � � ; ; ;

W � � � � � � � � � � ; $ ;

W " � � � ' � � � ! � ; ; �

� � � � � � � � 7 R �

� � ' � � � � � � � � � � " � � � � � ' � � � � ! � � �

; ; )

� � � � � � � � ! � � � � � � ' � � � � � � � � � � � � �

� � � � � � � � � � � � R �

� � � 0 � � 0 � � � � 7 � $

� � � � � � � � � � � � � � � ' � � � � � � � ; ;

� � � � � � � � � � � � � � � � � � � � � � ; .

� � � � � �→ � 7 � )

� � � � � 7 ) ."

7 � R

� � � � � � � � � � � ; ) $

� � � � � � � � � � � � � � � � � � � 0 � � ' � � � �

; . �

� � ' � � � � � � � � � � � � � � ! � � � � � � � ! �

� � � � � � � ; $ 7

� � � � � � � � � � � � � � � ! � � � � � ; 7 #

� � � � � � � � � � � < � � � � � �

� � � � � � � � � � 7 � $�

7 ; �

� � � � � � � � � � 7 R )

� � � � � � � � � � � � � � � � � � � � � � � � �

� 7

� � � " � � � � ; � �

� � 2 ; ;

� � < ; .

� � � � � 0 � � � � � � � �

� � � � 0 � � " � � � � � ; ) )

� � � � � � � � � � � � � � � � � � � � . )

W � � � � � � � � 0 � � � � � ; R .

� � � � ' ' � � � � � ) #"

; # $

W W � � � � � � � � � ! � ; 7 �

Page 147: Программирование: теоремы и задачи

7 # 7 � � � � � � � � � � 8 � � 5 � � � , �

� � � � � � � � � � � � � � � � � � � ' � � � � � ! �

� � � � � � � � . ;

� � � � � � � � " � � � � � 7 7 )

� � � � � � � � � � � � � ! � � � � � ; � )

� � � � � � � � � � " � � � � � � � � 7 7 )

� � � � ! � � � � ' � � � � � � � 7 ; �

� � � � � ' � ; � 7

W � 0 ' � � � � � � � � � ; � .

W � � � � � � � � � � � � ; ; R

W " � � � � � 0 � � � � � � � � � � � � ; � 7

W " � � ! � � � � � � � � � � � 0 � � � � ; � $

� � � � � � � � � ' � � � 0 � � � � � � � � � � 7 #

W � � ' � � � 0 � � � � � � � � � . R"

. )"

$ 7"

$ R

� � � � � � � R

� � � � � � � " � 0 ! � 7 #

- � � � � � � � � � � � � � � � � � ) �"

; . )

� � � � � � � � � � 0 � � � � � � � ) #

W " � � � � � � � � � � 7 � �

� � � � � � � � � � � � � � � � � � � ! �

� � � � � � � ; $ R

W � � � � � ' � � � � � ! � � � � � � � � � . ;

� � � � � � � � � � � � � � � � � � � � . )"

) 7

W � � � � � � � � � � � � � 7 )

W " � � � � � � � � . )

� � � � � � � � � � � � " � � � � � � � � � � � � R .

� � � � � � � � � � � � � � � � � � � � � � � � ) ;"

; 7 .

� � � � � ' ' � � � � � � � � � ' � � � � 7 �

� � ' ' � � � � � 7 7 $

� � ' � � � � � � � ' � � � � � � � � 0 � � � �

� � � � � � � � � � � � R R

W � � � � � � � � � � � � R .

� � ' � � � � � ' � � � � � � � � � � � �

� � � � � � � � � � � � � � 0 � � � � � � � � � R �

W � � � � � R �

W " � � � � � � � � . #

� � ' � � � � � ; ) $

W " � � � � � ; ) #"

; � 7"

; � ."

; � )

� � 0 � � � � � � � � � � � � � � � ' � � � � � ; $ $

� � 0 � � � � � � � � � � ! � � ! �

� � ! � � � ! � � ! � ; � #

� � � � � � � � � � � � � � ' � � � �"

; . 7"

7 . .

W � � � � � � � � ; ) ;

W � � � � � � � ; ; ."

; ) �

W ' � � � � � ! � . .

