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0 ∈ >-1 <l> 65 g j,=iA ����;k78j@7 �: × N → Z ==h �� �Eg�< 1J>��@< �2? �p,l<H1�j,=iA ��-;k78j@7 � : N → h�m3=< >c1|AMmeh�me< g@? ;=< � g@CJ<Hj`?/j6g�< �
�(0) = �
0� �
(�
+ 1) =�(�(�) � � ) :JL! Q
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(�� 2�j 2&'�< , ���=��6����\� , ���������86�'&�8# X '&0 . 01��6 { 5n' X ��<8#9��}3#9��5 N
_ ( e A , K b > b P1B d a ( = E � B d -T=�b1= b1=�b1/ &)(1K d a S (1& d 0�K�S 0 * " = b P+H�(1U#0 *�b+* " P�Bf]&�E P * MN0�"�2�P1(1G " / (10�K -R= " 0�G =)< & * "�0�" �
: → / B = B C b & *)<+*�bed b P�S * " =,Egk;91�� �@7J;k. 1|AM1���� m ,W. � � , a�< P�( d b h�1�� k,:gk; � m � 2�S *�b1= / "#H b1/I, " � " * (1U#K�B = " �d acb1= (+P1( d BRE *Jd 67B C d 0 4 0�B d 6�(0) = �
0� �
(�
+ 1) =�(�(�)) :
e G =I, C)M 6 * (�� b 0 d a S � , K�K b B =�b1/ &)(1K , 6 b P�( / BFE A = B *�bed b P1S * " = B &)A , * "+6I P b D�MND , 6 J P1(1G BRE =�bed/acbed ( H�S�D�(16%P�(1G acb H�BRE *�bed�� .9,2,21 Q 2�� 0 a "+0�"�� > @ c> ∗ b H+H < acbed " B &)A , I P b D�MND , 6%BbP1E 0�"+6c0�G =)< D�B *�bedlb P1S * (� b 0 d a S � , K�K b B =�b ]/ &)(1K , 6�2�� 0 a "�0�"�� >A@ ! ∗ 2 -f* 0 d P1(1G'K�P1(1&)(1U#K�B =�b P�(1U+K�B S *Jd ( d1/ U#( b G *)- 6 b &)A - 6B a�O & <%� (1G = d 0�( / U =�b K b * " E /Jd b 2NA b & b`a�* "+& d 0 *Jd a�,(d /Jd S * " *�b * M = O G+0 d a�4 =�b & d CR]K 4 = >A@ N 2'� ����������W� ���� %���1������� B P�S * " = acb C b & < K b CI"�K b+*Jd a�, 0 a (+P d < 2 * (
� b 0 d a S � , K�K b B =�b1/ &)(1K , 6 >A@ ! BFE =�bedia H b 0 d a S'P b & <1/ B d D�K b C)BRMN& , K b+* (16 <=hJ1�� �C`7 � acbed ,-mcAM1��J< > A ;k7�;91�� ��<�j|7 � D d b'-T=�b 0�U+0 * "�K b B C d 0 4 0�BRM = J > Q 2#S+P�(1G�(�# < D = M�]0 * (16 & D d � b G *)< *�b 0�G+0 *I, K b+*�b BRE =�bed 0�G =)< & * "�0�" g^ < C)B C)B 4 &I"�K b U)P b &HCI"+6 acbedK�( =�b1/Jd a S * " *�b 6 H+U+0�"�6 b P�( O�- &)B d acbed K d b , �85Mm��Jm m��@<HjJ,@m&< 2 * (1G'K�( =�b1/Jd a (1U b1=F*Jd ]a B d K -T= (1G * (1G.(+P�(�E (1G * ( C)B 4 &I"+K b BbD�D�G <+*�bed�* " = U�P b &HCI" " d /Jd b E * BR&I" 0�"+K b 0�E b* (1G�� b 0 d a (1U � , K�K b+* (16 B =�b1/ &)(1K , 6cP�"#D <%� B d b P�S *Jd 6 B C , 6 * &)B d 6 C)BRK�BbH d b`a�- 6d /Jd S * " * BT6 1|AM1���� m ,B< >|?BA m��@<HjJ,l?BA J�� Q�� < h8g �@<Hjcj�A`;0g �eg�� j�=8A=1��@;k.Jj6g�< � hJm&=�1|AM18> < h ;=m3=8A j|;k7#5=g�� � ? 1�1��@<Y5 ,l?BA>-1 </j@;k76A h��@7 �`m��`m��|< >E. m�� ?��Mm6Ak;91 <���.�,lhJm�� m&<iA A=1 m��|< j|;=m&<iA�� 1|AM1���� m ,B< > e * ( BfP1S1K�B = ( B /)<1O@d ( >AE C b P < &)(1G+K�BcK d b\d /)-Mb * (1G P+H�(1U * (1G * (1G 0�G = S�H�(1G
# P�&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�4N= 0�G =�b & *I, 0�BRM =�& J���� Q�� 1@A=1�� �`mk,l< > A�� m��|< j8,�A�� J > Q hJ1��� �Jg�< G�P�(�H�(�D�E 0 d K�BR6 j,=iAM1��-;k.8jeg�< � 1eh3A
�8m j8, �8A0g�� ==hJm��im ��? j-< ,Eg�� �(>c1e< � B G * S%C b * ( /Jd b+* G)P 4 0�(1G#K�B b G#0 * "+& < acbed C b * ( b P�( / BFE C)(1G#K�B 0 * (%B /)<1O@d (c! E 2b H+H < BRE =�bed@acbed-a�< P�M 6 P�&)( O@b1=)- 6 Wb1= - A�(1G#K�B�# b H#D�(1&�E C)K�(1G+6 & P�(1G%G)P1(�H+(�D�E � (1G =
*Jd 6 � acbed � 2 * S * BNK�P1(1&)(1U#K�B =�b G�P�(�H�(�D�E 0�(1G+K�BNK d b�* G#A b E b *Jd K , � ( � � � ) C -b* ( =F*�b 6/Jd b1/ (1A d a�<�(0 � � ) =
�( � ) = �
0�(1 � � ) =
�( � 0
� 0 � � ) = �1
�(� − 1 � � ) =
�( ��� −2
� � − 2 � � ) = ���−1�
(� � � ) =
�( � � −1
� � − 1 � � ) :J������ Q�� ,-m�� �e. ;=m3= 1|AM1���� m ,B< >2m&< m��@<HjJ,@m&< J > Q m��`7��Jg@?|,Eg �(=�j-< >+A ; �-A hJm j6gg�hJ1 �&� �c< > ��� 1eh�m��kg@? CMg�< ��< �8< m ;k.�; �BA�;k7 ��j,=iA ��-;k78j@7 � �
(� � � ) ;�B =Jd a S * BT& b 2�"c0�G+0�A -b*Jd 0�"
b1=�b1/ &)(1K d a S167(1& d 0�K�S16�� BfP b D�MWD d a�, b P�S / B d CI"
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Q ( 2 *Y,3.�/ ��0o#92&4�576�'&� #9��'&:&��� / ��6�4M'&2&'�< , ���=4BFE =�bed:b P�S *Jd 6?P�H - ( = C)BTK�BTH d b`a�- 6 0 *�b K b CI"�K b+*Jd a�<�acbed * " C)BTM &I" *Jd a�, P�H#"�&)( O (#]& d a�, 2 acbed C b7* " /Jd BR&)BTG =I, 0�(1G#K�B�0�B 51< C)(16 I /)4 C b P�BR& d (1& d 0 * (1U+K�B�0�B / U+(7P b & b ]/ BFE D�K b+*�b 29C)B a-d =)4 =F*�b 6�K�B * " =^a H b 0 d a�, b P1S / B d CI" * "�6 1@Ak;=< ,Egk;9165 gk;=< >+A`;k7J;91���;k7 �h���Aej@5 g=j@7 � 2 P�(1G A�&I"+0 d K�(+P�( d BFE * " K - C)( / ( * "+6�# /Jd P�H , 6 BfP b D�MWD , 6 & ;�& <1O (1G+K�B�( � ��� ) b1=F* E�D d b � + � 0 � b G * S * ( P b & <1/ B d D�K b 2 acbed�5`b 0�E � (1G#K�B * " = b P�S / B d CI"K�S = ( 0 * ( =pb1=�b1/ &)(1K d a S.(1& d 0�K�S * "�6 � ( � ��� ) >A@ Q,2 *Y, Z / '�� [*2�� g@? C`;0g A ;=<�7 j�=8A���@;k7Jj|7 �
( � ��� ) j|;=m&= ��1��|<P5 ,-m3<���h�m3=m��9? � gk;91e<-,Eg ;=< ��1|AM1���� m ,B< > ��� g@CJ<Hj`?/j6g�< ��(0 ��� ) = �
�( � + 1 ��� ) =
�( � ��� ) + 1g@?YAM1e<B1|A ;=< ,:gk;91c5=gk;=< >:. Z �i7�� 1��i. Z �c<H1 A �8m3=�� ;=m3=�� � ���
�( � ��� ) =
�( � � � ) :
� l�� [*2�� BRE A = (1G+K�B?K�B'BbP b D�MND , 2(D d b a�< C)B � ∈ N)(∀ � )[ � ( � ��� ) =
�( �� � )] :J N Q
� %�� [ 2 � = 0 � (∀ � )[ � (0 ��� ) =�( � � 0)] m b P1S / B d CI" b G *I, 6 * "+6 P�&)S *�b 0�"+6?BFE ]=�bed P < H d K�B BfP b D�MWD , j@;=m � 2�P�(1G acb H+BFE *�bed �Em67`5J7J;=< >E. * "�6 # a U#& d b 6 & BfP b D�MWD d a�, 6b P1S / B d CI"�6 * "+6%J N Q
� m67`5J7J;=< >E. � j|7 2 � = 0 � � (0 � 0) =�(0 � 0) 2 * B * & d K�K -R=�b
� m67`5J7J;=< >+A � h�1�1� �-< >+A � .0,21 �� BRA�S1K b 0 * B * " � m67`5J7J;=< >E. � h�1�1� �-< >E.Eh�A �5 g=j@7�(0 ��� ) =
�( �� 0)J � I � Q
acbed / BRE A = (1G+K�B b P � b G *I, S *Jd�(0 ��� + 1) =
�( � + 1 � 0)
K�B * ( = b P�H+S G)P1(�H+(�D d 0�K�S �( � + 1 � 0) =
�( � � 0) + 1
J 3'& d 0�K�S16 Q=�(0 ��� ) + 1
J�� I � Q= � + 1
JL3?& d 0�K�S16 Q=�(0 ��� + 1)
JL3?& d 0�K�S16 Q :e�� b G * S * (70�"�K�BRE ( - A�(1G#K�B 0�G+K�P+H+"+& 4 0�B d�* " � (�"�CI" *Jd a�, I P b D�MND , D d b * " � %�� [* "�6 a U+& d b 67BfP b D�MWD , 6
�=X '&0 . 01��6�Z�� 4��=' �� BTA�S1K b 0 * B * " = � X '&0 . 0P��6�4 VYX Z�k�#�� [
(∀ � )[ � ( � ��� ) =�( � � � )]J I � Q
acbed / BRE A = (1G+K�BI2�P < H d K�B � (�"�CI" *Jd a�, I P b D�MND , S *Jd(∀ � )[ � ( � + 1 ��� ) =
�( �� � + 1)] :
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(�� 2�j 2&'�< , ���=��6����\� , ���������86�'&�8# X '&0 . 01��6 { 5n' X ��<8#9��}3#9��5 9� m67`5J7J;=< >E. � j|7 2 �
( � +1 � 0) =�(0 � � +1)
B G * S - P1B *�bed-b P�S * " = b P1S / B d CI"* "�6 � %�� [ 5 2�S+P1(1G / BRE C b K�B?S *Jd D d b a�< C)B � 2 � ( � � 0) =
�(0 ��� )
� m67`5J7J;=< >+A � h�1�1� �-< >+A � .0,21 �� BRA�S1K b 0 * B * " � m67`5J7J;=< >E. � h�1�1� �-< >E.Eh�A �5 g=j@7�( � + 1 ��� ) =
�( �� � + 1)
J�� I � Qacbed / BRE A = (1G+K�B * ( � " * (1U#K�B = (
�( � + 1 ��� + 1) =
�( � + 1 � � + 1)
K�B * ( = B C , 67G�P�(�H�(�D d 0�K�S �( � + 1 ��� + 1) =
�( � ��� + 1) + 1
J 3'& d 0�K�S16 Q=�( � + 1 � � ) + 1
J I � Q= (
�( � � � ) + 1) + 1
J 3'& d 0�K�S16 Q= (
�( � ��� ) + 1) + 1
J I � Q=�( � + 1 ��� ) + 1
JL3?& d 0�K�S16 Q=�( �� � + 1) + 1
J�� I � Q=�( � + 1 � � + 1)
JL3?& d 0�K�S16 Q : a
>A@ 9 2g* ' , ' / 4 ,9[ � [N m K - C)( / (16 b P1S / B d CI"�6�P�(1G7A�&I"+0 d K�(+P�( d , 0 b K�B D d b7* " =b P1S / B d CI" * "+6%$?&)S *�b 0�"�6 > @ Q acb H�BRE *�bed �J<Hh �|. g�h�1�1� �c. 2 BfP1B d /I, A�&)B d < 0 * " acb1=/Jd b1O (1&)B *Jd a�- 6�2�J 5 (�"�CI" *Jd a�- 6 Q BbP b D�MND d a�- 6 b P�( / BFE C)B d 6 * "+6 � %�� [ 5 acbed�* (1G �=X '��0 . 0P��6��&l � 4��=' / ��5 * "�6�# a U#& d b 6 & BbP b D�MND , 6 * "�6 b P�S / B d CI"+6 B G * S BRE =�bed A b ]& b`a�* "�& d 0 *Jd a S BbP b D�MND d a�4N=�b P�( / BFE C)BTM = D d b P�&)( *)< 0�B d 6 * "�6?K�(1& O�, 6
(D d b a�< C)B � )(∀ � ) � ( � ��� ) �
S+P1MN6 BRE =�bed " J N Q 3 H+S�D�(16 P�(1G b G * S BRE =�bedEb P b & b E * " * ( D�E = B *�bed P�&)( O@b1=I, 6 b1=P1&)(10�P b C , 0�(1G#K�B =�b�b P1( / BRE C)(1G+K�B acb+* BRG#C)BFE b1=7* " = B d /Jd a�, P1BT&�E P * MN0�"
(D d b a�< C)B � )[ � ( � � 17) =
�(17 � � )] �
K�B?BfP b D�MWD , 0 * ( � � ��������4� " B &)A , I P b D�MND , 6'K�P�(1&)BFE =�b A�&I"+0 d K�(+P�( d "�C)BRE�D d b * " = b P�S / B d CI"P1&)( *)< 0�BRM =7* "�6?K�(1& O�, 6
(D d b a�< C)B � ∈ N) � (
�) �J Q Q
������������ ����� �����������(E�� 8 �������8 ���98 ������ 98 D d b =�b b P�( / BFE C)(1G#K�B7K�B?BbP b ]D�MND , a�< P�( d b P1&)S *�b 0�" * "�6?K�(1& O�, 6(∀ � )(∀ � ) � (
� � � ) �
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T ( 2 *Y,3.�/ ��0o#92&4�576�'&� #9��'&:&��� / ��6�4M'&2&'�< , ���=4D d b P b & <1/ B d D�K b 2)P1& - P1B d�=�b BbP d H - C)(1G+K�BWP�( d b � (
�)C b A�&I"�0 d K�(+P1( d , 0�(1G#K�B)2)P A 2
� (�) ⇐⇒ � (
� ��� ) (D d b 0�G#D a B a & d K -R= (�2�0 *�b C)BT&)S � ) �
� (�) ⇐⇒ � ( � � � ) (
D d b 0�G�D a B a & d K -T= (�2�0 *�b C)BR&)S � ) �� (�) ⇐⇒ (∀ � ) � (
� ��� ) �� (�) ⇐⇒ (∀ � ) � ( � � � ) �
,^b`a S1K�" acbedJa�< P1( d b P d (�P1BT&�E P�H+( a "�P�&)S *�b 0�" (∀ � ) � (�)D d b�* " = (+P1(�E b K�P1(1&)(1U#K�B
=�b�b P�( / BFE C)(1G#K�B b1= B C < & * " *�b S *Jd(∀ � ) � (
�) � ⇒ (∀ � )(∀ � ) � ( � ��� ) :
e BcK�BR& d a�- 6cP1BT& d P *)4 0�B d 6 * ( P d ( / U#0 a (�H+( K - &)(16.K d b 6.BbP b D�MND d a�, 6 b P1S / B d CI"�6BFE =�bed:b`a & d 514 6 " BfP d H+(�D ,.* "�6 acb+*)< H+H#"+H#"�67P�&)S *�b 0�"+6 * "+6 K�(1& O�, 6 J Q Q P�(1G / BFE ]A = B *�bed BTU a (�H b 2 acbed 0�G = BfP < D�B *�bed�* " = P1&)S *�b 0�" P1(1G%K b 6?B =)/Jd b1O�- &)B d ; d b / BTU * BT&)( P b & <1/ B d D�K b C)BTM &)(1U#K�B K d b 0�G =)< & * "�0�"7H d D�S * BT&)( ( d a BFE bpb P1S * " =
P1&)S10�C)BR0�"�2 b H#H < K�B'B C�E 0�(1GcB =)/Jd b1O�- &)(1G+0�BR6 d /Jd S * " * BR6 acbed P�(�H+H - 67B O@b &)K�(�D - 6 >A@ T 2�� ���������#������� ������������� ������ (1&�E � B *�bed K�B * " H�BbD�S1K�B = " �J<Hh �|.1|AM1���� m ,W.
� (0 � � ) = � + 1� (�
+ 1 � 0) = � (� � 1)
� (�
+ 1 � � + 1) = � (� � � (
�+ 1 � � )) D
JL9 Q
acbed D d b a�< C)B��W2N"�# * (1K , & � � : N → NJ * "�6.0�G =)< & * "�0�"�6 Q?* (1G @ : r�V i X.Y b3b
(1&�E � B *�bed K�B * " = B C�E 0�MN0�"� � ( � ) = � (
� � � ) :J T Q; d b P b & <1/ B d D�K b 2
� 0(� ) = � + 1
/ "+H b1/I, " � 0BRE =�bed "c0�G =)< & * "+0�"�� * (1G g�h�A ,Eg8AMm3= 0 * (1G#6 O G+0 d a (1U+6 b & d C)K�(1U+6
3 (1& d 0�K�S16 P1& - P1B d�=�b /Jd acbed (�H�(�D�"+C)BFE 2 acbed " b P�S / B d CI" - A�B d6b1= B C < & * " * ( B =)/Jd b ]O�- &)( = BbP�B d /I,(b P bed * BFE BbP1E a H+"+0�" * (1G�� b 0 d a (1U � , K�K b+* (16 B =�b1/ &)(1K , 6cD d b * ( =(1& d 0�K�S.0�G =)< & * "+0�"+6 � : N → S+P1(1G 6= N >A@ d 2�� 4����=' 2 +-m�j�<�j|;k79,:1�j�=8A=1��@;k7Jj-< 18>@?BA g@C8< j ?/jeg��BA JL9 Q �>�lg�<@18>��|< �:? �,B< 1���<�j|7 Z / "#H b1/I, d acb1= (+P1( d BRE *�bed|b P1S b`a & d 514 67K�E b./Jd K�BTH , 0�G =)< & * "�0�" j-X Z�<\#3��} [*2�� 0 * M * ( 0�U = (�H�( S�H�M =N* M = K�( = (1K�BbH 4N= 0�G =�b & *I, 0�BTM = 0 * (1G#6
O G#0 d a (1U#6�2 / "#H b1/I,� ∈ ⇐⇒ " � BFE =�bed 0�G =)< & * "+0�" ��� : N → N :
3?&�E � (1G#K�B K d b 0�G =)< & * "�0�" � : N → K�B BfP�E a H#"�0�" * (1G � b 0 d a (1U � , K�K b+* (16B =�b1/ &)(1K , 6�2�M 6?B C , 6
�(0) =
� �
����� �!�#"�$&%('*)�+!,!-!+/. �/0!"�$1 �/2 3!4/+!5/670 28�9#:�+/; +/<���,8��5!=#> "/>�2? "&@A4/+/0 2#>�2 4#0/>���0!6CB#0�3/+!,/"ED�:A4/=!- FG�H4/+�DI<�F/5!�#>�+J; � K#"('L�MJN FO:!>AF/5/.�4�PH+!9�D ?/Q!Q R D ?SLGTUM!V ' �
(�� 2�j 2&'�< , ���=��6����\� , ���������86�'&�8# X '&0 . 01��6 { 5n' X ��<8#9��}3#9��5 d
/ "+H b1/I, " *Jd K , � (0) BFE =�bed "c0�G =)< & * "�0�" * (1G%BbP�S1K�B = (1G12 � ( � ) = � + 1 acbed�(�
+ 1) =�(�(�)) �
S+P1(1G " *Jd K , �� =
�(� ) * "+6 0�G =)< & * "�0�"�6 � : → (1&�E � B *�bed D d b#a�< C)B
� : N → NK�B * " = B C , 6 b1=�b1/ &)(1K ,
�� (0) = � (1)
�� ( � + 1) = � (
�� ( � )) :
B = C - 0�(1G+K�B
� � =�(�) �
* S * B?( d 0�G =�b & *I, 0�B d 6 � � d acb1= (+P�( d (1U =7*Jd 6?B C , 67B C d 0 4 0�B d 6 � 0(
� ) =�( � ) = � + 1
� � +1(� ) =
�( � � )( � )
-b* 0 d P1(1G� � +1(0) =
�( � � )(0) = � � (1) �
� � +1(� + 1) = � � ( � ( � � )( � ))
= � � ( � � +1(� )) :
_ BTH d a�< C -b* (1G#K�B� (� � � ) = � � ( � )
acbed C b1=�b D�& <1O (1G+K�B b G *)- 6 *Jd 67B C d 0 4 0�B d 6�2� (0 � � ) = � 0(
� ) = � + 1
� (�
+ 1 � 0) = � � +1(0) = � � (1) = � (� � 1)
� (�
+ 1 � � + 1) = � � +1(� + 1) = � � ( � � +1(
� )) = � (� � � (
�+ 1 � � )) �
-b* 0 d P1(1G BRE =�bedlb`a & d 514 6%( d B C d 0 4 0�B d 6�2�D d b *Jd 6%(+P1(�E BT6 - P1&)BfP1B =�b / BFE C)(1G#K�B S *Jd- A�(1G = H�U#0�" m K�( =�b1/Jd a S * " *�b%* "�6'H�U#0�"+6 / BFE A = B *�bed K�B J /Jd P+H ,�Q BbP b D�MND , 0 * (��W2 , K�B * " =
P1&)(10�B a�*Jd a�, B O@b &)K�(�D , * (1G�� b 0 d a (1U � , K�K b+* (16�2�P�(1G BbD�D�G <+*�bed�* " K�( =�b1/Jd a S#]* " *�b%* "�670�G =)< & * "+0�"+6 � ( � ) = � � 2 � 0 a "�0�" � > @ d� am B &)A , I P b D�MWD , 6 acbed�* ( � b 0 d a S � , K�K b B =�b1/ &)(1K , 67BFE =�bed�5 b 0 d a�< # b C d 4 ]
K b+*�b%& D d b * (1G+6 O G#0 d a (1U#6 b & d C)K�(1U+6 P1(1G / B = K�P1(1&)(1U = =�b(b P�( / B d A * (1U = 2�P b & <K�S = ( = b1= - A�(1G+K�B a�< P1( d (70�G#D a B a & d K -R= (7(1& d 0�K�S * M = b & d C)K 4N= 0 * ( P�H b E 0 d (?K d b 6D�B =Jd a S * BT&I"�67C)BRMN&�E b 6�2�P A 2 * "�6?C)BRMN&�E b 670�G = S�H�M =
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O ( 2 *Y,3.�/ ��0o#92&4�576�'&� #9��'&:&��� / ��6�4M'&2&'�< , ���=4����� �p�Byl~:�*$e}��� > @ c> ∗ 2 B P1( / BRE C * B * ( � b 0 d a S � , K�K b B =�b1/ &)(1K , 6 > @ ! b P�S * " = B &)A ,
I P b D�MND , 6 VvX Z�<\#3��} [�� ; d bc* " = U�P b &HCI" * "+6 � : N × � → 2�C -b* (1G#K�B �
= {( � 0�;:;:;:�� ���
−1) | � 0�;:;:;:�� � �
−1 ∈ } (� ∈ N) �
� (� � � � � ) ⇐⇒ � = ( � 0
�;:;:;: � � � ) ∈ �+1
& �0 =
�( � ) & (∀ FHG �
)[ ��� +1 =�( ��� � F � � )]
acbed / BRE A = (1G+K�B?K�B'BbP b D�MND , * " = P�&)S *�b 0�"(∀ � )(∃! � ) � (
� � � � � ) :; d b7*�b * G#A b E b � � � 2 � = � (
� � � ) = ( � 0(� � � ) �;:;:;:�� � � ( � � � )) BFE =�bed " K�( =�b1/Jd a�,b`a (�H�(1G#C�E b J�K ,ia (1G#6 � + 1 Q D d bc* " = (+P1(�E b�d 0�A�U#B d " � (
� � � � � )2�C -f* (1G+K�B
�(� � � ) = � ⇐⇒ � = � � ( � � � ) :
� > @ ! ∗ 2�� BTA�C)BFE * B * ( � b 0 d a S � , K�K b B =�b1/ &)(1K , 6 > @ ! M 6 b C�E MNK b 2 acbed/ BFE C * B b P � b G * S * " = B &)A , I P b D�MND , 6
� > @ N 2�� BRE C * BcJ b P�S * " = B &)A , I P b D�MND , 6 Q S *Jd|a�< C)B K�")] a B = S 0�U = (�H+( O GI]0 d a�4N= � ⊆ N - A�B d BbH < A d 0 * ( K - H�(16
� > @ Q 2�� BRE C * B S *Jd D d b / U#(7P1& b D�K b+*Jd a (1U#6 b & d C)K�(1U+6�� ≥ 02��
021G)P < &)A�B db`a & d 514 6 -T=�b 6 O G#0 d a S16 b & d C)K�S16��12 *)-b* ( d (16'P�(1G D d b a�< P1( d ( = J P�& b D�K b+*Jd a S Q� 2
�=���
+ �� 0 ≤ G � :� P1B *�bed S *Jd:acbed�* ( BFE =�bed K�( =�b1/Jd a S�2�B O S10�( =� =
� − ��� 3 dEb & d C)K�(�E�� acbed BRE =�bed�* ( he7 ��? >2m J�� g _#dfZ\VAb+d QBacbed�* ( ==h�A��im <Hh�m J i VFX.Y�Zcb3w�V i QN* "+6 /Jd b E &)BT0�"+6 * (1G� /Jd bc* (1G�� 2 acbed�* (1G#670�G+K 5 (�H1E � (1G#K�B
� g _#d(� � � ) =
� � i X(� � � ) =
b P1S * (1G+6 B D�D�H d a (1U+6?S1&)(1G+6 \I P�E 0�"�6?BFE =�bed�5 (�H d a S.C b C - 0�(1G+K�B� g _�d
(� � 0) = 0 � i X
(� � 0) =
� �-b* 0 d P�(1G ( d 0�G =�b & *I, 0�B d 6 b G *)- 6.BFE =�bed (1& d 0�K -R= BT6.D d b S�H b *�b � � � acbedWd acb1= (#]P1( d (1U = P <1=F*�bc* " = B C�E 0�MN0�"�� =
��� g _#d(� � � ) +
i X(� � � )
� > @ 9 2�� d acbed (�H+(�D , 0 * B b1=�b1/ &)(1K d a (1U#67(1& d 0�K�(1U+6 * "+67K�(1& O�, 6
�(0 � � ) =
�1(� ) ��
(1 � � ) =�2(� ) ��
(�
+ 2 � � ) =�(�(� � � ) � � ( � + 1 � � ) � � � � ) �
J d Q
S+P1(1G%( d �1� �
2� � BFE =�bed / (10�K -R= BT670�G =�b & *I, 0�B d 670 * (1G+6 b & d C)K�(1U#6
����� �!�#"�$&%('*)�+!,!-!+/. �/0!"�$1 �/2 3!4/+!5/670 28�9#:�+/; +/<���,8��5!=#> "/>�2? "&@A4/+/0 2#>�2 4#0/>���0!6CB#0�3/+!,/"ED�:A4/=!- FG�H4/+�DI<�F/5!�#>�+J; � K#"('L�MJN FO:!>AF/5/.�4�PH+!9�D ?/Q!Q R D ?SLGTUM!V ' �
(�� 2�j 2&'�< , ���=��6����\� , ���������86�'&�8# X '&0 . 01��6 { 5n' X ��<8#9��}3#9��5 ]
� > @ T 2 m b`a (�H+(1G+C�E b � Z ��_Pb�Y :F: Z�(1&�E � B *�bed K�B * " =pb1=�b1/ &)(1K ,� 0 = 0 � � 1 = 1 � � � +2 = � � + � � +1
:J O QJLY Q � P�(�H�(�D�E 0 * B * " =7*Jd K , � 9
JU� Q � BRE C * B?S *Jd D d b a�< C)B �W2
� � +2 ≥ �� � S+P1(1G � =
1 +√
5
2:
m 5`b 0 d a�, P b & b+*I, &I"+0�"'BFE =�bed S *Jd ( � BFE =�bed K�E b b P1S *Jd 6 &�E � BT6 * "�6 / BRG * BR&)( 5�< C)K d b 6B C�E 0�M 0�"�6� 2 = � + 1 :J ] Q
� BFE C * B BfP�E 0�"�6 S *Jdcb1= (�� = 1−√
52
BFE =�bed " < H+H#" &�E ��b * "�6 J ] Q 2 * S * B)2�D d b�a�< C)B �� � =
��− �
�√
5:
� > @ d-2�� BRE C * B?S *Jd�* (cP�(�H�U%K�E b 0�G =)< & * "+0�" d acb1= (+P�( d BFE * (.0�U#0 * "+K b J�9 Q � > @ O 2�� P�(�H�(�D�E 0 * B * " =7*Jd K , � (3 � 2) � > @ ] 2 ; d b%*Jd 6 * (1K - 6 * (1G @ : r�V i X.Y b3b 2 / BRE C * B7S *Jd
� 1(� ) = � + 2 �
� 2(� ) = 2 � + 3 :
� > @ c> 8 2 � &)BFE * B -T=�b # a H+B d 0 * S & * U)P1(.D d bc* " = � 3(� )
� > @ c>�>32 � BFE C * B?S *Jd D d b a�< C)B � acbed � 2 � � ( � ) ≥ 1
� > @ c> ! 2 � BFE C * B'S *Jd-a�< C)B * (1K , � �.* "+6?0�G =)< & * "+0�"+6 * (1G @ : r+V i XcY�b3b BFE =�bed1�=�j|;k7��� 1�<@CMm&=Jjc1 2 / "#H b1/I,� G � � ⇒ � � ( � ) G � � ( � ) �
b P1S * ( (+P1(�E ( - P1B *�bed S *Jd � � ( � ) ≥ � �VYX Z�<8#9��} [�� � BFE C * B�K�B # /Jd P+H , BbP b D�MND ,�&S *Jd � � ( � ) G � � ( � + 1)
� > @ c>AN 2 � BFE C * B?S *Jd D d b S�H b%*�b � � � acbed � 2
� G � � ⇒ � � ( � ) G ��� ( � ) :VvX Z�<\#3��} [�� � BFE C * B?K�B�# /Jd P+H , BbP b D�MND ,�& S *Jd � � ( � ) G � � +1(
� ) � > @ c> Q,2 � BFE C * B?S *Jd D d b a�< C)B � acbed � 2
� � ( � � ( � )) G � � +2(� ) :
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> 8 ( 2 *Y,3.�/ ��0o#92&4�576�'&� #9��'&:&��� / ��6�4M'&2&'�< , ���=4��� � u�w,) {cx+'*$��c� � z �cz:�cwcx � }Yy/�i�����9�czEwe{e~2�#$ }��>AE' c>&2�Y, ������Z�5 2�� =�b 0�U = (�H�(�� P�H+B d (1K�BbH 4N= 1 0�G =�b & *I, 0�BRM = 0 * (1G#6 O G#0 d ]a (1U+6?BFE =�bed ����7������+E���� 8 � �,E���� ��� b1=�JLY Q m 0�G =)< & * "+0�"�� ( � ) = � + 1 * (1G g�h3Ak,:g8A=m&= b1=I,8a B d 0 * (�� JU� Q ; d b a�< C)B � acbed �12�"c0 *�b C)BR& , 0�G =)< & * "�0�"�� K�B *�b+5 H#" *)4N=
�� ( � 1
�;:;:;: � � � ) =�
b1=I,ia B d 0 * (� B = � = 02 * S * B.J�0�G#K 5 b+*Jd a�<�Q � 0 =
�12 / "+H b1/I, *�b G * E � (1G+K�B * "0�G =)< & * "�0�" # 0 K�B *�b+5 H#" *)4N=�& K�B * " JLK�( =�b1/Jd a�,�Q *Jd K , * "�6 J :)Q ; d b a�< C)B � acbed F 2 1 ≤ F ≤ �W2�" h��`m �:m �@.