W � � � � � � � � � � � � � � � ; R )

W � � � � � � � � � 7

W � � � � 0 � � ; � ."

; � )"

; � $

W � ' � � � � � 0 � � � � ; � �

W � � ' � � � � � ; ) )"

; ) #"

; � 7"

; � ."

; � )

W � � � ' � � � � � � � � � � � � � � � � � � � � � $

- � � � � � 7 ) .

- � � � � ' � � 7 � R

� � � � � ' � � � � � � � � � � � � � � � � � � � � ! � "

� � � � � � � � � � � � ; 7 R

� � � � � ' � � � � � � � � � � � � " � � ' � � � � � �

� � � � � � � � ' � � � ! � ; � $

� � � � � � � � � � � 0 � � � � � ; R .

� � � � � � � � � � � � � ! ; 7 �

� � � � � � � � � � � � � � R

� � � � � ' � � � � 7 $ �

� � � � � � � � � � � � � � � � � ' � ; ; R

� � � � � � � � � � � � � � � � � � � � � � � �

7 ; )

� � � � � � � � � � � � � � � � ' � � � � � � � � � � �

; . R"

; . �"

7 R #

� � � � � � ! � � ! � � � 0 � � � � ; � #

� � � � 0 � � ' � � � � � � � � � � � � � � � 7 $

W � � � � � � � � � � � � � � � ) )"

; . �

� � � � � ! � � � � � � � � � ; )

� � � � � � � � � � ; R #

� � � � � � � � � � � � � � ; � )

� � 0 � � � � � � � � � � � � � � � ! �

� � � � � � � � � � � � R $"

; 7 )

W W � � � � � ) �

� � 0 � � � � � � ; 7 � 7

W � � � ; 7 $ �

W " � � � � � < � � � � � � � � � 7 R #

� � � � � � � � � � 8 � � 5 � � � , � 7 # .

� � 0 � � � " � � � � � � � � � ! � � � � � � 7 ) ;

� � 0 � � � � � � � � � � � � � � � � ; )

� � 0 � � � � � � � � � � � � � � � � � � � � R 7"

; 7 7

� � � � � � � � � � � � � � � � � � 7 ; $

� � � � � � � � � � � � ' � � � � � � � � . �

� � � � � � � � � � ; 7 #

� � � � � � � � � � � ! � � � � � � ; $ �

W � � � � � � � � ; $ ;

W W " ' � � � � � � � � � ; $ R

W W " � � � � � � � � � � � ; $ R

� � � � � � � � � � � � � � � � � � � � ; ; �

� � � � � � � � � ! � � � � � � 7 ) ;

� � � � � � � � ; ; �

W " � � � � � � � � � � ; . R

� � � � � � � � � � � � � ! � � � � � � 0 � � � !

7 � R

� � � 0 � � " 0 � � � � � � � � � � #

� � � 0 � � " 0 � � � � � � � � � ; . #

� � � � � � � 7 $ �

� � � 0 � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � ; 7 #

W W � � � � � � � � � � � � � � � � � � � � �

; ; 7"

; . �"

; R #

� � � 0 � ! � � � � � ; � )

� ' � � � 7 $ �

� � � � � � � � � � � � � � � 7 ; )

� � � � � � � � � � � � � � � ' � � � � � ; $ �

� � � � � � 7 � �

W � � ' 7 � �

W � � � � � � � ! � 7 R )

W � � � � � � � � � � � � ! � 7 R )

W � � � � � � � � � � ! � 7 R )

� � � � � � � � � � � � � � � � � � 7 $ .

� � � � � � � � � � � � � � � � � � # $

� � � � � � " � � � � � � � � � � � � 7 R �

� � � � � � � � � � � � � � � � � ) )

� � � � � � � � � 7 R �

� � � � � � � � � � � � ' � � � � � ! � � � � � � � � �

. �

� � � � � ; ) $

W � ! � � ' � � � � 7 R )

� � � � � � � � � � � � O � $ 7

W ' � � � � � � $ R"

; ; R

W � � � ' � � � � � � � � $ ;

W " � � � � � � � � � � � � � � � � � � � � �

W � � � � � � � � $ 7"

$ #

W � � � � � � � � � � � � � � ; 7 $"

; ) .