��� ( � 1
�;:;:;:�� � � ) = � �b1=I,ia B d 0 * (�� $ b & b+* "+&)(1U+K�B7S *Jd " � 1
1
BFE =�bed " ;91�=8;=m ;=< >E. 0�G =)< & * "�0�" 0 * ( N2
� 11 ( � ) = � J w Q � �,E���� ����������� ����� ��������E)���� B = " � ] K�BTH , 6 � ( � 1
�;:;:;: � � � ) acbed ( d � 2��] K�BbH�BRE 670�G =�b & *I, 0�B d 6
�1( �� ) �;:;:;:�� � � ( �� )b1=I,ia (1G = 0 * (��%2�K�B �� = ( � 1
�;:;:;:�� � � ) 2 * S * B acbed "�( �� ) =
�(�1( �� ) �;:;:;:�� � � ( �� ))J > 8 Q
b1=I,ia B d 0 * (�� J�V Q�� �,E���� ����������� ����� ����7������+E��� �������������� fB = " ��] K�BTH , 6 � acbed "
(�
+ 2)] K�BbH , 6 � b1=I,ia (1G = 0 * (��%2 acbed2b1= " (
�+ 1)
] K�BTH , 6 � (1&�E � B *�bed:b P1S *Jd 6B C d 0 4 0�B d 6
{ �(0 � �� ) =
�( �� )�
( � + 1 � �� ) =�(�( � � �� ) ��� � �� ) �
J >�> Q* S * B acbed " � b1=I,8a B d 0 * (� �e G#K 5 b+*Jd a�< P�BR& d H b K 51<1= (1G#K�B 0 � b G * S * ( 0�A , K b* " = P�BR&�E P * M 0�"�� = 0
2�(+P�S * B * (.0�A , K b P b E & = B d�* ".K�(1& O�,{ �
(0) =�
(= � 0 )�( � + 1) =
�(�( � ) ��� ) :
1 ������������� �"!�#�$�%'&�(*),+ -*$.-0/ 1 2.3.4,576 8*3�891;:=< >'<,?A@'B�>�@�?A6 @�C�B <D:AE�$"+DF�$�G9&DF9H*F IKJ L &�M"N�-O+D%9P.M"L M$�%9&D(*) + -9$.-
Q: ISR → I
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(�� 2�*Y,3.�/ ��0o#92���5n'&2&'�< , ���=��6 { 5n����2&' ,3/ 4���#9��5 >P>
m 0�G =)< & * "�0�" � BRE =�bed ����7������(E)��� 8 ������������W� �� J 0 i Z\X%Z dfZ V i V : g�i ` Z V Q b1=b1=I,ia B d 0�B a�< C)B P1&)M * (�D�B =)4 6 a H�B d 0 * S 0�U = (�H+( 0�G =�b & *I, 0�BTM = 2 acbed ����7������(E)��� 8������������W� �� � ���Ψ2�D d b * G#A b E ( 0�U = (�H+( Ψ
# / (10�K -T= M =�& P�H+B d (1K�BbH 4N= 0�G =�b &R]*I, 0�BRM = 2 b1=�b1=I,8a B d 0�B a�< C)B P1&)M * (�D�B =)4 6 a H�B d 0 * S 0�U = (�H�(cP1(1G P�BR& d - A�B d * ( Ψ
� &I"+0 d K�(+P�( d (1U+K�B * (1G#670�G+K 5 (�H d 0�K�(1U+6
R� = { � | " � BRE =�bed P1&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�, } �R� (Ψ) = { � | " � BRE =�bed P1&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�, 0 * ( Ψ} �
-b* 0 d P1(1GR� = R� (∅) :
; d b P b & <1/ B d D�K b 2�" P1&)S10�C)BR0�"� ( � ��� ) = � + �
BFE =�bed P�&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�, BbP�B d /I,�d acb1= (+P1( d BRE *Jd 67B C d 0 4 0�B d 6� (0 ��� ) = � 1
1 ( � ) = � �� ( � + 1 ��� ) =
�( � ( � ��� ) � � ��� ) = � ( � ��� ) + 1 �
J > ! QS+P1(1G " 0�G =)< & * "�0�" � ( ��� � ��� ) =
�( � )
BFE =�bed P1&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�, BbP�B d /I,(1&�E � B *�bed K�B * ".0�U = C)BT0�"
�( � � � ��� ) =
�( � 3
1 ( � � � ��� )) :J >AN Q@ B'P b &)S1K�( d b A�& , 0�"%P1&)( 5 (�H 4N= K�P1(1&)(1U#K�B =�b / BRE C)(1G+K�B7S *Jd�* ( 0�U = (�H+( R� (Ψ)BFE =�bed@a H+B d 0 * S.D d b #f&I" * (1U#67(1& d 0�K�(1U+6 & P < 0�"�6 O U#0�BTM 6 $ A 2 b1=
�( � ��� ) =
�( � � � 1( � � � + 1) � � 2( ���� )) �
* S * B." � BRE =�bed P�&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�, 0 *Jd 6 � � �1� �
2J / "#H b1/I, 0 * ( 0�U = (�H+(
R� (� � �
1� �
2)Q 2�BfP1B d /I,
�( � ��� ) =
�( � 2
1 ( � ��� ) � � ∗1( � ��� ) � � ∗2( � ��� )) �J > Q QS+P1(1G
� ∗( � ��� ) =�( � 2
1 ( � ��� )) = � + 1J > 9 Q
� ∗1( � ��� ) =
�1( � 2
2 ( � ��� ) � � ∗( � ��� )) =�1(�� � + 1)
J > T Q� ∗2( � ��� ) =
�2( � 2
2 ( � ��� ) � � 22 ( � ��� ))J > d Q
-b* 0 d P�(1G " � (1&�E � B *�bed`b P1S *Jd 6 / (10�K -R= BT6 0�G =�b & *I, 0�B d 6 K�B /Jd b1/ (1A d a�- 6 B O-b &)K�(�D - 60�U = C)BR0�"�6 m BbP�S1K�B = "%P�&)S *�b 0�"cBRE =�bed�* B * & d K�K -T= "�2 b H#H < A�& , 0 d K�" >AE' ! 2g*-, Z / '�� [*2 +@m j�<8A=m��im R� (Ψ)
; �BA(h ���@;=m �Jg8A0? �\1|AM1���� m ,B< >|?BA j�= �A=1��@;k.Jj6g��BA j|;=mΨhJg �|< �>�lg�<W;=m
Ψ>-1 <lg@?PA=1 </h ���@;=m �Jg8A0? � > �ig�<Hj@;BA
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> ! ( 2 *Y,3.�/ ��0o#92&4�576�'&� #9��'&:&��� / ��6�4M'&2&'�< , ���=4j-X Z�<\#3��} [*2 B = 2�P A 2�" � (1&�E � B *�bed K�B P�&)M * (�D�B =I, b1=�b1/ &)(1K , b P1S *Jd 6 � � � ∈
R� (Ψ)2 * S * B)2 � � � ∈ � D d b a�< C)B P�&)M * (�D�B =)4 6 a H+B d 0 * S � P�(1G P1BT& d - A�B d * (
Ψ < & b 2 � ∈ �%2 D d b a�< C)BcP�&)M * (�D�B =)4 6 a H�B d 0 * S`� P�(1G P1BT& d - A�B d * ( Ψ < & b� ∈ R� (Ψ) a
; d b P b & <1/ B d D�K b 2�( d 0�G =�b & *I, 0�B d 6�( ����� ) = � + �
�( ��� � ��� ) =
�( � 3
1 ( ��� � ��� ) � � 33 ( ��� � ��� ))= � + �
BFE =�bed P�&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�- 6�2 acbed BbP�(1K -R= MN6�2 b1= C - 0�(1G+K�B�(0 ��� ) = 0 = � 1
0 ( � )�( � + 1 ��� ) =
�(�( � ��� ) � � ��� ) =
�( � ��� ) + ��J >AO Q
* S * B acbed " � ( � ��� ) BFE =�bed P�&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�, b H#H < 2�P�&)( O@b1=)4 6 JLK�B BbP b ]D�MND , 0 * ( � 2 b1= / B = O@b E = B *�bed P1&)( O-b1=)- 6�� Q 2 � ( � ��� ) = � · � 2 < & b�acbed ( P1(�H)]H b P+H b 0 d b 0�K�S16 BRE =�bed P�&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�, 0�G =)< & * "�0�" e * ( K - H#H�( = C bB O-b &)K�S10�(1G#K�B * " = > E' ! #b0 d MWP+"+H <%& >AE' N 2g*-, Z / '�� [*2 � h���Aej@5 g=j@7 � + � Z m�hJm�� � 1eh �e1 j-< 1ejJ,9A�� � · � >c1e<pm <g�h�A ,Eg8A9g���j�=8A=1��@;k.Jj6g�< �pg@?PA=1 </h ���@;=m �Jg8A0? � 1@A=1�� �`mk,l< > �����
� ># � ! = 1 · 2 · · · � 0! = 1( � + 1)! = � !( � + 1)
� ! ����( � ) = b1= ( � = 0) * S * B 0 b H+H d 4 6 � − 1
��(0) = 0���
( � + 1) = ��YN� � −· � = b1= ( � G � ) * S * B 0 b H+H d 4 6 � − � � −· 0 = �
� −· ( � + 1) =��
( � −· � )�-Q�
min( � ��� ) min( � ��� ) = � −· ( � −· � )� 9
max( � ��� ) max( � ��� ) = ( � + � )−· min( � ��� )� T | � − � | | � − � | = ( � −· � ) + ( � −· � )�vd1 �� � 0 = 1�� +1 = �� · �
3 deb P�( / BFE C)B d 6 * M = P1BT& d 0�0�( *)- &)M =�b P1S *Jd 6 P1&)( *)< 0�B d 6 0 � b G * S * ( B /)<1O@d ( BFE =�bedb P�H - 6�2 acbed C bc*Jd 6 b1O�, 0�(1G+K�B�D d b�b 0 a�, 0�B d 6
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(�� 2�*Y,3.�/ ��0o#92���5n'&2&'�< , ���=��6 { 5n����2&' ,3/ 4���#9��5 > N
>AE' Q 2�Y, ������Z�5 2 m 4��������������1��� �1� �� ���������#������� K d b 6 0�A - 0�"+6 � ( �� ) BRE ]=�bed "��� ( �� ) =
{1 � b1= � ( �� ) �0 � b H#H d 4 6 �J > ] Q
acbed "%0�A - 0�" � ( �� ) BRE =�bed ����7������(E)��� 8 ������������W� �� b1= " � � ( �� ) BFE =�bed P1&)M * (#]D�B =)4 6 b1=�b1/ &)(1K d a�, �_ ( E /Jd ( D d b 0�U = (�H b " A b & b`a�* "+& d 0 *Jd a�, 0�G =)< & * "+0�" * (1G� ⊆ N
BRE =�bed "��� ( � ) =
{1 � b1=�� ∈ � �0 � b H#H d 4 6 �JL!�8 Q
acbed * ( � BFE =�bed P1&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a S b1= " � � BRE =�bed P�&)M * (�D�B =)4 6 b1=�b1/ &)(#]K d a�, >AE' 9 2g*-, Z / '�� [*2 n A#7 � ( �� ) g@?YAM1e< h��3�@;=m ��g8AM? ��1|AM1���� m ,B< >E. j@� �=j|7 Z m <�
( �� ) >-1 < � ( �� ) h��3�@;=m ��g8AM? ��1@A=1�� �`mk,l< > ����j,=iAM1��-;k.8jeg�< � >c1e<�7 � ( �� ) m�� ?��9gk;91 <1eh��61�=8;����^,:g hJg �|< he; ?/jeg�< � Z�( �� ) =
{�( �� ) � b1= � ( �� ) ��( �� ) � b H#H d 4 6 �
;BA ;0g >-1 < 7 �( �� ) g@?YAM1e<Bh��3�@;=m ��g8AM? ��1|AM1���� m ,B< >E.��
j-X Z�<\#3��} [*2 �( �� ) = ��� ( �� )� ( �� ) + (1−· ��� ( �� )) � ( �� ) a@ B /Jd b1/ (1A d a�- 6.B O-b &)K�(�D - 6 b G *I, 6 * "�6%P�&)S *�b 0�"+6 / BRE A = (1G+K�BcS *Jd�* ( 0�U = (�H+(
* M = P�&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�4N= 0�G =�b & *I, 0�BRM = BFE =�bed:a H+B d 0 * S D d b (1& d 0�K�(1U#6 K�B� P�BR& d P *)4 0�B d 6�2�D d b a�< C)B � ≥ 2
>AE' T 2g*-, Z / '�� [*2 JLY Q � <@g@C`. � j,=iAM1��-;k.8jeg�< � >-1 <2j@� �=j6g�< ��g@?PA=1 <:h ���@;=m �Jg8A0? �1|AM1���� m ,B< > ���
�YO� � = � �=( � ��� ) = 1−· | � − � |
� ] � ≤ �� � G � �≤( � ��� ) = 1−· ( � −· � )��� ( � ��� ) = �
≤( � + 1 ��� )� > 8 Ni X
( � ��� ) i X(0 ��� ) = 0
i X( � + 1 ��� ) =
� + 1 �0b1=�� = 0 �i X( � ��� ) + 1 �b H+H d 4 6�2 b1= i X
( � ��� ) + 1 G � �0 �0b H+H d 4 6� >P>� � | � JL( � /Jd bed &)BFE * ( =���Q �
|(� ��� ) = 1−· i X ( �� � )
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>AQ ( 2 *Y,3.�/ ��0o#92&4�576�'&� #9��'&:&��� / ��6�4M'&2&'�< , ���=4� > ! � g _#d
( � ��� ) � g _�d(0 ��� ) = 0
� g _�d( � + 1 ��� ) =
0 �0b1= � = 0 �� g _#d
( � ��� ) + 1 �b H+H d 4 6�2 b1= i X( � + 1 ��� ) = 0 �
� g _#d( � ��� ) �0b H#H d 4 6
JU� Q _ ( 0�U = (�H+( * M = P�&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�4N= 0�A - 0�BTM = BFE =�bed|a H+B d 0 * S D d b* (1G+6 P1&)( *�b 0 d b`a (1U+6 * BTH+BR0 *)- 6 ¬ � ∨ � & � � ⇒ 2�P A 2 b1=
� ( �� ) ⇐⇒ � ( �� ) & � ( �� )acbed ( d � 2 � BFE =�bed P1&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�- 6�2 * S * B acbed " � BRE =�bed P�&)M * (�D�B =)4 6b1=�b1/ &)(1K d a�, J :)Q B = "%0�A - 0�" � ( �� ) acbed ( d 0�G =�b & *I, 0�B d 6 � 1( �� ) �;:;:;:�� � � ( �� ) BFE =�bed P1&)M * (�D�Bf]=)4 6 b1=�b1/ &)(1K d a�- 6�2 * S * B acbed "c0�A - 0�"
� ( �� ) ⇐⇒ � (�1( �� ) �;:;:;:�� � � ( �� ))
BFE =�bed P�&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�, J w Q B = " � ( F � �� ) BRE =�bed P1&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�, acbed
� ( � � �� ) ⇐⇒ (∃ F ≤ � ) � ( F � �� )� ( � � �� ) ⇐⇒ (∀ F ≤ � ) � ( F � �� ) �
* S * B acbed ( d � ( � � �� ) 2 � ( � � �� ) BRE =�bed P�&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�- 6 ; d bc*Jd 6 b P1( / BRE C)B d 6'P b & b P - K�P�(1G+K�B P < H d 0 *Jd 6 b 0 a�, 0�B d 6 >AE' d-2g* Z , �����=' 2 n A 7 j � �=j@7 � ( F � �� ) >c1e</7�j�=8A���@;k7Jj|7 � ( �� ) g@?YAM1e<Eh��3�@;=m ��Jg8A0? ��1|AM1���� m ,B< > ��� Z ;BA ;0g h��3�@;=m ��g8AM? ��1|AM1���� m ,B< > ��� g@?PA=1 <:>c1e<Bme<Bj@� �=j6g�< �
� ( �� ) ⇐⇒ (∃ F ≤ �( �� )) � ( F � �� )
� ( �� ) ⇐⇒ (∀ F ≤ �( �� )) � ( F � �� ) �>-1 < ;=m ? �J< mp,Eg G j@;k7 5��=j|7�;=m3= ≤ ��� hJgk;91e<*A`;=< 7 j@� �=j|7
� i Z\X%V( � ) ⇐⇒ ( � BRE =�bed P1& 4�* (16 b & d C)K�S16
g@?YAM1e<Bh��3�@;=m ��g8AM? ��1|AM1���� m ,B< >E.��>AE' O 2 3 * BTH+BR0 *I, 6 ���������������8 E)����4���� �������� �����98 (1&�E � B *�bed M 6
( �+F ≤ � ) � ( F � �� ) =
{ ( BbH < A d 0 * (16 F ≤ � *)-f* ( d (16'P1(1G � ( F � �� ) � b1= (∃ F ≤ � ) � ( F � �� ) �� + 1 b H#H d 4 6 :
>AE' ] 2g*-, Z / '�� [*2 <H1p>c5=g h ���@;=m �Jg8A0? � 1@A=1�� �`mk,l< >:.pj � �=j@7 � ( F � �� ) Z 7pj�= �A���@;k7Jj|7�( � � �� ) = ( �+F ≤ � ) � ( F � �� )
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(�� 2�*Y,3.�/ ��0o#92���5n'&2&'�< , ���=��6 { 5n����2&' ,3/ 4���#9��5 > 9g@?YAM1e<Bh��3�@;=m ��g8AM? � 1@A=1�� �`mk,l< >:. � � h8gk;91 <#A`;=</1|Ap7 �
( �� ) g@?PA=1 <lg�h�? j|7 � h ���@;=m �Jg �AM? ��1@A=1�� �`mk,l< >:. Z ;BA ;0g h��3�@;=m ��g8AM? � 1@A=1�� �`mk,l< >:.�g@?YAM1e<2>-1 < 7 j,=iA ��-;k78j@7�( �� ) = ( �+F ≤ �
( �� )) � ( F � �� ) (=�(�( �� ) � �� )) :
j-X Z�<\#3��} [ 0 * " = � 0 a "�0�" � >AE' c> 8 a>AE' c> 8 2 * Z , �����=' 2 � j�=8A���@;k7Jj|7
� � =_ F � (10 * S16 P1& 4�* (16 b & d C)K�S16g@?YAM1e<Bh��3�@;=m ��g8AM? ��1|AM1���� m ,B< >E.��
j-X Z�<\#3��} [*2 m � � (1&�E � B *�bed K�B * " = P1&)M * (�D�B =I, b1=�b1/ &)(1K ,�
0 = 2� �
+1 = ( � � ≤ � � ! + 1)[� � G �&� i Z\X%V
(�)] �
BbP�B d /I, JLBTU a (�H b 2�� >AE' ] Q D d b a�< C)B���2�G�P < &)A�B d P�& 4W* (16 b & d C)K�S16 � *)-b* ( d (16 P1(1G� G � ≤ �
! + 1 a
>AE' c>�>32 � 7��1� �������� ����� ����������� �1�7���� ;�B =Jd a�< 2 �+7��1� �������� ����� B = S16.0�G)]= S�H�(1G � 0�B -R=�b 0�U = (�H+( � BRE =�bed " * G+A b E b7-T=�b ]LP�&)(16b] -T=�b 0�G =)< & * "+0�" : � � � 2P1(1GcJ�C)BRMN&I" *Jd a�<�Q K b 6�BfP d * & - P1B d�=�b # b1=�b`a�*I, 0�(1G+K�B & -T=�b * G#A b E ( 0 * ( d A�BRE ( � ∈ �b P1S * ( >9� �8< >+A * (1G ( � ) D d b P b & <1/ B d D�K b 2�" 0�G =)< & * "+0�" P1(1G b1=F*Jd 0 * ( d A�BFEW0�Ba�< C)B B =I, H d a ( - H#H+" =�b * ( =�b & d C)K�S * "+6 *�b G * S * " *)< 6 * (1G e * " 0�G =)- A�B d b 2?C ba M /Jd a (+P�( d , 0�(1G+K�B P1(�H#H < 0�U = (�H b 0 * ( 0�U = (�H�( N
2�0�A�B / S = P <1=F*�b A�&I"+0 d K�(+P�( d 4 ]=F*�b 6.MN6 5 b 0 d a S BR&ID b H+BFE ( a�< P�( d b b P+H , a M /Jd a (+P1(�E "+0�" * (1G 0�G = S�H+(1G N∗ * M =J P�BbP�BR& b 0�K -R= M =�Q b`a (�H�(1G#C d 4 = b P1S b & d C)K�(1U+6�2�K�B?A�& , 0 d K�BT6 d /Jd S * " * BT6�2�MN6?B C , 6 m * G+A b E b a M /Jd a (+P�(�E "�0�"
〈 〉 : N∗ � N* (1G N∗ BFE =�bed ����7������(E)��� 8 ������������W� �� 2 b1= D d b�a�< C)B b`a (�H+(1G+C�E b O G#0 d a�4N=b & d C)K 4 = � 0
�;:;:;: � � � −12
� � G 〈 � 0�;:;:;: � � � −1〉 ( F*G �
) DJL! > Q"c0�A - 0�"
� V �( � ) ⇐⇒ � = 〈 � 0
�;:;:;: � � � −1〉D d b a�< P�( d b � 0
�;:;:;:�� � � −1JL!#! QBFE =�bed P�&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�, D d b a�< C)B �W2�"���] K�BbH , 670�G =)< & * "�0�"
� � ( � 0�;:;:;:�� � � −1) = 〈 � 0
�;:;:;:�� � � −1〉JL! N QBFE =�bed P�&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�, acbed G)P < &)A�(1G = P1&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�- 6.0�G)]=�b & *I, 0�B d 6'P�(1G d acb1= (+P1( d (1U =7*�b B C , 6
[ e(〈 � 0
�;:;:;: � � � −1〉) =�
0 i _��(〈 � 0
�;:;:;:�� � � −1〉 � F ) = (〈 � 0�;:;:;:�� � � −1〉) � = � � ( FHG �
)Y10 0�VAb3w(〈 � 0
�;:;:;:�� � � −1〉 ��� ) = 〈 � 0�;:;:;: � � � −1
��� 〉 :
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> T ( 2 *Y,3.�/ ��0o#92&4�576�'&� #9��'&:&��� / ��6�4M'&2&'�< , ���=4I P1E 0�"+67BRE =�bed J * BTA =Jd a�<�Q A�& , 0 d K�( =�b�b P bed *I, 0�(1G#K�B
[¬ � V � ( � ) ∨ F ≥ [ e( � )] � ⇒ [\e
( � ) = ( � ) � = 0 �b1= acbed ( d *Jd K - 6 [\e ( � )2( � ) �
BRE =�bed�<1= BTG 0�"+K b 0�E b 6 S *�b1= ( � / B = BFE =�bed:a M /Jd a S16b`a (�H�(1G#C�E b 6 , * ( F BFE =�bed K�BbD b H�U * BR&)( b P�S * ( K ,8a (16 * "�6 0�A�B *Jd a�, 6 b`a (�H+(1G)]C�E b 6 $ b & b+* "+&)(1U+K�B S *Jd K�B * " = JL! > Q 2�( dlb P bed *I, 0�B d 6 b G *)- 6c0�G = BfP < D�( =F*�bed�* " =b1=Jd 0�S * " *�b
� 0 � ⇒ ( � ) � G � :JL! Q Q3 d 0�G#K 5 (�H d 0�K�(�E � V � ( � ) � [ e ( � ) acbed 0 i _�� ( � )
P�&)( - &)A�( =F*�bedWb P�S * (1G#6 B D�D�H d ]a (1U+6�S1&)(1G#6 L I����cI=D��0I J b`a (�H�(1G#C�E b�Q 2 X I=D`_iN�� J�K ,ia (16 Qlacbed� K9Q� I�� NHFYQJD J P�&)( 5 (�H ,1Q >AE' c> ! 2 *Y, Z / '�� [*2 Eh �� �Eg�< h��3�@;=m ��g8AM? ��1@A=1�� �`mk,l< >:.(>9� �8< >2m hJm-?Y7Jj|7�;=m&=
N∗ Z j�= �8>-gM>��@< , �8A=1(7 � > � 1ejc< >:.���>9� �8< >2m hJm-?Y7Jj|7〈 � 0
�;:;:;: � � � −1〉 = ��� 0+10 · ��� 1+1
1 · · · � ��� −1+1�−1
JL!#9 Q,Eg 〈 〉 = 1
�
j-X Z�<\#3��} [ 0 * " = � 0 a "�0�" � >AE' c>AN� am a H b 0 d a�,\a M /Jd a (+P�(�E "�0�" * (1G N∗ / B = BFE =�bed # b P�( * BTH+BR0�K b+*Jd a�, & 2 acbed 0 *Jd 6b 0 a�, 0�B d 6 C b /Jd BR&)BTG =I, 0�(1G#K�B P d ( &)B b H d 0 *Jd a�- 6 a M /Jd a (+P�( d , 0�B d 6 P�(1G A�&I"�0 d K�(#]
P1( d (1U =F*�bed 0�B K�BTH -b* BT6 h�m �,= h �imJ>+A`;k7J;91�� 2 b H#H <�b P1S * " = < P1(���" * "�6 = h�m �8m �c< �j-< ,9A`;k7J;91�� P1(1G BRE =�bed * ( a U#& d ( C - K b K b 6�2 S�H+BR6 ( d P1&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�- 6a M /Jd a (+P�( d , 0�B d 6 * (1G N∗ BRE =�bed2d 0�( / U =�b K�BR6�2 � >AE' ! Q� B P�S / M acbed P - & b 0 *�b C)BR&)(+P�( d (1U+K�B'K d b 0�G#D a B a & d K -R= "%P1&)M * (�D�B =)4 6 b1=�b1/ &)(#]
K d a�, a M /Jd a (+P�(�E "�0�" 〈 〉 : N∗ � N2�P d C b1= S = S1A d�* " = a H b 0 d a�,
>AE' c>AN 2 *Y, Z / '�� [*2 Eh �� �@m3=8A^h ���@;=m �Jg8A0? � 1@A=1�� �`mk,l< > ��� j,=iAM1��-;k.8jeg�< �� � F>-1 <
� ∗ �Z ;��k;=me< g�� h�m3=
〈 � 0�;:;:;: � � � −1〉 ∗ 〈 � 0
�;:;:;:�� � � −1〉 = 〈 � 0�;:;:;:�� � � −1
� � 0�;:;:;: � � � −1〉
〈 � 0�;:;:;:�� � � −1〉 � F = 〈 � 0
�;:;:;:�� � � −1〉 ( F ≤ �) :
j-X Z�<\#3��} [ 0 * " = � 0 a "�0�" � >AE' c> Q� a>AE' c> Q,2 *Y, Z / '�� [ J �c��������� �� ����7������(E)��98 ������������ �12 n AWm < �
1Z �
2Z �
1>-1 < �2g@?YAM1e<Wh��3�@;=m ��g8AM? � 1@A=1�� �`mk,l< > ��� Z ;BA`;0g�h ���@;=m �Jg8A0? � 1|AM1���� m ,B< > ����g@?PA=1 <>-1 </m < �1>-1 < �
2hJm&= m�� ?��Mm6Ak;91 <@,:g ;=< �pg@CJ<Hj`?/j6g�< ���
�1(0
� �� ) =�1( �� )�
1(� + 1 � �� ) =
�1(�1(� � �� ) � � 2( �� �� ) ��� � �� )
�2(0
� �� ) =�2( �� )�
2(� + 1 � �� ) =
�2(�1(� � �� ) � � 2( �� �� ) ��� � �� ) :
j-X Z�<\#3��} [ 0 * " = � 0 a "�0�" � >AE' c> 9 a
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(�� 2�*Y,3.�/ ��0o#92���5n'&2&'�< , ���=��6 { 5n����2&' ,3/ 4���#9��5 >Hd
>AE' c> 9 2 *Y, Z / '�� [ J � �1����98.����7������+E����8 ������������ �12 n A^7 � g@?YAM1e<@h��3� �;=m ��g8AM? � 1@A=1�� �`mk,l< >:. Z ;BA`;0gph��3�@;=m ��g8AM? � 1|AM1���� m ,B< >E. g@?YAM1e<@>c1e<B7 � h�m3= m��9? � g �;91e<-,Eg ;k7eA�g@C&? j-�/j@7�( �� �� ) =
�(〈 � (0 � �� ) �;:;:;:�� � ( � −· 1 � �� )〉 ���� �� )
J -f* 0 d P1(1G � (0 � �� ) =�(〈 〉 � 0 � �� ) 2 � (1 � �� ) =
�(〈 � (0 � �� )〉 � 1 � �� ) 2 a H#P Q
j-X Z�<\#3��} [*2 $'& 4W*�b (1&�E � (1G+K�B'K�B�P1&)M * (�D�B =I, b1=�b1/ &)(1K , * " =�(0 � �� ) = 〈 〉 �
�( � + 1 � �� ) =
�( �� �� ) ∗ 〈 � ( � ( �� �� ) ���� �� )〉 �
acbed K�B *)< BbP b H+"+C)BRU#(1G+K�B?S *Jd "�( �� �� ) = (
�( � + 1 � �� )) d acb1= (+P1( d BRE * " = b P bed * (1U#K�B = " B C�E 0�MN0�" * BbH d a�< / BFE A = (1G#K�B K�B P�H , &I" BfP b D�MWD ,
0 * ( � S *Jd K�S = ( K d b 0�G =)< & * "�0�" d acb1= (+P1( d BRE * " =pb P bed * (1U+K�B = ".B C�E 0�MN0�" a
��� � �p�/yl~2�#$ }��>AE' c> T 2 � ��7������(E)� �18c������������W� ��� ��������������� BRE =�bed " * G+A b E b b`a (�H�(1G#C�E b
P�H+B d (1K�BbH 4N= 0�G =�b & *I, 0�BRM =�
= (�0� �
1�;:;:;: � � � )
*)-b* ( d b P�(1G D d b a�< C)B�� ≤ � b H#"�C)BTU+B d -R=�b�b P�S *�b B C , 6 J > Q m ��� BRE =�bed K d b b P�S *Jd 6 5 b 0 d a�- 6'P�&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�- 670�G =�b & *I, 0�B d 6
� � ��� � �
� JL! Q m � � (1&�E � B *�bed K�Bc0�U = C)BR0�" J > 8 Q 2 S+P�(1G ( d � � �
1�;:;:;:�� � � BRE =�bed 5`b 0 d a�- 6
P1&)M * (�D�B =)4 6 b1=)/ &)(1K d a�- 6 0�G =�b & *I, 0�B d 6 0�G =�b & *I, 0�B d 6 , BRK O-b1= E � ( =F*�bed 0 * " = b`a (#]H�(1G#C�E b �
0�;:;:;:�� � �
−1
J N Q m � � (1&�E � B *�bed K�B P�&)M * (�D�B =I, b1=�b1/ &)(1K , J >P> Q 2WS+P�(1G ( d � � � BRE =�bedW5`b 0 d ]a�- 6 P�&)M * (�D�B =)4 6 b1=)/ &)(1K d a�- 6 0�G =�b & *I, 0�B d 6 0�G =�b & *I, 0�B d 6 , BTK O@b1= E � ( =F*�bed 0 * " =b`a (�H�(1G#C�E b �0�;:;:;: � ���
−1
� > E' c>32 � BFE C * B S *Jd " * G#A b E b � : N � → NBRE =�bed P1&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�, b1=acbed K�S = ( =�b1= � =
� � BRE =�bed D d b a�< P1( d ( P�&)M * (�D�B =)- 6 b1=�b1/ &)(1K d a S P1&)S�D�& b K�K b(�0� �
1�;:;:;: � � � )
� > E' ! 2 � BFE C * B J acb+* BTG+C)BRE b1= 2 b P1S * (1G+67(1& d 0�K�(1U+6 Q S *Jd|b1=�( � ��� ) =
�(�1(� ) � � 2( � � � ) ��� )acbed ( d � � �
1� �
2BFE =�bed P�&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�- 6�2 * S * B acbed " � BFE =�bed P�&)M * (�D�B =)4 6b1=�b1/ &)(1K d a�,
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> O ( 2 *Y,3.�/ ��0o#92&4�576�'&� #9��'&:&��� / ��6�4M'&2&'�< , ���=4� > E' N 2 � BFE C * B S *Jd:b1= " � : N2 → N
BFE =�bed P1&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�, 2 * S * BP1&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�, BRE =�bed@acbed "
�( � ��� ) =
�( �� � ) :
� > E' Q,2 � BFE C * B?S *Jd D d b a�< C)B � ≥ 32�( d ��] K�BbH�BRE 670�G =�b & *I, 0�B d 6
min � ( � 1�;:;:;: � � � ) =
(.BTH < A d 0 * (16 * M =��1�;:;:;: � � �
max � ( � 1�;:;:;: � � � ) =
(.K - D d 0 * (16 * M = �1�;:;:;: � � �
BFE =�bed P�&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�- 6 � > E' 9 2 � BFE C * B S *Jd " B a C)B *Jd a�, 0�G =)< & * "+0�" � ( � ��� ) = �� JLK�B
00 = 1 Q BFE =�bedP1&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�, � > E' T 2 � BFE C * BcS *Jd ( d /Jd K�BbH�BRE 6 0�A - 0�B d 6 � ≤ � 2 � G � BRE =�bed P�&)M * (�D�B =)4 6b1=�b1/ &)(1K d a�- 6 � > E' d 2 � BFE C * B?S *Jd|b1= " � ( F ��� � � ) BFE =�bed P�&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�, 2 acbed
� ( �� � ) ⇐⇒ (∀ FHG � ) � ( F ���� � ) �* S * B acbed " � ( � � � ) BRE =�bed P1&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�,
� > E' O 2 � BFE C * B7S *Jd ( d 0�G =�b & *I, 0�B d 6 � g _�d ( � � � ) acbed i X ( � � � ) * (1G P+"+H�E a (1Gacbed G)P1S�H+( d P1(1G%BFE =�bed P�&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�- 6 � > E' ] 2 � BFE C * B S *Jd D d b a�< C)B �W2�G)P < &)A�B d P�& 4W* (16 b & d C)K�S16 � *)-b* ( d (16 P1(1G
� G � G �! + 1
J � =�b b P1S *�b P1(1&�E 0�K b+*�b BFE =�bed * (.P�M 6 G�P < &)A�(1G = < P1B d &)( d * (P�H , C)(167P1& 4�* ( dEb & d C)K�(�E 2 * ( H+BTD�S1K�B = (�� gM? �@79,:1�;=m&= � ==>��`g@? �`7 Q VvX Z�<\#3��} [��B = ( � ! + 1 / B = BRE =�bed P1& 4�* (16�2 * S * B a�< P�( d (16'P�& 4W* (16 b & d C)K�S16 � | � ! + 1
� > E' c> 8 2�� BRE C * B S *Jd b1= " /Jd K�BTH , 6 0�A - 0�" � ( � ��� ) acbed " 0�G =)< & * "+0�" � ( � )BFE =�bed P1&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�- 6�2 * S * B P�&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�, BRE =�bed^acbed "
0�G =)< & * "�0�"�( � ) = ( � � ≤ �
( � )) � ( � ��� ) :3 , ���-< j|;=m�� >|me<PABA����J<H1 < ���k;k7 � / U#( b & d C)K 4 = � ��� ≥ 1
BRE =�bed J O G+0 d a�< � Q (K�BTD b H+U * BT&)(16 b & d C)K�S16 P1(1G /Jd bed &)BFE acbed�* (1G+6 / U#(�2 acbed D d b * " = P�H#"�&)S * " *�b * (1G(1& d 0�K�(1U C -b* (1G#K�B
gcd( � ��� ) =
0 � b1=�� = 0 , � = 0 �(.K�BTD b H+U * BT&)(16 � *)-f* ( d (16'P�(1G� | � acbed � | �� b H+H d 4 6 :
JL!�T Q
� > E' c>P>&2�� BRE C * B�S *Jd "?0�G =)< & * "�0�" gcd( � ��� ) BRE =�bed P�&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�, � > E' c> ! 2 ; d b+* E / B = K�P�(1&)(1U+K�B =�b (1&�E 0�(1G#K�B * " = -R=)= ( d b # P1&)M * (�D�B =)4 6 b1=�b ]
/ &)(1K d a�, 6 a M /Jd a (+P1(�E "+0�"+6 * (1G N∗ & K�B * ( b P+H�S> ] > 2�P1&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�, 0�G =)< & * "+0�" 〈 〉 : N∗ � Nb1=F* E D d b *Jd 6 P1BT&�E P�H+( a BR6 J acbed P�(�H+H - 6 Q 0�G = C ,ia BT6 0 *Jd 6 (+P�(�E BR6 5`b 0�E 0 b K�B * ( =
(1& d 0�K�S��
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(�� 2�*Y,3.�/ ��0o#92���5n'&2&'�< , ���=��6 { 5n����2&' ,3/ 4���#9��5 > ]
� > E' c> N 2�� BRE C * BNS *Jd "$# a H b 0 d a�, a M /Jd a (+P1(�E "+0�" &�* (1G N∗ 0 * " = $?&)S *�b 0�" > E' c> !BFE =�bed P�&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�, � > E' c>AQ 2�� BRE C * B�S *Jd ( d B C , 6 / U+(%0�G =�b & *I, 0�B d 6 BRE =�bed P1&)M * (�D�B =)4 6 b1=�b1/ &)(1K d ]a�- 6
� � F ={〈 � 0
�;:;:;: � � � −1� 〉 b1= � = 〈 � 0
�;:;:;: � � � −1〉K�B F ≤ � �
0 � b H+H d 4 6 �
� ∗ � =
〈 � 0�;:;:;: � � � −1
� � 0�;:;:;:�� � � −1〉 � b1= � = 〈 � 0
�;:;:;:�� � � −1〉 �� = 〈 � 0
�;:;:;: � � � −1〉 �0 � b H#H d 4 6 :
� > E' c> 9 2�� BRE C * B * " = $?&)S *�b 0�" > E' c>AQ� � > E' c> T 2�� BRE C * B S *Jd D d b�a�< C)B�P1&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�, 0�G =)< & * "�0�" � ( � ) 2�"0�G =)< & * "�0�"
�( � ) =
∏ ���
�( � )
BFE =�bed P�&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�, � > E' c>Hd J � �1����98.����7������+E����8 ������������ ����� �(4 �)�(E#� 8�12�� BFE C * B S *Jdcb1=
" 0�A - 0�"��( � � �� ) BFE =�bed P�&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�,pacbed " � ( �� �� ) d acb1= (+P�( d BFE * " =d 0�( / G =�b K�E b� ( �� �� ) ⇐⇒ �
(〈 ��� (0 � �� ) �;:;:;:�� ��� ( � −· 1 � �� )〉 ���� �� ) �* S * B acbed " � ( � � �� ) BRE =�bed P1&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�,
� > E' c> O ∗ J�� 7 �������� �)��� ����7������+E����8 ������������� 2�b�VF` dfVAw i V : g�i `fZ _Pb Q 2�� BRE ]C * B�S *Jd D d b a�< C)B * &)B d 6 0�G =�b & *I, 0�B d 6 � ( � ) 2 � ( ��� � ��� ) acbed�� ( � ��� ) 2�G�P < &)A�B deb`a & d ]514 67K d b 0�G =)< & * "�0�" � ( � ��� ) P1(1G d acb1= (+P�( d BFE *Jd 67B C d 0 4 0�B d 6
�(0 ��� ) =
�( � )�
( � + 1 ��� ) =�(�( � ��� ( � ��� )) � � ��� ) Dacbed6b1= ( d�/ (10�K -T= BR6�0�G =�b & *I, 0�B d 6�BRE =�bed P�&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�- 6�2 * S * B P1&)M * (#]
D�B =)4 6 b1=�b1/ &)(1K d a�, BRE =�bed@acbed " � ( � ��� ) � > E' c> ] 2�� BRE C * B S *Jd G�P < &)A�B d P�&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�, 2 -T=�b ]LP�&)(16b] -T=�b 0�G)]
=)< & * "�0�" � : N × N → N2 *)-b* ( d b P�(1G�( � ��� ) ≤ ( � + � + 1)2 :
;�B =Jd a S * BT& b 2 / BRE C * B S *Jd D d b�a�< C)B � ≥ 22�G)P < &)A�B d P1&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�, 2
-R=�b ]�P1&)(16b] -R=�b 0�G =)< & * "�0�" � � : N�→ N
2 *)-b* ( d b P�(1G� � ( � 1
�;:;:;:�� � � ) ≤ � � ( � 1�;:;:;: � � � ) �JL! d Q
S+P1(1G * ( � � ( � 1�;:;:;:�� � � ) BRE =�bed P1(�H+G 4N= G+K�( 5 b C)K�(1U �
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!#8 ( 2 *Y,3.�/ ��0o#92&4�576�'&� #9��'&:&��� / ��6�4M'&2&'�< , ���=4� > E' !#8 2�� BRE C * B�S *Jd D d b�a�< C)B � ≥ 2
2 / B = G�P < &)A�B d�-T=�b ]LP�&)(16b] -T=�b 0�G =)< & * "+0�"�
: N�→ N
P�(1G =�b�d acb1= (+P1( d BRE * " = J�! d Q K�B P�(�H�G 4 = G#K�( 5`b C)K�(1U ≤ � − 1
� > E' ! >&2�� BRE C * B S *Jd G�P < &)A�B d P�&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�,pa M /Jd a (+P�(�E "�0�" b`a (#]H�(1G#C d 4 = 2 *)-f* ( d b P�(1G%D d b a�< C)B �W2 acbed S�H b%*�b �
1�;:;:;: � � � 2
〈 � 1�;:;:;:�� � � 〉 ≤ 2
�� � ( � 1
�;:;:;:�� � � ) �S+P1(1G * (%P1(�H+G 4N= G+K�( � � BRE =�bed�5 b C)K�(1U � � > E' !+! 2�� BRE C * B S *Jd D d b^a�< C)B a M /Jd a (+P�(�E "�0�" 〈 〉 : N∗ � N * M = b`a (�H+(1G+C d 4N=b P1S * ( N
2max{〈 � 1
�;:;:;: � � � 〉 | � 1�;:;:;: � � � ≤ � } ≥ 2
�(� � � ≥ 2)
� > E' ! N ∗ 2 J�Y Q � BRE C * B S *Jd a�< C)B * (1K , � � ( � ) * "�6 0�G =)< & * "+0�"+6 * (1G @ : r+V i ]X.Y b3b BRE =�bed P1&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�, JU� Q � BFE C * BNS *Jd D d b a�< C)B�P�&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�, 0�G =)< & * "�0�" � ( � 1
�;:;:;:�� � � ) 2G�P < &)A�B d@a�< P1( d ( � *)-f* ( d (cP1(1G�( � 1
�;:;:;:�� � � ) G ��� (max( � 1�;:;:;:�� � � )) ( ��� :;:;: � � � ∈ N) :JL! O Q
J :)Q � BRE C * B%S *Jd " 0�G =)< & * "+0�" * (1G @ : r�V i X.Y b3b � (� � � ) / B = BFE =�bed P�&)M * (�D�B =)4 6b1=�b1/ &)(1K d a�, VYX Z�<8#9��} [�� � BFE C * B�S *Jd�* (%0�U = (�H�( * M = � ] � �`1� , �8A �BA 0�G =�b & *I, ]0�BTM = J / "+H b1/I,�* M = 0�G =�b & *I, 0�BTM = P�(1G d acb1= (+P1( d (1U = * " = JL! O Q K�B a�< P�( d ( � Q BFE =�bed
P1&)M * (�D�B =)4 6 a H�B d 0 * S 3 d`b 0 a�, 0�B d 6 � >A@ c>P> � � >A@ c>AQ P b & - A�(1G = *�b b P b & b E * " *�b� , K�K b+*�b � > E' ! Q 2�� BRE C * B7S *Jd@b1= ( d 0�G =�b & *I, 0�B d 6
〈 〉1 � 〈 〉2 : N∗ � NBFE =�bed P�&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�- 6 a M /Jd a (+P�( d , 0�B d 6�2 * S * B G)P < &)A�B d P�&)M * (�D�B =)4 6b1=�b1/ &)(1K d a�, 0�G =)< & * "+0�"�� : N → N2 *)-f* ( d b P1(1G
�(〈 �� 〉1) = 〈 �� 〉2 ( �� ∈ N∗) :
��� ����|z�W}��l{@}Yy �!z(�czE�cwcx(� }Yy/�`� �#$iw@}Yy/�`���O� �cz:we{e~:�*$e}��3 b & d C)K�S16 � acb H�BRE *�bed �1? ��=9,-m�� h �`?@;=m�� b1= BRE =�bed P1& 4�* (16�2 acbed ( � + 2
BFE =�bedBbP1E 0�"+6 P�& 4W* (16 D d b P b & <1/ B d D�K b 2'( 5
BFE =�bed�/ E / G+K�(16 P1& 4�* (16�2 b H+H < (7 / B =
BFE =�bed � P < &)A�(1G =�b`a & d 514 6 * & d <1=F*�b P -R=F* B / E / G+K�( d P�& 4W* ( d K d a &)S * BR&)( d1* (1G 10002
( d B C , 6�2#( acb C -T=�b 6 � BTG#D b &)MNK -R= (16 K�B * ( = #bBfP1S1K�B = ( P�& 4W* ( & P1(1G * ( = b`a (�H�(1G#C)BFE ����� ����� ��������� ��������� ���������� ���!�"� �#� �%$ � �����%�&� ��'�������'�� ��'�� ����'������� ������� ���&� ���(�)� ����� ����*�� ������������� ����� ��������&�#� ���&�#� ����� ���#� � ��$�� ������� ��*�������*�� � �����%� ������&� �%���&� ����� �!�+�)� �"�����!�"��� �"$����!�"$�� �&�)�����&�#���$�� ������� ����� �%$&'�� $ ��� �%$ ��� $�� ���%$��&� $"�#� �%$&$��*&'�� �%* ��� *"�)���%*"�#� *"�#� �%*"�#� *"�#� �%*"�#� *&*����%*&*��
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(�� 2 � ��'&:&��� / ��6�%M'&2&'�< , ���=��6 { 5n��# , ��6 { 57����2&' ,9/ 4���#3��5 ! >m B d acb 0�E b S *Jd ==h �� �@m3=8A h8g�< � m < ;=m h��@.`5Mm����&? �-= ,@m <Wh��i?@;=me< BRE =�bed�-R=�b b P�S *�bb &)A bed S * BR& b b1= ( d a�*)< P1&)( 5 H , K b+*�b * "�6 b & d C)K�(1C)BRMN&�E b 6�2 acbed J�S+P1MN6 K b 6 H -R= B ( db & d C)K�(1C)BRMN&)(�E QW/ B = G�P < &)A�B d &)B b H d 0 *Jd a�, P1&)(10 / ( a E b S *Jd C b b P�( / B d A * BRE�0�U =F* (1K b B 6?G�P�(1C - 0�(1G+K�B'S *Jd P�& < D�K b+*Jd2d 0�A�U#B d 2 acbed@b 67(1&�E 0�(1G+K�B * "c0�G =)< & * "�0�"
���� =( F ] (10 * S16 / E / G+K�(16'P�& 4W* (16 b & d C)K�S16 �
-b* 0 d P1(1G J b P1S * ( = P1E =�b`acb�Q 2� �
0 = 3 � � �9 = 107 � � �34 = 881 :m 0�G =)< & * "�0�" � �� P1&)( O-b1=)4 6 d acb1= (+P1( d BRE * " =pb1=�b1/ &)(1K d a�, B C�E 0�M 0�"
� �0 = 3 �
� ��+1 =
�(� �� + 1) �
S+P1(1G�( � ) =
(.K d a &)S * BR&)(16 / E / G#K�(16'P1& 4�* (16 � ≥ � := ( � � ≥ � )
� i Z\X%V � ( � ) �acbed "c0�A - 0�"
� i Z XcV � ( � ) ⇐⇒ _ � BRE =�bed / E / G+K�(16'P�& 4W* (16BFE =�bed P�&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�, B G * S S1K�MN6 / B = 0�G = BbP < D�B *�bed S *Jd " � �� BFE =�bedP1&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�, J�S+P1MN6 0 * " = b P�S / B d CI" * "+6 b1=)< H+(�D�"�6 $'&)S *�b 0�"+6 > E' c> 8D d b?* "70�G =)< & * "�0�" � � Q 2�BbP�B d /I,7/ B = K�P�(1&)(1U+K�B =�b / BFE C)(1G#K�B�S *Jd " � ( � )
BRE =�bed P1&)M�]* (�D�B =)4 6 b1=�b1/ &)(1K d a�,�� a-d2b G * S BbP�B d /I, / B = C - &)(1G#K�B a�< P�( d ( O & < D�K b D d b.* ( =#bBfP1S1K�B = ( / E / G+K�( P1& 4�* ( & �B P�S * " = < H#H+" K�BT& d < 2W" � ( � )
K�P1(1&)BRE =�b G)P1(�H+(�D d ]0 * BRE�K�B * " = P�&)( O@b1=I, # 5 H b`a�4 / " b1=�b%�I,�* "�0�" & J w g X�� ` VIY i : e Q 2 S+P1(1G /Jd b1/ (1A d a�<BTH - D�A�(1G#K�B *Jd 670�G = C ,8a BR6
� i Z\X%V � ( � + 1) � � i Z\X%V � ( � + 2) � � i Z XcV � ( � + 3) �;:;:;:��K - A�& d 6 S * (1G 5 &)(1U#K�B a�< P1( d ( � + 1 + F P�(1G.BRE =�bed 2�P1& < D�K b+*Jd 2 / E / G+K�(16?P�& 4W* (16 ; d b P b & <1/ B d D�K b 2
�(6) =
�(7)
BbP�B d /I, ¬ � i Z\X%V � (6)=�(8)
BbP�B d /I, ¬ � i Z\X%V � (7)=�(9)
BbP�B d /I, ¬ � i Z\X%V � (8)=�(10)
BbP�B d /I, ¬ � i Z\X%V � (9)=�(11)
BbP�B d /I, ¬ � i Z\X%V � (10)= 11
BbP�B d /I, � i Z XcV � (11) D
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!+! ( 2 *Y,3.�/ ��0o#92&4�576�'&� #9��'&:&��� / ��6�4M'&2&'�< , ���=4< & b acbed " � �� G�P�(�H�(�D�E � B *�bed MN670�G =I, C)M 6�2 2
���0 = 3 � ���1 =
�(���0 + 1) � :;:;: ������ =
�(���� −1 + 1) :
$ b & < * " = G�P�( *Jd K�" *Jd a�, * "+6 ( = (1K b 0�E b 2 " /Jd b1/Jd acb 0�E b * "�6 5 H b`a�4N/ (1G+6 b1=�b ]�I,#* "+0�"+6cBFE =�bed E 0�MN6 * ( 5`b 0 d a S * BT&)( 0�G#0 *�b+*Jd a S 0 * " = acb+*�b 0 a BRG , b H#D�(1&�E C)K�M =0 * (1G#6 O G#0 d a (1U#6 b & d C)K�(1U#6 �m B O@b &)K�(�D , * "�6%( / "+D�BRE O G#0 d a�< 0 * " = B d 0 b D�MND ,#bK�BT& d a�4 = 0�G =�b & *I, 0�BTM =�& P�(1G / B = b P1( / E / (1G = P <1=F*�b%*Jd K , 2 acbed C b * " = (1&�E 0�(1G+K�Bb G+0 * "�& < 0 � b G * S * ( BTG+&)U * BR&)(cP�H b E 0 d ( >HG c>&2�Y, ������Z�5 2�� E��1� �� ���������#������� J 0�Y i dfZ Y�[�� g b : dfZ\_�b Q J b P1S * ( hJg��&? mg�<Hj,A��Jm3= � 0 * ( h8g��1? m ;=< ,E?BA , g@C�A��8m&= � Q
�: ��� �
BFE =�bed " * G#A b E b 0�G =)< & * "�0�"�
: � 0 → � ( � 0 =S?_#X.Y�Zcb
(�) ⊆ � ) �
S+P1(1G * ( � 0acb H+BFE *�bedN* ( hJg��&? m j�< �8>��-< j|7 � J w�_+XcY#Z b _�� : _Pb V ifh VAb : V Q'* "�6 � d 0�( / U =�b K b 2�" * G#A b E b 0�G =)< & * "+0�"
�: � → � ∪ {⊥}
S+P1(1G ⊥ J P <+* (16�2 ��_#d db_#X Q BRE =�bedEa�< P1( d ( JL0�G+K 5`b+*Jd a�< BfP d H+BTD�K -T= ( Q b1=F*Jd a BFE K�B = (P1(1G / B = b1=I,ia B d 0 * ( � �I &)K�" = BTU+(1G#K�B * " = # *Jd K ,�& � ( � ) = ⊥ MN6 ���∈ S?_#X.Y�Zcb ( � ) 2acbed H - K�B S *Jd " � 1 hJmJ> �,?YA9g�< 0 * ( � b1= � ( � ) = ⊥ 2 B =)4 b1= � ( � ) ∈ � 2 * S * B?" �j�= �8>���?PA0g�< 0 * ( � 3 e G+K 5 (�H d a�< 2�C -b* (1G#K�B
�( � )↓ ⇐⇒ � ∈ S'_+XcY#Z b ( � ) ⇐⇒ �
( � ) ∈ � ��( � )↑ ⇐⇒ ���∈ S'_+XcY#Z b ( � ) ⇐⇒ �
( � ) = ⊥ �acbed B d /Jd a�< D d b ��] K�BTH+BFE 6 K�BR& d a�- 6 0�G =�b & *I, 0�B d 6 � � � : N�� N
0 * (1G+6 O G#0 d a (1U#6b & d C)K�(1U+6�2�( �� ) 0 ⇐⇒ �
( �� )↓ &�( �� ) 0 ��
( �� ) G �( �� ) ⇐⇒ �
( �� )↓ &�( �� )↓ &
�( �� ) G �
( �� ) �
2 rD] W*)�Y7W*J N &DMfW*M*)�M'+ -9)DV9$�F*% T J j +�N�+DFsW*) j v�)�M T�T MfW*F*%srD- T N F*G')Dv.-9$�J +DF*&lW"L &�M9a�Mf+D[�&r'N r�G T [�&cW*)�]�+D[�&^$"+ -9&^M*)DP"V M*%D+DF*G^+DF*%�J,rDM*� L F*%^\*M*$�L uDJA+DM"N�$"+DF*&OW*F9H*G M'W*F'+DJZH9J,$ T M'+�N a j +DJ,)DFM9H9v j )�N i T F +DF*% }=a j $.a�N &�F*%^+DF*% )�M'+DF*$�iDY,& -*~Zm
3 ��X@93.@� "4Z895=5xW*F*%^J,N $.V'v�M9v�J M*%�+ j +DF*&OJZ&�M9H'H*M9a"+�N a j F*)�N $ T�j +D[�& T JZ)'N a�]�&c$�%9&DM*) + V9$�J,[�&M*&�M*��YZ)�J7+DM"N'$"+DFXM*&,+�N a�J,L T JZ&�F ⊥ [�U T N M }7M*&,+�N a�J,N T JZ&�F'W*F"L -*$.-�+ -9UxM'W j a.H*N $.-9U ~Zm ��N M�W*M*)D(*r�J,N v T M�_i�M T W*F*)�F*G'$�M T Jc&DM iDY,$�F*% T J v�N {�M*%�+DYZU +�N U�� - T J N ]�$�J N U
⊥ = � %'a�M'\9-'+,+ j U��F'W j +DJ T JZ)'N a.V�$�%9&D(*) + -9$.- Q
: N � N J L &�M"N�-^+D%'P.M"L Mq/�F9H*N a.V*E�$�%9&D(*) + -9$.-Q
: N → N ∪ { ����������� ����! } �a�M"N j W*F*%pJZ) T -9&�JZG9F*% T JO+ -9& +�N T V Q
(3) = � %'a�M'\9-'+,+ j U�[�U Q(3)↑_ J7W*J N rDVpF � %�a�M'\*-�+,+ j U�r�JZ&
J L &�M"N M*)�N i T�j U /�J,L &DM"N H j ��F*UDEAm
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(�� 2 � ��'&:&��� / ��6�%M'&2&'�< , ���=��6 { 5n��# , ��6 { 57����2&' ,9/ 4���#3��5 ! Na H#P �m b`a & b E b P�BR&�E P * M 0�".K�BR& d a�, 670�G =)< & * "�0�"�6?BFE =�bed "
� ( � ) = ⊥ ( � ∈ � )JL!�] QK�B-S'_+XcY#Z b
(�) = ∅ P1(1G b P1( a H�E = B d D d b a�< C)B BRE 0�( / ( J � Q 2�B =)4&a�< C)B J�0�G =I, CI"+6�2
#b(�H d a�, &fQ 0�G =)< & * "+0�" � : � → � BFE =�bed K�BR& d a�, 0�G =)< & * "�0�"�2�K�B S?_#X.Y�Zcb ( � ) = � ; d b / U#( ��] K�BTH+BFE 67K�BT& d a�- 670�G =�b & *I, 0�B d 6 � � � : � � � 2�C -f* (1G+K�B� v � ⇐⇒ (∀ �� )[ � ( �� )↓ � ⇒ �
( �� ) =�( �� )] :J N 8 Q
m 0�A - 0�" � v � BFE =�bed J�BRU a (�H b 2 � >HG ! Q ,Eg �@< >:.��8< c;91-C`7 0 * (.0�U = (�H�(( ��� � ) = { � | � : ��� � }
S�H�M = * M = K�BR& d a�4N= 0�G =�b & *I, 0�BTM = K�B P�B / E ( B d 0�S / (1G�� acbed P1B / E ( *Jd K 4 = � 2/ "+H b1/I,� v � � [
� v �&� v �
] � ⇒ � v � � [� v �
&� v �
] � ⇒ �=� :
>HG ! 2��p������E)��� ����� ����7������+E����8 �������������� B G * (�E ( d * BTH+BR0 *)- 6 BR&)K�")]= BRU#( =F*�bed D d b K�BT& d a�- 6 0�G =�b & *I, 0�B d 6 K�B * ( O G#0 d a S * &)S+P1( * (1G�G�P�(�H�(�D d 0�K�(1U * (1G#6 b1= 2�P A 2 � � � : ��� � acbed � : � 2 � � 2 * S * B'D d b S�H b%*�b � ∈ � � � ∈ � 2�(�( � ) � � ( � )) = � ⇐⇒ (∃ � � � ∈ � )[
�( � ) = � &
�( � ) = � &
�( � � � ) = � ] D
acbed|b1=�(0 � �� ) =
�( �� )�
( � + 1 � �� ) =�(�( �� �� ) ��� � �� )
acbed ( d � � � BFE =�bed K�BT& d a�- 6 0�G =�b & *I, 0�B d 6 0 * ( 0�U = (�H�( * M =�b & d C)K 4N= N2 * S * BI2�D d b
S�H b%*�b �� �� � � ∈ N2
�( �� �� ) = � ⇐⇒ (∃ � 0
�;:;:;: � � ∈ N)J N9> Q
[ � 0 =�( �� )
& (∀ F � 0 G F ≤ � )[ � � = �( � � −· 1
� F −· 1 � �� )]& � = � ] :
$'S1& d 0�K b b G * (1U * (1G%(1& d 0�K�(1U%BRE =�bed S *Jd�( �� )↑ � ⇒ (∀ � )[ � ( � � �� )↑] �
BbP�B d /I, 2�D d b a�< C)B � 2�(cG)P1(�H+(�D d 0�K�S16 * "+6 *Jd K , 6 � ( � � �� ) # b1=)< D�B *�bed & * BbH d a�< 0 * ( =G�P�(�H�(�D d 0�K�S * "�6 � (0 � �� ) =�( �� )
@ B b G * (1U#6 * (1G#6'(1& d 0�K�(1U+6�2�K�P�(1&)(1U+K�B =�b BfP1B a�* BFE = (1G#K�B * ( = (1& d 0�K�S * (1G 0�G)]= S�H�(1G R� (Ψ) * M = Ψ
] h��3�@;=m ��g8AM? ��1@A=1�� �`mk,l< >@?BA j�=8A=1��@;k.Jj6g��BA 0 * " = P1BT&�E P * MN0�"P1(1G * ( Ψ
P�BR& d - A�B d K�BT& d a�- 6%0�G =�b & *I, 0�B d 6 -m -R=)= ( d b / B =.- A�B dBd /Jd b E * BT&)( B =)/Jd b ]O�- &)( = 2 b H+H < A�&)B d <%� B *�bed D d bc* ( 5`b 0 d a S.(1& d 0�K�S > G Q�
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! Q ( 2 *Y,3.�/ ��0o#92&4�576�'&� #9��'&:&��� / ��6�4M'&2&'�< , ���=4>HG N 2�Y, ������Z�5 2 m EI����4���� �������� ����� * "�6?K�BT& d a�, 670�G =)< & * "+0�"+6 � ( F � �� ) BRE ]=�bed "cK�BR& d a�, 0�G =)< & * "�0�"
�( �� �� ) = ( �+F ≥ � )[
�( F � �� ) = 0]
J N ! Q=( BbH < A d 0 * (16 � ≥ � 2 *)-f* ( d (16'P1(1G
�( ��� �� ) = 0 & (∀ F )[ � ≤ F*G ��� ⇒ �
( F � �� ) 0] �S+P1(1G12�G)P1B = C)G+K�E � (1G#K�B)2
�( F � �� ) 0 ⇐⇒ D d b a�< P1( d ( � � � ( F � �� ) = � + 1 :
; d b P b & <1/ B d D�K b 2�(1)↑ � ⇒ ( �+F ≥ 1)[
�( F ) = 0]↑�
b`a S1K�" a-d b1= � (2) = 0
>HG Q 2�Y, ������Z�5 2 m * G#A b E b K�BT& d a�, 0�G =)< & * "+0�" � : N�
� NBRE =�bed E)�����4���� �1� � � ������������W� �� J , � ] b1=�b1/ &)(1K d a�, 2 � ] i V : g�i ` Z V Q 0 * ( 0�U = (�H�( K�BR& d a�4N=
0�G =�b & *I, 0�BRM = Ψ2 b1= " � b1=I,8a B d 0�B a�< C)B 0�U = (�H+( K�BT& d a�4 = 0�G =�b & *I, 0�BRM = P1(1G
P1BT& d - A�B d+* ( Ψ acbed BFE =�bed P1&)M * (�D�B =)4 6 a H�B d 0 * S acbed8a H�B d 0 * S'D d b BbH b A d 0 * (+P1(�E "+0�" e G+K 5 (�H d a�<R � (Ψ) = { � | " � BFE =�bed � ] b1=�b1/ &)(1K d a�, 0 * ( Ψ} �
R � = R � (∅) :_ (c0�U = (�H+( R � (Ψ) * M = Ψ
] BTH b A d 0 *Jd a�< b1=�b1/ &)(1K d a�4N= K�BR& d a�4N= 0�G =�b & *I, 0�BTM =BFE =�bed@a H+B d 0 * S.D d b P�&)M * (�D�B =I, b1=�b1/ &)(1K , b P1S * ( = (1& d 0�K�S * (1G�2 < & b
R� (Ψ) ⊆ R � (Ψ) D" b1=F* E 0 * &)( O " 0�G#K�P1BT&�E H+" ��" / B = d 0�A�U#B d 2�S+P�M 6 C b / (1U+K�BI2 b H+H < b G * S / B = BFE =�bedP1&)( O-b1=)- 6 *)4 & b '@ P�(1&)(1U+K�B S1K�MN6 =�b / BRE C)(1G+K�B b K - 0�M 6 * " =�a H�B d 0 * S * " *�b * (1GR � (Ψ)
D d b * ( = * BTH+BR0 *I,.* "�6�# /Jd b`a H <1/ M 0�"�6 & 2�P1(1G BRE =�bedld /Jd b E * BT& b 0�"+K b1=F*Jd a S160 *Jd 6?B O-b &)K�(�D - 6 * (1G%0�B'K�BR& d a�- 6 0�G =�b & *I, 0�B d 6 >HG 9 2�Y, ������Z�5 2 m �1����� � ����7���� * & d 4N= 2 / (10�K -R= M = K�BT& d a�4 = 0�G =�b & *I, 0�BTM =
( �� ) 2 � ( �� ) 2 � ( �� ) 2�BRE =�bed ".K�BT& d a�, 0�G =)< & * "+0�"�( �� ) = b1= ( ( �� ) = 0) * S * B � ( �� ) b H#H d 4 6 � ( �� )J NPN Q
=
�( �� ) � b1=� ( �� ) = 0 ��( �� ) � b1=� ( �� )↓ & ( �� ) 6= 0 �
⊥ � b1=� ( �� ) = ⊥ �K�B * "c0�G = C ,8a ".0�U#D a H d 0�"+6
�( �� )↓ ⇐⇒ [ ( �� ) = 0 &
�( �� )↓ ] ∨ [ ( �� )↓ & ( �� ) 0 &
�( �� )↓ ] :
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(�� 2 � ��'&:&��� / ��6�%M'&2&'�< , ���=��6 { 5n��# , ��6 { 57����2&' ,9/ 4���#3��5 !+9$ b & b+* "+&)(1U+K�B S *Jd "?0�U�D a H d 0�" * "�6 /Jd b`a H <1/ M 0�"�6 b P bed * BFE * "70�U�D a H d 0�" * (1G ;0g=j@; ( �� ) b H#H < / B = b P bed * BRE * " 0�U�D a H d 0�" acbedN* M = / U#( *Jd K 4N= � ( �� ) acbed � ( �� ) D d bP b & <1/ B d D�K b 2
b1= (0 = 0) * S * B � b H+H d 4 6 ⊥ = �+:>HG T 2g*-, Z / '�� [*2 <H1 >c5=g j,<iAMm �8m ,Eg �@< >@?BA j,=iAM1��-;k.8jeg��BA
ΨZ ;91 R� (Ψ)>-1 < R � (Ψ)
g@?PA=1 <:>��`g�< j|; ��c<H1��8< 18>�����1�/j@7 �j-X Z�<\#3��} [*2 ; d b%* ( / (10�K -R= (.(1& d 0�K�S
�( �� ) = b1= ( ( �� ) = 0) * S * B � ( �� ) b H+H d 4 6 � ( �� ) �
(cP�& 4W* (16 P1B d & b 0�K�S167BRE =�bed =�b C - 0�(1G#K�B)2�K�B�P1&)M * (�D�B =I, b1=�b1/ &)(1K , 2�1(0
� �� ) =�( �� ) ��
1( F + 1 � �� ) =�( �� ) �
acbed =�b P�&)(10�P b C , 0�(1G+K�B =�b / BRE C)(1G+K�B'S *Jd�( �� ) =
�1( ( �� ) � �� ) �J N Q Q
-b* 0 d P1(1G b1=" �� � � � ∈ R� (Ψ)2 * S * B � ∈ R� (Ψ) ⊆ R � (Ψ)
B G * S / B / (1G�H�BTU+B d 2� 0 a "+0�" � > G Q 2 acbed|b1=F* � b G * (1U%C -b* (1G#K�B)2 /Jd b1/ (1A d a�< 2
�0(0
� � � � ) = � ��
0( F + 1 � � � � ) = � ���� (0 � �� ) = 0 �
� � ( F + 1 � �� ) =�( �� ) �
��� (0 � �� ) = 0 �� � ( F + 1 � �� ) =
�( �� ) ��
1( F � �� ) = �0(1−· F ����� (1−· F � �� ) ����� (1−· (1−· F ) � �� ) �
acbed�*)4 & b " J N Q Q - P�B *�bed BRU a (�H b 2 � 0 a "�0�" � >HG Q� a>HG d-2'�������������� ���@8�E�1��� ���(E�� 81 �m 0�G =)< & * "+0�" * (1G @ : r+V i XcY�b3b%J > @ T Q / B =
BFE =�bed P1&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�, 2 acbed�/ B = BFE =�bed P1&)( O-b1=)- 67J *)4 & b�QBb1= BFE =�bed BTH b A d ]0 *Jd a�< b1=�b1/ &)(1K d a�, acbed " � 0 a "�0�" � > E' c> O ∗ P1&)(10 O�- &)B d�-R=�b b`a S1K�" P b & <1/ B d D�K b0�G =)< & * "�0�"�6 D d b * " = (+P�(�E b / B = BFE =�bed BRU a (�H�( =�b / BRE C)(1G+K�B S *Jd BRE =�bed P1&)M * (�D�Bf]=)4 6 J , BTH b A d 0 *Jd a�<�Q b1=�b1/ &)(1K d a�, 3 d 0�G =�b & *I, 0�B d 6 b G *)- 6�2 S1K�M 6�2 BFE =�bed #bG)P1(#]H�(�D�E 0 d K�BR6 & 2�BfP1B d /I, ( dEb1=�b1/ &)(1K d a�- 6%B C d 0 4 0�B d 6 P1(1G *Jd 6 acb C)(1&�E � (1G = b P�( / E / (1G =b H+D�(1&�E C)K�(1G#6�D d b7* ( = G�P�(�H�(�D d 0�K�S * M = *Jd K 4 = * (1G#6�2+P A 210 *Jd 6 B 0 a�, 0�B d 6�� > @ Tacbed � > @ O1 \m BfP1S1K�B = "%$?&)S *�b 0�" / E = B d@b`a S1K�" -R=�b � a�< P�M 6 d /Jd S * G�P�( � *)-b* ( d (P b & <1/ B d D�K b >HG O 2g*-, Z / '�� [*2 � g�� 1B�B<Hj@;=meh�m ?P78j@7
�( � � �� ) = ( �+F ≥ � )[
�( F � �� ) = 0]
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� ( �� �� ) = b1= (�( �� �� ) = 0) * S * B � b H#H d 4 6 � ( � + 1 � �� ) �J N 9 Q
�`7 �e1��`.��JLY Q � J N 9 Q <Hj � < g�< �-< 1�A��e1 ;91 �� �� 1|Ap5��=j-m&=9,:g � :=
� �JU� Q n Ap7 J N 9 Q <Hj � < g�< �-< 1�A��e1 ;91 � � �� ,Eg > hJm <H1 � Z ;BA`;0g � v � �
j-X Z�<\#3��} [*2 J�Y Q $?& - P1B d�=�b / BRE C)(1G+K�B'S *Jd D d b S�H bc*�b � � �� 2�( �� �� ) = b1= (
�( �� �� ) = 0) * S * B � b H#H d 4 6 � ( � + 1 � �� ) �J N T Q
acbed C)BTM &)(1U#K�B *Jd 6 * &)B d 6?P�BR& d P *)4 0�B d 6�2�( �� �� )↑� � ( � � �� ) = 0 � � ( � � �� ) 0
m b H , C)B d b * "�6 J N T Q BFE =�bed P1&)( O-b1=I, 6%0 *Jd 6 / U+( P�& 4W* BT6 b P � b G *)- 6 �e * " =%* &�E * "P1BT&�E P * MN0�"�2 b1=
(∀ F ≥ � )[�( F � �� )↑∨ � ( F � �� ) 0] �
* S * B)2�P�&)( O@b1=)4 6�2�( � � �� ) = ⊥ =
�( � + 1 � �� ) �
-b* 0 d P1(1G P < H d - A�(1G+K�B d 0�S * " *�b \_ BbH d a�< 2 b1= 2�D d b a�< P1( d ( � � 2�( � � �� ) = 0 & (∀ F ≥ � )[ FHG ��� ⇒ �
( F � �� ) 0] �* S * B � ( �� �� ) =
�( � + 1 � �� ) = � 2 acbed�- A�(1G#K�B P < H d2d 0�S * " *�b
JU� Q $'& - P�B d =�b / BRE C)(1G+K�B?S *Jd@b1= D d b S�H b%*�b �� ��� 2� ( �� �� ) = b1= (
�( �� �� ) = 0) * S * B � b H#H d 4 6 � ( � + 1 � �� ) �
* S * B)2�D d b a�< C)B �� acbed S�H bc*�b ��acbed � ≥ � 2�( � � �� ) = ��� ⇒ � ( �� �� ) = � DJ�� Q
A�&I"�0 d K�(+P1( d (1U#K�B'BbP b D�MND , 0 * " /Jd b1O (1& < � − � e * " � %�� [ 2 � = � 2 acbed|b K - 0�MN6�2 b P1S * ( = (1& d 0�K�S * "+6 � 2
�( � � �� ) = � = � ( � � �� ) :
e * ( � X '&0 . 0P��6�Z�� 4��=' 2 � ��acbed � ( � � �� ) 02 -f* 0 d P1(1G
� =�( � � �� ) =
�( � + 1 � �� ) J b P1S * " = J N 9 Q Q
= � ( � + 1 � �� ) J�BbP G�P�(1C 2b1O (1U � − ( � + 1) = ( � − � ) − 1 Q��
= � ( � � �� ) J b P1S * " = J�� Q Q : a
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(�� 2 � ��'&:&��� / ��6�%M'&2&'�< , ���=��6 { 5n��# , ��6 { 57����2&' ,9/ 4���#3��5 ! dB G * S * ( P b & <1/ B d D�K b BFE =�bed^d /Jd b E * BR& b 0�"+K b1=F*Jd a S�2 S+P�M 6 / BFE A = B d S *Jd " B C�E ]
0�MN0�"cJ N 9 Q P1(1G7K�( d <%� B d�=�b (1&�E � B d1* " =�*Jd K , � ( � � �� ) b P1S * " = BfP1S1K�B = " � ( � +1 � �� )J � Q 0 * " = P�& b D�K b+*Jd a S * " *�b B a�O & <%� B dW* ( =.5`b 0 d a S b H+D�S1& d C)K�( * "�6 # 5 H b`a�4N/ (1G+6b1=�b%�I,#* "+0�"+6 & D d b P b & <1/ B d D�K b 2 D d b =�b G�P�(�H�(�D�E 0�(1G+K�B * " = *Jd K , � (2 � �� ) 2 B O-b &R]K�S � (1G+K�B * " = B C�E 0�M 0�"cBbP b1= B d H#"�K�K -T=�b 2� (2 � �� ) = � (3 � �� ) = · · · = � ( � � �� ) = � �
K - A�& d 6�S * (1G J E 0�M 6 Q�5 &)(1U#K�B a�< P1( d ( � *)-b* ( d ( P�(1G � ( � � �� ) = 021(+P1S * B C - &)(1G+K�B
S *Jd � (2 � �� ) = � _ ( P b & <1/ B d D�K b / BFE A = B d S *Jd " 0�A - 0�" b1=)< K�BR0 b 0 * " = # b1=�b ]/ &)(1K , &:acbed#* ( = #fG�P�(�H�(�D d 0�K�S & P1(1G G�P�( / BFE C b K�B 0 * (%J ��� Q�* (1G > @ N 2+B O-b &)K�S � B *�bedP1(�H+U BTG+&)U * BR& b b P1S * " =�a H b 0 d a�, P�BR&�E P * M 0�" * "�6 P�&)M * (�D�B = (1U+6 b1=�b1/ &)(1K , 6�2 acbedBFE =�bed�* ( a H�B d / EWD d b * ( = (1& d 0�K�S * "�6%C)BTK�BTH d b`a�, 6 a H < 0�"�6 * M = J�D�B =Jd a�4 = 2�K�BR& d ]a�4 =�Q 1@A=1�� �`mk,l< >@?BA j�=8A=1��@;k.Jj6g��BA P�(1GcC b./)4 0�(1G+K�B'0 * ( BfP1S1K�B = ( a B O�< H bed (
��� � �p�/yl~2�#$ }��� > G c>32 � BFE C * B?S *Jd D d b S�H+BR6 *Jd 67K�BT& d a�- 670�G =�b & *I, 0�B d 6 � � � : � � � 2
�=� � ⇒ (∀ � ∈ � )[
�( � )↓ ⇐⇒ �
( � )↓ ]J�Y Qacbed
�=� ⇐⇒ (∀ � ∈ � � � ∈ � )[
�( � ) = � ⇐⇒ �
( � ) = � ] :J � Q� > G ! 2 � BFE C * B?S *Jd D d b S�H+BR6 *Jd 67K�BT& d a�- 670�G =�b & *I, 0�B d 6 � � � � � : ��� � 2� v � � [
� v �&� v �
] � ⇒ � v � � [� v �
&� v �
] � ⇒ �=� :
� > G N 2 p?BTM &)(1U#K�B * (1G#67(1& d 0�K�(1U#6�( � ���� � ) = b1= ( � = 0) * S * B ��b H+H d 4 6 � ��
1(�) =
�(� � � ( � ) � � ) ��
2(�) = b1= (
�= 0) * S * B � ( � ) b H+H d 4 6 �
S+P1(1G " �(�)BRE =�bed|a�< P�( d b K�BT& d a�, 0�G =)< & * "+0�" \B H#"�C)BTU+B d ".B C�E 0�MN0�"
�1(�) =
�2(�)
D d b a�< C)B � ��J � 4 0 * B b P�S / B d CI" , b1=F*Jd P b & <1/ B d D�K b Q� > G Q,2�I CI"#D , 0 * B D d b+* E�".P�& 4W* " d /)-=b D d b * " = b P�S / B d CI" * "+6 > G T / B / (1GI]
H�BTU+B d 2 acbed /)4 0 * B *Jd 6'H�BfP * (1K - &)B d BT6 * "�6?0�MN0 *I, 6 b P�S / B d CI"+6
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BFE =�bed:a H+B d 0 *)- 6 D d b (1& d ]0�K�(1U#6?K�B �
+ 1P1BT& d P *)4 0�B d 6�2 * "+67K�(1& O�, 6
�( �� =
�1( �� ) � b1=�
1( �� ) = 0 ��2( �� ) � b1=�
1( �� ) 0 & 2( �� ) = 0 �
· · ·�� ( �� ) � b1=�
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�� +1( �� ) b1=�
1( �� ) 0 �;:;:;:��� � −1( �� ) 0 �� � ( �� ) 0 :
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� ( � + 1 � 0) = 2 � + 1 �� ( � + 1 ��� + 1) = 3� ( � ��� ) + � ( � � � ) + 2 �acbed G)P1(�H+(�D�E 0 * B * " =7*Jd K , � (3 � 2) D d � b G *I,%* " = � JU� Q � BRE C * B?S *Jd "%K�( =�b1/Jd a�, H+U+0�" b G * (1U * (1G%0�G+0 *I, K b+* (16'BRE =�bed P�&)M * (�D�B =)4 6b1=�b1/ &)(1K d a�, J , BTH b A d 0 *Jd a�< b1=�b1/ &)(1K d a�, 2 b1= b G * S.BFE =�bed BTG a (�H+S * BR&)( Q
� > G d 2 ; d b * G+A b E BT6 K�BR& d a�- 6 0�G =�b & *I, 0�B d 6 � ( �� ) 2 � ( � ��� � �� ) 2 / BRE C * B S *Jd "b1=�b1/ &)(1K d a�, B C�E 0�MN0�"� ( �� �� ) = b1= ( � = 0) * S * B � ( �� ) b H#H d 4 6 � (� ( � −· 1 � �� ) ��� −· 1 � �� )J NPd Q
- A�B dlb`a & d 514 6%K d b H+U+0�"�2�0�G�D a B a & d K -T=�b * " K�BT& d a�, 0�G =)< & * "+0�" � P�(1G (1&�E � B *�bedK�B * " = P1&)M * (�D�B =I, b1=�b1/ &)(1K ,
�(0 � �� ) =
�( �� ) ��
( � + 1 � �� ) =�(�( � � �� ) ��� � �� )) :
� > G O J�� ���������1�7�����8 ����� 5 ��� �,E ���� �12 � BRE C * B S *Jd " B C , 6 b1=�b1/ &)(1K d a�,B C�E 0�M 0�" - A�B d K�( =�b1/Jd a�, H+U+0�"�2 * "c0�G =)< & * "+0�" � ( � ��� ) = gcd( � ��� )
� ( � ��� ) =
0 � b1= � = 0 , � = 0 �� ( �� � ) � b H#H d 4 6�2 b1=�� G � �� � b H#H d 4 6�2 b1=�� | � �� ( �� i X ( � ��� )) � b H#H d 4 6 :
J N�O Q
� > G ] 2 � 4 0 * B7P b & b1/ BFE D�K b+*�b K�BR& d a�4N= 0�G =�b & *I, 0�BTM = � ( �� �� ) *)-f* ( d M = P1(1G"cB C�E 0�M 0�" J N 9 Q�=�b BbP d /)- A�B *�bed�J > Q _ " a B =I, K�BT& d a�, 0�G =)< & * "+0�" � M 6?K�S = "%H�U#0�" J�! Q @ S = ( K�E b H�U#0�"�2�P�(1G =�b BFE =�bed (�H d a�, 0�G =)< & * "+0�" J N Q � P�B d &)BT6 * (%P�H , C)(16?H+U+0�B d 6 � > G c> 8 ∗ 2�� BFE C * B7S *Jd / B = G)P < &)A�B d K�BR& d a�, 0�G =)< & * "�0�" � ( � � �� ) *)-f* ( d b P1(1G%"B C�E 0�M 0�" J N 9 Q�=�b.- A�B d2b`a & d 5�4 6 / U+(cH�U#0�B d 6
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(�� 2 � ��'&:&��� / ��6�%M'&2&'�< , ���=��6 { 5n��# , ��6 { 57����2&' ,9/ 4���#3��5 !P]� > G c>P> ∗ 2 JLY Q � BRE C * B7S *Jd " b1=�b1/ &)(1K d a�, B C�E 0�MN0�"
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^ I�� B��8B�� 3 !
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s+����� $iw@}Yy/�`� � �9'*$J�:w,$`�� b & b`a�* "+& d 0 *Jd a S � acbed D d b K b 6�2 a U#& d ( � P b & <1/ B d D�K b K�BT& d a�, 6 < H#D�B 5 & b 6?BRE ]
=�bed " �E1 j-< >E.��8m ,W.�;k7 ��1��@<Y5 ,W7�;=< >:. ��
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= ( � 0 � 1 � � 1 �;:;:;:�� ��� ) �
S+P1(1G * (� BFE =�bed 0�U = (�H+( 0 � 1 ∈ acbed 0 6= 1 acbed D d b F = 1 �;:;:;:���� 2�"� � :
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0�B d 6 BRE =�bed " �02 P�(1G BFE =�bed ����� �� JLBfP1B d /I, *�b / (10�K -R=�b � � �� BFE =�bed (�H d a�- 6
0�G =�b & *I, 0�B d 6 Q 2 acbed EF��E)�������(E#� 8 * "+6 �02 / "+H b1/I, K�BR& d a�- 6 < H+D�B 5 &)BT6 * "�6 K�(1& O�, 6
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(Z � 0 � 1 � + � − � ·) � (Q � 0 � 1 � + � − � · � ÷) �S+P1(1G
Z = {0 � 1 � −1 � 2 � −2 �;:;:;: } BFE =�bed ( d J�C)B *Jd a (�E acbed b & = " *Jd a (�E Qlb`a�- & bed ( d b & d CR]K�(�E 2 acbedQ =
{ �� | � ��� ∈ Z ��� 6= 0
}
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) � �����#��� ���� @ B a�< C)B K�BT& d a�, < H+D�B 5 & b� 0�G+0�A�B * E � (1G#K�B * " = * G)P d a�, D�H 4 0�0 b � =
�(
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Y i Z d ^( ��� ) =
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( � 1 �;:;:;:���� � ) (Y i Z d ^
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A�BFE b J , j�< ,��:m � 1 Q * (1G Σ2�D d b P b & <1/ B d D�K b ( d
0=(0� 3 � 2 � � 01 � 1�
3�� = ������� � �
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0=(0� 3 � 2 �8� 01 � 1�
3�� = ������� � �
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* "�6 � Q 2 acbed2b1= * ( A�&I"+0 d K�(+P�( d (1U+0 b K�B?D d b.* " = d 0�S * " *�b b1=)< K�BR0 b 0�B?H - C)B d 6 C bP1&)( acb H+(1U+0�B 0�U�D�A�G+0�" e G+A =)< C b A�&I"�0 d K�(+P1( d (1U#K�B D�& < K�K b+*�b b P1S * ( BbH+H#" =Jd a Sb H O�<+5 " * (cD d b =�b ( = (1K <%� (1G#K�B'0�U#K 5 (�H b acbed H - C)B d 6�2
� ≡ �0�
1 · · ·� �
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3 d #f0�G =F*�b`a�*Jd a�< 0�M 0 *)- 6 & B a�O & < 0�B d 6 * "+6 � BRE =�bed H - C)B d 6 b P�S * ( b H O�<+5 " * (acbed /Jd b A�MN&�E � ( =F*�bed 0 * (1G+6 A��`m3=�� acbed�*Jd 6 E �1��� ���(E#� 8 ����7�� 2 / "+H b1/I, H - C)B d 6 * "+6K�(1& O�, 6
� = �S+P1(1G%( d � acbed � BRE =�bed S1&)( d 3 d ������� J dfV i X%` Q * "+6 � b P�( * BTH+(1U =.* ( BTH < A d 0 * ( 0�U = (�H+( H - C)BTM = Y, ��� K�B
*Jd 67B C , 6 d /Jd S * " * BR6 J ` > Q ^ < C)B � ∈ J acbed|d /Jd b E * BT& bc*�b 0 acbed 1 QWacbed@a�< C)B b+* (1K d a�, K�B *�b+5 H+" *I, � �BFE =�bed S1&)( d J `�! Q B = ( d H - C)B d 6 � 1
2 :;:;: 2 � � � BRE =�bed S1&)( d 2 * S * B'S1&)(167BRE =�bed@acbed "%H - CI"� � ( � 1
�;:;:;:�� � � � )J ` N Q B = ( d H - C)B d 6 � 1
� � 2� � 3
BRE =�bed S1&)( d 2 * S * B'S1&)(167BRE =�bed@acbed "%H - CI"( �� ( � 1 = 0) �� � � 2
������� � � � 3)3 (1& d 0�K�S16 b G * S16?B a�O & <%� B *�bed 0�G = (+P *Jd a�< acbed K�B * " = # d 0�( / G =�b K�E b%&� :≡ � | � � | � � ( � 1
�;:;:;:�� � � � ) | ( �� ( � 1 = 0) �� � � 2������ ��� � 3)
J Q3> QP1(1G /Jd b+5�<%� B *�bed
����� �!�#"�$&%('*)�+!,!-!+/. �/0!"�$1 �/2 3!4/+!5/670 28�9#:�+/; +/<���,8��5!=#> "/>�2? "&@A4/+/0 2#>�2 4#0/>���0!6CB#0�3/+!,/"ED�:A4/=!- FG�H4/+�DI<�F/5!�#>�+J; � K#"('L�MJN FO:!>AF/5/.�4�PH+!9�D ?/Q!Q R D ?SLGTUM!V ' M!M
N�Q J 2gL #32&��6�4M'&2&'�< , ���=4"%H - CI" � BRE =�bed S1&)(16 b1= BFE =�bed K - H�(16 * (1G 2 , b+* (1K d a�, K�B *�b+5 H#" *I, 2 ,b1= BRE =�bed�* "+6 K�(1& O�, 6 � � ( � 1
�;:;:;: � � � � ) S+P�(1G ( d � 1�;:;:;: � � � � BRE =�bed S1&)( d 2, b1= BFE =�bedW* "�6.K�(1& O�, 6 ( �� ( � 1 = 0) �� � � 2������� � � � 3)
2 S+P�(1G ( d� 1� � 2
� � 3BRE =�bed S1&)( d
$ b & b+* "+&)(1U+K�B'S *Jd " a B =I, H - CI" � / B = BFE =�bed S1&)(16 3 S1&)(16 � BFE =�bed ��� E#��� ����8 b1=�acb K�E b b+* (1K d a�, K�B *�b+5 H#" *I, / B = BTK O@b1= E � B *�bed
0 * ( = � 2 acbed ��������8 b1= ( d K�S = BR6 b+* (1K d a�- 6?0 *�b C)BT& - 6 P1(1G J E 0�M 6 Q BTK O@b1= E � ( =F*�bed0 * ( = � BFE =�bed�* ( 0 , * ( 1 $ b & b1/ BFE D�K b+*�b 0 * " = � (
�0)
� (0) : a H+B d 0 * S16�2 b D = S16 � � (13) : a H+B d 0 * S16�2�S1A d|b D = S16 ��8� (� 1) : b D = S16�2�S1A d-a H�B d 0 * S16 �
( �� ( � 1 = 0) �� � 3 ������� � � 1) :(1U * B a H+B d 0 * S167(1U * B b D = S16
3 b1=�b1/ &)(1K d a S16 (1& d 0�K�S16 * M = S1&)M = /Jd acbed (�H�(�D�BRE1BfP b D�MWD d a�- 6 b P�( / BFE C)B d 6 d /Jd (#]*I,#* M =7* M = S1&)M =�! @ N 2�� 4����=' J I P b D�MWD , 0 * (1G#6?S1&)(1G+6 Q 2 � j@; � Σ
j�<8A=m��im����@C=g��BA 1eh3A ;=m1 � � �l7J;=m ;k7 � �(
);��k;=me< m hJm&=
� 1�;:;:;:�� � � � ∈ Σ � ⇒ � �� ( � 1
�;:;:;: � � � � ) ∈ Σ �� � � � � ∈ Σ � ⇒ ( �� ( � = 0) �� � � ������� � � � ) ∈ Σ :
JLY Q n A ;=m ΣhJg �|< �>�lg�< A��`g�� ;=< ��1c;=m ,B< > ��� j@;91c5=g ������>-1 <E,Egk;91�� �@7J;���� Z ;BA`;0g;=m
Σh8g �@< �>�Eg�<*A �8m3=�� ;=m3=�� A��`m3=����
JU� Q n A ;=m ΣhJg �|< �>�lg�<�A��`g��(;=< � 16;=mk,l< > ��� j|;9165 g ����� Z ;BA ;0g ;=m
ΣhJg �|< �>�lg�<
A �8m3=�� ;=m3=�� > �ig�<Hj@;=m3<�� A��`m3=����JU� Q n Ap;=m Σ
hJg �|< �>�lg�< ;=m0Z ;=m
1>c1e<HA �ig�� ;=< ��1c;=m ,B< > ��� ,Egk;91�� �@7J;���� Z ;BA`;0g;=m
Σh8g �@< �>�Eg�<*A �8m3=�� ;=m3=���1�`A=m&< � A�� m&= ���
j-X Z�<\#3��} [*2 J�Y Q _ ( Σ d acb1= (+P�( d BFE *Jd 670�G = C ,ia BT6cJ _ > Q � J _=N Q b P1S * " = G�P�S#]C)BR0�"�2 acbed BfP1(1K -T= M 6�P1BT& d - A�B d�* ( BbH < A d 0 * ( 0�U = (�H�( Y, ��� P1(1G d acb1= (+P�( d BFE b G *)- 6*Jd 670�G = C ,8a BR6 B O�,�= (1G#K�B *�b J � Q acbed J :)Q D d bc* " = � 0 a "+0�" ��! @ c>� a$?MN6.K�P1(1&)(1U#K�B =�b / BFE C)(1G#K�B.S *Jd " H - CI" ( � (1) / B = BRE =�bed S1&)(16 � m b G+0 * "�& ,b P1S / B d CI" 5 b 0�E � B *�bed 0 * ( BbP�S1K�B = ( * BRA =Jd a S b H+H < 0�"�K b1=F*Jd a S � , K�K b 2 S+P�(1G "
H - CI" � BRE =�bed � A .8jc< m 1�� �B< > A(;�,/.0,21 * "+6?H - CI"�6 � b1=� ≡ ��� K�B�� 6≡ � � � 6≡ �
! @ Q,2�� 4����=' J @ ( =�b1/Jd a�, b1=�b D = MN0 d K�S * " *�b S1&)M =�Q 2 J�Y Q <H1 >c5=g A��`m � Zm 1��|<P5 ,�A�� g , �@1@A@? j6g��BA ;k7 �p1��@<Hj@;0g �@. ��hJ1����8A 5 g=j@7 ���(�ej|;=m6A � g@?PA=1 <H?Hjcm�� ,:g;=mcA 1��@<Y5 ,9A�g , �@1|A ?Hjeg��BAp;k7 � ��g@CJ< ���h�1����8Ak5=g=j|7 ���
)�6j@;=mcA � �JU� Q�� 1@A��8AM1 � A .8jc< m 1��;�/< >+A ;�,W.9,:1 A��`m3=��kg8A�g@?YAM1e<HA��`m����
J :)Q n A 7 ���@C`7 � ≡ �1 · · ·
� � g@?PA=1 < A�� m�� Z ;BA`;0gp< j@��<kg�<:18>��|< �:? � ,l<H1 1 h�A ;=< �h8g �@<Hh ; ?/j6g�< � J ` > Q 2�J `�! Q . J ` N Q j@;=mcA m��|< j8,�A ; �BA A����BA �
����� �!�#"�$&%('*)�+!,!-!+/. �/0!"�$1 �/2 3!4/+!5/670 28�9#:�+/; +/<���,8��5!=#> "/>�2? "&@A4/+/0 2#>�2 4#0/>���0!6CB#0�3/+!,/"ED�:A4/=!- FG�H4/+�DI<�F/5!�#>�+J; � K#"('L�MJN FO:!>AF/5/.�4�PH+!9�D ?/Q!Q R D ?SLGTUM!V ' M
J � 2 � # , ��6 { 5n%&��01#�� , #�5 N 9j-X Z�<\#3��} [ 0 * " = � 0 a "�0�" ��! @ ! a$?&)( O-b1=)4 6 *)4 & b 2W" H - CI" ( � (1) / B = BFE =�bed S1&)(16�2WB O S10�( = - A�B d P1BT& d 0�0�S * BR&)BT6b & d 0 * BT& - 6 b P�S / B C d - 6?P b &)B = C - 0�B d 6 _ ( � , K�K b ! @ Q BbP1E 0�"+6 /Jd acbed (�H+(�D�BFE 1|AM1���� m ,B< >2m&< � m��@<HjJ,@m&< � 0�G =�b & *I, 0�BTM =
0 * (1G#6?S1&)(1G+6 ! @ 9 2�� 4����=' J B =�b1/ &)(1K , 0 * (1G+6?S1&)(1G#6 Q 2 < 1 >c5=g j,<iAMm �8m >-1 < �8m j8, � �A0g���j,=iAM1��-;k.8jeg�< �
Φ1� Φ�2 (�
= 0 � 1 �;:;:;: ) Z ==h �� �Eg�<@,-mcAM1��J< >:. j�=8A���@;k7Jj|7Φ :
Y, ��� → ;��k;=me< 1 hJm&=
D d b"� ∈ � Φ( � ) = Φ1(� ) � Φ( � � ) = Φ1(
��� ) �Φ( � � ( � 1
�;:;:;: � � � � )) = Φ� �2 ( � � � Φ( � 1)
�;:;:;:�� Φ( � � � )) �Φ( �� ( � = 0) �� � � ������ ��� � ) = Φ3
2(�� � Φ( � ) � Φ( � ) � Φ( � )) :
j-X Z�<\#3��} [ 0 * " = � 0 a "�0�" ��! @ Q� aB =)< H�(�D b^b P�( * BTH - 0�K b+*�b d 0�A�U+(1G = P�(1G /Jd acbed (�H�(�D�(1U = b1=�b1/ &)(1K d a (1U#6 (1& d 0�K�(1U+6
0 *�b 0�U = (�H b%* M = a H�B d 0 *)4N= acbed�* M =pb D =)4 = S1&)M = 2 � 0 a "+0�" ��! @ 9 $ b & b+* "+&)(1U+K�B S *Jd " j�<8Ak;91-C`7 * "+6 � (
)P�&)(10 /Jd (1&�E � B *�bed P�H , &)MN6 b P1S * (.0�U)]
= (�H�( acbed�* ( H+B C d H+S�D d ( ( � 1 �;:;:;:���� � )2 / "#H b1/I, / B = B C b & *)<+*�bedEb P1S *Jd 6 BR&)K�")]
= BFE BT6 �1�;:;:;:�� � � b G *)4N=?* M = 0 *�b C)BR& 4N= 0 * " = 3 d BR&)K�" = BFE BT6 * M = �
1�;:;:;: ��� �
G�P�B d 0 - &)A�( =F*�bed 0 * " j@70,21ejc< m��im ��? 1 * "+6 � (
)2�MN67B C , 6
! @ T 2�� �+�1��� �� � ��������� �(
) � �����,7��1� �� ������������������ ������ �) m�1�� J Y�[ g V Q�� ���� ( � ) * (1G * G+A b E (1G ��� E#��� �����%������� � 0 * " K�BT& d a�, < H+D�B 5 & b (1&�E � B *�bed b1=�b1/ &)(1K d a�< b P1S *Jd 6�B C , 6�0�G = C ,ia BT6�21K�B BfP�E a H#"�0�" * (1G � , K�K b+* (16�! @ 9D d b a H+B d 0 * (1U+67S1&)(1G#6 J S ` > Q� ����� ( � ) = � 2�D d b � ∈ J S `�! Q B = � ��� � ( � 1) = � � D d b � = 1 �;:;:;: � � � 2 * S * B
� ��� � ( � � ( � 1�;:;:;:�� � � � )) =
� � ( � 1�;:;:;:�� � � � ) J S ` N Q B = � ����� ( � ) = � ��� ����� ( � ) = ��acbed�� ����� ( � ) = 2 * S * B
� ��� � ( �� ( � = 0) �� � � ������ ��� � ) = �� ( � = 0) �� � �@������� � � $ b & b+* "+&)(1U+K�B S *Jd ( (1& d 0�K�S16 K�P1(1&)BRE =�b�b P1( /)4 0�B d�� ��� � ( � )↑2 B O S10�( = *�b
# / (10�K -R=�b%& * "+6 < H#D�B 5 & b 6 BbP d * & - P�B *�bed =�b BRE =�bed K�BT& d a�- 6 0�G =�b & *I, 0�B d 6 b1=�1(1)↑
2 * S * B � ����� ( � 1(1)) =�1(1) = ⊥ 2 / "+H b1/I,�� ����� ( � 1(1))↑ ; d b * ( BbP�S1K�B = ( � , K�K b A�&)B d b%� S1K b 0 * B * " = -R=)= ( d b * "�6 ;@==h`< >E. �(1@Ak;=< >-16; �j|;91 j|7 �
! @ d 2'� �#�1� ��������� ������� ����7��� ; d b a�< C)B / U+( S1&)(1G#6 � 2 a�< C)B b+* (1K d a�, K�B ]*�b+5 H#" *I, � acbed@a�< C)B'S1&)( � 2�C -f* (1G+K�B� {� :≡ � } ≡ * ( b P�( *)- H+BR0�K bc* "�6 b1=F*Jd acb+*)< 0 *�b 0�"�6 * "+6 �%K�B � 0 * ( = S1&)( � �
����� �!�#"�$&%('*)�+!,!-!+/. �/0!"�$1 �/2 3!4/+!5/670 28�9#:�+/; +/<���,8��5!=#> "/>�2? "&@A4/+/0 2#>�2 4#0/>���0!6CB#0�3/+!,/"ED�:A4/=!- FG�H4/+�DI<�F/5!�#>�+J; � K#"('L�MJN FO:!>AF/5/.�4�PH+!9�D ?/Q!Q R D ?SLGTUM!V ' M R
N T J 2gL #32&��6�4M'&2&'�< , ���=4acbed D�B =Jd a S * BR& b 2�D d b �� =
�1�;:;:;:�� � � 2 �� = � 1
�;:;:;: � � � 2
� { �� :≡ �� } ≡ � { � 1 :≡ � 1} · · · {� � :≡ � � } :J Q ! Q
; d b P b & <1/ B d D�K b 2�1(�5���
1){� 1 :≡ � } ≡ �1(�5� � ) � �
2(�5���
1){� 1 :≡ � ��� 5 :≡ � } ≡ �2( � � � )
! @ O 2�� 4����=' 2 J�Y Q n A 7 � g@?YAM1e<27/,9A6Ak7�1c;=m ,B< >E.W,:gk;91 ��|7�;k. hJm&= g , �@1@A@?��9g �;91e<-j@;=mcA�A�� m � >-1 < Q � g@?PA=1 < > �ig�<Hj@;BA�� A�� m���;��k;=m < m�� hJm&= � ��� � ( � ) = � ∈ Z;BA ;0g� ����� ( � { � :≡ � }) = � ����� ( � {� :≡ � }) :
� Q g8A�< >+A`;0g � 1 Z 1@A A��`g�� m < 16;=mk,l< > ���p,Egk;91�� �@7J;���� hJm&= g , �@1|A ? �0mcA ;91e< j|;=m6AA��`m � g@?PA=1 < j@;k7 ��? j|;91 �
1�;:;:;: � � � >-1 < m < � 1
�;:;:;:�� � � g@?PA=1 < > �ig�<Hj@;=m ? A�� m <;��k;=me< me</h�m3=� ��� � ( � 1) = �
1 ∈ �;:;:;: ��� ��� � ( � � ) = � � �;BA ;0g� ����� (
� {� 1 :≡ � 1�;:;:;: � � � :≡ � � }) = � ����� (
� { � 1 :≡ �1�;:;:;: � � � :≡ ��� })
j-X Z�<\#3��} [*2 I U a (�H#"�2�K�B?BfP b D�MWD , 0 * ( = S1&)( � 2 � 0 a "+0�" ��! @ T am G)P1S1C)BT0�" � ����� ( � ) = � ∈ 0 * ( J�Y Q * (1G � , K�K b+* (16 J acbed " b1=)< H�(#]
D�" G)P1S1C)BT0�" 0 * ( JU� Q Q / B = K�P1(1&)(1U = =�b P b & b H+B d O C)(1U = 2 BbP�B d /I, * ( � , K�K b / B =d 0�A�U#B dlb P b & b E * " *�b b1=�� ����� ( � )↑2 � 0 a "+0�" ��! @ d1 3 d P d ( 0�"+K b1=F*Jd a�- 6%B O-b &R]K�(�D - 6 * (1G � , K�K b+* (16 S1K�MN6 BFE =�bed S *�b1=�*�b � 1
�;:;:;: � � � BRE =�bed b+* (1K d a�- 6 0 *�b C)BT& - 6�1�;:;:;:�� � � ∈ 2�D d b *Jd 6 (+P�(�E BR6�2 51-f5 bed b 2�" G)P1S1C)BT0�" d 0�A�U#B d 2 acbed:b G *I, BRE =�bed "
P1BT&�E P * MN0�" * (1G � , K�K b+* (16 P1(1G A�&I"�0 d K�(+P1( d (1U#K�B D d b =�b b1=�b C - 0�(1G#K�B *Jd K - 6%0�BS1&)(1G+6 P1(1G / B = BRE =�bed@a H�B d 0 * (�E 2�MN67B C , 6 ! @ ] 2�� �+�1��� �� � ���������=�
(
) � �����,7��1� �� ������������������ �� � � �) �� ������ � ����� J Y�[ g Y)dbZ _Pb 2�Y#` `fZ h b�X%VAb�d Q 0 * ( BFE =�bed " * G+A b E b 0�G =)< & * "�0�"
�: { � 0 ��� 1 �;:;:;:�� } →
P1(1G b1=�b C -f* B d 0�B a�< C)B b+* (1K d a�, K�B *�b+5 H+" *I, K d b *Jd K , �( � � ) 0 * ( acbed^b1=S�H�BT6 ( d2b+* (1K d a�- 6 K�B *�b+5 H+" *)- 67P1(1G.BTK O@b1= E � ( =F*�bed 0 * ( = S1&)( � BFE =�bed 0 * ".H1E 0 *�b
�1�;:;:;:�� � � 2�C -b* (1G#K�B
� ��� � ( � � � ) = � ����� ( � {� 1 :≡ �(�1)�;:;:;:�� � � :≡ �
(� � )}) :J QPN Q
@ B * ( = a H b 0 d a S.0�G+K 5 (�H d 0�K�S acbed�* " = (1&)(�H�(�D�E b%* "�6'H�(�D d a�, 6�2 � � |= � = � ⇐⇒ � ����� ( � � � ) = � ����� ( � � � )
⇐⇒ " � ������������� E * " = B C�E 0�MN0�" � = � D d b%* " = � �
����� �!�#"�$&%('*)�+!,!-!+/. �/0!"�$1 �/2 3!4/+!5/670 28�9#:�+/; +/<���,8��5!=#> "/>�2? "&@A4/+/0 2#>�2 4#0/>���0!6CB#0�3/+!,/"ED�:A4/=!- FG�H4/+�DI<�F/5!�#>�+J; � K#"('L�MJN FO:!>AF/5/.�4�PH+!9�D ?/Q!Q R D ?SLGTUM!V ' M �
J � 2 � # , ��6 { 5n%&��01#�� , #�5 N1d
-b* 0 d P1(1G12�P d ( b1=�b H�G *Jd a�< 2 � � |= � = � ⇐⇒
[ � ��� � ( � � � )↑ acbed�� ��� � ( � � � )↑]
, [ � ��� � ( � � � ) = � ∈ acbed�� ��� � ( � � � ) = � ] :I P1E 0�"+67C -f* (1G+K�BI2
|= � = � ⇐⇒ J D d b a�< C)B � Q � � |= � = �⇐⇒ "cB C�E 0�MN0�" � = � BRE =�bed �)� ������� 0 * " = :
! @ c> 8 2'� ��������� ��E)�� �)����� ���� ������������� � P1MN6 0�G = "�C�E � B *�bed 0 * " H+(�D d a�, 20�P <1=Jd b D�& <1O (1G+K�B # D�& b K�K b+*Jd a�< 0�M 0 * (1U+6 & S1&)(1G#6 acbed B C d 0 4 0�B d 6 e * " = P�& < CI"A�&I"�0 d K�(+P1( d (1U#K�B *�b 0�G = "�C d 0�K -R=�b K b CI"+K b+*Jd a�< 0�U#K 5 (�H b�b1=F* E D d b *�b * G)P d a�<* (1G+6 d 0�( / U =�b K b 2�P A 2 � ��� �;:;:;: D d b b+* (1K d a�- 6 K�B *�b+5 H#" *)- 6 b1=F* E�D d b �
1���
2�;:;:;: 2
� � � � + � · ��� �;:;:;: D d b 0�G =�b & * "+0 d b`a�- 6 0 *�b C)BR& - 6 b1=F* E D d b �1�;:;:;: 2 a H#P MI P1E 0�"+6
P b & b H+BFE P�(1G+K�B , B d 0 < D�(1G+K�B P1BT& d 0�0�S * BR&)BT6.P b &)B = C - 0�B d 6 acbed # a B = (1U+6 A 4 &)(1G+6 &b1=pb G * S /Jd BTG a (�H+U = B d�* " =�b1=)< D = MN0�"�2 acbed D�& <1O (1G#K�B J�D d b P b & <1/ B d D�K b�Q( � + � ) · � b1=F* E�D d b · (+( � ��� ) � � )
m # D�& b K�K b+*Jd a�< 0�MN0 *I,�& -0a�O & b 0�" * "�67J N 9 Q J�K�B * ( acb+*)< H#H+"#H�( H+B C d H+S�D d ( Q BFE =�bed� ( � 1 ��� 2 �;:;:;: ��� � +1)
= ( �� ( � 1(� 1 ��� 2 �;:;:;:���� � +1) = 0) �� � � 1 ������ ��� � ( � ( � 1) ��� 2 �;:;:;:���� � +1))B =)4 " b P�H+(1G+0 * BRG#K -R= " * G�P�(+P1(�E "+0 , * "�6?0 * " = � (�
0� � ��� ) BFE =�bed
� ( � � �� ) = b1= (�( � � �� ) = 0) * S * B ��b H+H d 4 6 � ( � ( � ) � �� ) �J QPQ Q
P1(1G%(+P�M 0 /I, P�( * B?BRE =�bed P d ( BTG b1=)< D = MN0 * "
s+��� �p�Byl~:�*$e}����! @ c>&2�� BRE C * B *�b K - &I" JU� Q acbed J :)Q * (1G � , K�K b+* (16?! @ N1 ��! @ ! 2�� BRE C * B * ( � , K�K b @ ( =�b1/Jd a�, 6 B =�b D = M 0 d K�S * " *�b 6 � &)M = ! @ Q� ��! @ N J @ ( =�b1/Jd a�, b1=�b D = MN0 d K�S * " *�b D d b B C d 0 4 0�B d 6 Q 2�� BFE C * BNS *JdJb1= ( d � 1
� � 12
� 2� � 2
BFE =�bed S1&)( d�* "+6 � ( ) acbed� 1 = � 1 ≡ � 2 = � 2
* S * B � 1 ≡ � 2acbed � 1 ≡ � 2
��! @ Q 2�� BRE C * B * ( � , K�K b ! @ 9 ��! @ 9 2�� d b+* G)P 4 0 * B acbedW/ BFE C * B � , K�K b+*�b b1=)< H+(�D b * (1G ! @ 9 P�(1G /Jd acbed (#]
H�(�D�(1U = (1& d 0�K�(1U+6?0 * (1G#6 a H�B d 0 * (1U#6 acbed�* (1G#6 b D = (1U#67S1&)(1G#6 ��! @ T 2�� BRE C * B * ( � , K�K b ! @ O1
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NPO J 2gL #32&��6�4M'&2&'�< , ���=4��! @ d-2�� 4 0 * B P b & <1/ B d D�K b S+P�(1G * ( J�Y Q�* (1G � , K�K b+* (16c! @ O / B = d 0�A�U#B d 2
BbP�B d /I,�� ��� � ( � )↑ ��! @ O 2�� BRE C * B%S *Jd D d b a�< C)B S1&)( � 2 b1= " H1E 0 *�b �
1�;:;:;: � � � P�BR& d - A�B d S�H�BT6
*Jd 6 b+* (1K d a�- 67K�B *�b+5 H+" *)- 6'P�(1G%BRK O-b1= E � ( =F*�bed 0 * ( = � 2 * S * B�(�1) =
�(�1)�;:;:;:�� � (
� � ) =�(� � ) � ⇒ � ����� ( � � � ) = � ����� ( � � � ) :
��! @ ] 2 B = S�H+BR6 ( d b+* (1K d a�- 6 K�B *�b+5 H+" *)- 6.P1(1G BTK O@b1= E � ( =F*�bed 0 * ( = S1&)( �BFE =�bed 0 * ".H1E 0 *�b �1�;:;:;:�� � � 2 " � BRE =�bedEb P�( * E K�"�0�" 0 * " = acbed ( d � 1
�;:;:;:�� � �BFE =�bed S1&)( d 2 *)-b* ( d ( d P�(1G� ����� ( � 1
� � ) = �1 ∈ �;:;:;:���� ����� ( � � � � ) = ��� ∈ � �
* S * B� ����� ( � {� 1 :≡ � 1
�;:;:;:�� � � :≡ � � } � � )
= � ��� � ( � {� 1 :≡ �1�;:;:;:�� � � :≡ � � } � � ) :
s�� � � �czE�cwcx(�B~ y/zB}l��%:x �|x(' }��*���E�e�� b G * S * ( B /)<1O@d ( C b B d 0 b D < D�(1G+K�B * " = �1������������� �1� �� ������������������ ��
* "�6 D�H 4 0�0 b 6 � ( )2�P�(1G 0�G#0�A�B * E � B d # acb1= ( =Jd a�- 6 & J�BTH < A d 0 * BT6 Q H+U+0�B d 6�K�B a�< C)Bj�<�j|;k79,:1p1|AM1���� m ,B< >|?BABg@C8< j ?/jeg��BA 2+0 * " K�BT& d a�,'< H+D�B 5 & b acbedib P�( / E / B d 1 � ��m �
� ?P5 ,-m3=�� D d b%* ( = G)P1(�H+(�D d 0�K�S b G *)4N=7* M = H+U+0�BRM = � b 0 d a S16?K b 6?0 * S1A�(16?BFE =�bed=�b (1&�E 0�(1G+K�B * "%C)BTK�BTH d b`a�, a H < 0�" * M = J�D�B =Jd a�<�Q ������������W� ����� E)�1� � ��� ��� ������#����(E@7�� 0 * " = * G+A b E b K�BR& d a�, < H#D�B 5 & b 2 acbed:d /Jd b E * BT& b 2 5�-b5`bed b 2 * " = �
02
acbed =�b�b P�( / BFE C)(1G#K�B *Jd 6 5`b 0 d a�- 6 * "+6 d /Jd S * " * BR6 ! E' c>&2�� �����������������1� �� ��� ������� �
(
) � �����#��� ���� ; d b =�b A�&I"�0 d K�(#]P1( d , 0�(1G#K�B * " = � ( )
M 6 P1&)(�D�& b K�K b+*Jd a�, D�H 4 0�0 b 2 acb+*�b &)A ,+=.* " = BTK�P�H+(1G * E ]� (1G+K�B'K�B ���������#�������������@8 (EI��� ���1��� �18
� �0� � �
1�;:;:;: (
�= 0 � 1 �;:;:;:�� Y i Z d ^ ( �
�� ) =
�) �
< P1B d &)BR6 * ( P�H , C)(16 D d b�a�< C)B / G =�b+*I, P+H�B d (1K - H+B d b � B P1S * "'0�G =F*�b`a�*Jd a�,'< P�(���"�2( d 0�G =�b & * "�0 d b`a�- 6'K�B *�b+5 H#" *)- 6 / B = C)BRA�MN&�E � (1G = b P�S *Jd 6?0�G =�b & * "+0 d b`a�- 6'0 *�b C)B ]& - 6 �
1�;:;:;:���� � P1(1G%( = (1K <%� (1G =?*Jd 6 / (10�K -T= BR67K�BR& d a�- 6 0�G =�b & *I, 0�B d 6 �
1�;:;:;: � � �
* "�6 ( d S1&)( d (1&�E � ( =F*�bed K�B * " =pb1=�b1/ &)(1K ,� :≡ � | � � | � � ( � 1
�;:;:;:�� � ��� ) | ��� ( � 1
�;:;:;:�� � � )| ( �� ( � 1 = 0) �� � � 2
������� � � � 3)acbed�- A�(1G = S�H�BT6 *Jd 6 d /Jd S * " * BT67P1(1G - A�(1G+K�B b P1( / BRE C)B d � K�( =�b1/Jd a�, b1=�b D = MN0 d K�S#]* " *�b 2 a H#P � P�M 6 acbed P�& d = 2�( S1&)(16 � BRE =�bed ��������8 b1= ( d K�S = BT6 b+* (1K d a�- 60 *�b C)BT& - 6'P�(1G JLE 0�MN6 Q BRK O-b1= E � ( =F*�bed 0 * ( = � BRE =�bed�* ( 0 , * ( 1
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J � 2�j 2&'�< , ���=4;6�'&� � X ������0P������Z�5 N ]
� �1��� �� �������������� ���E � ��(7���� BFE =�bed " * G#A b E b B C�E 0�M 0�"cS1&)M =�(�1�;:;:;:�� � � ) = � �
0 * ( H�B C d H�S�D d ( ( � 1 �;:;:;: ��� � � ��0� � �
1�;:;:;: ) S+P�(1G " � BFE =�bed 0�G =�b & * "�0 d b`a�, K�B *�b ]
5 H#" *I, ( d �1�;:;:;: � � � BRE =�bed /Jd b1O (1&)B *Jd a�- 6 K�B *�b C)U * (1G+6 b+* (1K d a�- 6 K�B *�b+5 H+" *)- 6 acbed ( � BFE =�bed:b D = S16 S1&)(16 * "+6 � (
)0 * ( = (+P�(�E ( / B = BRK O-b1= E � ( =F*�bedEb+* (1K d a�- 6
K�B *�b+5 H+" *)- 6 < H+H+BR6 b P1S *Jd 6��1�;:;:;:�� � � Mm B C�E 0�MN0�" acb H+BFE *�bed �@7�;k. b1= acb K�E b
0�G =�b & * "�0 d b`a�, K�B *�b+5 H+" *I,./ B = BTK O@b1= E � B *�bed 0 * ( = S1&)( � ; d b P b & <1/ B d D�K b J acbed K�B b P�H+(+P1( d "+K -R= (.0�G#K 5 (�H d 0�K�S Q "�( � ) = �� ( � = 0) �� � 1 ������ ��� 0
BFE =�bed &I" *I, B C�E 0�MN0�"c0�B a�< C)B'K�BR& d a�,c< H+D�B 5 & b "�( � ) = �� ( � = 0) �� � 0 ������� � � � (
�(���
( � )))BFE =�bed|b1=�b1/ &)(1K d a�, J b H#H < S1A d &I" *I,�Q B C�E 0�MN0�" * "+6 �
0 acbed "
�( � ) =
�( � )
/ B = BFE =�bed:b1=�b1/ &)(1K d a�, B C�E 0�MN0�"�2�BfP1B d /I, " K�B *�b+5 H+" *I,� BTK O@b1= E � B *�bed 0 *�b / B C d <acbed S1A d 0 *�b b & d 0 * BT& < _ BTH d a�< 2 ������������W� ��� ��������������� * "�6%K�BT& d a�, 6 < H+D�B 5 & b 6 BFE =�bed�* ( * GI]
A b E ( 0�U+0 * "�K b b1=�b1/ &)(1K d a�4N= B C d 0 4 0�BRM =( � 0)
�0( �� 0) = � 0
( � � )�� ( ��� ) = � �
J � Q
S+P1(1G�( d 0�G =�b & * "+0 d b`a�- 6 K�B *�b+5 H+" *)- 6 �0�;:;:;: � �
�BRE =�bed+/Jd b1O (1&)B *Jd a�- 6 K�B *�b C)U * (1G#6acbed BRE =�bed ( d K�S = BR6�0�G =�b & * "�0 d b`a�- 6 K�B *�b+5 H#" *)- 6 P1(1G BRK O-b1= E � ( =F*�bed 0 * (1G#6 S1&)(1G#6
�0�;:;:;:�� �
� 3 d B C d 0 4 0�B d 6 * (1G � acb H�(1U =F*�bed J b1=�b1/ &)(1K d a (�E Q ���1������� * M =
0�G =�b & * "�0 d b`a�4 = K�B *�b+5 H+" *)4 = �0�;:;:;: � �
�
m 5 b 0 d a�,�d /)-=b * "�6 G)P1(�H+(�D d 0 *Jd a�, 6 0�"�K b 0 d (�H�(�D�E b 6 * "�6 � ( )BFE =�bed S *Jdea�< C)B
P1&)S�D�& b K�K b � b1=F*Jd 0 * ( d A�E � B d J acbed G)P1(�H+(�D�E � B d Q K d b K�BT& d a�, 0�G =)< & * "+0�"� � : �
��� ( b1= Y i Z d ^ ( � � ) =
� � )0�B a�< C)B'0�U+K 5 (�H�( � � P1(1G%(1&�E � B *�bed|b P � b G * S�2 -f* 0 d P�(1G%( d �
0�;:;:;:�� �
�* ( < >-1@A=m �hJm < m&<iA 2 / "+H b1/I,
( � �
0�;:;:;:�� �
� ) |=� � ( �� � ) = � � ( F = 0 �;:;:;:�� � ) :
$?& d = (1&�E 0�(1G+K�B b G#0 * "+& <c* " =pb1=F*Jd 0 * ( d A�E b� 7→ (
�0�;:;:;: � �
� )C)BRMN&)(1U+K�B'K�BR& d a�< P b & b1/ BFE D�K b+*�b 3 (1& d 0�K�S16
�( � ��� ) =
�( � )
J �1Q
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Q 8 J 2gL #32&��6�4M'&2&'�< , ���=4BFE =�bed@b P�S.K�S = (16 * (1G P1&)S�D�& b K�K b%* "+6 �
02�P�(1G%(1&�E � B d JL&I" *)<�Q * " /Jd K�BbH , 0�G =�b &R]
* "�0 d b`a�, K�B *�b+5 H+" *I, � \m 0�"�K b 0 d (�H�(�D�E b 21P1&)( O-b1=)4 67C b P1& - P1B d�=�b K b 6 /)4 0�B d�( � ��� ) = � �����
0(�( � )) = � + 1 :
m B C�E 0�M 0�"�( �� �� ) = b1= (
�( �� �� ) = 0) * S * B � b H#H d 4 6 �
(�( � ) � � )J �
2Q
BFE =�bed P1&)S�D�& b K�K b * "+6 (�
0� � ) S+P1(1G �
: N�+1 � N
2 acbed G)P < &)A�(1G = P b & b ]/ BFE D�K b+*�b � D d b *Jd 6 (+P1(�E BT6 " B C�E 0�MN0�" J �
2Q K�P�(1&)BFE =�b - A�B d P1(�H#H - 6 H�U#0�B d 6�2
� 0 a "+0�"�� > G ] S1K�MN67D d b a�< C)B � 2�G)P < &)A�B d BTH < A d 0 * ".H�U#0�" * "+6.J �2Q b P1S * " =
$?&)S *�b 0�" > G O 2�"�( � � �� ) = ( �+F ≥ � )[
�( F � �� ) = 0] �
acbedcb G *I, BRE =�bed " H+U+0�" � P1(1G C b b1=F*Jd 0 * ( d A�E 0�B d 0 * (%0�U+K 5 (�H�( � " G�P�(�H�(�D d 0 *Jd a�,0�"+K b 0 d (�H+(�D�E b%* "+6 � (
�0� � ) _ BTH d a�< 2�C)BRMN&)(1U+K�B acbed�* ( B C , 6 * B * & d K�K -T= ( P b & <1/ B d D�K b P1&)(�D�& < K�K b+* (16�2�K�B
* ( K�( =�b1/Jd a S.(1& d 0�K�S�( � ) =
�( � ) :J �
3Q
m B C�E 0�MN0�" J �3Q d acb1= (+P�( d BFE *�bed-b P�S%S�H�BT6 *Jd 6'K�BT& d a�- 6'0�G =�b & *I, 0�B d 6 " G)P1(�H+(�D d ]0 *Jd a�, 0�"�K b 0 d (�H�(�D�E b C b b P1( /)4 0�B d�* " = a B =I, K�BT& d a�, 0�G =)< & * "�0�"
�( � ) = � ( � ) = ⊥ �
/ "+H b1/I, P < H d�* " = BbH < A d 0 * "cH+U+0�" * "�6 J �3Q 2�S+P1MN6 acbed 0 * (%P b & <1/ B d D�K b J �
2Q @ B *)< b P�S b G *)< *�b P1&)( acb+*�b & a�*Jd a�< 2�P�&)(1A�M &)(1U#K�B 0 * ( = b G#0 * "+&)Sc(1& d 0�K�S * "+6
G�P�(�H�(�D d 0 *Jd a�, 6 0�"�K b 0 d (�H�(�D�E b 6 * "+6 � ( ) m 5 b 0 d a�, d /)-Mb BRE =�bed�=�b b1=F*Jd 0 * ( d ]
A�E 0�(1G#K�B�0�B a�< C)B b1=�b1/ &)(1K d a S7P1&)S�D�& b K�K b � acbed a�< C)B K�B *�b+5 H#" *I, � ��* (1G � K d b ��4����� 2 P�(1G =�b G�P�(�H�(�D�E � B d:a�< P�( d b K�BR& d a�, 0�G =)< & * "+0�" � � 2 acbed P�& - P�B d P�& 4W*�b=�b (1&�E 0�(1G+K�B *Jd B =)= (1(1U+K�B'K�B #fK�"�A b1=I,�& p b./)4 0�(1G+K�B * ( = b G+0 * "�&)S.(1& d 0�K�S.0�B/ U+(.0 *)<1/Jd b ! E' ! 2�Y, ������Z�5 2��p��� ������ (EI��� � ���(E17�� J d i Y�b�` Z dfZ\_�b.` ^�` dfVFX Q BFE =�bed " * GI]
A b E bc* & d <1/�bT = (
� � → ��� ) �S+P1(1G J�Y Q _ ( � BFE =�bed K�")] a B = S 0�U = (�H�(�2�( d >c1c;91 j|; j6g�< � J�` dTY)dbVF` Q * (1G T JU� Q _ ( → BFE =�bed K d b./Jd K�BbH , 6 j � �=j@7�,Egk; ��E1 j|7 � J d i Y b�`fZ dbZ _Pb i VF[ Y)dbZ _Pb Q 0 * (�� J :)Q _ ( � ⊆ � BRE =�bed�* ( 0�U = (�H�( * M = ;0g �J,21c;=< >|?BA J dfV i X%Zcb�Y�[ Q acb+*�b 0 *)< 0�BRM = 2acbed D d b a�< C)B * BR&)K b+*Jd a�, acb+*)< 0 *�b 0�"�2
� ∈ ��� ⇒ (∀ � ′)[ � 6→ � ′] :J Q 9 Q
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@ d b acb+*)< 0 *�b 0�".P1(1G d acb1= (+P�( d BFE * " = J Q 9 Q^b H#H < / B = BRE =�bed�* BR&)K b+*Jd a�, acb H+BFE *�bed�kmcA 7 �� P1B *�bed S *Jd ( d:acb+*�b 0 *)< 0�B d 6 *�b C d = (1K�(1U =F*�bed 0�B * &)B d 6 acb+* "#D�(1&�E BR6 '*Jd 6* BR&)K b+*Jd a�- 6�2 *Jd 6 < D�( = BT6�2 acbed�*Jd 6 �@A6A�< ,Eg�� 2 / "#H b1/I, b G *)- 6 P�(1G - A�(1G =c* (1G#H < A d ]0 * ( = K�E b�� BbP�S1K�B = "�� acb+*)< 0 *�b 0�" �_ ( T BFE =�bed ��� �1�����������1� ��� J =F* B * BT&)K d =Jd 0 *Jd a S�2w�VRdbV i XcZ b�Z\` dbZ :)Q 2 b1= a�< C)B acb+*)< 0 *�b 0�" � - A�B d�* (cP�(�H�U%K�E b BbP�S1K�B = "�2 / "#H b1/I,
[ � → � ′ & � → � ′′] � ⇒ � ′ = � ′′ :J Q T Q; d b P b & <1/ B d D�K b 2 - 0 * M
� →1� ⇐⇒ � � � � →2
� ⇐⇒ � =�
+ 1 DJ Q1d Q* ( 0�U+0 * "�K b (N � →1
� {0}) BRE =�bedeb1=�bed *Jd ( a & b+*Jd a S�21B =)4 * ( 0�U+0 * "�K b (N � →2� {0})BFE =�bed|bed *Jd ( a & b+*Jd a S
! E' N 2�� ����������������� 3 g �|< >+A���==hJm��im �-< j8,�A�� * (1G 0�G+0 *I, K b+* (16NK�B *�b+51< 0�BRM =T BFE =�bed@a�< C)B�P1BfP1BT& b 0�K -T= ( ,-mcAMmeh�c;=< J b`a (�H+(1G+C�E b�Q
�= ( � 0 → � 1 → · · · → � � )J QPO Q
0 * ( ��� ��e79,:1(� � →) ( � BFE =�bed ;0g ��,:16;=< > A�� b1= " * BTH+BRG *�b E b acb+*)< 0 *�b 0�" � �BFE =�bed�* BT&)K b+*Jd a�, 2 acbed �kmcAMm�� b1= "
� � BFE =�bed'< D�( = " �� hJg�< �`m�� J b P1( a H�E = M = 2w�Z V ibh VAb�d Q G)P1(�H+(�D d 0�K�S16?BRE =�bed@a�< C)B < P1B d &)( K�( = (+P <+*Jd
�= ( � 0 → � 1 → · · · ) :
_ ( ,/. >|m�� � B = S16'P�BbP�BR& b 0�K -R= (1G12�K�BT& d a (1UcG)P1(�H+(�D d 0�K�(1U J Q1O Q BRE =�bed � + 1
_ (c0�U+0 * "�K b K�B *)<+5 b 0�"�6 T = h�m �8m �,? � g�< * " K�BR& d a�, 0�G =)< & * "+0�" � :� � �!b1= 2
D d b S�H bc*�b � ∈ ��2 � ∈ � 2�( � ) =
� ⇐⇒ (∃( � 0 �;:;:;:�� � � ) ∈ � (T ))[ � 0 = � & � � =�] �J Q ] Q
S+P1(1G� (T ) = * ( 0�U = (�H+( * M =7* BT&)K b+*Jd a�4 = G�P�(�H�(�D d 0�K 4 =?* (1G T :JL9�8 Q
� P1B *�bed S *Jd�( � )↓ ⇐⇒ (∃( � 0 �;:;:;: � � � ) ∈ � (T ))[ � 0 = � ] :
^ < C)B bed *Jd ( a & b+*Jd a S 0�U#0 * "+K b K�B *�b+51< 0�BRM = G)P1(�H+(�D�E � B dBb`a & d 514 6cK d b K�BT& d a�,0�G =)< & * "�0�" � :
� � � P�(1G (1&�E � B *�bedlb P�S * " = J Q ] Q I P d P+H - ( = 2 b1= * ( T BFE =�bedbed *Jd ( a & b+*Jd a S�2 * S * B�D d b a�< C)B acb+*)< 0 *�b 0�" � G)P < &)A�B d@b`a & d 514 6 -R=�b 6 * BR&)K b+*Jd a S16�2< D�( = (16 ,c< P�B d &)(16%J h��@. �-7 � Q G�P�(�H�(�D d 0�K�S16: _+X 0 ( � ) = : _+X 0
T ( � ) = ( � → � 1 → � 2 �;:;:;: )JL9 > QP1(1G / B = K�P�(1&)BFE =�b BfP1B a�*�b C)BFE 2 acbed P1(1G%(1&�E � B *�bed K�B * " =pb1=�b1/ &)(1K ,
� 0 = � �
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K�B *�b+5�< 0�BTM = T 2�P�& - P�B d�=�b BRK�P+H�(1G * E 0�(1G#K�B * ( T K�B a�< P�( d ( * &)S+P�( B d 0 b D�MND , 60 * ( d A�BRE M =pb P�S * ( � acbed B C b D�MND , 6 *Jd K 4 = 0 * ( � ! E' Q 2�Y, ������Z�5 2'� �������������� ��4����� J�` V � g VAb�dfZ Y�[�XcY : e�Zcb�V Q K�B h8g��1? m g�< �j�A��8m&= * ( 0�U = (�H�( � acbed h8g��1? m�;=< ,E?BA , g@C�A��8m&= * ( 0�U = (�H�( � BRE =�bed " * G#A b E b
P1B =F*)<1/�bM = T (
Z b 0 g d � _ g d 0 g d ) = (� � → ��� � Z b 0 g d � _ g d 0 g d ) �S+P1(1G * ( T = (
� � → ��� )BRE =�bed 0�U#0 * "+K b K�B *�b+5�< 0�BTM = 2 acbed BbP d P�H - ( =�
J w Q Zcb 0 g d : � → ��2�BFE =�bed j,=iA ��-;k78j@7 g�< j�A��8m&= J Zcb 0 g d � g b : dbZ _Pb Q J�V Q _ g d 0 g d : � → � 2�BRE =�bed j,=iA ��-;k78j@7�g@C�A��8m&= J _ g d 0 g d � g b : dfZ\_�b Q m T (
Z b 0 g d � _ g d 0 g d ) G�P�(�H�(�D�E � B d�* "%K�BT& d a�, 0�G =)< & * "+0�" � : � � � b1= 2�D d b S�H b*�b � ∈ � 2 � ∈ � 2JL9#! Q �
( � ) = �
⇐⇒ (∃( � 0 �;:;:;:�� � � ) ∈ � (� � → ��� ))[ � 0 =
Z b 0 g d( � ) &
_ g d 0 g d( � � ) = � ] :
m K�"+A b1=I, acb H+BFE *�bed 1e<P;=< mJ>��`1c;=< >E. b1=�* ( T BFE =�bed6bed *Jd ( a & b+*Jd a S�2 acbed 0 � b G *I,?* " =P1BT&�E P * MN0�" G)P1(�H+(�D�E � B d * " K�BT& d a�, 0�G =)< & * "�0�" � : � � � P�(1G (1&�E � B *�bed b P�S* " = J�9+! Q � B =)4 K d b b1=�bed *Jd ( a & b+*Jd a�, K�"+A b1=I, K�P�(1&)BFE =�b K�" = G)P1(�H+(�D�E � B dcacb K�E bK�BR& d a�, 0�G =)< & * "�0�" ! E' 9 2'�������������� ���@8 ��4���� �@81 ; d b a�< C)B%P1&)S�D�& b K�K b � * "+6 � (
)2 (1&�E ]
� (1G+K�B -R=�b 0�U#0 * "+K b K�B *�b+51< 0�BRM = T (�
) = T (� � )
MN67B C , 6 JLY Q 3 d@acb+*�b 0 *)< 0�B d 6 * (1G T (
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�0:;:;: � � −1 :
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1
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−1* "+6 � d acb1= (+P�( d (1U =�*Jd 6 B C , 6�0�G = ]C ,ia BT6
• a�< C)B � � BFE =�bed 0�G =�b & * "+0 d b`a S 0�U#K 5 (�H+( JL0 *�b C)BR&)S , K�B *�b+5 H#" *I,�Q 2 , a H+B d ]0 * S16?S1&)(16�2 , * (.B d /Jd a S 0�U#K 5 (�H+( ?
2 acbed• a�< C)B � � BFE =�bed|b+* (1K d a�, 0 *�b C)BT& < J / "+H b1/I, � � ∈ Q
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J 0�Y�`f` Q � � :� → �
: � � ( � ∈ )
J VT] : Y�[\[ Q 2�D�B =Jd a�< � � � : �� � → �:� � ( �� ) �
J�VT] : Y#[ [ Q 2�D d bc* " = �0
� �: � � → �
: � + 1�
� ��: � � → �
: � −· 1�
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J : _+X 0 Q � �( � 1
�;:;:;:�� � � ) :� → � � � 1 · · · � � :
�JU� i Q � b1= ( � = 0) * S * B � b H+H d 4 6 � :
� → � � � ? � :�
J � i 8 Q � � � ? : 0� → � � :
�J � iH> Q � � � ? : � + 1
� → � � :�
• 3 d G�P�(�D�BTD�& b K�K -T= BR6�H - C)B d 6�BFE =�bed b G *)- 6 P�(1G b H+H <%� (1G = 0�B a�< C)B K�B *)<+5`b 0�" • �� = �
1�;:;:;: � � � BRE =�bed ��] <1/�b b+* (1K d a�4 = 0 *�b C)BT& 4 =
• e * " = g@C1�@;0g �@< >:. > �@.8j@7 J�VT] : Y#[ [ Q 2 � =� � BRE =�bed�/ (10�K -R= "'K�BT& d a�, 0�G =)< & * "+0�"
* "�6 K�B'Y i Z d ^(� � ) =
� � =�
• e * " = g=j��@;0g �|< >E. >��|.Jj|7 J�Z ] : Y�[\[ Q 21" � � BRE =�bed ��] K�BbH , 6?0�G =�b & * "+0 d b`a�, K�B *�b ]5 H#" *I, * (1G P1&)(�D�& < K�K b+* (16 � P1(1G%(1&�E � B *�bed2b P�S * " = B C�E 0�M 0�" � � ( �� ) =
� � • e * " ,:gk; �:1ej@7 j�<8Ak5=g=j|7 � J : _+X 0 Q " � BRE =�bed 0�G =�b & * "+0 d b`a S?0�U#K 5 (�H+(%J�0 *�b ]C)BR& <., K�B *�b+5 H+" *I,1Q K�B'Y i Z d ^ ( � ) =
� * �z2&'�6�'�5F( 2'@ B *�b+5�< 0�B d 6 * (1G%0�G#0 *I, K b+* (16 T (
� � )
� P1B = C)G+K�E � (1G#K�B S *Jd -R=�b 6 S1&)(16 � BRE =�bed�a H+B d 0 * S16 b1= / B = P1BT& d - A�B d b+* (1K d a�- 6K�B *�b+5 H+" *)- 6�2 acbed P b & b+* "�&)(1U#K�B�0�B a�< C)B acb+*)< 0 *�b 0�" * ( B d /Jd a S%0�U#K 5 (�H+( � � - A�B db`a & d 514 6 K d b BRK O�<1=Jd 0�" 3 d@acb+*�b 0 *)< 0�B d 6 * M = T (
� � )BRE =�bed ( d E /Jd BT67D d b S�H b
*�b P1&)(�D�& < K�K b+*�b * "+6 K�BT& d a�, 6 < H+D�B 5 & b 6 2 D d � b G * S acbed�*Jd 6 acb H+(1U+K�B acbed>-16;91ej@; ejeg�< � ;k7 � ; d b P b & <1/ B d D�K b 2�"%H - CI"
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0BFE =�bed@acbed ( d
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, b`a S1K�" acbed " b P�H+(1U+0 *�b+* ":
JU� Q 3 d�* BT&)K b+*Jd a�- 6 acb+*�b 0 *)< 0�B d 6 * (1G T (�
)BRE =�bed ( d H - C)B d 6 * "+67K�(1& O�, 6
: �
/ "+H b1/I,\b G *)- 6 P�(1G / B = - A�(1G = 0�U+K 5 (�H b 0 *�b b & d 0 * BR& < * (1G � � acbed K�S = ( K�E b0 *�b C)BT& < 0 *�b / B C d < � H�BT6 ( d K�"�A b1=)- 6 P�(1G C b 0 * "+&�E � ( =F*�bed 0 * ( T (
�)C b
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( : � ) = ��:J :)Q m 0�A - 0�" K�B *)<+5 b 0�"�6 * (1G T (
�)(1&�E � B *�bed 0�B BbP *)< P�BR& d P *)4 0�B d 6 b P�S * ( =
$7E =�b`acb @ B *�b+5�< 0�BTM = > 2 / "#H b1/I, "� → � ′ d 0�A�U+B d/b1= BFE =�bed B d /Jd a�, P1BT&�E P * MN0�"a�< P1( d b 6 D�& b K�K , 6 * (1G $7E =�b`acb $ b & b+* "+&)(1U+K�B S *Jd ( d K�B *�b+51< 0�B d 6 J VT] : Y�[\[ Q BFE =�bed
( d K�S = BT6'P�(1G # acb H�(1U =�& *�b./ (10�K -T=�bc* "+6 2 acbed -f* 0 d B C b & *)4 =F*�bed|b P�S * " = 2B =)4 ( d K�B *�b+5�< 0�B d 6 J Z ] : Y#[ [ Q BRE =�bed ( d K�S = BR6WP�(1G�B C b & *)4 =F*�bed8b P1S * ('0�G�D a B a & d K -T= (P1&)S�D�& b K�K b � _ ( 0�U+0 * "�K b T (
�)BFE =�bed P�&)( O@b1=)4 6 bed *Jd ( a & b+*Jd a S
; d b�a�< C)B ��] K�BTH , 0�G =�b & * "+0 d b`a�, K�B *�b+5 H+" *I, * (1G � 2�" �������������� �� ��4�����T (
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( �� ) = � ⇐⇒ ���( �� ) = �JL9 Q Q
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0BFE =�bed "c&I" *I,
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J 0�Y�`f` Q� �
: 3J � Q
�: 4
J � Q: 5
� :&4��=' U 2 3 G)P1(�H+(�D d 0�K�S16 * (1G 2 + 3 b P1S * ( P�&)S�D�& b K�K b�( F � � ) = b1= ( F = 0) * S * B �(b H#H d 4 6 � (
�(��
( F ) � � ))
* S * B?(.G�P�(�H�(�D d 0�K�S16 * (1G T (�
)
�: � → �
(�( � )) : → � �
( � ) : → � � � :
→ � �: � → �
: � + 1 → : � + 2
/ BFE A = B d S *Jd D d b a�< C)B � 2 �( � ) = � + 2
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Q T J 2gL #32&��6�4M'&2&'�< , ���=4; d b%-R=�b P d (cB =)/Jd b1O�- &)( = P b & <1/ B d D�K b 2�P b & b+* "�&)(1U#K�B S *Jd " P�&)S10�C)BT0�" (1&�E � B *�bedb1=�b1/ &)(1K d a�< 0 * " = �
0b P1S * (cP�&)S�D�& b K�K b K�B * "cK�( =�b1/Jd a�, B C�E 0�M 0�"
�( F � � ) = b1= ( F = 0) * S * B � b H#H d 4 6 � (
�(��
( F ) � � )) :e * ( e A , K b N / BRE A = (1G+K�B * ( = G)P1(�H+(�D d 0�K�S * "�6 *Jd K , 6 �
(2 � 3) = 5
! E' T 2�Y, ������Z�5 2 �T�������������� ���@8c E)�1� ���@8%���������#����(E#� 8��W�(4 ���(E�� 8 ����������������� � ��] K�BbH , 6 K�BT& d a�, 0�G =)< & * "+0�" � :
�� BRE =�bed ������������W� ��
J i V : g�i ` Z V Q 0 * " K�BR& d a�, < H+D�B 5 & b J , ] ������������W� �� Q b1= � =� D d b a�< P�( d (
P1&)S�D�& b K�K b � * "�6 acbed|a�< P�( d b 0�G =�b & * "+0 d b`a�, K�B *�b+5 H#" *I, � * (1G � 2 / "+H b1/I,b1=�( �� ) = � ⇐⇒ � � ` �
( �� ) = �K�B * ( 0�G#K 5 (�H d 0�K�S * "+6.J�9 Q Q I O S10�( = " 0�B d & < K�B * " = (+P1(�E b b P b & d C)K�(1U#K�B *Jd 6B C d 0 4 0�B d 6 B = S16 P�&)(�D�& < K�K b+* (16 / B = b H#H <%� B d�* ( = (1& d 0�K�S * M = K�BT& d a�4 = 0�G =�b &R]*I, 0�BRM = � 2 7 ;@=>�E1-?H1 ,Eg �@< >:. j�=8A���@;k7Jj|7 �
: �
� g@?YAM1e< � 1|AM1���� m ,B< >E.1|Ap>-1 <B,9A6A=m6A�1@A�==hJm��im ��?��9gk;91 <�1 h�A(>eh�me< m#1@A=1�� �`mk,l< > A h �-A ���`1k,|,:1#;k7 � 2/ "+H b1/I,�b1= � ( �� ) =
�0( �� ) K�B �
0* ( a U#& d ( 0�U+K 5 (�H�( * (1G � 2�� 0 a "+0�"��! E' !
p -f* (1G+K�B
R(
) = { � : ��� | " � BFE =�bed ] b1=�b1/ &)(1K d a�, } :
m * G#A b E b 0�A - 0�" � ( �� ) 0 * ( BFE =�bed ] �������������� �� b1= ".A b & b`a�* "+& d 0 *Jd a�,* "�6 0�G =)< & * "+0�" J > ] Q BFE =�bed ] b1=�b1/ &)(1K d a�, 2 acbed * ( * G+A b E ( 0�U = (�H�( � ⊆ BFE =�bed ] b1=�b1/ &)(1K d a S b1= " A b & b`a�* "+& d 0 *Jd a�, * (1G 0�G =)< & * "�0�" J�!#8 Q BRE =�bed ]b1=�b1/ &)(1K d a�, ; d b?* " =�< H+D�B 5 & b �
0acbed+*Jd 6 BfP1B a�*)< 0�B d 6 * "�6 P1(1G?K b 6�B =)/Jd b1O�- &)(1G =�d /Jd b E * BT& b 2
C b D�& <1O (1G+K�BR = R(N � 0 � 1 � � � �� ) = { � : N
�� N | " � BRE =�bed �
0] b1=�b1/ &)(1K d a�, } �acbed D d b a�< C)B'0�U = (�H�(%P+H�B d (1K�BTH 4 = K�BR& d a�4N= 0�G =�b & *I, 0�BRM = 0 * ( N
2JL9#9 Q R(Ψ) = { � : N
�� N | " � BFE =�bed ( � 0
� �1�;:;:;:�� � � )
] b1=�b1/ &)(1K d a�,D d b a�< P�( d BR6 �
1�;:;:;:�� � � ∈ Ψ} �
-b* 0 d P1(1G R = R(∅) ; d b K�BT& d a�- 6 0�G =�b & *I, 0�B d 6�2 0�A - 0�B d 6 acbed 0�U = (�H b 0 * (1G+6 O G#0 d a (1U#6 b & d C)K�(1U#6/ B =^b1=�b1O�- &)(1G#K�B &I" *)<?* " = < H#D�B 5 & b �
02 / "+H b1/I, K�B # b1=�b1/ &)(1K ,�& B =)= (1(1U+K�B # �
0]
b1=�b1/ &)(1K , &
s�� � �p�/yl~2�#$ }����! E' c>32�_ d K�BT& d a (1U+6 G)P1(�H+(�D d 0�K�(1U#6 - A�(1G = *�b 0�G#0 *I, K b+*�b K�B *�b+5�< 0�BTM = J Q1d Q 2acbed P�( d BR67K�BT& d a�- 670�G =�b & *I, 0�B d 6?G�P�(�H�(�D�E � (1G = �
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J � 2�j 2&'�< , ���=4;6�'&� � X ������0P������Z�5 Qod
��! E' ! 2 � BFE C * B?S *Jd " * G#A b E b K�BR& d a�, 0�G =)< & * "�0�" � ( �� ) BRE =�bed ] b1=�b1/ &)(1K d a�,b1= acbed K�S = ( = b1= G)P1(�H+(�D�E � B *�bed|b P1S a�< P1( d ( b1=�b1/ &)(1K d a ScP�&)S�D�& b K�K bc* "�6 ��! E' N 2 ; d bc*�b%* &�E b�b1=�b1/ &)(1K d a�< P�&)(�D�& < K�K b+*�b 0 * " = �
0
�( � ) =
�(
�( � ))J �
1Q
�( � ) =
�( � ( � ))J �
2Q
� ( � ) = � �
�( � ��� ) =
�1(
�( � ��� ) ��� ) :J �
3Q
�1(� ��� ) = � �
J > Q $'( d BR6?K�BR& d a�- 6 0�G =�b & *I, 0�B d 6 *�b d acb1= (+P�( d (1U = �JL! Q $'( d BR6 K�BR& d a�- 6 0�G =�b & *I, 0�B d 6 G�P�(�H�(�D�E � (1G = 2 acbed 0�B *Jd�/Jd b1O�- &)(1G = ( d G�P�(#]
H�(�D d 0�K�(�E * (1G+6 ���! E' Q ∗ 2 J > Q � BRE C * B S *Jd D d b a�< C)B P�&)S�D�& b K�K b � 0�B K d b m��-< >E. < H+D�B 5 & b
2 acbedca�< C)B ��] K�BbH , 0�G =�b & * "�0 d b`a�, K�B *�b+5 H#" *I, � P�(1G BTK O@b1= E � B *�bed 0 * ( � 2 / B =G�P < &)A�B d < D�( = (16?G�P�(�H�(�D d 0�K�S16%J�! E' N Q * "+67K�(1& O�, 6
�: � 1
�;:;:;:�� � � → � 1 → · · · → � � :J�� QJL! Q � BFE C * BcS *Jd/b1= " BRE =�bed K�BR& d a�, < H#D�B 5 & b 2N" � BFE =�bed ��] K�BTH , 6�2N0�G =�b &R]
* "�0 d b`a�, K�B *�b+5 H+" *I, a�< P�( d (1G P�&)(�D�& < K�K b+* (16 � 0 * " = acbed ( P1BfP1BT& b 0�K -T= (16G�P�(�H�(�D d 0�K�S16 J�� Q BRE =�bed�< D�( = (16�2 * S * B." * BTH+BRG *�b E b\acb+*)< 0 *�b 0 , * (1G BRE =�bed * "+6K�(1& O�, 6
� � � : � 1�;:;:;: ��� � � �
S+P1(1G " � � BFE =�bed K d b�b P1S *Jd 6 / (10�K -T= BR6�2 K�BT& d a�- 6 0�G =�b & *I, 0�B d 6 * "+6 2 acbed� � ( � 1 �;:;:;:�������� )↑
��! E' 9 2 3?&�E 0 * B -T=�b b1=�b1/ &)(1K d a S.P�&)S�D�& b K�K b P�(1G =�b G�P�(�H�(�D�E � B d�* ".0�G =)< &R]* "�0�" * (1G @ : r�V i X.Y b3b > @ T
��! E' T 2 3?&�E 0 * B -T=�b b1=�b1/ &)(1K d a S?P1&)S�D�& b K�K b � 0 * " = BfP -0a�*�b 0�" (�
0� � � � ��� )* "�6 �
0P�(1G =�b G�P�(�H�(�D�E � B d�* "7K�BT& d a�, 0�G =)< & * "+0�" � P�(1G?(1&�E � B *�bed b P1S *Jd 6 � � � ���
K�B O MNH d b 0�K -T= " b1=�b1/ &)(1K , 2 � >AE' c>AO ∗ ��! E' d 2 � BFE C * B?S *Jd " #fB C�E 0�M 0�" &
�( � ) =
{0 � b1= � ( � )↑��( � ) + 1 � b H+H d 4 6 :
/ B = K�P�(1&)BFE =�b.* G)P1(+P�( d "�C)BRE�0 * " = � 2 b1O (1U P1& 4�*�b B CI"+D , 0�B * B *Jd P1& - P1B d�=�b�b P1(#]/ B d A * BRE
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Q1O J 2gL #32&��6�4M'&2&'�< , ���=4��! E' O 2�B P�( / BFE C * B , /)4 0 * B b1=F*Jd P b & b1/ BFE D�K b+*�b D d b *Jd 6 B C , 6 / U#( P1&)( *)< 0�B d 6 JLY Q ; d b a�< C)B?K�BT& d a�,.< H#D�B 5 & b acbed@a�< C)B �
0 ∈ 2�"c0 *�b C)BT& , 2�K�( = (1K�BbH , 60�G =)< & * "�0�" � ( � ) = �
0BRE =�bed ] b1=�b1/ &)(1K d a�,
JU� Q ; d b a�< C)B b & d C)K�S �0 ∈ N
2#" "'0 *�b C)BT& , 2+K�( = (1K�BTH , 6 0�G =)< & * "�0�" � ( � ) = �00 * (1G#6 O G#0 d a (1U#67BRE =�bed|b1=�b1/ &)(1K d a�,
! E' d-2�Y, ������Z�5 2 JLY Q _ ( * G#A b E ( 0�U = (�H�( � ⊆ BRE =�bed >��`g�< j|;BA �c<H1 ;91�8m j8, �8A=1 * "�6?K�BR& d a�, 6 < H#D�B 5 & b 6 , ] ��� E#��� ��� b1= 0 � 1 ∈ � 2 acbed[ � 1
�;:;:;: � � � � ∈ � acbed � � ( � 1�;:;:;: � � � � ) = � ] � ⇒ � ∈ � � ( F = 1 �;:;:;: ��� ) :JU� Q 3'&�E � (1G+K�B b1=�b1/ &)(1K d a�<
� (0)= � ∪ {0 � 1} �
� ( � +1)= � ( � ) ∪ { � � ( � 1
�;:;:;:�� � ��� ) | � 1�;:;:;: � � ��� ∈ � ( � ) � F = 1 �;:;:;:���� } :
m -T= M 0�" � =⋃∞� =0 �
( � ) BFE =�bed�* (c0�U = (�H+( P�(1G ���������(EF����� b P1S * ( � 0 * " = ��! E' ] 2 � BFE C * B�S *Jd D d b a�< C)B�K�BR& d a�, < H#D�B 5 & b acbed � ⊆ 2 * ( � BFE =�bed�* (
BTH < A d 0 * ( ] a H�B d 0 * S.G�P�(10�U = (�H�( * (1G P�(1G P1BT& d - A�B d�* ( � 2 / "+H b1/I, � ⊆ � 2* ( � BFE =�bed ] a H�B d 0 * S�2 acbed D d b a�< C)B � ⊆ 2 b1= � ⊆ � acbedN* ( � BFE =�bed ] a H�B d 0 * S�2 * S * B � ⊆ �
��! E' c> 8 ∗ 2�� BRE C * B%S *Jdlb1= � ⊆ acbed " K�BR& d a�, 0�G =)< & * "�0�" � : �
� BFE =�bed ] b1=�b1/ &)(1K d a�, 2 * S * B * ('0�U = (�H�( � P1(1G P b & < D�B *�bedib P1S * ( � BFE =�bedJa H�B d 0 * SD d bc* " = � 2 / "#H b1/I,
[ � 1�;:;:;: � � � ∈ � � � ( � 1
�;:;:;: � � � ) = � ] � ⇒ � ∈ � :
s�� ����'ByO�@w �2{et@{cz�y/zB} $ ��� W}��B{,$i� ���@�#$ }��e *�b BfP1S1K�B =�b./ U+(�2�C)BRK�BbH d b`a�< C)BTM & , K b+*�b./ E = (1G+K�B -R=�b # / (1K d a S & A b & b`a�* "�& d ]
0�K�S * M = K�BT& d a�4 = 0�G =�b & *I, 0�BTM = P�(1G%G�P�(�H�(�D�E � ( =F*�bed|b P�S *�b 0�G+0 *I, K b+*�b K�B *�b ]51< 0�BRM = T (
� � )2�P�(1G C b K b 6 /)4 0�_+G ='*Jd 6 5`b 0 d a�- 6 d /Jd S * " * BR6 * M = ] b1=�b1/ &)(#]
K d a�4 = K�BR& d a�4N= 0�G =�b & *I, 0�BTM = ge * ( *)- H+(16 * (1G.B /�b1O E (1G.C b 0�G#H#H - C)(1G+K�B b G *)- 6*Jd 6 d /Jd S * " * BR6'D d b%*�b 0�A�B *Jd a�< 0�U = (�H b R acbed R(Ψ)
0 * (1G#6 O G#0 d a (1U#6 b & d C)K�(1U#6 ^ H+B d / E�D d bc*Jd 6 b P1( / BRE C)B d 67BFE =�bed�* ( B C , 6�2 b P+H�S! G c>&2 � 4����=' 2 <H1 > 65 g^,:g �|< >+A = h�m �8m �c<HjJ,9A
�0 :�
0 → �1 :�
1 → · · · → � � :� �j|;=m j�<�j|;k79,:1 T (
� � )>c1e< ���@C=g�< � � ∗ Z � ∗ ;��k;=me< g�� h�m3= 7
� ∗ �0 :�
0� ∗
A=1�g@?PA=1 <:>c1c; j|;91 j|7 Z m� ∗ �
0 :�
0� ∗ → � ∗ �
1 :�
1� ∗ � → · · · → � ∗ � � :
� � � ∗
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J � 2 � 0o6&� , Z /�[&/ ';6�'&�8#9��%&:&��� / #�5n�3l���#3��5 Q ]
g@?YAM1e<Eg�h�? j|7 �^,:g �|< >+A�� = h�m �8m �c<HjJ,9A�� ;=m&= T (� � )
�� hJgk;91e<*A`;=</1@A m
�0 :�
0 → �1 :�
1 → · · ·g@?YAM1e<^1 hJmJ> �,?YA �BA==hJm��im �-< j8,�A�� >-1 < 7 � ∗ �0 :
�0� ∗ g@?PA=1 < >-16; ej@;91ej@7 Z ;BA`;0g1eh�m8>���?PA��BA = h�m �8m �c<HjJ,9A��pg@?PA=1 <:>c1e<Bm
� ∗ �0 :
�0� ∗ → � ∗ �
1 :�
1� ∗ → · · ·
j-X Z�<\#3��} [*2'@ BNBbP b D�MND , 0 * ( � ≥ 02 * ( � , K�K b BRE =�bed�* B * & d K�K -T= (70 * " 5�< 0�"
� = 02 b P1S * " = G�P�S1C)BR0�" e * (cBbP b D�MND d a S 5+, K b 2 / BRA�S1K b 0 * B'S *Jd " b`a (�H�(1G#C�E b� ∗ �
0 :�
0� ∗ → � ∗ �
1 :�
1� ∗ → · · · → � ∗ � � :
� � � ∗
BFE =�bed K�BR& d a S16?G)P1(�H+(�D d 0�K�S16�2�B C)B *)<%� (1G#K�B C)BRA�MN& d 0 *)< *Jd 6?BfP *)< P�BR& d P *)4 0�B d 6 P1(1G/Jd acbed (�H�(�D�(1U =7* "cK�B *)<+5`b 0�"
� � :� � → � � +1 :
� � +1acbed BFE =�bed P1&)( O-b1=)- 6 S *Jd " E /Jd b D�& b K�K ,�b P1S * ( = $7E =�b`acb > BbP1E 0�"+6 /Jd acbed (�H+(�D�BFE* "cK�B *)<+5`b 0�"
� ∗ � � :� � � ∗ → � ∗ � � +1 :
� � +1� ∗
_ ( / BRU * BR&)( 0�G+K�P - & b 0�K b 0�G =)< D�B *�bed K�B?B O@b &)K�(�D , * (1G P�& 4W* (1Gc0 * (1G+6?K�BR& d ]a (1U+6?G�P�(�H�(�D d 0�K�(1U+6�
0 :�
0 → �1 :�
1 → · · · → � � :� � ( � ∈ N) a
! G ! 2�� # � ,3[ �=' J � ������������� �1� �� 5 ��� ������������� ���98 �(
)�12 <H1 ;@=>�E1-? m1|AM1���� m ,B< >+A h �-A ���`1k,|,:1 � j@;k76A
= ( � 0 � 1 � � 1 �;:;:;:�� � � ),Eg�j�=8A=1��@;k7Jj-< 1 �> ��� ,:gk;91 ��|7�;���� �
0�;:;:;: � �
�Z 5��k;=m3= ,Eg
= ( � �
0�;:;:;: � �
� ) = ( � 0 � 1 � � 1 �;:;:;:�� ��� � �0�;:;:;:�� �
� ):
� hJgk;91e<*A`;=< �c<H1 > 65 g >��`g�< j|;BA�A�� m � j|;=m��`g@CJ< �&A �c< m � 1 �;:;:;: � ��� Z �0�;:;:;:�� �
��
JLY Q n A � ��� � ( � )↑Z ;BA`;0g m6= h�m �8m �c<HjJ,9A�� : _+X 0T ( � : )
;=m3= T (� � )
,:g1��;�/< >E. >-16; ej@;91ej@7 � :g@?PA=1 < h8g�< � m�� . ��m6A=m�� J < & b acbed|b P�( a H1E = B d Q Z >-1 <
JU� Q 1|A � ��� � ( � ) = � Z ;BA ;0g m�==hJm��im �-< j8,�A�� : _+X 0T ( � : )
;=m&= T (� � )
,:g1��;�/< >E. >-16; ej@;91ej@7 � :j,=��J> �,?YA9g�<@,Eg ;0g ��,:16;=< >:.p>c1c; j|;91 j|7
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J 0�Y�`f` Q: �
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� � � 1:;:;: � � � :
p?BTM &)(1U#K�B * &)B d 6'P1BT& d P *)4 0�B d 6 JL!#Y Q ; d b a�< P�( d ( ��2 � ��� � ( � � )↑2�(+P1S * B acbed � ��� � ( � )↑ B = * ( �7BRE =�bed , ���-< j|;=mJ ≤ � � Q K�B b G *I,.* " = d /Jd S * " *�b 2 * S * B.J�K�B BfP�E a H#"�0�" * "+6 BbP b D�MND d a�, 6 G�P�S1C)BR0�"�6 Q
( G)P1(�H+(�D d 0�K�S16 : _#X 0 ( � : ) b &)A�E � B d K�B *�bc5�, K b+*�b� � ( � 1
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� � � 1:;:;: � � � :
J�BbP b D G�P Q
� � � 1:;:;: � ��� −1 : � ��� J�BbP b D G�P Q
� � � 1
:;:;: � � : � � +1 · · · � � �B P�S * " = BbP b D�MND d a�, G�P�S1C)BR0�"%P < H d 2�(.G�P�(�H�(�D d 0�K�S16
: _+X 0 ( � � : ) = � � : → �1 :�
1 → · · ·BFE =�bed�< P�B d &)(16�2�B O S10�( = � ��� � ( � � )↑2 acbed-b P�S * ( � , K�K b ! G c> 2 < P1B d &)(16'BFE =�bedcacbed( G)P1(�H+(�D d 0�K�S16
� � � 1:;:;: � � −1 � � : → � � � 1
:;:;: � � −1�
1 :�
1 → · · ·P1(1G%0�G = BfP < D�B *�bed S *Jd ( : _#X 0 ( � : )
BFE =�bed < P�B d &)(16 JL!!� Q � P < &)A�(1G = 0 * ( d A�BFE b �
1�;:;:;: � ����� 0 * ( *)-b* ( d b P�(1G � ��� � ( � � ) = � �
D d b � = 1 �;:;:;: � � � 2 b H+H < � � ( � 1�;:;:;: � ����� )↑ Ye�� b G *I, * " = P�BR&�E P * M 0�"�2 b P�S * " =
BbP b D�MND d a�, G�P�S1C)BR0�" acbed K�B BbP1E a H+"+0�" P < H d+* (1G � , K�K b+* (16 ! G c> ( G)P1(�H+(�D d 0�K�S16: _+X 0 ( � : ) b &)A�E � B d K�B *�b%5�, K b+*�b
� � ( � 1�;:;:;:�� � ��� ) :
J : _+X 0 Q� � � 1
:;:;: � � � :J�BbP b D G�P Q
� � � 1
:;:;: � ��� −1 : � ��� J�BbP b D G�P Q
� � : � 1�
2 · · · � � �acbed B /)4 (.G�P�(�H�(�D d 0�K�S16?0 *�b K b+*)< J < D�( =�b�Q B O S10�( = � � ( � 1
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� � ( � 1�;:;:;: � � ��� )) = � 2 -b* 0 d P�(1G�G)P < &)A�(1G = 0 * ( d A�BFE b �
1�;:;:;:�� � ���
0 * ( K�B � ��� � ( � � ) = � � D d b � = 1 �;:;:;:�� � � acbed � � ( � 1�;:;:;: � � � � ) = � 8B P�S
* " = BbP b D�MND d a�, G�P�S1C)BR0�" acbed K�B BfP�E a H#"�0�" * (1G � , K�K b+* (16�! G c> 2�( G)P1(�H+(�D d 0�K�S16: _+X 0 ( � : ) *)4 & b BRE =�bed ( B C , 6
� � ( � 1�;:;:;:�� � ��� ) :
J : _#X 0 Q� � � 1
:;:;: � � � :JLBfP b D G)P Q
� � � 1
:;:;: � ��� −1 : ����� JLBfP b D G)P Q
� � : � 1�
2 · · · � � �:� � ( � 1
�;:;:;:�� � ��� )
P1(1G%BRE =�bed�* ( � " * (1U#K�B = ( J N Q B = � ≡ � � ( � 1
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� � ( � 1�;:;:;:�� � � ) :
J : _+X 0 Q� � � 1
:;:;: � � :
p?BTM &)(1U#K�B * &)B d 6'P1BT& d P *)4 0�B d 6 S+P1MN6 acbed 0 * " = JL! Q J N Y Q ; d b a�< P�( d ( ��2 � ��� � ( � � )↑2�(+P1S * B acbed�� ��� � ( � )↑
J N � Q � P < &)A�(1G = 0 * ( d A�BFE b �1�;:;:;: � � � 0 * ( *)-f* ( d b P�(1G � ��� � ( � � ) = � �
D d b � = 1 �;:;:;:�� �W2 b H#H < � � ( � 1�;:;:;:�� � � )↑
J N :)Q � ��� � (� � ( � 1
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BRE =�bed ( B C , 6 � � ( � 1
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� � � 1:;:;: � � :
J�BbP b D G)P Q
� � � 1:;:;: � � −1 : � � J�BbP b D G)P Q
� � : � 1
�2 · · · ��� (
(1& d 0�K�S16 * "�6 � � ) :
� � ( � 1�;:;:;: � � � )
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BbP�B d /I, (.G�P�(�H�(�D d 0�K�S16 * "+6 � � ( �� ) b &)A�E � B d K�B *�bc5+, K b+*�b� � : �� (
Z ] : Y#[ [ )� � ( �� :≡ �� }) :_ b J�Y QWacbed JU� Q *)4 & b 0�G = BfP < D�( =F*�bed S *Jd
� � ( �� :≡ �� }) : → � 1 → · · · → : � ⇐⇒ � ��� � (� � ( �� :≡ �� })) = ���
-b* 0 d P1(1G� � ( �� ) = � ⇐⇒ � ��� � (
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* (cP�&)S�D�& b K�K b�( � 1
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�( �� ) = � � � �
( �� ) = 0 � �( �� ) = 1 :
; d b * " /Jd b`a H <1/ M 0�"�2�" G)P1S1C)BT0�" K b 6 / E = B d P1&)(�D�& < K�K b+*�b ��
2 � ��acbed � �* "�6 acbed 0�G#D a B a & d K -R= BT6 0�G =�b & * "+0 d b`a�- 6 K�B *�b+5 H#" *)- 6 2 � acbed � 0 � b G *)<.*�bP1&)(�D�& < K�K b+*�b 2 acbed P1& - P1B d�=�b acb+*�b 0 a BTG < 0�(1G+K�B -T=�b =)- (cP�&)S�D�& b K�K b � P1(1G =�b(1&�E � B d@a�< P�( d b # O & - 0 a-d b%& K�B *�b+5 H#" *I, � 2 -b* 0 d P�(1G
���( �� ) = b1= ( � � ( �� ) = 0) * S * B � � � ( �� ) b H#H d 4 6 � � � ( �� ) �
S+P1(1G.( d�/ BFE a�* BT6 O-b1= BR& 4N= (1G = *�b P�&)(�D�& < K�K b+*�b P1(1G G)P1(�H+(�D�E � (1G = *Jd 6 ��
� ��acbed � �
�
\^ <1= ( =F*�b 6'K�BR& d a�- 6 b H O@b+5 " *Jd a�- 6 b H+H b D - 6'0 *Jd 6'0�G =�b & * "�0 d b`a�- 6'K�B *�b ]5 H#" *)- 6 * M = �
�� � � � � � 2�K�P�(1&)(1U+K�B =�b G)P1(1C - 0�(1G#K�B7S *Jd:b G *)<.*�b P�&)(�D�& < K�K b+*�b
/ B = - A�(1G = a ( d =)- 670�G =�b & * "+0 d b`a�- 67K�B *�b+5 H#" *)- 6�2 acbed C -b* (1G#K�B�
=�
� +� � +
� � + { �( �� ) = b1= ( ( �� ) = 0) * S * B � ( �� ) b H+H d 4 6 � ( �� )} �
S+P1(1G K�B � + � B =)= (1(1U+K�B * " 0�G�H+H+(�D , S�H+M = * M = (1& d 0�K 4N= 0 *�b / (10�K -T=�b P�&)(#]D�& < K�K b+*�b _ ( � BRE =�bed P1&)S�D�& b K�K b 2NB O S10�( =�a�< C)B%0�G =�b & * "�0 d b`a�, K�B *�b+5 H+" *I,(1&�E � B *�bed|b`a & d 5�4 67K d b.O (1& < $ b & b+* "+&)(1U+K�B'S *Jd
� ( �� ) = �� ( �� ) �b P�H < BfP1B d /I, a�< C)B?G)P1(�H+(�D d 0�K�S16
: _+X 0 �� ( : �� ) = : �� → �
1 :�
1 → · · ·* (1G �
�
BRE =�bedWacbed G�P�(�H�(�D d 0�K�S16 * (1G � 2 acbed BbP�(1K -R= MN6 J b P1S * " = bed *Jd ( a & b+* E b* M = P1&)(�D�& b K�K <+* M =�Q BFE =�bed ( K�S = (16%G�P�(�H�(�D d 0�K�S16%0 * ( � P�(1G b &)A�E � B d K�B * " =acb+*)< 0 *�b 0�" : �� 2 / "+H b1/I,
: _+X 0 �� ( : �� ) = : _#X 0 � ( : �� ) D
< & b � = ��
2 acbed * ( E /Jd (�2 51-f5 bed b 2 d 0�A�U+B d acbed D d b *�b 0�U#K 5 (�H b � acbed � _ BTH d a�< 2 b P1S * ( p?B 4 &I"+K b ! G !�2�" ��� d acb1= (+P�( d BFE * " = B C�E 0�MN0�" P1(1G * " = (1&�E � B d0 * ( � 2 acbed -f* 0 d
�( �� ) = b1= ( � ( �� ) = 0) * S * B � � ( �� ) b H#H d 4 6 � � ( �� )
= b1= ( � � ( �� ) = 0) * S * B � � � ( �� ) b H+H d 4 6 � � � ( �� ) :m b P1S / B d CI"cD d bc* "c0�U = C)BT0�"cBFE =�bed P b &)S1K�( d b 2 � 0 a "�0�" ��! G c>� a! G Q 2g* Z , �����=' 2 < 1#> 65 g ,:g �|< >E. �� ��g ��`1
= ( � 0 � 1 � � 1 �;:;:;:�� � � )>-1 < ;@=>�E1-? g�� �:
��� Z � : � �� Z
[� ∈ R(
) &
� ∈ R( � � )] � ⇒ � ∈ R(
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�d acb1= (+P�( d (1U = * ( 0�U#0 * "+K b � b P1S * (7p'B 4 &I"�K b
! G !12 < & b�b & a BRE =�b / BRE C)(1G+K�B.S *Jd 1|A me< ;@= �:1 ? g�� � ′0�;:;:;:�� � ′
�< >-1@A=m hJm < m&<iA ;=< �g@C8< j ?/jeg�< � ;=m&= � Z ;BA`;0g
� � ( �� ) = ��� ⇒ � ′� ( �� ) = � ( F = 0 �;:;:;:�� � ) :� P1(1C -f* (1G+K�B S *Jd ( d � ′
0�;:;:;:�� � ′
�d acb1= (+P1( d (1U = *Jd 6 B C d 0 4 0�B d 6 * (1G � 2 acbed C)BRM�]
&)(1U+K�B *Jd 6 / (1K - 6
= ( � �
0�;:;:;: � �
� )� ′ = (
� � ′0�;:;:;:�� � ′
� ):
B P�S * ( p'B 4 &I"�K b ! G !12|C - &)(1G#K�B S *Jd D d b�a�< C)B a H�B d 0 * S S1&)( � 0 * ( H�B C d H�S�D d ((�1�;:;:;: � ��� � �
0�;:;:;:�� �
� )2 b1= � ��� � ( � ) = � 2 * S * B ( G�P�(�H�(�D d 0�K�S16 : _+X 0 ( � : )* "�6 T (
�)BRE =�bed�* BR&)K b+*Jd a S16 K�B * BbH�BTG *�b E b acb+*)< 0 *�b 0�" * " = : � p b / BRE C)(1G+K�BK�B?BfP b D�MWD , 0 * ( � 2�S *Jd D d b a�< C)B � acbed D d b a�< C)B � 2b1= � : → �
1 :�
1 → · · · � � −1 :� � −1 → : ����* S * B � ��� � ′( � ) = ��:JL9�T Q
e * " = B d /Jd a�, P1BT&�E P * MN0�" � ≡ � � ( � 1�;:;:;: � � � ) 2 b G * S b P1( / E / B d
� � ( �� ) = ��� ⇒ � �����′(
� � ( �� )) = ��� ⇒ � ′� ( �� ) = � �P1(1G%BRE =�bed�* ( � " * (1U#K�B = ( ; d b%* " = b P1S / B d CI" * "�6 JL9�T Q B C)B *)<%� (1G#K�B * "cK�(1& O�, * (1G � 2 acbed�* (.BbP d A�BFE &I"+K b
BFE =�bed * B * & d K�K -T= ( J�S+P1MN6 0 * " = b P�S / B d CI" * (1G ! G ! Q 0�B S�H�BT6 *Jd 6 P1BT& d P *)4 0�B d 6B a�* S16 b P�S * " =
� � ( � 1�;:;:;: � � � ) �D d bc* " = (+P1(�E b ( G)P1(�H+(�D d 0�K�S16 - A�B d�* ".K�(1& O�,
� � ( � 1�;:;:;:�� � � ) :
� � � 1 · · · � � : � � � 1 · · · � � −1 : ���
� � : � 1 · · · � �� � { �� :≡ �� } : : �
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J � 2 � 0o6&� , Z /�[&/ ';6�'&�8#9��%&:&��� / #�5n�3l���#3��5 9+9m BbP b D�MND d a�, G�P�S1C)BR0�" *)4 & b 0�G = BbP < D�B *�bed S *Jd
� ��� �′( � 1) = �
1�;:;:;: ��� �����
′( � � ) = � � ��� ��� �′(� � { �� :≡ �� }) = �
BbP�B d /I, ( d G�P�(�H�(�D d 0�K�(�E�P�(1G b1=F*Jd 0 * ( d A�(1U = 0 � b G *)- 6 *Jd 6 *Jd K - 6 - A�(1G = K d a &)S * BR&)(K ,ia (16 �%& b
� �����′(
� � ( � 1�;:;:;: � � � )) =
� ′� ( � ����� ′( � 1)�;:;:;: ��� �����
′( � � ))=
� ′� ( � 1�;:;:;:�� � � )
= � �����′(� � { �� :≡ �� }) = � �
S+P1(1G " * BbH�BTG *�b E b B C�E 0�M 0�"cB a�O & <%� B d|b`a & d 514 6 * " = G�P�S1C)BR0�" ′ |= � � ( �� ) =
� � : a
s�� � �p�/yl~2�#$ }����! G c>32 � BFE C * B S *Jd "'0�U = C)BR0�" J > 8 Q ] b1=�b1/ &)(1K d a�4N= K�BT& d a�4 = 0�G =�b & *I, 0�BTM =
BFE =�bed ] b1=�b1/ &)(1K d a�, ��! G ! 2 ; d b a�< C)B K�BT& d a�, < H+D�B 5 & b = ( � 0 � 1 � � 1 �;:;:;:�� � � ) acbed * G#A b E BR6�
: ��� 2 �
: � �� 2[� ∈ R(
) &
� ∈ R( � � )] � ⇒ � ∈ R(
) �S+P1(1G
( � � ) = ( � 0 � 1 � � 1 �;:;:;:�� � � � � ) BFE =�bed "cBbP -Ma�*�b 0�" * "+6 K�B * " = �
��! G N 2 � BFE C * BcS *Jd/b1= ( d � ( �� ) acbed � ( �� ) BRE =�bedWb1=�b1/ &)(1K d a�- 6.0�A - 0�B d 6.0 * "K�BR& d a�,c< H+D�B 5 & b 2 * S * B ] b1=�b1/ &)(1K d a�- 67BFE =�bed@acbed ( d 0�A - 0�B d 6
� 1( �� ) ⇐⇒ ¬ � ( �� )� 2( �� ) ⇐⇒ � ( �� ) & � ( �� ) �� 3( �� ) ⇐⇒ � ( �� ) ∨ � ( �� ) :
��! G Q,2 � BFE C * B?S *Jd " -R= MN0�" � ∪ � 2�" * (1K , � ∩ � 2 acbed " /Jd b1O (1& <� \ � = { � ∈ � | ���∈ � }
/ U+( ] b1=�b1/ &)(1K d a�4N= 0�G = S�H�M = BRE =�bed|b1=�b1/ &)(1K d a�< 0�U = (�H b � BFE C * B BfP�E 0�"�6 S *Jd�*�b K�( = (10�U = (�H b {0} acbed {1} BFE =�bed ] b1=�b1/ &)(1K d a�< 0�B a�< C)B
K�BR& d a�,c< H+D�B 5 & b ! G T 2�Y, ������Z�5 2 ; d b a�< C)B � : � 2�" EI������������� � �
: N × � * "�6 � (1&�E � B *�bed K�B * " =pb1=�b1/ &)(1K ,� 0( � ) = � � � � +1( � ) =
�(� �
( � )) :��! G 9 ∗ 2�� BRE C * B.S *Jd/b1= ( d � � � : � BRE =�bed ] b1=�b1/ &)(1K d a�- 6.K�BR& d a�- 6
0�G =�b & *I, 0�B d 6�2 * S * B ] b1=�b1/ &)(1K d a�, BRE =�bed@acbed "�( � ) =
� �( � ) S+P�(1G � = ( � � ≥ 1)[
� �( � ) = 1] :
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9PT J 2gL #32&��6�4M'&2&'�< , ���=4s�� ����cz:�cwcx � }Yy/�i� �#$`w@}Yy/�i�\�O� �cz:we{e~:�*$e}����l{cx �@���^�@� }Yy/x �|�� P1B = C)G+K�E � (1G#K�B * ( = (1& d 0�K�S J�9+9 Q
R(Ψ) = { � : N�� N | " � BRE =�bed ( � 0
� �1�;:;:;: � ��� )
] b1=�b1/ &)(1K d a�,D d b a�< P�( d BR6 �
1�;:;:;:�� � � ∈ Ψ}
* M = K�BT& d a�4 = 0�G =�b & *I, 0�BTM = P�(1G BRE =�bed b1=�b1/ &)(1K d a�- 6 0 * ( * G#A b E ( 0�U = (�H+( ΨP�H+B d (1K�BbH 4N= 2�K�BR& d a�4N= 0�G =�b & *I, 0�BTM = 0 * (1G#6 O G#0 d a (1U#6�2 acbed 0�G�H+H - D�(1G#K�B'0�B -R=�bC)B 4 &I"�K b *Jd 6 5`b 0 d a�- 6 d /Jd S * " * BR6 * (1G R(Ψ)
P1(1G 0�G =)< D�( =F*�bed^b P�S *�b D�B =Jd a�<C)BRMN& , K b+*�b 0 *�b%/ U#( P�&)(�"+D�(1U#K�B =�b B /)<1O@d b B G *)- 6 d 0�A�U+(1G = 2 51-f5 bed b 2 acbed D d b * (0�U = (�H�(
R = R(∅) = R(�
0)* M = J b P1S�H+G *�b�Q b1=�b1/ &)(1K d a�4 = 2�K�BT& d a�4 = 0�G =�b & *I, 0�BTM = 0 * ( N
!�S c>32 � # � ,9[ �=' 2 +-m j,<iAMm �8m R(Ψ)
g@?YAM1e< h ���@;=m �Jg8A0? ��> �ig�<Hj@;BA >c1e<l> �ig�< �j|;BA �-< 1 g�� 1B�B<Hj@;=meh�m ?P78j@7 Z �k;=j-</h�m3=R � (Ψ) ⊆ R(Ψ) �>-1 < Z < �J<H1-?Y;0g �`1 Z > 65 g�g�� 1B�B<Hj@;=< > 1|AM1���� m ,B< >E.�,Eg �@< >:. j�=8A���@;k7Jj|7 g@?YAM1e< 1@A=1 �
� �`mk,l< >:. �j-X Z�<\#3��} [*2 _ ( 0�U = (�H�( R(Ψ)
P�BR& d - A�B dW*Jd 6 P�&)( 5 (�H - 6�2 *Jd 6%0 *�b C)BT& - 6 0 acbed1 acbed *Jd 6 / (10�K -T= BR6 � ( � ) acbed �� ( � ) BfP1B d /I,�a�< C)B R(
�0� 0 � 1 � � 1 �;:;:;: � � � )
K�B�1�;:;:;:�� �
� ∈ Ψ - A�B d|b G *)- 6 *Jd 6 d /Jd S * " * BT6 b P�S * (c$?S1& d 0�K b ! G N1 _ " = P�&)M * (�D�B =I, a H+B d 0 * S * " *�b * (1G R(Ψ) * " = b1O�,+= (1G+K�B D d bc< 0 a "+0�"�2 ��! S c>� B & a BRE =�b / BRE C)(1G+K�B?S *Jd�* ( 0�U = (�H+( R(Ψ) * M =pb1=�b1/ &)(1K d a�4 = 0 * ( Ψ
K�BR& d a�4N=0�G =�b & *I, 0�BRM = BFE =�bed@a H+B d 0 * S.D d b BbH b A d 0 * (+P1(�E "+0�"�2 / "#H b1/I, b1= � ∈ R(Ψ) acbed
�( �� �� ) = ( �+F ≥ � )[
�( F � �� ) = 0] �
* S * B � ∈ R(Ψ) B P�S * " = $'&)S *�b 0�" >HG O 2 " � BRE =�bed " BbH < A d 0 * " H�U#0�" * "+6b1=�b1/ &)(1K d a�, 6?B C�E 0�M 0�"�6
�( �� �� ) = b1= (
�( �� �� ) = 0) * S * B � b H#H d 4 6 � ( � + 1 � �� ) �
P1(1G J b P�S K�S = " * "+6 Q BFE =�bed P1&)S�D�& b K�K b 2 < & b J b P1S * (Kp'B 4 &I"�K b ! G 9 Q 2'" �G�P�(�H�(�D�E � B *�bed`b P � b G * S * ( P1&)S�D�& b K�K b�acbed � ∈ R(Ψ∪{ � }) b H+H < b1= � ∈ R(Ψ)
2* S * B R(Ψ ∪ { � }) = R(Ψ) b P1S * ( ! G Q� a
s�� � �p�/yl~2�#$ }����! S c>&2�� BRE C * B'S *Jd-b1= ( d � ( �� ) acbed � ( ������ �� ) BRE =�bed@b1=�b1/ &)(1K d a�- 6?0 * ( Ψ acbed" �( �� �� ) (1&�E � B *�bedWb P � b G *)- 6 K�B * " = P1&)M * (�D�B =I, b1=�b1/ &)(1K , J >P> Q 2 * S * B acbed "
�( � � �� ) BFE =�bed|b1=�b1/ &)(1K d a�, 0 * ( Ψ
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J�� 2�j 2&'�< , ���=��6 { 57��# , ��6 { 57����2&' ,9/ 4���#3��57� / �&��57$�������6��&l�5 9 d��! S ! ∗ 2�� BFE C * B S *Jd "'K�( =�b1/Jd a�, (�H d a�, H�U#0�" * "+6 B C�E 0�MN0�"+6�J ∗ Q 0 * " = < 0 a "�0�"� >HG c>�> ∗ BFE =�bed@b1=�b1/ &)(1K d a�, 2 b H+H < " J ∗ Q�- A�B d@acbed H+U+0�B d 6 P1(1G / B = BFE =�bed@b1=�b1/ &)(#]K d a�- 67K�BT& d a�- 670�G =�b & *I, 0�B d 6
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^ I�� B��8B�� 3 N
� �K � �g� � �� � � � � � �7�7� �� �K � ��� � �� � � � �
B P�S *�b a B =F* & d a�< b P1( * BbH - 0�K b+*�b 0 � b G * S * ( ^ B O�< H bed (cP�"#D <%� (1G = ( d 0�"+K b1=F*Jd ]a S * BR&)BT6 B O@b &)K�(�D - 6 * "+6 C)BRMN&�E b 6 b1=�b1/ &)(1K , 6 0 * (1G#6 O G+0 d a (1U+6 b & d C)K�(1U#6 P1(1GC b B a C - 0�(1G#K�B?0 * ( BfP1S1K�B = ( ^ B O�< H bed ( 8I /)4 C b C - 0�(1G+K�B *Jd 6 b P b & b E * " * BT6 K b ]CI"�K b+*Jd a�- 6 5�< 0�B d 6cD d � b G *)- 6 *Jd 6 B O@b &)K�(�D - 6�2 acbed C b C)BRK�BbH d 4 0�(1G#K�B * " 0�A - 0�"b1=)< K�BR0 b 0 * " = # b1=�b1/ &)(1K ,�&:acbed#* " = #bG)P1(�H+(�D d 0 d K�S * " *�b%& � * ('P�BR&�E O "�K�( n ?P;k70,21T ��eK � � ��� � KMFHD`_
�+����� z �cx(�@}Yyl~ �WxEw �/~ y/zl} z %Ez:w9!��,�Bt:�lte�� b G * S * ( B /)<1O@d ( C b / BRE C)(1G+K�B * ( B C , 6 5`b 0 d a S C)B 4 &I"�K b 2 P1(1G BRE =�bed C)BTK�Bf]
H d b`a ScD d bc* (.C - K b K b 6 N�@ c>32�� E.�������� � �������1� ��98 � ����� 98�������� ����� 7�� �����98� � [\VFVAb�V� lh�����
�lg�<Bh��3�@;=m ��g8AM? ��1|AM1���� m ,B< >E. j�=8A���@;k7Jj|7 �( � )Z >-1 <Bh��3�@;=m ��g8AM? ��1|AM1���� m ,B< > ���j@� �=j6g�< � � � ( � � � 1
�;:;:;:�� � � ��� ) (� ≥ 1)
>c1e< j,=iAM1��-;k.8jeg�< � � �� ( � � � 1 �;:;:;:�� � � )(� � � ≥ 1)
Z �c<H1 ;=< ��meh�m ? g���<Hj � <�m3=8A�;91�g@C`. ���JLY Q � ;@=>�E1-?H1 Z � � ,Eg��@. �B,:g �|< >E. j�=8A���@;k7Jj|7 � ( �� ) j|;=m&= � �(=Jjc< >|m3<�� g@?PA=1 <|1@A=1 �� �`mk,l< >:. 1|A >c1e<-,9A6A=m6A 1|A ==h �� �Eg�<:>eh�me< m���1��@<Y5 ,9A�� � ;��k;=m < m�� h�m3=
�( �� ) =
�( � � � � ( � � �� ��� )) ( �� = �
1�;:;:;:�� � � ∈ N) :JL9 d Q
JU� Q < 1�A��e1 ;91 � Z � Z �� = � 1 �;:;:;:�� � � >c1e< �� = �1�;:;:;:�� � � Z
�( � � � � +
� ( � � �� � �� ��� )) =�
( � � � � ( � �� ( � � �� ) � �� ��� )) :JL9 O Q� hi<Hh � ��m6A Z m </j,=iAM1��-;k.8jeg�< � � �� ( � � �� ) g@?PA=1 < �8A=1 � h � m�� � �8AM1 �B P�S * " = >-1@A=m6A�< >E.^,-m�� �e. J�9 d Q 0�G =)< D�B *�bed S *Jd-a�< C)B b1=�b1/ &)(1K d a�, K�BR& d a�, 0�G)]
=)< & * "�0�" # P b & < D�B *�bed & b P1S P1&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�- 6.0�G =�b & *I, 0�B d 6.K�B J�K�S = ( QK�E b 5 H b`a�4N/ " b1=�b%�I,#* "+0�"�2 acbed B d /Jd a S * BR& b 2�S *Jd6a�< C)B b1=�b1/ &)(1K d a�, K�BR& d a�, 0�G =)< &R]* "�0�"cBFE =�bed BbH b A d 0 *Jd a�<�b1=�b1/ &)(1K d a�, \I P�E 0�"�6�2 b1= C - 0�(1G+K�B
� � � ( �� ) =�
( � � ��� ( � � �� ��� )) �JL9�] Q
9P]
T#8 U 2=VvX ������0P��������Z /�[&/ ';6�'&�\'&2&' X ��6 , ��������Z /�[&/ '* S * B'"cB C�E 0�MN0�" J�9 d Q 0�G = BbP < D�B *�bed S *Jd 2�D d b a�< C)B �W2�" b`a (�H+(1G+C�E b
� �0��� �
1�;:;:;: �
b P b & d C)K�BRE7S�H�BT6 *Jd 6���] K�BbH�BRE 6 b1=�b1/ &)(1K d a�- 6 K�BR& d a�- 6 0�G =�b & *I, 0�B d 6�2 K�B *)-b* ( d (* &)S+P1(cP�(1G " (
�+ 1)
] K�BbH , 67K�BT& d a�, 0�G =)< & * "�0�"� � ( � � �� ) = � � � ( �� ) =
�( � � � � ( � � �� ��� ))
=�b BFE =�bedeb1=�b1/ &)(1K d a�, 3 b & d C)K�S16 � acb H+BFE *�bed >9� �8< >+A�� * "�6�K�BR& d a�, 6�0�G =)< & * "+0�"+6� � � 2 acbed BRE =�bed BTU a (�H+( =�b / (1U#K�B7S *Jd2a�< C)B b1=�b1/ &)(1K d a�, K�BT& d a�, 0�G =)< & * "+0�" - A�B dP1(�H#H�(1U#6 a M /Jd a (1U+6�2 � 0 a "+0�" � N @ ! m 5 b 0 d a�,\d /)-Mb D d b * " = b P1S / B d CI" BFE =�bed =�b�a M /Jd a (+P1( d , 0�(1G#K�B K�B b & d C)K�(1U#6
JLS+P�M 6 0 * ( > E' c>P> Q�*�b b1=�b1/ &)(1K d a�< P1&)(�D�& < K�K b+*�b?* "+6 �0acbed+* (1G+6 * BT&)K b+*Jd a (1U+6
G�P�(�H�(�D d 0�K�(1U+6 * M =pb1=�b1/ &)(1K d a�4 = K�"+A b1=)4 = 2 -f* 0 d P1(1G "c0�A - 0�"� � ( � � � 1
�;:;:;: � � � ��� ) ⇐⇒ ( � BRE =�bed@a M /Jd a S16 a�< P�( d (1GJ T�8 Qb1=�b1/ &)(1K d a (1U P1&)(�D�& < K�K b+* (16 � 2acbed ( � BRE =�bed@a M /Jd a S16* BR&)K b+*Jd a (1U%G�P�(�H�(�D d 0�K�(1U * (1G �0 * " = BRE 0�( / ( �
0 : � 1�;:;:;:�� � �
=�b BRE =�bed P�&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�, 2 acbedN=�b G�P < &)A�B d K d b P�&)M * (�D�B =)4 6 b1=�b1/ &)(#]K d a�, 0�G =)< & * "�0�" � ( � )
2 *)-f* ( d b P�(1G b1= ( � BFE =�bedca M /Jd a S16 * BT&)K b+*Jd a (1U G)P1(�H+(�D d ]0�K�(1U12 * S * B
�( � ) =
" *Jd K , B C)S / (1G b P1S * ( = J * BT&)K b+*Jd a S Q G�P�(�H�(�D d 0�K�S �,:J T > QB P � b G * (1U+6 * (1G+6.(1& d 0�K�(1U+6.0�G =)< D�B *�bedWb K - 0�M 6 * ( JLY Q acbed /Jd BRG a & d = E � B *�bed * (= S�"�K < * (1G m 0�"�K b 0�E b * (1G a�< P�M 6 * BTA =Jd a S * BR&)(1G J � Q C b B CI"#D�"�C)BRE�0 * " P�(1&)BFE b e * (1G+6 G)P1(�H+(�D d 0�K�(1U#6cP1(1G P�& - P�B d =�b�a�<1= (1G#K�B.C b A�&I"�0 d K�(+P1( d , 0�(1G#K�B.BbP b ]
= B d H#"�K�K -T=�b a M /Jd a (1U+6 b`a (�H�(1G#C d 4 = J >AE' c>�> Q ; d b * " = b P�H+(1U+0 * BRG#0�" (1& d 0�K -T= M =* U�P�M = 2�C -f* (1G+K�B
( � ) � ��
= (( � ) � )� � ( � ) � �
� �� = (( � ) � )
�) �� a H�P �
��i ` d( � ) = ( � )0
� [ Y�` d( � ) = ( � ) [\e ( � )−· 1
�-b* 0 d P1(1G
��i ` d(〈 � 0
�;:;:;: � � � −1〉) = � 0 = * (cP1& 4�* ( 0 * ( d A�BRE ( * "+6 � 0�;:;:;:�� � � −1
�[ Y�` d
(〈 � 0�;:;:;: � � � −1〉) = � � −1 = * ( * BTH+BRG *�b E ( 0 * ( d A�BRE ( * "+6 � 0
�;:;:;:�� � � −1�
acbed(〈〈0 � 2〉 � 〈1 � 0〉〉)0 � 1 = 2 � (〈〈0 � 2〉 � 〈1 � 0〉〉)1 � 0 = 1 :
I P1E 0�"+6 C b A�&)B d b 0 * (1U+K�B K�BR& d a�- 6 * BTA =Jd a�- 6 d /Jd S * " * BT6 * M = a M /Jd a�4 = b`a (�H+(1G+C d 4N=�
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U � 2 "v'&2���2&��6�4;��� , $�4;6�'&�\' X ' , ��k�� [ � [ T >
N�@ ! 2�� 4����=' 2 < 1(;k7eA^>��e1 j-< >E.p>9� �8< >2m hJm-?Y7Jj|7 J > E' c>P> Q Z m <lg@Ci. ��j,=iAM1����;k.8jeg�< � g@?YAM1e<Bh��3�@;=m ��g8AM? ��1|AM1���� m ,B< > �����XcY ��` V �
(�) = max{〈 � 0
�;:;:;:�� � � −1〉 | � � � 0�;:;:;:�� � � −1 ≤ � }J T#! Q
J T N Q ` V h( � � F � � )
=
{〈( � ) � � ( � ) � +1
�;:;:;: � ( � )�−· 1〉
� b1= � V � ( � ) & 0 ≤ F*G � ≤ [ e( � ) �
0 � b H+H d 4 6 � hi<Hh � ��m6A Z �c<H1 A � 1(;91
� � F � � Z`fV h
( � � F � � ) ≤ � �>-1 </1@A F 0. � G [ e
( � )Z ;BA ;0g ` V h
( � � F � � ) Go��
j-X Z�<\#3��} [ 0 * " = � 0 a "�0�" � N�@ c># aj-X Z�<\#3��} [M/ �&� � # .�, 4��=' / ��5 U � 2 ( 2 m 0�B d & < (1& d 0�K 4 = P�(1G b`a (�H�(1G#C)BFEa M /Jd a (+P�( d BFE *�b?5`b 0 d a�< 0�U = (�H b b1=F*Jd a B d K -T= M = P1(1G b1=�b`a U�P * (1G = 0 * "7C)BTM &�E b7* "+6
P1&)(�D�& b K�K b+*Jd a�, 6 D�H 4 0�0 b 6 �� � &I"�0 d K�(+P1( d (1U#K�B * " = a H b 0 d a�,(a M /Jd a (+P�(�E "�0�"b`a (�H�(1G#C d 4 = 2 acbed�* ( 0�G+K 5 (�H d 0�K�S [�
] �D d b * ( =pa M /Jd a S * (1G b1=F*Jd a B d K -R= (1G �
/ "+H b1/I, ( d (1& d 0�K�(�E�K b 6 / E = (1G = /Jd b1/ (1A d a�< -T=�b ]LP�&)(16b] -T=�b 0�G =�b & *I, 0�B d 6[ ] � : � � N
D d b�a�< C)B 5 b 0 d a S 0�U = (�H�( � P�(1G.C b�a M /Jd a (+P�( d , 0�(1G+K�B =B =F* E D d b [ ] �C b D�& < ]
O (1G#K�B [ ] �2�S+P�(1G%" a M /Jd a (+P1(�E "+0�" * (1Gc0�G = S�H+(1G � (1&�E � B *�bed 0 * ( (1& d 0�K�S F 2�D d b
F = 1 �;:;:;: 6 �� �p�� �������� ge U#K O M =�b K�B * (1G+6 (1& d 0�K�(1U#6 * "+6 � ( � 0)
0 * ( ! @ !12 C -f* (1G+K�BP1& 4�*�b
[��� ]1 = 〈0 � 0 � F 〉 � [�]1 = 〈0 � 1 � � 〉 � [ � ]1 = 〈1 � 1 � 0〉 �
[��
]1 = 〈1 � 1 � 1〉 � [� �� ]1 = 〈1 � � � 2 + F 〉 �
acbed D d bc*�b G�P�S�H�( d P b (1A *)4 0�U+K 5 (�H b��� �� �������� ��� � ( ) = ?
A�&I"�0 d K�(+P1( d (1U#K�B * (1G+6 b & d C)K�(1U+6〈2 � 0〉 �;:;:;:�� 〈2 � 7〉 �
/ "+H b1/I, [ �� ]1 = 〈2 � 0〉 2 [ �� � ]1 = 〈2 � 1〉 2 :;:;: 2 [?]1 = 〈2 � 7〉 $ b & b+* "+&)(1U+K�B S *Jd ( a M /Jd a S16 * (1G * G#A b E (1G b & d C)K�(1U � BRE =�bed K�BbD b H�U * BR&)(16 b P�S* ( = �W2�P A 2
[1]1 = 〈0 � 1 � 1〉 = 2 · 32 · 52 = 450 DP1&)( O-b1=)4 6 / B = K b 6 B =)/Jd b1O�- &)B d B /)4 " b P�( * BTH+BR0�K b+*Jd a S * " *�b'* "�6 a M /Jd a (+P1(�E "+0�"+6
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T+! U 2=VvX ������0P��������Z /�[&/ ';6�'&�\'&2&' X ��6 , ��������Z /�[&/ '� � � ��E#� 8� ; d b a�< C)B H - CI" J b`a (�H+(1G+C�E b�Q
� ≡ �0�
1 · · ·� �
b P1S.0�U#K 5 (�H b JL0�G+K�P�BR& d H b K 5 b1= (1K -T= (1G acbed�* (1G � � � Q 2�C -f* (1G+K�B
[�
0�
1 · · ·� � ]2 = 〈[ � 0]1
� [ � 1]1�;:;:;:�� [ � � ]1〉 :
; d b P b & <1/ B d D�K b 2[ � (� 1)]2 = 〈[ � ]1 � [(]1 � [� 1]1 � [)]1〉 = 〈〈1 � 1 � 0〉 � 〈2 � 4〉 � 〈0 � 0 � 1〉 � 〈2 � 5〉〉 �
[� 21(�1� 0)]2 = 〈[ � 21]1 � [(]1 � [� 1]1 � [0]1 � [)]1〉 = · · · :
I d /Jd a S * BT& b 2 ( (1& d 0�K�S16 b G * S16 b1=�b C -f* B d a M /Jd a (1U#6 0 * (1G+6 A�� m&= � * "+6 � 2�P1(1GBFE =�bed H - C)B d 6 b P � b G *)<c*�b 0�U#K 5 (�H b � ����������������� 3 d|b & d C)K�" *Jd a�- 67K�B *�b+5 H+" *)- 6 � � acbed ( d|b & d C)K�" *Jd a�- 670 *�b C)BT& - 6�.BFE =�bed 0�U+K 5 (�H b 2 b H+H < BRE =�bediacbed S1&)( d K ,8a (1G+6 1
� * 0 d+- A�(1G =�/ U+( /Jd b1O (1&)B *Jd a�- 6a M /Jd a (+P�( d , 0�B d 6�2�M 6?0�U+K 5 (�H b acbed MN6?S1&)( d 2 acbed P1& - P1B d�=�b P�&)(10 - A�(1G+K�B =�b K�" =*Jd 670�G#D�A - (1G+K�B
[� � ]1 = 〈0 � 0 � F 〉 � [� � ]2 = 〈〈0 � 0 � F 〉〉[�]1 = 〈0 � 1 � � 〉 � [
�]2 = 〈〈0 � 1 � � 〉〉 :
; d b P b & <1/ B d D�K b 2 [0]1 = 〈0 � 1 � 0〉 = 2 · 32 · 5 = 902�B =)4 [0]2 = 〈90〉 = 291
� W�������������� ���@8 E�1��� ���(E�� 81 m * G+A b E b J * G�P d a�,1Q B C�E 0�MN0�" acb C)(1&�E � B *�bedeb P�S* ( b & d 0 * BR&)S acbed�* ( / B C d S * "+670 a�- H+(16�2�P1(1G%BRE =�bed S1&)( d C -f* (1G+K�B
[�( �� ) =
�]3 = 〈[ �
( �� )]2 � [ � ]2〉 :� �������������� � � �������������������� �B = � = ( � 0
�;:;:;: � � � )2 * S * B a�< C)B � � BFE =�bedB C�E 0�M 0�"�2 acbed C -b* (1G#K�B
[�
]4 = 〈[ � 0]3�;:;:;: � [ � � ]3〉
:� � ������� �����(E�� 81 ;_ b 0 * ( d A�BFE b 0 * ( b & d 0 * BR&)S 0 a�- H+(16 K d b 6 acb+*)< 0 *�b 0�"�6
BFE =�bed�,pa H+B d 0 * (�E�S1&)( d�, 0�G =�b & * "�0 d b`a�< 0�U#K 5 (�H b ,7* ( � � � 2�B =)4 *�b 0 * ( d A�BFE b 0 * (/ B C d S * "+670 a�- H�(16?BFE =�bed|b & d C)K�" *Jd a�- 670 *�b C)BT& - 6%J / "+H b1/I, b & d C)K�(�E Q p -b* (1G#K�B [�
0�;:;:;:�� � � −1 :
�0�;:;:;: � � �
−1]5 = 〈〈[ � 0]′ �;:;:;: � [ � � −1]
′〉 � 〈[� 0]1�;:;:;: � [ � � −1]1〉〉 �
S+P1(1G
[� � ]′ =
{[� � ]1 � b1=?* ( � � BFE =�bed 0�G =�b & * "+0 d b`a S.0�U#K 5 (�H+( ,%* ( � � � �[� � ]2 � b H#H d 4 6�2 / "#H b1/I, b1= ( � � BRE =�bed@a H�B d 0 * S16?S1&)(16 :
I /)4 P1& - P1B d�=�b P�&)(10 - C)(1G+K�BI2�BbP�B d /I, ( dcb & d C)K�" *Jd a�- 6'0 *�b C)BT& - 6 a M /Jd a (+P�( d (1U =F*�bedM 6?0�U#K 5 (�H b 0 * ( / B C d S 0 a�- H+(16 acbed M 6?S1&)( d 0 * ( b & d 0 * BR&)S D d b P b & <1/ B d D�K b 2
[2 : 2]5 = 〈〈〈〈0 � 1 � 2〉〉〉 � 〈〈0 � 1 � 2〉〉〉 :
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U � 2 "v'&2���2&��6�4;��� , $�4;6�'&�\' X ' , ��k�� [ � [ T N
$ b & b+* "+&)(1U+K�B S *Jd S *�b1= ( � BRE =�bed a M /Jd a S16 K d b 6 acb+*)< 0 *�b 0�"+6 � :� 2 * S * B (
��i ` d( � )
BRE =�bed-a M /Jd a S16 * (1G b & d 0 * BT&)(1U 0 a�- H�(1G#6 � 2 acbed (%[ Y�` d ( � )BRE =�bed-a M /Jd a S16
* (1G / B C d (1Uc0 a�- H+(1G+6 � ; d b =�b BfP b H#"�C)BTU+0�(1G+K�B S *Jd " 0�G =)< & * "+0�" � 7→ [
�]5BRE =�bed�-R=�b ]�P1&)(16b] -R=�b 0 * (
0�U = (�H�( acb+*�b 0 *)< 0�BTM = 2�P1& - P1B d'=�b P b & b+* "+& , 0�(1G+K�B S *Jd acb1=)-T=�b 6 b & d C)K�S16 / B =BFE =�bed *�b G * S1A�&)( =�b�a M /Jd a S16 0�G =�b & * "�0 d b`a (1U 0�G+K 5 S�H�(1G , * (1G � � � acbed BfP�E 0�"�6a M /Jd a S16 a H�B d 0 * (1U S1&)(1G b G * S d 0�A�U+B d BbP�B d /I, * ( P1& 4�* (70�U#K 5 (�H+( a�< C)B a H+B d 0 * (1US1&)(1G � BRE =�bed�-T=�b�b P�S *�b
� � �� � � � ��� � (
K�B a M /Jd a S 22 -b* 0 d P�(1G.D d b a�< C)B S1&)( � 2 ��i ` d
([�
]2)
2 B =)4�b1= ( � BFE =�beda M /Jd a S16?0�G#K 5 S�H+(1G�2 * S * B ��i ` d( � ) ≤ 2
� ������������������� ; d b a�< C)B b`a (�H�(1G#C�E b acb+*�b 0 *)< 0�BRM = � = ( � 0 �;:;:;:�� � � )
2C -b* (1G#K�B
[ � 0 �;:;:;: � � � ]6 = 〈[ � 0]5 �;:;:;: � [ � � ]5〉 :B G * S a M /Jd a (+P�( d BFE S�H�BT6 *Jd 6 b`a (�H+(1G+C�E BT6 acb+*�b 0 *)< 0�BRM = 2 acbed�d /Jd b E * BT& b * (1G#6* BR&)K b+*Jd a (1U#6 G�P�(�H�(�D d 0�K�(1U+6 * "+6 K�"+A b1=I, 6cD d b (+P1( d ( /I, P1( * B b1=�b1/ &)(1K d a S P�&)S#]D�& b K�K b e�� b G * S * ( 0�"+K�BFE ( ( d (1& d 0�K�(�E'J T�8 Qpacbed J T > Q BFE =�bed b G#0 * "+&)(�E 2 acbed D d b * (
K - &)(16 J�Y Q�* (1G C)BTM & , K b+* (16 b & a BRE =�b / BRE C)(1G+K�B S *Jd " 0�A - 0�" � � ( � � �� ��� ) acbed "0�G =)< & * "�0�" � ( � )BRE =�bed P�&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�- 6
_ ( / BTU * BT&)(.BRE =�bed-b P�H+S D d b+* E 2�K�B�P1&)(10�B a�*Jd a�, # b P1( a M /Jd a (+P1(�E "+0�" & * M = (1& d ]0�K 4N= 2 b & a BRE =�b C - 0�(1G#K�B
�( � ) =
[\Y#` d( � )1
�0�2
J T Q QJ�� 0 a "�0�" � N @ N Q ; d b * ( P�& 4W* (�2 / E = (1G#K�B K d b 0�B d & < b P�S 5 (�"+CI" *Jd a (1U#6 (1& d 0�K�(1U#6 P�&)M * (�D�B =)4 6b1=�b1/ &)(1K d a�4 = #f0�G =�b & *I, 0�BRM = acbed 0�A - 0�BTM = b P�( a M /Jd a (+P�(�E "�0�"�6 & P�(1G acb+*�b H , ]
D�(1G = K�B * ( � " * (1U+K�B = ( 3 d P1BT& d 0�0�S * BR&)BT6 b P�S *Jd 6 b P bed * (1U+K�B = BR6 b P�( / BFE C)B d 6P1&)M * (�D�B = (1U#6 b1=�b1/ &)(1K d a S * " *�b 6 BRE =�bedWb P�H - 6�2 acbed D d � b G *)- 6cP�(1G / B = BFE =�bed C bP b & b C - 0�(1G#K�B?K�BT& d a�< 0�A�S�H d b K�B *)<c* ( = acb+*)< H+(�D�(
B & @ B * ( � ) ⇐⇒ ( � BFE =�bed@a M /Jd a S16 a�< P�( d b 6 � �⇐⇒ � = 〈0 � 0 � ( � )2)〉
B & e *�b C ( ) ⇐⇒ ( BRE =�bed@a M /Jd a S16 b & d C)K�(1U⇐⇒ = 〈0 � 1 � ( )2〉
e G = @ B * ( � ) ⇐⇒ ( � BFE =�bed@a M /Jd a S16 a�< P1( d b 6 � ��⇐⇒ �
= 〈1 � ( � )1 � ( � )2〉 & (�)1 ≥ 1 & (
�)2 ≥ 2
e G = e *�b C ( � ) ⇐⇒ ( � BFE =�bed@a M /Jd a S16 * (1G � , * (1G ��⇐⇒ �
= [ � ]1 ∨ � = [���
]1
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T Q U 2=VvX ������0P��������Z /�[&/ ';6�'&�\'&2&' X ��6 , ��������Z /�[&/ 'e G = e G#K 5 ( � ) ⇐⇒ e G = @ B * ( � ) ∨ e G = e *�b C ( � )Y i Z d ^
(�) =
" P�H+B d (1K - H�B d bc* (1Gc0�G =�b & * "+0 d b`a (1U%0�G+K 5 S�H�(1G �J b1= ( � BFE =�bed@a M /Jd a S1670�G+K 5 S�H�(1G Q
= (�)13'&)(16
( � ) ⇐⇒ ( � BFE =�bed@a M /Jd a S167S1&)(1G^ H13'&)(16
( � ) ⇐⇒ ( � BFE =�bed@a M /Jd a S16 a H�B d 0 * (1U%S1&)(1G⇐⇒ 3'&)(16
( � ) & (∀ F*G [ e( � ))¬ B & @ B * (( � ) � )
e G = C�3?&)(16 ( � ) ⇐⇒ ( � BFE =�bed@a M /Jd a S167S1&)(1G � �� ( � 1�;:;:;: � � � )
⇐⇒ 3'&)(16( � ) &
e G = @ B * ( ��i ` d( � ))
� P1(�3'&)(16( � � � ) ⇐⇒ ( � BFE =�bed@a M /Jd a S167G)P1(1S1&)(1G * (1G%S1&)(1G%K�B a M /Jd a S �
⇐⇒ 3'&)(16( � ) &
3'&)(16( � ) & (∃ F � � ≤ [\e
( � )[ FHG �& � =
`fV h( � � F � � )]
B P+H13'&)(16( � ) ⇐⇒ ( � BFE =�bed@a M /Jd a S167S1&)(1G � �� (� 1 �;:;:;:���� � )
⇐⇒ e G = C�3?&)(16 ( � ) & (∀ � Go� )[� P1(�3'&)(16
( � � � ) � ⇒ B & @ B * ( � )]I C�E 0�MN0�"
( � ) ⇐⇒ ( � BRE =�bed@a M /Jd a S16 b1=�b1/ &)(1K d a�, 67B C�E 0�MN0�"+6⇐⇒ � = 〈 ��i ` d
( � ) � [ Y�` d ( � )〉&B P+H13'&)(16
(��i ` d
( � )) &3?&)(16
([ Y�` d
( � ))
& (∀ FHG [ e([\Y#` d
( � )))[B & @ B * (( [ Y�` d ( � )) � )
� ⇒ (∃� G [ e(
��i ` d( � )))[(
[\Y#` d( � )) � = (
��i ` d( � ))
�]]$?&)(�D�&
( � ) ⇐⇒ ( � BRE =�bed@a M /Jd a S16 P1&)(�D�& < K�K b+* (16⇐⇒ � V �
( � ) &[\e
( � )
0 & (∀ FHG [ e( � ))[
I C�E 0�M 0�"(( � ) � )]
& (∀ F*G [ e( � ))(∀� G [\e
(( � ) � � 1)[e G = @ B * (( � ) � � 1 �
�) � ⇒ (∃ � G [\e
( � ))[( � ) � � 1 ��
= ( � ) ��0�0]]
^ b+* ( � ) ⇐⇒ (�BRE =�bed@a M /Jd a S16 acb+*)< 0 *�b 0�"+6
⇐⇒ � = 〈( � )0 � ( � )1〉& (∀ FHG [ e
(( � )0)
[e G = e G#K 5 (( � )0 � � ) ∨ ( � )0 � � = [?]1 ∨
^ H�3?&)(16(( � )0 � � )]
& (∀� G [\e(( � )1)
B & e *�b C (( � )1 � � )_ BT&)K ^ b+* ( � ) ⇐⇒ (
�BRE =�bed@a M /Jd a S16 * BT&)K b+*Jd a�, 6 acb+*)< 0 *�b 0�"+6
⇐⇒ ^ b+* ( � ) &[ e
(( � )0) = 0 &[\e
(( � )1) = 1B =F* ( � � � � � ) = ���
* ( b P1( *)- H�BT0�K b%* "�6 b1=F*Jd acb+*)< 0 *�b 0�"+6 * "�67K�B *�b+5 H#" *I, 6K�B a M /Jd a S � = [ � � ] K�B * "c0 *�b C)BT& < � 0 * ( = S1&)( K�B a M /Jd a S �$'H B =F* ( � � � � � ) = ���* ( b P1( *)- H�BT0�K b%* "�6 b1=F*Jd acb+*)< 0 *�b 0�"+6* M = K�B *�b+5 H#" *)4N= K�B a M /Jd a (1U+6 ( � )0 �;:;:;:�� [ Y�` d ( � )
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U � 2 "v'&2���2&��6�4;��� , $�4;6�'&�\' X ' , ��k�� [ � [ T+9K�B *Jd 670 *�b C)BR& - 6 ( � )0
�;:;:;:�� [ Y�` d ( � )0 * ( = S1&)(.K�B a M /Jd a S �@ B * B P+H13'&)(1G ( � ) = ��� 〈[� �1]1�;:;:;:�� [� � � ]1〉 ( b1= � = [
� �(� �
1
�;:;:;:���� � � )]2)@ B *)<+5`b 0�" ( � � � � � ′) ⇐⇒ $'&)(�D�&( � ) &
^ b+* ( � ) &^ b+* ( � ′)
& � → � ′0 * _ T (
�)D d b � = [
�]4
� P�(�H( � ��� ) ⇐⇒ ( � BRE =�bed@a M /Jd a S16 * BR&)K b+*Jd a (1U%G�P�(�H�(�D d 0�K�(1U
0 * ( T (�
)D d b � = [
�]4
⇐⇒ $'&)(�D�&( � ) &
� V �( � ) &
[\e( � )
1
& (∀ FHG [ e( � )−· 1)
@ B *)<+5 b 0�" ( � � ( � ) � � ( � ) � +1)
&_ BR&)K ^ b+* ( [\Y#` d ( � ))
� � ( � � �� ��� ) ⇐⇒ $'&)(�D�&( � ) &
� P1(�H( � ��� )
&��i ` d
(��i ` d
( � )) = 〈 ��i ` d(
��i ` d(
��i ` d( � )))〉
&[ Y�` d
(��i ` d
( � )) = [ � 1�
2 · · · � � ]2m P1& 4�* " K�")]LP�&)( O@b1=I, 6 b P1S / B d CI" 0 � b G * S * ( =^acb+*)< H�(�D�( BFE =�bed "7P1&)M * (�D�B =I, 6b1=�b1/ &)(1K d a S * " *�b%* "�670�A - 0�"�673?&)(16 ( � )
2�P1(1G%BFE =�bed@acbed "%P d ( / U#0 a (�H#" � ����� � j � �=j@7
3?&)(16( � ) ⇐⇒ ( � BRE =�bed@a M /Jd a S167S1&)(1GJ T#9 Q
g@?YAM1e<Bh��3�@;=m ��g8AM? ��1|AM1���� m ,B< >E. n h3A��kg�< Ci7 2 B P�S * ( = (1& d 0�K�S * (1G#6�2�( d S1&)( d BFE =�bed�* BR0�0 < &)M = K�(1& O�4 = 2 / "+H b1/I,
3'&)(16( � ) ⇐⇒ 3'&)(16
1( � ) ∨ 3'&)(162( � ) ∨ 3?&)(16
3( � ) ∨ 3?&)(164( � )
J T�T Q0�B b1=F*Jd 0 * ( d A�E b K�B b G *)- 6 *Jd 6'P�BR& d P *)4 0�B d 6�2�S+P�(1G
3'&)(161( � ) ⇐⇒ � = [ � � ]2 D d b a�< P1( d b K�B *�b+5 H+" *I, � �
⇐⇒ � = 〈〈0 � 0 � ( � )0�2〉〉 �3'&)(16
2( � ) ⇐⇒ � = [�]2D d b a�< P1( d ( = b & d C)K�S �
⇐⇒ � = 〈〈0 � 1 � ( � )0�2〉〉 :
e *Jd 6.P d ( / U#0 a (�H+BR6 * BbH�BTG *�b E BT6 / U+( P�BR& d P *)4 0�B d 6�2 * ( b1= K d b H - CI" BFE =�bed S1&)(16B C b & *)<+*�bed:b P�S * ( b1= K�BT& d a�< J D =I, 0 d b�Q�* K , K b+*)< * "�6 BFE =�bed S1&)( d 2 acbed:b G * S K b 6/ E = B d�/ U#( d 0�( / G =�b K�E BT6�P�(1G7P1& - P1B d�=�b d acb1= (+P1( d (1U =F*�bed-b P�S * (1G+6 a M /Jd a (1U#6 ; d b* " = P�BR&�E P * M 0�" * "�6 /Jd b`a H <1/ MN0�"+6�2�G)P1(�H+(�D�E � (1G+K�B 3?&)(16
4( � ) ⇐⇒ � = [ b1= ( � = 0) * S * B � b H+H d 4 6 � ]2D d b S1&)(1G#6 � � � � �
⇐⇒ (∃ � ��� �� G � )[3'&)(16
( � ) &3?&)(16
( � ) &3?&)(16
( )& � = [( �� (]2 ∗ � ∗ [= 0) �� � ]2 ∗ � ∗ [ ������� � � ]2 ∗ ∗ [)]2]
:
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TPT U 2=VvX ������0P��������Z /�[&/ ';6�'&�\'&2&' X ��6 , ��������Z /�[&/ '3 d2b1=Jd 0�S * " * BT6 � ��� �� G � /Jd acbed (�H�(�D�(1U =F*�bed BbP�B d /I,�b1= ( � BRE =�bed|a M /Jd a S167B = S16S1&)(1G
� ≡ ( b1= ( � = 0) * S * B � b H#H d 4 6 � ) �* S * B7( d S1&)( d � � � � � BFE =�bed D =I, 0 d b.* K , K b+*�b.* (1G � 2 acbed�-b* 0 d ( d|a M /Jd a (�E * (1G#6BFE =�bed K d a &)S * BR&)( dN* (1G � b P1S * " 5 b 0 d a�, d /Jd S * " *�b * "+6 0�G =)< & * "+0�"+6c`fV h ( � � F � � )0 * ( � , K�K b N @ ! _ BTH d a�< C)BRMN&)(1U+K�B * " =?* BbH�BTG *�b E b P�BR&�E P * M 0�"�2�P1(1G BFE =�bed�/ G#0 a (�H+S * BR&I"%BbP�B d /I,
( #f0�U = C)B * (16 & S1&)(16?K�B a M /Jd a S � P b & < D�B *�bed|b P1S�� G)P1(1S1&)(1G#6�2�D d b./Jd <1O (1& b � 3?&)(16
3( � ) ⇐⇒ � = [�( � 0
�;:;:;: � � � −1)]2D d b S1&)(1G#6 � 0�;:;:;:�� � � −1
acbed ��] K�BbH - 670�U#K 5 (�H+( � �S+P1(1G K�B #f0�U#K 5 (�H+( & B =)= (1(1U+K�B a�< P�( d b 0�G =�b & * "�0 d b`a�, K�B *�b+5 H+" *I, , K�E b b P�S *Jd 60 *�b C)BT& - 6 � 2 ��� \B = ( � d acb1= (+P1( d BRE b G *I, * " 0�G = C ,ia "�2 * S * B�K�P1(1&)(1U#K�B =�b G�P�(#]H�(�D�E 0�(1G+K�B * ( =^a M /Jd a S * (1G 0�G+K 5 S�H�(1G � 2 [ � ]1 = ( � )0
acbed�* " = P�H+B d (1K - H�B d < * (1G�
= ( � )0�1 I P�E 0�"�6 D = M &�E � (1G#K�B S *Jd ( d a M /Jd a (�E * M = S1&)M = � 0
�;:;:;: � � � −1BFE =�bed K d a &)S * BR&)( d * (1G � 2 B O S10�( = b G * (�E?( d S1&)( d BFE =�bed D =I, 0 d b * K , K b+*�b * (1G�( � 0
�;:;:;:�� � � −1) � * 0 d " d 0�( / G =�b K�E b P1(1G d acb1= (+P�( d BFEW" 0�G = C ,ia " 3?&)(16
3( � )P b E & = B d�* "cK�(1& O�,3?&)(16
3( � ) ⇐⇒ � V �( � ) &
e G = e G#K 5 (( � )0)
& (∃ � 0�;:;:;:�� � ( � )0 � 1−· 1)[(∀ F*G ( � )0
�1)3?&)(16
( � � )& � = 〈( � )0〉 ∗ [(]2 ∗ � 0 ∗ [ � ]2 ∗ · · · ∗ � ( � )0 � 1−· 1 ∗ [)]2]
:B G * S?C)G#K�E � B d P1(�H+U * " = d 0�( / G =�b K�E b P�(1G d acb1= (+P1( d BRE+" 0�G = C ,ia "?3'&)(16
4( � )2#B a�* S16b P1S *Jd 6 * BbH1E * 0�BR6�# :;:;: & P�(1GcP�& - P�B d 51-f5 bed b =�b *Jd 6 b P�( O U�D�(1G+K�BI2 acbed D d � b G * S
* ( 0 a (+P1S (1&�E � (1G#K�B *Jd 6 B C , 6�2 5 (�"�CI" *Jd a�- 6 J P1&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�- 6 Q 0�G =�b &R]*I, 0�B d 6
�(0 � � � � ) = 〈( � )0〉 ∗ [(]2 ∗ ( � )0�
(�+ 1 � � � � ) =
�(� � � � � ) ∗ [ � ]2 ∗ 〈( � ) � 〉
�( � � � ) =
�(( � )0
�1� � � � ) ∗ [)]2
:B = _ ( � )0
BFE =�bed a M /Jd a S16 a�< P�( d (1G ��] K�BTH+(1U+6�0�G =�b & * "�0 d b`a (1U?0�G+K 5 S�H�(1G12 ( � )0 =[�]12 acbed D d b F*G � (
( � ) � = [ � � ]2 BFE =�bed@a M /Jd a S16 a�< P1( d b 6'H - CI"�6�2 * S * B�(0 � � � � ) = [
�( � 0]2
� �(1 � � � � ) = [
�( � 0
� � 1]2�;:;:;: �
acbed�* BbH d a�< 2�( � � � ) = [
�( � 0
� � 1�;:;:;:�� � � −1)]2
:� P1B *�bed S *Jd "c0�G = C ,ia "%D d bc* " = 3?&)(16
3( � ) * BbH d a�< P b E & = B d�* ".K�(1& O�,3?&)(163( � ) ⇐⇒ � V �
( � ) &e G = e G#K 5 (( � )0)
& (∃ � ≤ X.Y ��`fV �( � ))[(∀ FHG ( � )0
�1)3?&)(16
(( � ) � )& � =
�( � � � )] :
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U � 2 "v'&2���2&��6�4;��� , $�4;6�'&�\' X ' , ��k�� [ � [ T d
B =?*)4 & b b1=F*Jd acb+*�b 0 *I, 0�(1G+K�B b G *)- 6 *Jd 6 d 0�( / G =�b K�E BT6'D d b%*Jd 6 *)- 0�0�BT& d 6 P1BT& d P *)4 ]0�B d 6 0 * " = J T�T Q 2�P b & < D�(1G#K�B K d b d 0�( / G =�b K�E b P�(1G d acb1= (+P1( d BRE *�bedEb P1S * " 0�A - 0�"3?&)(16
( � ) acbed P1(1G�0�G = BfP < D�B *�bed BRU a (�H b K�B�P�H , &I" P�&)M * (�D�B =I,�b1=�b1/ &)(1K , D d b 0�A - ]0�B d 6 J � 0 a "+0�" � > E' c>Hd Q S *Jd "c3'&)(16 ( � )BFE =�bed P�&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�, ; d bc=�bacb+*�b D�& < ��(1G#K�B a-d � b G *I,7* " H�BfP * (1K - &)B d b JLK d bpacbed�O�*)< 0 b K�B MN6 B /)4 Q 2�C -f* (1G+K�B
�( � � � ) ⇐⇒ 3?&)(16
1( � ) ∨ 3?&)(162( � )
∨[ � V �
( � ) &e G = e G#K 5 (( � )0)
& (∃ � ≤ X.Y ��`fV �( � ))[(∀ FHG ( � )0
�1)[(
� )( � )� = 1]
& � =�( � � � )
]
∨(∃ � ��� �� G � )[( � ) � = 1 & ( � ) � = 1 & ( � ) � = 1)
& � = [( �� (]2 ∗ � ∗ [= 0)]2 ∗ � ∗ [ ������ ��� ]2 ∗ ∗ [)]2]�
acbed BfP b H#"�C)BTU+(1G#K�B JLK�B * " =pb1=)< H�G#0�" P�(1G - A�(1G+K�B ,�/ " a�<1= B d Q S *Jd 2�D d b a�< C)B � Q 23'&)(16
( � ) ⇐⇒ �(〈 �������� (0) �;:;:;:�� �������� ( � −· 1)〉 � � ) �
P1(1G 0�G = BbP < D�B *�bed JLS+P�M 6 0 * " = � 0 a "�0�" � > E' c>Hd Q S *Jd "%3'&)(16 ( � )BFE =�bed P1&)M * (�D�Bf]
=)4 6 b1=�b1/ &)(1K d a�, J � , K�K b�Q am P1&)M * (�D�B =I, 6 b1=�b1/ &)(1K d a S * " *�b * M = P1BT& d 0�0�( *)- &)M = b P1S *Jd 6 < H+H+BR6 0�A - 0�B d 6acbed 0�G =�b & *I, 0�B d 6 * (1G acb+*�b H�S�D�(1G BRE =�bed P1&)( O-b1=I, 6�2 , * (1G#H < A d 0 * ( = BTU a (�H#"�2
B a�* S16 b P1S * " = @ B *)<+5`b 0�" ( � � � � � ′) P1(1G A�&)B d <%� B *�bedca�< P1( d b 0 a�- ��" acbed�* " = b1O�, ]= (1G+K�B'D d b < 0 a "�0�"�2 � N�@ Q ∗ b C�E � B d�=�b�acb+* BR&ID b 0 * BFE�( b1=�b D =)4 0 * "+6 *Jd 6?H�BfP * (#]K - &)B d BR6 * (1G�H < A d 0 * ( = B = S16 b P � b G * (1U+6 * (1G+67G)P1(�H+(�D d 0�K�(1U#6�2�D d b =�b�acb+*�b H <+5 B dP1BT&�E * E = (16 P1&)S a B d *�bed ; d b'* (7K - &)(16 JU� Q * (1G C)BRMN& , K b+* (16�2�P1& - P1B d1=�b G�P�(�H�(�D�E 0�(1G+K�B b P1S * ( = a M /Jd a S
� a�< P1( d (1G%0�G#0 *I, K b+* (16 � P�(1GcG)P1(�H+(�D�E � B d�* "cK�BR& d a�, 0�G =)< & * "�0�"�( � 1
�;:;:;:���� � � � 1�;:;:;:�� � � ) = �
�( ��� �� )
acbed�/ (10�K -T= (1G+6 b & d C)K�(1U+6 �� = � 1 �;:;:;: � � � 2 * ( = a M /Jd a S � �� ( � � �� ) a�< P1( d (1G < H#H�(1G0�G#0 *I, K b+* (16 ��� P1(1G%G�P�(�H�(�D�E � B d�* ".K�BT& d a�, 0�G =)< & * "+0�"�( �� ) =
�( �� � �� ) DJ T d Q
acbed BRE =�bed P1&)( O-b1=)- 6 S *Jd:b1= * ( P1& 4�* ( 0�G =�b & * "+0 d b`a S 0�U#K 5 (�H+( * (1G � BRE =�bed�* (� acbed�* ( � BRE =�bed|acbed = (1U#&ID d b 0�G =�b & * "�0 d b`a�, K�B *�b+5 H#" *I, J P1(1G / B = (1&�E � B *�bed 0 * (� Q 2 * S * B b & a BRE =�b P1&)(10�C - 0�(1G+K�B'0 * " =pb &)A , * (1G � * ( = (1& d 0�K�S J T d Q � 0 * M
� � +�
� ( � 0 �;:;:;:���� � −1 ��� � �;:;:;: ��� � +�−1) =
�0" P�& 4W* " B C�E 0�MN0�" * (1G 0�G+0 *I, K b+* (16 � K�B a M /Jd a S �
� P�B *�bed J b P�S * " = a M /Jd ]a (+P1(�E "+0�" Q 2�S *Jd ( a M /Jd a S16 * "+670�G =�b & * "+0 d b`a�, 6?K�B *�b+5 H+" *I, 6 P1(1G%(1&�E � B *�bed 0 * " =B C�E 0�M 0�"cK�B a M /Jd a S ( � ) � * (1G � BFE =�bed
[� ��� � ] =
�1( � ) = ( � ) � � 0 � 0 �
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T O U 2=VvX ������0P��������Z /�[&/ ';6�'&�\'&2&' X ��6 , ��������Z /�[&/ 'acbed B d /Jd a S * BR& b 2�( a M /Jd a S16 * "�6 0�G =�b & * "�0 d b`a�, 6 K�B *�b+5 H+" *I, 6�P�(1G (1&�E � B *�bed 0 * " =P1& 4�* ".B C�E 0�MN0�" * (1G � BRE =�bed
[� � +
�� ] =
�1( � ) = ( � )0
�0�0 Dacbed ( d2a M /Jd a (�E * M = b & d C)K�" *Jd a�4N= K�B *�b+5 H#" *)4N= 0 * " = P1& 4�* " B C�E 0�M 0�" G)P1(�H+(�D�E ]
� ( =F*�bed|b P1S * " = P�&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�, 0�G =)< & * "�0�"[��] =
�2( � � F ) = ( � )0
�0�2+2
� ( F = 0 �;:;:;: � � +� − 1) :
$ b & b+* "+&)(1U+K�B'S *Jd ( b & d C)K�S16� ∗( � ) = 〈1 � � � max{( � ) � � 0 � 0 � 2 + 1 | F*G [\e
( � )}〉BFE =�bed/a M /Jd a S16 0�G =�b & * "+0 d b`a�, 6 K�B *�b+5 H#" *I, 6cP�(1G BFE =�bed K�BbD b H�U * BR&)(16 b P�S * (1G#6a M /Jd a (1U#6 S�H+M =.* M = 0�G =�b & * "+0 d b`a�4N= K�B *�b+5 H#" *)4N= P1(1G BRK O-b1= E � ( =F*�bed 0 * ( � 2acbed BbP�(1K -R= MN6 / B = BTK O@b1= E � B *�bed 0 * ( � � P�B *�bed S *Jd J b1= ( � BFE =�bed a M /Jd a S16P1&)(�D�& < K�K b+* (16 Q 2 * S * B'( � " * (1U+K�B = (16 a M /Jd a S16?G�P�(�H�(�D�E � B *�bed K�B * " =� �� ( � � �� ) = 〈〈 � ∗( � ) � [(] � � 2( � � � ) � [ � ] �;:;:;:�� � 2( � � � +
� − 1) � [)]〉 �〈� 1( � ) � [ � 1]1 � [ � ]1 �;:;:;: � [ � ]1 � [ � � ]1
� [)]1〉〉 ∗ � :e * " =pb1=F* E C)B * "%P1BT&�E P * MN0�" J�S *�b1= ¬ $'&)(�D�& ( � ) Q b & a BRE =�b C - 0�(1G#K�B
� �� ( � � �� ) = 〈0 � � � �� 〉 �B a K�B *�b H+H+BRG#S1K�B = ( d�* ( D�BTD�( = S16 S *Jd|acb K�E b�b`a (�H�(1G#C�E b.* "�6 K�(1& O�, 6 〈0 � � � �� 〉 / B =BFE =�bed@a M /Jd a S16'P�&)(�D�& < K�K b+* (16 _ BTH d a�< 2�( d 0�G =�b & *I, 0�B d 6 � �� ( � � �� ) BRE =�bed P1&)( O-b1=)4 6 -T=�b ]LP�&)(16b] -T=�b 2 b P1S * ( =
(1& d 0�K�S * (1G#6 aN�@ N 2 * Z , �����=' 2 � ,Eg �@< >:.\j�=8A���@;k7Jj|7 � : N
�� N
g@?YAM1e< 1|AM1���� m ,B< >E.;BA ;0g >-1 <@,�AcAMm 1|A�g@?PA=1 <lg�� 1B�B<Hj@;=< > 1@A=1�� �`mk,l< >:. �j-X Z�<\#3��} [*2 3 d BbH b A d 0 *Jd a�<\b1=�b1/ &)(1K d a�- 6 K�BT& d a�- 6 0�G =�b & *I, 0�B d 6 BFE =�bed b1=�b ]
/ &)(1K d a�- 6 b P1S * ( $?S1& d 0�K b ! S c> 2 acbedW* (np'B 4 &I"�K b ^ b1= ( =Jd a�, 6 @ (1& O�, 6%P�&)(#]O@b1=)4 6�0�G = BbP < D�B *�bed S *Jdea�< C)B b1=�b1/ &)(1K d a�, K�BR& d a�, 0�G =)< & * "�0�" BFE =�bed BTH b A d 0 *Jd a�<b1=�b1/ &)(1K d a�, aI /)4 P1& - P1B d =�b P b & b+* "�& , 0�(1G#K�B S *Jd " acb1= ( =Jd a�, K�(1& O�, JL9 d Q 0�P <1=Jd b / E = B d
* ( = b P�( * BTH+BR0�K b+*Jd a S * BR&)( b H+D�S1& d C)K�( D d b * ( = G)P1(�H+(�D d 0�K�S K d b 6.0�G =)< & * "+0�"+6�2P1(1G 0�B.P�(�H+H - 6 P�BR& d P *)4 0�B d 6 (1&�E � B *�bed BRU a (�H b J acbed P d ( O G#0 d a�<�Q K�B # D�B =Jd a�,�&b1=�b1/ &)(1K , b G * S16 < H+H+M 0 * B BRE =�bedcacbed (%H�S�D�(16 P1(1G / B =?,#*�b1= P�&)( O@b1=)- 6 b P�S * ( =(1& d 0�K�S * "+67S *Jd "c0�G =)< & * "�0�" * (1G @ : r�V i X.Y b3b BRE =�bed BTH b A d 0 *Jd a�<�b1=�b1/ &)(1K d a�, B P�S * " = < H#H+" K�BR& d < 2 * ( p?B 4 &I"+K b N�@ c> BRE =�bed�5 b 0 d a S'D d b * " K�BTH -b* " * M = b1=�b ]
/ &)(1K d a�4 = K�BT& d a�4 = 0�G =�b & *I, 0�BTM = 2 acbedid /Jd b E * BT& b D d b�b P�( / BFE C)B d 6 ,W7 � 1@A=1�� �`mk,l< > A �;k7J;91�� 21P1(1G BFE =�bed6acbed ( d 0�"+K b1=F*Jd a S * BR&)BT6'B O@b &)K�(�D - 6 * "�6�C)BRMN&�E b 6 b1=�b1/ &)(1K d a�4 =0�G =�b & *I, 0�BRM = 2WS+P1MN6%C b / (1U#K�B 0 * ( BfP1S1K�B = ( a B O�< H bed ( �I /)4 P1BT& d (1& d � S1K b 0 * B0�B / U#(cP1(1&�E 0�K b+*�b P�(1G%BFE =�bed P b & b1/ B d D�K b+*Jd a�<c* "�6?A�& , 0�"�6 * (1G
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�( � � � ) ⇐⇒ (∃ � ) � 1( � � � ��� )
⇐⇒ $'&)(�D�&( � ) &
" b1=�b1/ &)(1K d a�, K�"+A b1=I, K�B a M /Jd a S �* BT&)K b+* E � B d 0 * " = BFE 0�( / ( �
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�( � ) =
{ � � ( � ) + 1 � b1= � ( � � � )0 � b H+H d 4 6