W � � � � � � � ! � � � � � $ #"

; . ;

W W � � � � � � � � � � � � � ; R 7

W � � � � � � � � � 7

W " � � � � � � � � � � � � � � � �

� � � � � � 7 $ R

� � � � � � � ! � � � � � � � ! " 0 � � � � � � #

� � � � � � � � � � � � � � ) �"

; . )

� � ! � � � � � � � � � � � ! � ; $ �

� � � � # )

W " ' � � � � � � � � � � # #

W � � � � � � � ! � 0 � ' � � � � ; R �

W " � � � � � 0 � � � � � � � � � � � � # �

W " � � ! � � � � � � � � � � � 0 � � � � # #

� � � � � � ! � � � � � � � � � � � � ; R .

� � � � � � � "� ! � � � � � � ! � � � � � � � � #

W " � ! � � � � � � � � #

W " � � � � � � � � � � � � � � � � � � � � ; ; $

� � � � � � � � � � � � � � ; # ;

W � � 0 � � � � � � � � ; # ;

� � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � ; ; #

� � � � � � � ; $ �

� � � � � � � � � � ' � � � � � ; $ R"

; $ �

� � � � � � � � ! � � � ! � � � ; $ �

� � � � � � � � ! � � � � � � � ; � )

� � � � � � � � ! � � � � � � ! " � � � � � ) $

� � � � � � � 6 � � � � � � ; # ;

� � � � � � � � 7 R )

W � � � � � � � � � � � � 7 � )

� � � � � � � � � � � � � � � � � � � � � � � � ; 7 $"

; ) .

� � � � � � � � � � � - � � � � � � ; . )

Page 148: Программирование: теоремы и задачи

7 # R � � � � � � � � � � 8 � � 5 � � � , �

� � � � � � � � � � � � � � � � � � � � � � � � � �

) �"

; . �

� � � � � ' � � � � � � � ' � � � � � 7 7 $

� � � � � � � � � ; ;

W " � � � � � � � � � � � � � � � � � � � � ; ; �

� � � 0 � � � � � � � � � � � � � ) #

� � � � � � � � � � � � � � ' � � � � � � � � � � � �

; ;"

; . �"

7 . R

W W "� ! � � � � � � ! � � � � � � � � ; ;

� � � � ' � � � � � � � � � ; R �"

; $ R

� � � ' � �

� � � � � � � � � � � � � � � � ; R )

� � � � � � � � � ' � � � � � � � � . $

W � � � � � � � � � � � 7 ; $

� � � � � � � � � � � � � � " � � � � � � � � � � � � �

� � � � � � � � � ; . #

W W " � � � � � � � � � � � � � � � � � � � �

; ; �

� � � � � � � � � � ' 7 ; ;�

7 ; .

� � � � � � � � � � 7 ; $

W " � � � � � � � � � � � � � � � � � � � � � � 7 7 7"

7 7 R

� � � � � � � � � � � 7 ; $

W " � � � � � � � � � � � ' � � � � � � � 7 7 7

W � � � � � ! � � � � ' � � � � � � � � 7 ; �

� � � � � � � � � � � � � � � � � � � � � 7 7 �

W � � � � � � � � � � � � � 7 7 7

� � � � � � � � � � � � � � � $ #"

; . ;

W W � � � � � � � � � � � � � ; R 7

� � � ' ! � � � � � � � � � ) �

� � � ! � � � � � � � � � � � � ; �

� � � � � � � ! ; � #"

; # ;

W W " � ! � � � � � � � � ; # $

6 � � � � � � � � � � � � � ; # ;

� � � � � � � � � � � � � � � � � � � � � � � �

� � � � � ; � )

� � � � � � � � � � � ! 7 � $

� � � � � � � � � � � � � � � � � � � � . )

� � � � � < � � � � � � � ) R"

) �"

; . �

W � � � � � � � � � � � � � � 7 �

W � � 0 � � � � � � ) �

W � � � � � � � � � ) �

W � � � � � � � � ! � � � � � � � � ) $

� � � � � � � �

� � � � � � ' 7 ; .�

7 ; )