C b.,�*�b1=pb1=�b1/ &)(1K d a�, < & b 2�D d b a�< P�( d ( � acbed S�H bc*�b"��( � ) = �
�( � ) = � � ( � ) + 1 �
P1(1G%BRE =�bed <+* (+P1(cD d b � = � a
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� � ( �� ) =�( �� � �� ) :
_ ( b P1( *)- H�BT0�K b J P1(1G�0�G#A =)<^acb H+BFE *�bed ;=m � gM? �@79,:1 � �� Q BFE =�bed 0�"�K b1=F*Jd a S BbP�B d /I,0�G = BbP < D�B *�bed S *Jd P�(�H+H - 6 < H+H+BR6�2�P�(�H�G)P�H+( a S * BR&)BT6 acb+*�b 0 a BTG - 6 P�&)(�D�& b K�K <+* M =BFE =�bed BfP�E 0�"�6'P�&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�- 670 * (1G#6 a M /Jd a (1U+6 N�@ 9 2 *Y, Z / '�� [ J�$ b & <1/ B d D�K b�Q 2 � h ���@;=m �Jg8A . ��1|AM1���� m ,W.�g@?YAM1e< h ���@;=m �Jg �AM? � 1@A=1�� �`mk,l< >:.�j|;=m&= �^> � �J< >|m3<�� Z �i7�� 1��i. � �c<H1 >c5=g � Z = h���;�lg�<lh ���@;=m �Jg8A0? �1|AM1���� m ,B< >E. j,=iA ��-;k78j@7
� ( � � � );��k;=m <H1 h�m3= 1|A
�(0 � �� ) = � � � ( �� )�
( � + 1 � �� ) = � � +2� (�( � � �� ) ��� � �� ) �;BA ;0g
�( � � �� ) = � � +1
� (� � � )(
�� �� ) :j-X Z�<\#3��} [*2 m K�BT& d a�, 0�G =)< & * "+0�"
�(0 � � � � � �� ) = � � ( � � �� )
�( � + 1 � � � � � �� ) = � � +2( � � � ( �� � � � � �� ) ��� � �� )
BFE =�bedEb1=�b1/ &)(1K d a�, 2�BbP�B d /I, * ( R BFE =�bed:a H+B d 0 * S D d b P1&)M * (�D�B =I, b1=�b1/ &)(1K , 2 < & bb1=�b1/ &)(1K d a�, BRE =�bed@acbed "�( � � � ���� �� ) =
�( � � � � � � �� ) �
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�( �� � � � � �� ) =
�( � � � ��� � �� )
= � � +3� ( � � � ���� �� )= � � +1�
2� +1(� � � � � )
( � � �� ) �
acbed�* ( � " * (1U#K�B = ( d 0�A�U#B d K�B * "c0�G =)< & * "�0�"� ( � � � ) =
� 2�+1(
� � � � � ) : a
�+��� �p�Byl~:�*$e}��� N @ c>&2�� BRE C * B * ( � , K�K b N�@ !�2 acbed BbP1E 0�"+6cS *Jd�* ( #fBbP d P�H - ( =�& / B = d 0�A�U#B d
D d b a�< P�( d b P�&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�, a M /Jd a (+P�(�E "�0�" * (1G N∗ � N @ ! 2�� BRE C * B S *Jdia�< C)B b1=�b1/ &)(1K d a�, K�BR& d a�, 0�G =)< & * "�0�" � ( �� ) - A�B d�< P1B d &)(1G+6* ( P�H , C)(16 a M /Jd a (1U+6�2 / "#H b1/I, G)P < &)A�(1G =c< P1B d &)( d�* ( P+H , C)(16 b & d C)K�(�E � *)-b* ( d ( d
P1(1G �= � � �
� N @ N 2�� BRE C * B * " = J T Q Q � N @ Q ∗ 2�� BRE C * B S *Jd " 0�A - 0�" @ B *)<+5 b 0�" ( � � � � � ′) BFE =�bed P�&)M * (�D�B =)4 6 b1=�b ]
/ &)(1K d a�, JL3 d H�BfP * (1K - &)B d BT6 b G * (1U * (1G G)P1(�H+(�D d 0�K�(1U BRE =�bedW* S10�( P1(�H#H - 6%P1(1G/ B = BRE =�bed B O@d a�* S =�b acb+*�b D�& b1O (1U = S�H+BR6 �_ ( � " * (1U#K�B = ( BFE =�bed K d b BRU�D�H�M *F* "B C , D�"�0�" * "+6 b &)A d * B a�* ( =Jd a�, 6 * "+6 b P�S / B d CI"+6�2 acbed " acb+* BT&ID b 0�E b K�BR& d a�4N=�b P�S*Jd 6'P�BR& d P *)4 0�B d 6?P�(1G - A�(1G =7* ( K�BTD b H+U * BT&)( B =)/Jd b1O�- &)( = Q
� N @ 9 2�� BRE C * B S *Jd6a�< P�( d b P�&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�, 0�G =)< & * "+0�" � (�) / E = B dD d b a�< C)B � -T=�b1=pa M /Jd a S * "+6 * (1K , 6 * (1G @ : r�V i X.Y b3b � � ( � )
� N @ T 2�� BRE C * B S *Jd " 0�U = C)BT0�" b1=�b1/ &)(1K d a�4 = K�BT& d a�4 = 0�G =�b & *I, 0�BRM = BFE =�bedP1&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�, 0 * (1G#6 a M /Jd a (1U#6�2�K�B * " = B C , 6%J�D d b P b & <1/ B d D�K b�Q -R= ]= ( d b D d b a�< P�( d b P�&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�, 0�G =)< & * "�0�" � ( � � � � � )
2� �� ( � �
� � � )( �� ) = � 2� ( �
� �( �� ) ���
�� ( �� )) :
� N @ d-2�� BRE C * BcS *Jd G�P < &)A�B dBb1=�b1/ &)(1K d a�, K�BT& d a�, 0�G =)< & * "�0�" � ( � ) P1(1G / B =- A�B d (�H d a�,�b1=�b1/ &)(1K d a�, BfP -0a�*�b 0�"�2 / "#H b1/I, � v � / B = d 0�A�U#B d D d b acb K�E b (�H d a�,b1=�b1/ &)(1K d a�, 0�G =)< & * "+0�"
��� ��� x � !P{et(�Wz ��������� �������������
3 5`b 0 d a S16 0 * S1A�(16 K b 6�0 � b G * S * ( B /)<1O|d ( BRE =�bed�=�b BfP1B CI"+D , 0�(1G#K�B acbed J * (1G)]H < A d 0 * ( = K�BT& d a�<�Q =�b./Jd acbed (�H�(�D , 0�(1G#K�B * (cP�BR&�E O "�K�(
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�1
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2
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�: N�� N
g@?PA=1 < ==hJm��im ��? j-< ,W7 1@A >-1 <@,�AcAMmcA 1@A g@?PA=1 </1@A=1�� �`mk,l< >:. �_ ( B E * "+K b G ` J G e g�i : e1]a` g�i Zcb h ` e�VF`fZ ` Q�/Jd b+* G�P 4 CI" a B b P1S * ( = B K�BR& d acb1= S
@ [ _Pb �F_ G e g�i : e acbedN* (�� &)B *�b1= S @ [\Y�b7` g�i Zcb h * ( > ] N T�2 b1= B C < & * " *�b 2 acbed 0�B/Jd b1O (1&)B *Jd a�- 67K�(1& O�- 6 m 0�G = BbP b D�MND ,
� b1=�b1/ &)(1K d a�, � ⇒ � G)P1(�H+(�D�E 0 d K�"D d bc* " = * G#A b E b K�BT& d a�, 0�G =)< & * "�0�" � BRE =�bed " ;0gk; �|< ,2, �8Ak7p>c1c;0g3<853=8A=j@7 * (1G B d ]*I, K b+* (16�2 acbed 2WD�B =Jd a�< 2 C)BTM &)BRE *�bed P1&)( O-b1=I, 6�2 acb+* BTG+C)BRE b1= b P1S * ( = (1& d 0�K�S * "+6b1=�b1/ &)(1K d a S * " *�b 6 * (1G�H < A d 0 * ( = 0 , K�BT& b J b1= S1A d�* ( > ] N T Q BFE =�bed BRU a (�H�( =�bB a H <+5 (1G#K�B *Jd 6 b1=�b1/ &)(1K d a�- 6 K�"�A b1=)- 6 * (1G (1& d 0�K�(1U ! E' 9 MN6 # d / B b+*)- 6 B a�/ (1A - 6 &"+H+B a�* &)( =Jd a�4N= G)P1(�H+(�D d 0 *)4N= 2 ,\acbed MN6 P�& b D�K b+*Jd a (1U+6 G�P�(�H�(�D d 0 *)- 6 K�B P�&)S#]0 5`b 0�" 0�B # b P1BT& d S1& d 0 * " K =I, K�" & Rm 0�"+K b 0�E b * (1G bed *I, K b+* (16 5 &�E 0 a B *�bed 0 * " =m3=�j-< 1ej@;=< >:. * "+6 >-16;0g3<i5&=iAMj|7 2
� G�P�(�H�(�D�E 0 d K�" � ⇒ � b1=�b1/ &)(1K d a�, :_ ( B E * "�K b G e g�i : e1]a` g�i Zcb h / B = BfP d /)- A�B *�bedcb G+0 * "�& , b P�S / B d CI"�2�BfP1B d /I, *�b G * E ]
� B d�* " /Jd bed 0�CI" *Jd a�, -T=)= ( d b * "�6 = h�m �8m �c<Hjc< ,�A ;k7�;91�� K�B * " = b G+0 * "�& , JLK b CI"+K b+*Jd a�,1Q-R=)= ( d b.* "+6 1|AM1���� m ,B< >+A`;k7J;91�� @ B < H#H b H�S�D d b 2 / B = K�P�(1&)BFE =�b BRE =�bed 5=gM? �@79,:1 2-b* 0 d P1(1G b P�S * " = acb C b & < K b CI"�K b+*Jd a�, 0 a (+P d <?- A�B d#* " = G)P1S10 *�b 0�" m��@<HjJ,@m&< ; d b=�b acb+*�b H <+5 (1G#K�B * ( = S�"�K < * "�6�P�& - P�B d�=�b B C)B *)< 0�(1G+K�B * (c&)S�H�( P1(1G7P b E � (1G = ( d(1& d 0�K�(�E�0 *�b K b CI"+K b+*Jd a�<�� P�M 6 �8< >-1 < m��im ��m&<iA ;91e< acbed jeg ;=<,� �@7Jj-< ,Eg3<�m3=8A ; d b?-T=�b�a H b 0 d a S P b & <1/ B d D�K b 2 * ( g ,��E1��&A6A K d b 6 J b 6 * " = P�(1U+K�B Q 1 h��@. � h8g �@< m �
� . � � a�<+* M b P1S * ( D�& <1O "+K b K d b 6 C)B *Jd a�, 6�2�0�G = BRA�(1U#6 0�G =)< & * "�0�"�6 � (1&�E � B *�bed
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d ! U 2=VvX ������0P��������Z /�[&/ ';6�'&�\'&2&' X ��6 , ��������Z /�[&/ 'M 6?(�H�( a H , &)MNK b 2
I K 5 ( � ) =
∫ �
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�( � ) � �(:J T�] Q
3 (1& d 0�K�S16�2 5�-b5`bed b 2 / B = BFE =�bed8b G#C b E &)B * (16 � b1= BRE A�BNBfP d H+(�D , 2#( a�< C)B�P�&)M * (1B *I, 6O ( d * " *I, 6 * M = K b CI"+K b+*Jd a�4 = C b - C)B * B
I K 5 ( � ) = 2D d b a�< C)B P1BT& d (1A , � 2 / E = ( =F*�b 6?BTU a (�H#" J acbed H b1= C b 0�K -T= " Q H+U+0�"%0 *�b K d 0 < P�&)(#]5 H , K b+*�b * (1G (�H+( a H+"+&)M *Jd a (1U H+(�D d 0�K�(1U � m b C�E b * (1G 0�M 0 * (1U (1& d 0�K�(1U J T�] QBFE =�bed S *Jd�/ E = B d�* " 0�MN0 *I,c*Jd K , D d b *�b 0�"�K b1=F*Jd a�< P b & b1/ BRE D�K b+*�b 2�P A 2 b P � b G)]* S = G)P1(�H+(�D�E � (1G+K�B%S *Jd�* ( BTK 5 b1/ S = a U a H+(1G K�B b`a�* E =�b BRE =�bed � 2 2 a�<+*Jd P1(1GBbP b H+"+C)BRU#B *�bed K�B?K�B * & , 0�B d 6 @ B < H+H b H+S�D d b 2N" P�& 4W* " b P b E * "+0�" b P�S -R=�b K b CI"+K b+*Jd a S (1& d 0�K�S a�< P�( d b 6/Jd bed 0�CI" *Jd a�, 6 -R=)= ( d b 6 BRE =�bed S *Jd P1& - P1B d =�b j,=9, ���BA0g@?l,Eg(;91 m < >-g@?H1 h�1��`1��kg@? � �,:16;91 2 b G *)< P1(1G b P1( / E / (1G = B =)/Jd b1O�- &)( = 0 * " =?-T=)= ( d b ; d b * ( B E * "�K b G `?2 b G * SB a�O & <%� B *�bed M 6
> 65 g >��e1 j-< >E. Z ==hJm��im ��? j-< ,W7 j,=iA ��-;k78j@7 g@?PA=1 </1@A=1�� �`mk,l< >:. ZJ d 8 Qacbed 2?0 * " = BbP�(1A , K b 6 P d b 2 / B = b K O@d 0 5 " * BRE *�bed� P - & b b P1S * " K�BbD < H+" H1E 0 *�bJ P�&)M * (�D�B =)4 6 Q�b1=�b1/ &)(1K d a�4 = 0�G =�b & *I, 0�BRM = 0 * ( B /)<1O|d ( > E acbed�*Jd 6 d 0�A�G#& - 6d /Jd S * " * BT6 a H�B d 0 * S * " *�b 6 * "�6 a H < 0�"+6 * M = b1=�b1/ &)(1K d a�4N= K�BR& d a�4N= 0�G =�b & *I, 0�BTM =P1(1G - A�(1G+K�B / BFE C)B d 2�G)P < &)A�B deacbed "7BRK�P�B d &�E b 0�A�B / S = B 5�/ (1K ,�=F*�b A�&)S = M = - &)BRG =�b 6P1(1G / B = b P -T/ M 0 b1= acb1=)-R=�b # b1=F*Jd P b & <1/ B d D�K b%& � a�< P1( d b 0�G =)< & * "+0�" P1(1G =�bC)BRMN&)BFE *�bed J D�B =Jd a�<\b P1S * " K b CI"�K b+*Jd a�,(a ( d = S * " *�b�Q G�P�(�H�(�D�E 0 d K�" acbed P�(1G =�bK�" = BFE =�bed|b1=�b1/ &)(1K d a�, $?MN6'K�P1(1&)(1U#K�B S1K�M 6 =�b BFE K b 0 * B 5�-b5`bed ( d S *Jd�/ B = C b 5 &)BTC)BFE b1=F*Jd P b & <1/ B d D�K b
* (1G B d *I, K b+* (16 G ` 0 * (.K - H#H�( = � B 6?B C)B *)< 0�(1G+K�B * " = a H b 0 d a�, H+U+0�" * (1G b1=F* E ]0 * ( d A�(1G%P�&)( 5 H , K b+* (16 D d b.* " =%-R=)= ( d b * (1G.BRK 5`b1/ (1U�2�P1(1G 5 b 0�E � B *�bed 0 *Jd 6 B C , 6* &)B d 6 5`b 0 d a�- 6 # /Jd bed 0�C , 0�B d 6 & D d b�b P�H - 6 P1BT& d (1A - 67S+P1MN670 * ( e A , K b Q J�Y Q _ ( BTK 5 b1/ S = (1&)C)(1U P b & b H+H#"+H+S�D�& b K�K�(1G K�B P+H�BTG+& - 6 � acbed � BRE =�bed * (
D d = S1K�B = ( � � JU� Q B = �
1 ⊆ � 2 / "+H b1/I, " b P+H , P�BR& d (1A , �1BRE =�bed K - &)(16'K d b 6 < H#H+"+6 � 2 * S * B
I K 5 ( � 1) ≤I K 5 ( � )
J :)Q B = " b P�H , P1BT& d (1A , � BRE =�bed " # -R= MN0�" &'/ U#( b P�H 4 = P1BT& d (1A 4 = �
1acbed �
2JLS+P�M 6?0 * ( � d < D�& b K�K b Q Q 2 * S * B I K 5 ( � ) =I K 5 ( � 1) +
I K 5 ( � 2)
B P � b G *)< *�b * &�E b # b C d 4 K b+*�b%& D d b * " =c-R=)= ( d b * (1G BRK 5`b1/ (1U b P�H 4 = P�BR& d (1A 4N= 2( * U�P�(16 J TP] Q K�P�(1&)BFE =�b b P1( / B d A * BFE b G+0 * "�& <?/ BRE A = (1G+K�BI2 / "#H b1/I, 2�S *Jd ( J T�] Qg@?YAM1e<Em ,�AcAMm��p; �-A hJm�� 1|A c5=g=j|7 � h � 1 �k,21c;=< >2m&<�1��@<Y5 ,@m&< I K 5 ( � )
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m C)BRK�BbH d b`a�, b1=)< H+G+0�" * "+6 -R=)= ( d b 6 * "+6 = h�m �8m �c<Hjc< ,�A ;k7�;91�� P�(1Gc( / "#D�BFE 0 * " =b1=)< H�(�D�" /Jd acbed (�H+S�D�"�0�" * (1G B d *I, K b+* (16 G ` / S1CI" a B b P1S * ( = ` g�i Zcb h 2 P�(1G C)Bf]a E = "�0�B?K�B * " =pb P+H , acbed P�&)( O@b1=I,c/Jd b E 0�CI"�0�" 6 S *JdG)P1(�H+(�D d 0 d K�S * " *�b =
K�"�A b1=Jd a�, G�P�(�H�(�D d 0 d K�S * " *�b,:J do> Qe * " 0�G =)- A�B d b 2N( ` g�i Zcb h S1& d 0�B b G#0 * "+& < K d b 0�G#D a B a & d K -R= " a H < 0�" b1O "+&I"�K - ]= M = J acbed P�(�H�U b P+H 4N=�Q K�"�A b1=)4N= P�(1Gc0 , K�BT& b O�- &)(1G = * ( S = (1K <c* (1G�2 ,W7 �E1@A����� �eKMF D _ -R/ MN0�B d 0�A�G+& < BfP d A�B d & , K b+*�b S *Jd m < = h�m �8m �c<HjJ,@m-?2meh�me< 1ej�`.JhJm`;0g � >-16;91 �j >-g3= j-< ,W7 � ��,W7 �E1@Ak. � �0h � m j-m ,@m < ?BAMmcA ;91e< � 1eh3A >eh�me< 1 ,/7 �:1|A . � � KMFHD`_ acbed 2* BTH d a�< 2 /Jd b+* U�P�M 0�B * " = B d acb 0�E b S *Jd D d b a�< C)B 0�G =)< & * "�0�" � d 0�A�U#B d " d 0�( / G =�b K�E b" � BFE =�bed G)P1(�H+(�D�E 0 d K�" ⇐⇒ " � BRE =�bed G�P�(�H�(�D�E 0 d K�" b P1S.K�"�A b1=I, ` g�i Z b h :
; d b =�b 0�G#K�P�H#"�& 4 0�(1G#K�B * " = d 0 * (1&�E b 2�( G e g�i : e BFE A�B ,+/ "cP�&)( * BFE = B d H�E D�(.P1&)M�]* U * BT& bc* ( # b E * "�K b%& J ` e�VF`fZ ` Q S *Jd 2�D d b a�< C)B � 2
" � BRE =�bed G�P�(�H�(�D�E 0 d K�" ⇐⇒ " � BFE =�bed � ] (1&�E 0 d K�" �S+P1(1G # � ] (1&�E 0 d K�" & 0�"�K b E = B d #fG�P�(�H�(�D�E 0 d K�" b P�S a�< P1( d ( = S1&)( J P1&)S�D�& b K�K b�Q & * "+6* G�P d a�, 6?D�H 4 0�0 b 6 * (1G � ] H+(�D d 0�K�(1U P�(1G%BFE A�B?B d 0 b D < D�B d 3 d2d 0�( / G =�b K�E BT6" � BRE =�bed ` g�i Z b h G�P�(�H�(�D�E 0 d K�"
⇐⇒ " � BFE =�bed � ] (1&�E 0 d K�"⇐⇒ " � BRE =�bed|b1=�b1/ &)(1K d a�,
b P1( / BRE A * " acb1= 0�A�B / S =^b K - 0�MN6 K�B *)< J b P1S * (1G+6 ` g�i Zcb h acbed � [ VFVAb�V Q 2 acbed�-Ma H+B d ]0 b1= * ( = a U a H+( P�(1G ( /I, D�"�0�B 0 * " /Jd b+* U�P�M 0�" * (1G B d *I, K b+* (16 G e g�i : e1]a` g�i Zcb h0 * "cK�(1& O�, N�E' c>� _ ( b1= " b1=)< H+G+0�" * "�6 #bK�"+A b1=Jd a�, 6 G�P�(�H�(�D d 0 d K�S * " *�b 6 & P1(1G -T/ M 0�B.(n` g�i ]
Zcb h b1=)- &)A�B *�bed 0 * ( U ��(16cP+H , &)(1G+6 /Jd acbed (�H�S�D�"+0�"+6 * (1G B d *I, K b+* (16 G ` J�S+P1MN6" b1=)< H+G+0�" P�(1G 0 a-d b D�& b1O�, 0 b K�B /Jd acbed (�H�(�D�BRE * " = (1&)C)S * " *�b * (1G (1& d 0�K�(1U * (1GBRK 5`b1/ (1U.D d b b P+H - 67P1BT& d (1A - 6 Q BRE =�bedEb K O@d H�BbD�S1K�B = ( 3�P1MN0 /I, P1( * BI2 / B = C b * " =BbP b1=�b H <+5 (1G+K�B B /)4 2 b1= acbed BRE =�bed@b C d (1C b U#K b 0 * ( * ( P�S10�( P1B d 0 *Jd a�, "�A�BRE b`a S1K�"0 , K�BR& b 2�U+0 * BR& b b P1S * S10 b A�&)S =Jd b acbed S�2 *JdW- A�B d BbP d * BRG#A�C)BFE 0 * ( K�B *�b C)U 0 * " =P�H#"�&)( O (1& d a�, B =F* � b G * (1U�21C b / BRE C)(1G+K�B�S *Jd " K�"+A b1=Jd a�, G�P�(�H�(�D d 0 d K�S * " *�b K d b 6K�BR& d a�, 6 0�G =)< & * "�0�"�6 � 0�G = BfP < D�B *�bed * " = b1=�b1/ &)(1K d a S * " *�b * "+6 � 2 B O S10�( = "/ (10�K -T= ".K�"+A b1=I,�d acb1= (+P�( d BFE b 0�C)B =)- 0 *�b+* BR6 #b0�G = C ,8a BR6 acb+*�b 0 a BRG b 0 d K�S * " *�b 6 & 2
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d Q U 2=VvX ������0P��������Z /�[&/ ';6�'&�\'&2&' X ��6 , ��������Z /�[&/ 'P1(1G d acb1= (+P�( d (1U =F*�bedcb P1S%S�H b *�b #fK�( =F*)- H b G)P1(�H+(�D d 0�K�(1U & P�(1G - A�(1G = B d 0 b A�C)BRE 2acbed JLS+P�M 6.C b G�P�(10 * "+&�E C)(1G+K�B Qpb P1S S�2 *Jd #fK�"�A b1=)- 6 & C b�acb+*�b 0 a BRG b 0 * (1U = 0 * (K - H#H�( = N�E' ! 2'�������������� �� �(7��1� �������� ����� K d b 6 b1O "�&I"+K -R= "+67K�"�A b1=I, 6%JL! E' Q Q
M = (� ��� � → � Z b 0 g d � _ g d 0 g d )K�B?0�U = (�H b B d 0�S / (1G N
� acbed B C)S / (1G NBRE =�bed (+P1( d b1/I, P1( * B a M /Jd a (+P�(�E "�0�" J -T=�b ]
P1&)(16b] -R=�b 0�G =)< & * "+0�" Q :�
� N* M =�acb+*�b 0 *)< 0�BTM = * "+6 M 0 * (1G#6 b & d C)K�(1U#6�2 *)-b* ( d b P�(1G ( d B C , 6 0�G =�b & *I, 0�B d 6�20�A - 0�B d 6 acbed 0�U = (�H b.=�b BFE =�bed|b1=�b1/ &)(1K d a�<
�� = [ � ] = { � | (∃ � ∈ � )[ ( � ) = � ]}
�� = [ � ] = { � | (∃ � ∈ � )[ ( � ) = � ]}
� → �� ′ ⇐⇒ � � � ′ ∈ � � & −1( � ) → −1( � ′)
⇐⇒ (∃ � � � ∈ � )[ ( � ) = � & ( � ′) = � ′ & � → � ′]Zcb 0 g d� ( �� ) = ( Zcb 0 g d ( �� ))
_ g d 0 g d� ( � ) =
{ _ g d 0 g d( −1( � )) � b1= � ∈ �
��
0 � b H#H d 4 6 :N�E' N 2�� # � ,3[ �=' 2 � c5=g ,Eg �@< >:. j,=iA ��-;k78j@7 h�m3= = h�m �8m �,? � gk;91e< 1eh3A(1��e7 �-7 �, �8A 7�,W7 �E1@Ak. h�m3=pg�h`< ���>�Egk;91e</1@A=1�� �`mk,l< >:. > � �J< >|meh�m ?P78j@7 g@?YAM1e<B1|AM1���� m ,B< >E.��j-X Z�<\#3��} [*2 m b P�S / B d CI"cBFE =�bed�-R=�b K d a &)S.K - &)(16 * "+6 b P1S / B d CI"�6 * (1G�p?BTM & , ]
K b+* (16 ^ b1= ( =Jd a�, 6 @ (1& O�, 6 N @ c>� p -b* (1G#K�B� ( �� ��� ) ⇐⇒ ( � BRE =�bed@a M /Jd a S167G)P1(�H+(�D d 0�K�(1U 0 * " = BFE 0�( / ( ��
⇐⇒ � V �( � ) & ( � )0 =
Z b 0 g d� ( �� )
& (∀ F*G [\e( � ))[ F + 1 G [\e
( � ) � ⇒ ( � ) � → � ( � ) � +1]
& ( � ) [\e ( )−· 1 ∈ ��:
m 0�A - 0�" � ( �� ��� ) BRE =�bedib1=�b1/ &)(1K d a�,^b P�S * " = G�P�S1C)BR0�"�2 acbed8b1= " / (10�K -T= " K�"+A b1=I,G�P�(�H�(�D�E � B d�* ".K�BT& d a�, 0�G =)< & * "�0�" � 2 * S * B�( �� ) =
_ g d 0 g d�
((� � � ( �� ��� )
)[\e
( )−· 1
) �-b* 0 d P1(1G " � ( �� ) BFE =�bed|b1=�b1/ &)(1K d a�, am d /)-Mb *)4 & b BRE =�bed S *Jd8b1= " K�"�A b1=I, M JLK�BN0�U = (�H b B d 0�S / (1G acbed B C)S / (1G *�b N
�acbed N Q BFE =�bed # acb+*�b 0 a BTG < 0 d K�" & 2 * S * B ( d:acb+*�b 0 *)< 0�B d 6 * "+6 M P�& - P�B dN=�b BFE =�bed# P�BbP�BR& b 0�K -R=�b%& b1=F*Jd a BRE K�B =�b 2'P�(1G K�P1(1&)(1U = =�b�a M /Jd a (+P�( d "�C)(1U = b P+H < 0 * (1G#6O G#0 d a (1U#6 b & d C)K�(1U+6 JLS+P�M 6 0 * " =\b P�S / B d CI" * (1G N�@ c> Q 2 -f* 0 d P�(1G ( d�5`b 0 d a�- 60�A - 0�B d 6 =�b BRE =�bed|b1=�b1/ &)(1K d a�- 6�2�K�B < H+H b H�S�D d b%* ( b E * "�K b
a�< C)B acb+*�b 0 a BTG < 0 d K�"cK�"+A b1=I, BfP d /)- A�B *�bed2b1=�b1/ &)(1K d a�, a M /Jd a (+P�(�E "�0�"
����� �!�#"�$&%('*)�+!,!-!+/. �/0!"�$1 �/2 3!4/+!5/670 28�9#:�+/; +/<���,8��5!=#> "/>�2? "&@A4/+/0 2#>�2 4#0/>���0!6CB#0�3/+!,/"ED�:A4/=!- FG�H4/+�DI<�F/5!�#>�+J; � K#"('L�MJN FO:!>AF/5/.�4�PH+!9�D ?/Q!Q R D ?SLGTUM!V ' V
U�� 2 ��������������6�Z�5�� X ������0P������Z�576�'&�8'&2&' X ��6 , ��������Z /�[&/ ' d 9acbed " 5 b 0 d a�, P b & b+*I, &I"�0�" BRE =�bed S *Jd * ( B E * "�K b G ` 0�G =)< D�B *�bed b G+0 * "�& <�b P �b G * S * ( b E * "+K b acbed�* " 5`b 0 d a�,c/Jd b E 0�CI"�0�" * (1GY` g�i Z b h J d9> Q
��� ��� � �l�:x �:}Yy �E� ��%:x �|x(' }��*���E� y/zl} z(�cz+%:x:y/w@}�� }/���2{et@{cz_ ( B E * "+K b G ` BfP d acb H+BFE *�bed 0�G+A =)< b H#H < S1A d (1G+0 d b 0 *Jd a�< D d b * " = b P1( O G#D ,b G+0 * "�& 4N=�b P�( / BFE C)BTM = 2 / "#H b1/I, G�P�(10 * "+&�E � (1G+K�B7S *Jd@a�< P1( d b K�BR& d a�, 0�G =)< & * "+0�"
BFE =�bed b1=�b1/ &)(1K d a�, A�MN&�E 6 =�b * ( b P�( / BFE C)(1G#K�B)2 K�S = ( K�B * " /Jd bed 0�CI" *Jd a�, P�BR& d ]D�& b1O�, a�< P�( d (1G b H+D�(1&�E C)K�(1G P�(1G * " = G)P1(�H+(�D�E � B d 3 d # b K b & * MNH - 6 & J * BTK�P - H d ]a BR6 Q BbP d a H , 0�B d 6 b G * (1U * (1G'BFE / (1G#6�K�P1(1&)(1U = 5�-b5`bed b =�b b P1( O BRG#A�C)(1U = K�B a�< P�( d bBbP d P1&)S10�C)B * ".BR&ID b 0�E b 3 d 0�"�K b1=F*Jd a�- 6?B O-b &)K�(�D - 6 * (1G B d *I, K b+* (16 G ` BRE =�bed 0 * " = b P�S / B d CI" 1��6A 7J;=< �>|?BA 1 hJm`;0g��`g=jJ, 6; �BA 2 1|A = h�m �8m �c<Hjc< ,�A ;k7�;91�� J b�_�b1] : _#X 0 g dbY ��Z [\Z d ^ Q 0�G =�b & *I, 0�BTM =
, 1@A=1 hJmJ>��@<Hjc< ,�A ;k7�;91�� J g b3w�V : Zcw�Y!��Z\[ Z d ^ Q 0�A - 0�BRM = 2�0�G =I, C)MN6 0 *Jd 6 H - C)B d 6 b P�Sa�< P1( d ( P�BbP�BR& b 0�K -R= ( b H O�<+5 " * ( ; d b * " = BRU a (�H+" /Jd b+* U)P1MN0�" *)-b* ( d M =�b P1( * B ]H�BT0�K <+* M = B d 0 < D�(1G#K�B * " = B C , 67(1&)(�H�(�D�E bN1G c>&2'�������������1����������������� � 0 * M Σ = { � 0
�;:;:;: � ��� } P1BfP1BT& b 0�K -T= ( J�K�"�]a B = S Q b H O�<+5 " * (�2 acbed
[ � 0:;:;: � � −1] = 〈[ � 0]
�;:;:;:�� [ � � −1]〉( a M /Jd a S16 * G#A b E b 6.H - CI"+6 � = � 0:;:;: � � −1 ∈ Σ∗ 2 S+P1(1G [ � � ] = F �m * G#A b E b
��] K�BbH , 6 0�A - 0�" J , # P�&)S 5 H#"�K b%&fQ � 0 * ( Σ∗ BFE =�bed 1eh�m8>�� ? j-< ,W7 J w�V : Z w�Y ��[ V Q ,g�h`< ��<�j-< ,W7 J�` _+[ Y!��[\V Q@b1= G)P < &)A�B dib1=�b1/ &)(1K d a�, 0�G =)< & * "�0�" � ( � 1�;:;:;:�� � � ) *)-b* ( d bP1(1G
� ( � 1�;:;:;:�� � � ) ⇐⇒ � ([ � 1]
�;:;:;:�� [ � � ]) = 1 �b H+H d 4 6'" � BRE =�bed 1|AM1eh�m8>�� ? j-< ,W7 J g b3w�V : Z w�Y ��[ V Q , 1@A0g�h3? �,=i;k7 J g b�` _+[ Y!��[\V Q _ ( 5 b 0 d a S?BR&ID b H+BFE ('D d b'* " = b P�S / B d CI" b1=�b P1( a & d 0 d K�S * " *�b 6 0�"�K b1=F*Jd a�4N= P�&)(#]
5 H#"�K <+* M = BRE =�bed ( A b & b`a�* "�& d 0�K�S16 * M =�b P1( a &�E 0 d K�M = 0�A - 0�BRM = 0 * ( Σ∗ K�B a�< ]P1( d (�# P1&)S * G)P1( 0�G#K 5 (�H d a (1U?G�P�(�H�(�D d 0�K�(1U & J Xc_ow�VF[�_���` ^�X�� _#[\Z :�: _+X 0 g dTY)dbZ _Pb Q 2< K�BR0 b 2 / "#H b1/I, A�M &�E 6 b1=�b1O (1& < 0�B a M /Jd a (+P�( d , 0�B d 6 �e * ( BbP�S1K�B = ( B /)<1O@d ( C bBbP d 0 a (+P , 0�(1G+K�B�P�BR& d H#"#P *Jd a�<.* ( a H b 0 d a S J acbed|d 0 * (1& d a�< P�& 4W* ( Q P1&)S * G)P1( 0�G+K1]5 (�H d a (1U G)P1(�H+(�D d 0�K�(1U12 *Jd 6NP1BT&�E O "+K�BR6 ,/7 �:1|A ��� ` g�i Z b h� �I /)4 C b B d 0 < D�(1G#K�B -R=�b/ BRU * BR&)( a H b 0 d a S7P b & <1/ B d D�K b P�(1G7( O BFE H+B *�bed 0 * ( = � _+` dI21( (+P�(�E (16 b P -R/ B d C)B acbed* (cP�& 4W* ( b P1( *)- H�BT0�K b b1=�b P1( a & d 0 d K�S * " *�b 6?0 *�b acb C b & < K b CI"�K b+*Jd a�< 2 - C)M b P�S* "cK b CI"�K b+*Jd a�, H�(�D d a�, acbed�* "cC)BRMN&�E b G�P�(�H�(�D d 0 d K�S * " *�b 6 N1G ! 2��p��� ������ ����������� � 98 J i V�� i Z dfZcb h , `fVFX%Z ] ` e g V?` ^�` dbVFX Q BRE =�bed / (1K ,� = (Σ ��� 0 → �
0�;:;:;:����
� →�� ) = ({ � 0
�;:;:;:�� � � } ��� 0 → �0�;:;:;:����
� →�� )�
S+P1(1G * ( 1 � � �l7J;=m ΣBFE =�bed P1BfP1BT& b 0�K -T= ( 0�U = (�H�(�2 acbed 0 * (1G+6 >-1@ABA6A0g��p,:gk; �
�E1 j|7 � � � → � � ( d�� � ��� � BFE =�bed P1BfP1BT& b 0�K -T= BR6 b`a (�H�(1G#C�E BR6 b P1S * ( Σ2 / "+H b1/I,
����� �!�#"�$&%('*)�+!,!-!+/. �/0!"�$1 �/2 3!4/+!5/670 28�9#:�+/; +/<���,8��5!=#> "/>�2? "&@A4/+/0 2#>�2 4#0/>���0!6CB#0�3/+!,/"ED�:A4/=!- FG�H4/+�DI<�F/5!�#>�+J; � K#"('L�MJN FO:!>AF/5/.�4�PH+!9�D ?/Q!Q R D ?SLGTUM!V ' V R
d T U 2=VvX ������0P��������Z /�[&/ ';6�'&�\'&2&' X ��6 , ��������Z /�[&/ 'K - H#" * (1G Σ∗ �� 6�0�G =I, C)MN6�2 acb H+(1U+K�B b G *)- 6 *Jd 6 b`a (�H+(1G+C�E BT6 ���@C=g�< � 2 acbed BfP d * & - ]P1(1G#K�B acbed�* " = >-g8Ak.����@C`7 � Y_ ( 0�U#0 * "+K b � (1&�E � B d�* " = B C , 6 �E1 j-< >E. j@� �=j|7,Egk; ��E1 j|7 � 0 * ( Σ∗ � → � � ⇐⇒ (∃ � 1
� �2������� ∈ Σ∗)[ � → � &
�=�
1� �
2 &�
=�
1� �
2]�
/ "+H b1/I, � → � � b1= "�� P b & < D�B *�bed-b P1S * " = � K�B * " = b1=F*Jd acb+*)< 0 *�b 0�" a�< P1( d (1GK - &)(1G#6 * "+6 � K�B a�< P�( d b H - CI" � 2 -f* 0 d P1(1G * ( � → � =�b BRE =�bedEacb1= S =�b 6 ; d bP b & <1/ B d D�K b 2�0 * (.0�U#0 * "+K b K�B *�b D�& b1O�, 6
� = ({ � ��� } � � → �%� � � → � � ) �- A�(1G#K�B *Jd 6?K�B *�b D�& b1O�- 6
� → � �%� → � �%�%� → � � � � � :m J P�H , &I"+6 Q j � �=j@7 ,Egk; ��E1 j|7 � * (1G � BFE =�bed " #bK�B *�b+5 b+*Jd a�, a H+B d 0 * S * " *�b%&W* "+6
→ � 2� →∗� � ⇐⇒ � → � � 1 → � · · · → � � � −1 → � �
⇐⇒ (∃ � 0�;:;:;: � � � )[ � 0 =
�
& (∀ F*G �)[� � → � � � +1 &
� � =�] �
-b* 0 d P�(1G 0 * ( P b & <1/ B d D�K b � →∗� � � � � 2 b H+H < � 6→∗� � � � J�� 0 a "�0�" � N1G c> Q -mH - CI" � BRE =�bed ��E�������1� �� 0 * ( � b1= / B = G)P < &)A�B d � *)-f* ( d b P�(1G�� → � � 2
� � = { � | (∀� )[� 6→ � � ] �
JLS+P�M 6 " � 0 * (%P b & <1/ B d D�K b�Q $ b & b+* "+&)(1U+K�B�S *Jd 0 � b G *)<%*�b D�B =Jd a�< 0�G+0 *I, K b+*�bb1=�b D�& b1O�, 6�2 / B = /Jd b A�MN&�E � (1G+K�B # * BT&)K b+*Jd a�- 6 & acbed # < D�( = BT6 & H - C)B d 6�2 / "+H b1/I, acb ]H�(1U#K�B'S�H�BT6 *Jd 6 < D�( = BT6?H - C)B d 6 * BT&)K b+*Jd a�- 6 _ BTH d a�< 2�D d b � ⊆ Σ∗ 2 * ( 0�U+0 * "�K b � BFE =�bed ��� �1�����������1� ��� � ��� ��2 b1=
[� ∈ � &
� → � � &� → � � ′] � ⇒ �
=� ′ ∈ � �
-b* 0 d P�(1G * (7P b & <1/ B d D�K b / B = BFE =�bedebed *Jd ( a & b+*Jd a S 0 * (?P+H , &)BR6 { � ��� }∗ 2 b H+H < BFE =�bedbed *Jd ( a & b+*Jd a S 0 * ( { � }∗ _ b b1=�b1/ &)(1K d a�< 0�G#0 *I, K b+*�b K�B *�b+5�< 0�BTM = T (
� � )BFE =�bed 2 5 b 0 d a�< 2�0�G+0 *I, ]
K b+*�b b1=�b D�& b1O�, 6�27K�S = ( P�(1G /Jd b A�M &�E � (1G#K�B *Jd 6 * BT&)K b+*Jd a�- 6 acb+*�b 0 *)< 0�B d 6 0�B# < D�( = BT6 &pacbed # * BR&)K b+*Jd a�- 6.P1(1G b P1( / E / (1G = *Jd K , & 2 acbed J * ( 0�"�K b1=F*Jd a S * BT&)( Q ( dacb+*�b 0 *)< 0�B d 6 BFE =�bed H - C)B d 6 b P1S -R=�b < P�B d &)( b H O�<+5 " * ( � BfP1B d /I, P�& - P�B dW=�b 0�G)]K�P1BT& d H <+5 (1G+K�B 0 * ( Σ
S�H b�*Jd 6 0 *�b C)BT& - 6�2 / "+H b1/I, S�H+(1G+6 * (1G#6 O G+0 d a (1U+6 b & d C)K�(1U#60 * " = P�BR&�E P * M 0�" =
�0P1(1G K b 6.B =)/Jd b1O�- &)B d _ ( BfP1S1K�B = (�2 * BTA =Jd a S b H+H <
5 b 0 d a S b P1( *)- H�BT0�K b P b & b`a�< K�P * B d|b G * S * ( BRK�P�S /Jd ( K�B * " = a M /Jd a (+P1(�E "+0�"� = � 11 · · · 1︸ ︷︷ ︸�
� ( � ∈ N) :J d ! Q
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U�� 2 ��������������6�Z�5�� X ������0P������Z�576�'&�8'&2&' X ��6 , ��������Z /�[&/ ' dPd
N1G N 2�� # � ,3[ �=' 2 � c5=g 1|AM1���� m ,B< >E. Z ,:g �|< >E. j�=8A���@;k7Jj|7 � : N�� N
==hJm ��im ��?��9gk;91 <W1 h�A > hJm < m j,<Jj@;k70,21 1|AM1� � 1��e. � � Z jeg >eh�me< m J P1BfP1BT& b 0�K -T= ( Q j�< �A=m��im#j�= ,��(A �;�BA
ΣhJm&= hJg �|< �e1 ,�� |A9g�< ;91 g�< �8< > �j,<9,��Em��e1 � � 1 � � � :
Z ,Eg�;k76Ag@?Hjcm��Jm�� 7→ � : � 1
�2 · · · � � �
�`7 �e1��`.�( � 1
�;:;:;:�� � � ) = � ⇐⇒ � : � 1�
2 · · · � � →∗� : � :� hi<Hh � ��m6A Z ;=m � g@?PA=1 </1 <Y;=< m8>�� 16;=< > A j@;=m j,<iAMm �8m � >c1c;91 j|; j6g��BA �
�= { � :
� | * (�� � / B = BRK O-b1= E � B *�bed 0 *Jd 6'H - C)B d 6 � � � } :J d N Qp b P b & b H+BFE ��(1G+K�B * " = b P1S / B d CI"�2NP�(1G b P bed * BFE * " = J�S1A dN* B * & d K�K -R= " Qpacb+*�b ]
0 a BTG , 0�G+0 * "�K <+* M = b1=�b D�& b1O�, 6 K�B B d /Jd a�- 6 d /Jd S * " * BR6�2 -T=�b D d b a�< C)B b1=�b1/ &)(1K d a SP1&)S�D�& b K�K b N1G Q 2 � � ����� �������� ��������������8 � ���E@7�� ����� ��W���� ����E18� ; d b(a�< C)Bc0�U)]
0 * "+K b�b1=�b D�& b1O�, 6� = (Σ � � 0 → � 0
�;:;:;: � � � → � � )J dHQ Q0 * ( 0�U = (�H+( H - C)BTM = Σ∗ b P�S * ( P1BfP1BT& b 0�K -T= ( b H O�<+5 " * ( Σ
2 - 0 * M ∼ � "�BbH < A d 0 * "0�A - 0�" d 0�( / G =�b K�E b 6?0 * ( Σ∗ *)-f* ( d b P1(1G>#
� � ∼ � � � 2�D d b F ≤ ��2! � ∼ � � � ⇒ � � � ∼ � � � � 2�D d b S�H b%*�b � � � ∈ Σ∗ 2
acbed D d b a�< C)B � ∈ Σ∗ 2 - 0 * M
[[ � ]] = { � ∈ Σ∗ | � ∼ � � }" > ��ej@7 < j-m��-=8A=1 ,H? 1�� * (1G �
\_ (.0�U = (�H�( * M = a H < 0�BRM =�d 0�( / G =�b K�E b 6[[ � ]] = {[[ � ]] | � ∈ Σ∗}
BFE =�bed " 79,B< mk, ��J1 J�` VFXcZ h#i _ g 0 Q P�(1G P b & < D�B *�bed b P1S * ( � 2 acbed " ∼ � acb H�BRE ]*�bed " j@� �=j|7 , * ( h���A ��|79,:1�< j�A`;k7J;91���� �@CMg��BA J � _ i w 0 i _���[ VFX Q D d � b G *I, * " ="�K d (1K <1/�b 3 d S1&)( d BRE =�bed 2�P1&)( O-b1=)4 6�2 b P1S * " =�< H+D�B 5 & b�acbed K�P1(1&)(1U =�=�b?/Jd acbed (�H�(�D�"+C)(1U = 2b H+H <./ B = C bc* ( a�<1= (1G#K�B b G * S.B /)4 N1G 9 2�� # � ,3[ �=' JU� X%Z\[ � _#` d Q 2 Eh �� �Eg�<cj�<�j|;k79,:1 1|AM1� � 1��e. � � ;��k;=m < m hJm&=7 j@� �=j|7
� →∗� ���g8Apg@?YAM1e< 1eh�m8>�� ? j-< ,W7 Z >c1e<^;=m�h���A ��|79,:1 <Hj,A ;k7�;91�� � �@CMg��BA
� ∼ � ��c<H1 ;k76Ap70,l< m , ��81
[[ � ]]g@?PA=1 </1@A0g�h3? �,=i;=m �
j-X Z�<\#3��} [*2�� 0 * M � � = (Σ � � 0 → � 0�;:;:;:�� � � → � � )
-R=�b 0�U+0 * "�K b b1=�b ]D�& b1O�, 6'P�(1GcG)P1(�H+(�D�E � B d�* " =pb1=�b1/ &)(1K d a�, 2�K�BT& d a�, 0�G =)< & * "�0�"
�( � ) = 0 · � � � 1(
� � � ��� )
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d�O U 2=VvX ������0P��������Z /�[&/ ';6�'&�\'&2&' X ��6 , ��������Z /�[&/ 'b P1S * (�p'B 4 &I"�K b NPG N 2 -b* 0 d P1(1G
�( � )↓ ⇐⇒ �
: � →∗� :� D
- P�B *�bed S *Jd "�0�A - 0�" � →∗� � / B = BFE =�bed8b P�( a &�E 0 d K�"�2�D d b+* E b1= ,#*�b1= 2 * S * B J�BRU a (�H b�Qacbed�* (%P1B / E ( (1& d 0�K�(1U { � | � ( � )↓} C bc,#*�b1=pb1=�b1/ &)(1K d a S�2�P1(1G / B = BRE =�bed 2 N @ Q� ; d b.=�b 0�G+K�P�BR& <1= (1G#K�B'S *Jd "c0�A - 0�" ∼ � BbP1E 0�"+6 / B = BRE =�bed|b P�( a &�E 0 d K�"�2�P b & b ]
* "�&)(1U#K�B P�& 4W*�b S *Jd� ∼ � � ⇐⇒ (∃ � 0
�;:;:;:�� � � )[ � = �0 & � � = �
& (∀ FHG �)[ ��� = ���
+1 ∨ ��� → � ��� +1 ∨ ��� +1 → � ��� ]] �BRU a (�H b 2�0 * " = acb+* BRU#C)G = 0�" ⇒ b P1S * ( = (1& d 0�K�S * "�6 ∼ � J�BbP�B d /I, " 0�A - 0�" 0 *�b/ B C d < BFE =�bed 0�A - 0�" d 0�( / G =�b K�E b 6 Q 2 acbed 0 * " =�acb+* BRU#C)G = 0�" ⇐ K�B BbP b D�MND , 0 * ( � � P1B *�bed S *Jd|b & a BRE =�b./ BFE C)(1G#K�B?S *Jd J�K�B�� S+P�M 6?0 * " = J d N Q QJ d 9 Q (∀ F*G �
)[ ��� = ���+1 ∨ ��� → � ��� +1 ∨ ��� +1 → � ��� ]
& �0 ∈ � & � � = :
�
� ⇒ �0 = :
� ∨ �0 →∗� :
� �P1(1G%0�G = BfP < D�B *�bed S *Jd D d b � ∈ ��2
� ∼ � :� ⇐⇒ � = :
� ∨ � →∗ :�
acbed6b P1( a H+BFE B d�* " = b P�( a & d 0 d K�S * " *�b * "�6 ∼ � S+P�M 6 acbed P�& d = \_ BbH d a�< 2 / BRE A = (1G+K�B* " = J d 9 Q K�B'BfP b D�MWD , 0 * (��W2 acbed K�B * B * & d K�K -T= " 5�< 0�"�2 b1O (1U D d b � = 0
" J d 9 Q/ "+H 4 = B d S *Jd �
0 = :� � ⇒ �
0 = :� _ ( BbP b D�MND d a S 5+, K b BFE =�bed BfP�E 0�"�6
* B * & d K�K -T= ( b1= D d b�a�< P�( d ( F 2 � � = � �+12 , b1= 2�D d b a�< C)B F�G �W2 � � → � � � +1
B P�S * " = < H#H+" K�BT& d < 2�" acb+*)< 0 *�b 0�" :
� BRE =�bed�* BT&)K b+*Jd a�, 2 < & b / B = K�P1(1&)BRE =�bd 0�A�U#B d " ���+1 → � ��� D d b�a�< C)B F 2 acbed -b* 0 d b P1(1K -T= B d " P1BT&�E P * MN0�" P�(1G D d ba�< P1( d ( , ���c<Hj@;=m F 2
���+1 → � ��� �acbed BfP1(1K -T= M 6�2�BbP1E 0�"+6�2
���+1 → � ��� +2
�acbed@b P � b G *)<%*�b./ U#( acbed�* " =pbed *Jd ( a & b+* E b%* (1G � 0 * (�� - P�B *�bed S *Jd ��� = ���
+22
P1(1G7K�B * " = BfP b D�MWD d a�, G�P�S1C)BR0�" 0�G+K�P+H+"+& 4 = (1G =�* " =�b P�S / B d CI"�21D d b+* E�K�P1(1&)(1U#K�B=�b�b1O-bed & - 0�(1G+K�B * ( � �
+1b P1S * " / (10�K -R= " b`a (�H�(1G#C�E b a
B P�S *�b P1(�H#H < K b CI"�K b+*Jd a�< P1&)( 5 H , K b+*�b P�(1G - A�(1G = b P1( / B d A * BFE b1= BfP�E H+G *�bb1=�b1O�- &)(1G+K�B'B /)4 K�S = ( / U+( N1G T 2 � � ����� ���1���� ��������������8 � ���E@7�� ����� �� ����E@8 J dbe�V ��_ i w 0 i _���[ VFX
� _ i�h#i _ g 0�` Q ; d b a�< C)B P1BfP1BT& b 0�K -T= ( b H O�<+5 " * ( Σ = { � 0�;:;:;: � � � } 2 - 0 * M
Σ � = Σ ∪ { � −10�;:;:;: � � −1� }
S+P1(1G *�b � −10�;:;:;:�� � −1� acbed = (1U#&ID d b 0�U#K 5 (�H b acbed D d b�a�< C)BN0�U#0 * "+K b�b1=�b D�& b1O�, 6
� = (Σ � � ( � 0� � 0)
�;:;:;:�� ( � � � � � ))
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U�� 2 ��������������6�Z�5�� X ������0P������Z�576�'&�8'&2&' X ��6 , ��������Z /�[&/ ' d ]
0 * ( Σ∗� 2 - 0 * M ' � "cBTH < A d 0 * "c0�A - 0�" d 0�( / G =�b K�E b 6?0 * ( Σ∗� *)-b* ( d b P�(1G>#
� � ' � � � 2�D d b F ≤ ��2! � � � −1 ' � � 2�S+P�(1G � " a B =I, H - CI"�2N�
� ' � � � ⇒ � � � � � ∼ � � � � � � 2�D d b S�H b%*�b � � � ∈ Σ∗ 2acbed D d b a�< C)B � ∈ Σ∗� 2 - 0 * M
[[ � ]] � = { � ∈ Σ∗� | � ' � � }" > ��ej@7 < j-m��-=8A=1 ,H? 1�� * (1G �
\_ (.0�U = (�H�( * M = a H < 0�BRM =�d 0�( / G =�b K�E b 6[[ � ]] � = {[[ � ]] � | � ∈ Σ∗� }
BFE =�bed " m , ��81 J h�i _ g 0 Q P�(1G P b & < D�B *�bed/b P1S * ( � 2 acbed " 0�A - 0�" d 0�( / G =�b K�E b 6' � acb H+BFE *�bed�* (%P1&)S 5 H+"+K b <Hj,A ;k7�;91�� ���@C=g��BA D d � b G *I, * " = (1K <1/�b � P�M 6 acbed D d b?*Jd 6 "�K d (1K <1/ BT6�2�( d S1&)( d P1&)( O-b1=)4 6 P�&)( - &)A�( =F*�bedeb P�S * " =�< H+D�B ]
5 & b acbed K�P1(1&)(1U = =�b /Jd acbed (�H+(�D�"�C)(1U = 2 b H+H < / B = C bc* ( a�<1= (1G#K�B b G * S.B /)4 N1G d-2�� # � ,3[ �=' J �� �� _ Z r+_ 2�� �E _1_�b Q 2 lh���;�lg�< j�<�j|;k79,:1 1@A=1 ���`1��e. �
� ;��k;=me< m(hJm&= ;=m(h �-A � �@70,21 <Hj,A ;k7�;91�� � �@CMg��BA��-< 1�;k7eApmk, ��J1[[ � ]] �
g@?PA=1 <W1|A9g �h�? ��=8;=m �B G * S * ( C)B 4 &I"�K b - A�B d 0�"+K b1=F*Jd a�< P1(1&�E 0�K b+*�b D d b * " C)BTM &�E b (1K <1/ M = acbed
* " = b H#D�B 5 & d a�, * (+P1(�H+(�D�E b 2 d /Jd b E * BR& b�* (�# P1&)S 5 H+"+K b�*�b C d = S1K�"�0�"�6 &�* (+P1(�H+(�D d a�4 =P1(�H#H b P+H�( *I,�* M = P�BbP�BR& b 0�K -R= "+6 /Jd < 0 *�b 0�"+6 N1G O 2�� # � ,3[ �=' J � � � � � � ��� ���1���� ������� ���� ��� �12 +@m h���A ��|79,:1 1|A��Jm �j8, �8A=m � < m��@1|A ;=< > A h�m �,=�?BA =9,-m
� ( � 1�;:;:;: � � � ) =
∑
� 1+···+ � � ≤ � � 1
� � � �%�� �� � 11 · · · � � ��
j6g � ,Egk;91�� �@7J;���� ,Eg�j,=iA ;0g��`g=j@;�����j|;=mZ = { :;:;: � −1 � 0 � 1 �;:;:;: } �>�lg�<l18> � � 1 < g��� ?��9g�� g@?YAM1e</1@A0g�h3? �,=i;=m �
_ ( C)B 4 &I"+K b b G * S b P1( / BRE A * " a B b P1S * ( =�� � � Y�dfZ � Y#` V Z : e 2'K�B *)< b P�SBR&ID b 0�E b7* M = �'Z [ Y i ^ � g d b�Y#X 2 � g [ Z Y���_ ��Zcb�` _Pb acbed � Y i dbZ bvS?Y Z\`R2 acbed E 0�MN6 P�Bf]& d 0�0�S * BR&)( b P1S a�< C)B < H#H�(�21BTK�P -T/ M 0�B * " C)BTM &�E b�b1=�b1/ &)(1K , 6�MN6 5 b 0 d a�, K - C)( / (K�B?0�"�K b1=F*Jd a�- 67B O@b &)K�(�D - 670 *�b acb C b & < K b CI"+K b+*Jd a�<
��� � �p�/yl~2�#$ }��� NPG c>32 ; d b P1( d BT6�H - C)B d 6 �
1�
2 · · · � � d 0�A�U+B d " 0�A - 0�" � →∗� � 1�
2 · · · � � D d b* ( 0�U+0 * "�K b b1=�b D�& b1O�, 6 � = ({ � ��� } � { � → �%� � � → � � } �
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O 8 U 2=VvX ������0P��������Z /�[&/ ';6�'&�\'&2&' X ��6 , ��������Z /�[&/ '
· · · � � � · · ·↑ � � � H - C)B d 6� � 6= �
��� 6= �
�� �
′ +1−−−−−→ � ′ · · · � � ′ � · · · � � � � → � � ′ � ′ �
↑ � � → � � ′ � ′
� ′
�� �
′ 0−−−−→ � ′ · · · � � ′ � · · · � � � � → � � ′ � ′ �↑ � � → � � ′ � ′� ′
�� �
′ −1−−−−−→ � ′ · · · � � ′ � · · · � � � � � → � � ′ � � ′ �↑ � � � → � � ′ � � ′� ′ � � � → � ′
�� ′ �
� → � ′
* �z2&'�6�'�5MJ 2 3 d K�B *�b+51< 0�B d 6 * "�6 b1=�bed *Jd ( a & b+*Jd a�, 67K�"�A b1=I, 6g` g�i Zcb h�
��� ���Ut�Bz �c�i� � ���� ���3 d #bK�"+A b1=)- 6 & P1(1G B d 0 , D b D�B (v` g�i Zcb h K�P1(1&)(1U = =�b (1& d 0 * (1U = M 6'0�G+0 *I, K b+*�bb1=�b D�& b1O�, 6?J � N S c> ∗ Q 2 b H#H <pb P�S 0�B 5`b 0�K�S7D d b7* " =�d 0 * (1&�E b?* (1G+6 acbed BfP1B d /I, BTK1]
O@b1= E � ( =F*�bed 0�B P1(�H#H < 5 d 5 H�E b 2 *Jd 6 B d 0 < D�(1G#K�B B /)4 K�B * ( =^a H b 0 d a S * (1G+6�(1& d 0�K�S N S c>32*Y, ������Z�5 2�� ��4����� � � �� ��� BRE =�bed K d b./ (1K , (Σ � � ��� )
S+P1(1G J > Q _ ( Σ
BFE =�bed P�BbP�BR& b 0�K -R= ( 0�U = (�H�( j�= ,��(A �;�BA 2+P1(1G'P1BT& d - A�B d�* ( B d /Jd a S >@g8A�Aj�< ,��:m �8m�J # H�BTG a S16?A 4 &)(16 &fQ
J�! Q _ ( � BFE =�bed K�")] a B = S�2�P1BfP1BT& b 0�K -T= ( 0�U = (�H�( J�BR0�M * BT& d a�4 =�Q >c1c;91 j|; j6g��BA 2K�B
Σ ∩ � = ∅ J N Q _ ( � BRE =�bed ( h3?YAM18>c1�� �E1 j-< >|?BAB,Egk;91�� j6g��BA 2 -R=�b J P�BbP�BR& b 0�K -R= ( Q 0�U = (�H+(b P1S%P1B =F*)<1/ BR6 * "�67K�(1& O�, 6�
� �′ �−−−−→ � ′J d T Q
S+P1(1Gc( d � acbed � ′ BRE =�bed2acb+*�b 0 *)< 0�B d 6�2 *�b �(acbed�� ′ BRE =�bed 0�U+K 5 (�H b 2 acbed ">+?PAk7Jj|7 � BFE =�bed 0 2 1 , −1
e * " = B d a S =�b P1(1G � MWD�& b1O E � B d (7` g�i Zcb h 2 0�B a�< C)B 5�, K b * (1G G)P1(�H+(�D d 0�K�(1U* "�6 " K�"+A b1=I, BRE =�bed 0�B K d b 0�G�D a B a & d K -T= " acb+*)< 0 *�b 0�" � acbed b1=F*Jd K�B * MWP1E � B dK d b < P1B d &I" J P�&)(16 *Jd 6 / U+( acb+* BRG#C)U = 0�B d 6 Q # *�bed = E b%& K�B?P1BfP1BT& b 0�K -T=�b * ( P�H , C)(16K�"�] a B =)< 0�U+K 5 (�H b D�& b K�K -T=�b P <1= M * "�6�2 / "+H b1/I, K d b 0�G =)< & * "�0�"
�: Z = {· · · � −2 � −1 � 0 � 1 � 2 � · · · } → Σ
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U � 2 � [ :&'&2 { 5 ����� ���� O3>
S+P1(1G �( � ) = �
D d b #f0�A�B / S = S�H b%& *�b � ∈ Z f �|.��@7 � >-16; ej@;91ej@7 BFE =�bed "
* G+A b E b%* & d <1/�b( � � � � F ) �S+P1(1G." K�"�A b1=I,.5 &�E 0 a B *�bed 0 * " C - 0�" F ∈ Z
-m K�"�A b1=I, S1K�MN6�# / B = C - &)B d & P1(1G5 &�E 0 a B *�bed 0 * " = *�bed = E b 2 5 H - P1B d K�S = ( * ( #f(1& b+* S 0�U+K 5 (�H�( &�� =
�( F ) 2 acbed�-f* 0 dK�P1(1&)(1U#K�B =�b P1BT& d D�& < ��(1G#K�B * " = P+H , &I" acb+*)< 0 *�b 0�"%K�B?(+P�( d b1/I, P�( * B'H - CI"
� �− � +1 · · · � � −2
� �−1 � � � � � +1 · · · � � + � −1
J d�d Q0 * ( b H O�<+5 " * ( Σ ∪ ��2 S+P�(1G * ( (1& b+* S 0�U#K 5 (�H+( BFE =�bed * ( � � acbed � � =
�(�)D d b F − � G � G F +
� b & a BFE *�b � acbed � =�b BRE =�bedlb & a B *)< K�BTD < H b -b* 0 d P1(1G�(�) = �
D d b � ≤ F − � , � ≥ F +� acbed " H - CI" =�b P�BR& d H b K 51<1= B d S�H b *�b
K�"�] a B =)< 0�U+K 5 (�H b 0 * " =7*�bed = E b P b & b1/ BFE D�K b+*�bc* "+6%J dPd Q BFE =�bed�*�b� � � �
��� � � � � � � � �
S+P1(1G *�b./ U+( * BbH�BTG *�b E b P�&)(10 /Jd (1&�E � (1G = * " = E /Jd b P+H , &I" acb+*)< 0 *�b 0�" m / & < 0�" * "+6 K�"�A b1=I, 6 acb C)(1&�E � B *�bed�b P1S * " K d b , P�BR& d 0�0�S * BT&)BR6 5`b 0 d a�- 6
K�B *�b+5�< 0�B d 6 * "+6cK�(1& O�, 6 J d T Q P1(1G K�P�(1&)BFE =�b G)P < &)A�(1G = 0 * ( = P�E =�b`acb acbed P1(1Gb &)A�E � (1G = K�B b G *)< *�b � acbed� 2 acbed (1&�E � B *�bed 0�G = (+P *Jd a�< 0 * ( = $7E =�b`acb ! � 0 * "/ BRU * BR&I" 0 *I, H+" K�B * ( 0�G#K 5 (�H d 0�K�S * (1G ` g�i Zcb h 2 acbed 0 * " = * &�E * " MN6 0�U+0 * "�K bK�B *�b+5�< 0�BTM = 0 * (.0�U = (�H�( acb+*�b 0 *)< 0�BRM =
{ � � � � � | � � � ∈ Σ∗ � � ∈ � } :B =�b H�G *Jd a S * BR& b 2�" / & < 0�"cK d b 6?K�"�A b1=I, 6g` g�i Zcb h - A�B d M 6?B C , 6 J > Q B = / B = G�P < &)A�B d�5 b 0 d a�, K�B *)<+5`b 0�" P1(1G =�b b &)A�E � B d K�B *�b � acbed� 2 * S * B "acb+*)< 0 *�b 0�"cBRE =�bed�* BR&)K b+*Jd a�, acbed ( G)P1(�H+(�D d 0�K�S16?0 *�b K b+*)< J�! Q B = G�P < &)A�B d K�B *)<+5 b 0�"
�� �
′ 0−−−−→ � ′
K�B a E = "�0�" � = 02 * S * B " K�"+A b1=I, #bBfP d H - D�B d & K d b *)-f* ( d b K�B *)<+5`b 0�"�2 b H�]
H <%� B d�* ( � 0 * ( � ′ acbed|b H+H <%� B d-acb+*)< 0 *�b 0�" b P�S * " = � 0 * " = � ′ J N Q B = G�P < &)A�B d K�B *)<+5 b 0�"�
� �′ 1−−−−→ � ′
K�B a E = "�0�" � = 12 * S * B " K�"+A b1=I, #bBfP d H - D�B d & K d b *)-f* ( d b K�B *)<+5`b 0�"�2 b H�]
H <%� B d�* ( � 0 * ( � ′ 2 b H+H <%� B d@acb+*)< 0 *�b 0�" b P�S * " = � 0 * " = � ′ acbed@a-d = BRE *�bedK d b C - 0�" / B C d < J Q Q B = G�P < &)A�B d K�B *)<+5 b 0�"
�� �
′ −1−−−−−→ � ′
K�B a E = "�0�" � = −12 * S * B " K�"�A b1=I, #fBbP d H - D�B d & K d b *)-b* ( d b K�B *)<+5`b 0�"�2b H+H <%� B d#* ( � 0 * ( � ′ 2 b H#H <%� B d`acb+*)< 0 *�b 0�" b P�S * " = � 0 * " = � ′ acbedia-d = BRE *�bedK d b C - 0�" b & d 0 * BT& <
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O ! U 2=VvX ������0P��������Z /�[&/ ';6�'&�\'&2&' X ��6 , ��������Z /�[&/ 'lh�m �8m �c<HjJ,9A�� * "�6�K�"�A b1=I, 6 BRE =�bed " * G#A b E b J < P1B d &I" "7P�BbP�BR& b 0�K -R= " QWb`a (�H�(1G#C�E b
� 0 � � 1 �;:;:;: � � � �;:;:;:S+P1(1G a�< C)B � � BFE =�bed P�H , &I"+6 acb+*)< 0 *�b 0�"�2 acbed " � � +1BFE =�bed�* ( b P�( *)- H+BR0�K b * "+6
/ & < 0�"+6 * "+67K�"+A b1=I, 670 * " = � � � P1MN6'P <1=F*�b 2�C -f* (1G+K�B� →∗ � ⇐⇒ (∃ G)P1(�H+(�D d 0�K�S16 )[ � � � 1 �;:;:;:�� � � −1
� � ] :m K�"�A b1=I, BRE =�bed 1e<P;=< mJ>��`1c;=< >E. b1= D d b a�< C)B acb+*)< 0 *�b 0�" � acbed 0�U#K 5 (�H+( � G�P < &R]A�B d�* ( P1(�H+U K d b B =F* (�H , �
� �′ �−−−−→ � ′ 0 * ( P�E =�b`acb * "+6 (Σ � � ��� )
P1(1G b &)A�E � B dK�B *�b � 2 � ; d b * ( = G�P�(�H�(�D d 0�K�S J�K�BR& d a�4N=�Q 0�G =�b & *I, 0�BRM = 0 * (1G#6 O G#0 d a (1U#6 b & d C)K�(1U#6A�&)B d b%� S1K b 0 * B 0�G =�b & *I, 0�B d 6%B d 0�S / (1G acbed B C)S / (1G�2 acbed " P d ( 0�G = "�C d 0�K -R= " BfP d ]H�(�D , D d � b G *)- 67BRE =�bed ( d
Z b 0 g d( �� ) =
� ` @ � `11 · · ·︸ ︷︷ ︸�
1+1
� 11 · · ·︸ ︷︷ ︸�2+1
� · · · � 11 · · ·︸ ︷︷ ︸� � +1
�
_ g d 0 g d(� � S
11 · · ·︸ ︷︷ ︸� +1
� ) = � �J d O Q
S+P1(1G � ` @ � ` acbed � � S BRE =�bed 0�G�D a B a & d K -T= BR67BR0�M * BT& d a�- 6 acb+*�b 0 *)< 0�B d 6 N S ! 2�� E.��������� � ;@=>�E1-?H1 ,Eg �@< >:.�j,=iA ��-;k78j@7 �
: N�
� Ng@?PA=1 < J bed ]
*Jd ( a & b+*Jd a�< , b1=�bed *Jd ( a & b+*Jd a�<�Q � �eKMF D _ � = h�m �8m �,?Hjc< ,/7 ;BA`;0g >c1e<:,9A6A=m6A 1|Apg@?PA=1 <1|AM1���� m ,B< >E.��p b P b & b H�BRE ��(1G#K�B acbed�b G *I, * " =\b P1S / B d CI"�2 b H#H < P b & b+* "�&)(1U#K�B S *Jd " K d bacb+* BRU#C)G = 0�" * "�6 d 0�( / G =�b K�E b 6 P1(1G /Jd b+* G�P 4 = B *�bed 0 * ( C)B 4 &I"�K b BFE =�bed * B * & d K1]
K -R= " BbP�B d /I, " b P�H+(1U+0 * BR&I" a M /Jd a (+P1(�E "+0�" * M =�acb+*�b 0 *)< 0�BTM = K d b 6.K�"�A b1=I, 6` g�i Z b h J�M 6cH - C)B d 6 b P1S -T=�b P�BbP�BR& b 0�K -R= ( b H O�<+5 " * ( Q BFE =�bed J P�&)( O@b1=)4 6 Q b1=�b ]/ &)(1K d a�, 2 acbed BfP�E 0�"�6 b1=�b1/ &)(1K d a�- 6?BFE =�bed ( dca H b 0 d a�- 6?0�G =�b & *I, 0�B d 6?B d 0�S / (1G acbedB C)S / (1G7J d O Q D d � b G *I, * " = a M /Jd a (+P1(�E "+0�"�2 < & b^a�< C)B ` g�i Zcb h ] G)P1(�H+(�D�E 0 d K�" K�BT& d a�,0�G =)< & * "�0�" BRE =�bed2b1=�b1/ &)(1K d a�,�b P1S * " = $?&)S *�b 0�" N�E' N1 m b P1S / B d CI" * "�6 < H#H+"+6acb+* BRU#C)G = 0�"�6 b P bed * BRE * " = acb+*�b 0 a BTG , K�"+A b1=)4 = ` g�i Z b h K�B B d /Jd a�- 6 d /Jd S * " * BR6acbed BRE =�bed BbP1E P1( = "
��� � �p�/yl~2�#$ }��� N S c> ∗ 2�� BFE C * B7S *Jd D d b a�< C)B7K�"+A b1=I, ` g�i Zcb h = (Σ � � ��� )
G�P < &)A�B d 0�U)]0 * "+K b�b1=�b D�& b1O�, 6
� = (Σ ∪ � ∪ { � � } � →)K�B *�b BfP d P�&)S10�C)B *�b J O & - 0 acb�Q 0�U+K 5 (�H b � acbed 2 *)-f* ( d (%P�(1G D d b S�H+BR6 *Jd 6 H - C)B d 6� � � ′ 2 � � � ′ 0 * ( b H O�<+5 " * ( � acbed S�H+BR6 *Jd 6%JLBT0�M * BR& d a�- 6 Q acb+*�b 0 *)< 0�B d 6 � � � ′
� � � →∗� � ′ � ′ � ′ ⇐⇒ (∃ � � � )[ � � � � →∗��
�� � � ′ �
�� ] :
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U � 2 � [ :&'&2 { 5 ����� ���� OPN
I P d P�H - ( = 2 b1= " BRE =�bed2bed *Jd ( a & b+*Jd a�, 2 * S * B * ( � BRE =�bed2bed *Jd ( a & b+*Jd a S 0 * ( 0�U)]= (�H�( H - C)BTM = � � � b P�S * ( Σ ∪ � ∪ { � � } 0 *Jd 6 (+P1(�E BT6 BTK O@b1= E � B *�bed b`a & d 514 6K d b�acb+*)< 0 *�b 0�" VvX Z�<\#3��} [�� � &I"+0 d K�(+P�( d , 0 * B * " = P b & b H+H b D ,c* "+67P b & < 0 *�b ]0�"+6 J d�d Q D d b P�H , &)B d 6 acb+*�b 0 *)< 0�B d 6 P1(1G b &)A�E � B d K�B � acbed�* BTH+B d 4N= B d K�B 2 acbedP1&)(10�C - 0 * B 0 *Jd 6 5 b 0 d a�- 6�K�B *�b+51< 0�B d 6 * "+6 * " 0�M 0 *I, / & < 0�" 0 *�b # <`a & b%& � acbed 2�P A 2 � → � �
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^ I�� B��8B�� 3 Q
� ����)� nW� � �!� ��� � ��� �� � � ��� K � �
m * G+A b E b 0�A - 0�" � ( �� ) J�0 * (1G#6 O G#0 d a (1U#6 b & d C)K�(1U#6 Q BRE =�bed ������������W� �� b1= "A b & b`a�* "�& d 0 *Jd a�, * "�6 0�G =)< & * "�0�" BRE =�bed b1=�b1/ &)(1K d a�, 2 acbed�-R=�b 0�U = (�H+( � ⊆ NBFE =�bed �������������� ��� b1= " J�(�H d a�,�Q A b & b`a�* "�& d 0 *Jd a�, 0�G =)< & * "+0 , * (1G
� � ( � ) =
{1 � b1= � ∈ � �0 � b H#H d 4 6 �
BFE =�bed b1=�b1/ &)(1K d a�, Ke�� b G * S * ( ^ B O�< H bed ( C b K�BbH�B *I, 0�(1G+K�B *Jd 6 5 b 0 d a�- 6 K b ]CI"�K b+*Jd a�- 6 d /Jd S * " * BT6 * M = b1=�b1/ &)(1K d a�4N= JLK�BT& d a�4 =�Q 0�G =�b & *I, 0�BRM = 2�0�A - 0�BRM =�acbed0�G = S�H�M = 2 b H+H <^acbedI*Jd 6 79,B< 1|AM1���� m ,B< > ��� j@� �=j6g�< � S+P1MN6 acbedI*�b 1@A=1�� �`mk,l< > 1 hJ1 ��|<P5 ,/7J; �j�<8A=m��e1 2 *�b�b P+H�(1U#0 * BT& b K�")] G)P1(�H+(�D�E 0 d K b K b CI"+K b+*Jd a�<�b1=F*Jd a BFE K�B =�b _ b 5`b 0 d a�< BR&ID b H+BFE b BRE =�bed " 0 *Jd 5 b & ,#a H+B d 0 * S * " *�b * (1G 0�G = S�H�(1G R J�! G N 2! G Q 2 !�S c> Q acbed�* (�p?B 4 &I"+K b N @ c> ^ b1= ( =Jd a�, 6 @ (1& O�, 6 acbed B P b &�E C)K�"+0�"+6 * (1G� [ VFVAb�V e *Jd 6 BfP d a H , 0�B d 6 * (1G-p?BTM & , K b+* (16 N�@ c> C b b P+H�(+P�( d , 0�(1G+K�B * ( 0�G#K 5 (�H d 0�K�S�2
P b & b H+BFE P�( =F*�b 6 * ( = <1= M / BFE a�* "��( �� ) = � � � ( �� ) =
�( � � ��� ( � � �� ��� ))
J P�(1G�2�0�G =I, C)MN6�2�BRE =�bed�<1= BRG 0�"�K b 0�E b 6 b H+H <�acbed�/ "#H 4N= B *�bedcb P1S * " = BFE 0�( / ( �� =�1�;:;:;:�� � � Q 2 acbed A�&I"+0 d K�(+P�( d 4 =F*�b 6 0�B K�BT& d a�- 6 P1BT& d P *)4 0�B d 6 * ( = B =�b H#H b`a�*Jd a S
0�G#K 5 (�H d 0�K�S * (1G � [\VFVAb�V#2
{ � }( �� ) = � � � ( �� ) =�
( � � � � ( � � �� ��� )) :J d ] QB G * S 5 (�H�BTU+B d # * G)P1(�D�& b1O@d a�<%& J�H d D�S * BR&)( d�<1= M acbed a�<+* M / BRE a�* BR6 Q 2 b H#H <\acbed5 (�"+C < 0 * " acb+*�b1= S�"+0�"cK�BT& d a�4 =pb P�S *Jd 6 b P1( / BRE C)B d 6�2�D d b+* E * (+P�(1C)B * BRE�#f0 * ( E /Jd (BbP1E P1B / ( &�* ( h���A � � 1 ,2,21 � acbed�*�b ��g��Jmk, �8A=1 �� 8_ BbH d a�< 2�C b A�&I"+0 d P�(+P1( d , 0�(1G#K�BBbP1E 0�"+6 * ( 0�G+K 5 (�H d 0�K�S
� = { � | ��( � )↓}J O 8 Q
D d bc* (%P1B / E ( 0�U#D a H d 0�"+6 * "+6 b1=�b1/ &)(1K d a�, 67K�BR& d a�, 670�G =)< & * "�0�"�6?K�B a M /Jd a S �
O 9
O T i 2 j 2&'�< , ���=��6�%M' X ' , ��k�� [&/ %;��l�2�����'� ����� � }�z �cz:