� � � � � � � � � � � � � � � 7 ; R

� 0 ! � � � � � � � � � � � � � � � � ' � ! � 7 R )

W � � � � � � � � � � � � � � � � � ' � ! � 7 R $

�F G H G � � � I � A � B

� ' � � � � � � � � � � � � � � " �

7 . R

� � � � � " �

� #"

; $ R"

7 � $"

7 � �

� � � � P � U N "

7 $

� � � ' � � "�

V

7 )

"7 � )

� � � � � � � � � "�

R �

� � � � � � � � � � � "2

. #

"7 7 R

� � � � � � N M " �

7 � �

� � � � � � "�

R ;

� � � ' � � � � � N Q �� N "�

; # R

� � � � � � ' � U �� � Q"

J

7 � �

� � � � � N �� "J

7 )

". ;

". R

"R �

"7 � �

� � � � � "�

; # )

� � � � � � N� � " � �

� #

"7 � �

2 � � � � � � � J � � L M N � " �

; ."

7 ;"

. �"

7 � �

2 � � J � � " �

7 R R"

7 � �

2 � � � � � � � � � � � " J

� #"

7 � �

2 � � � � � � � � "�

R �

� � � � � � � "�

<

; R �

� � � � � � � "2

; R

< � � � � � � � � M � � � � " �

) R"

) �"

; . �

< � � � � � U M " J

; ) #

< � � � � � � � N �� � "T

7 � �

< � � � � � N � � M "

7 � $"

7 � #

< � ! � � � � � � � " �

7 )"

7 $"

. $"

; � ."

; � R"

7 � �

V � � ' � � "�

7 . R

V � " � ' � � "�

7 � �

V � � 0 � � � � � � � � N � � " �

7 � �

V � � � � � � � � L �"

7 � �

V � � � � � � � � " � � ' � � � ; R �

� � � � � � � � � � � � � � � � � " P

7 � $"

7 � #

� � � � # � � � � � "

$

R"

7 ;"

; ) #

� � � � � � � � � M % � � � " � 7 � �

� � � � � � � � � � � �� %� N O� � M "�

7 R R

"

7 � �

- � � � � � � S � � � � " P

R"

; . )

� � � � � � � � ' �� � � O � � Q"

7 R R"

7 � �

� � � � � � � � %� M "�

7 � �

� � � � � � M � " � ; $ R

"7 � $

"7 � �

� � � � � � � � M � N � � % � M � ; . ;

� � � � � � � � � � � � " �

J

� #"

; $ R"

7 � $"

7 � �

& � � � ' � � � " � �

7 ; .

� � � � ( � � N� "�

.

"; . ;

"; R 7

� � � � � � � � ( � � � N � � M " �

� #"

7 � �

� � � � � � � � � � � � " �

; # )"

7 ; ."

7 ; R

� � � � � � � � � M N � � � "�

7 $

Page 149: Программирование: теоремы и задачи

� , � � � � � � � � � � �

- � � � � � � � � � � � � � � � � ? � � � � � � � � � 2 � � �

� � � � � � � � � � � � � � � � � � � � � �

2 � 0 � � � � " � � � � � � �

� � � � � �

� � � � � � � � � � � � � � � �� � � � � � � � � �

� � � � ! � " � � # ! � $ � % � � � � � � � � � �

& � ' ( ! � % �

×

) � * + * , �

- . ( ! � ! � / � � ! � �� � # ! � $ � / � � ! � �� � # � 0� � 12 � �

3 ' ! 4 � � � � 5 6 � �7 ! 6 ! � �

� � � ! � � 0 $ � " � 8 � 6 � " 6 � � � � � � ' !

� � ' � ' 9 " � � � ( ! � � ( ! � # � 6 � � � � : ' ! � � " ! ��

� � ) � � � 2

8 � 6 " !2- � 0 $ ; � < = 0 ! $ � " 6 < � � ' � 2 � � �3 � 0� � � � > � � > � � �

? � � � # ! � ! � � � � � " 9 @ � ! � � � A " � " " � � � B 3 � � � ' ! / � C D ! . 6 ! E F �

� � ) � ) ) 2

8 � 6 " !2

G . : 6 < � � ' � 2 % �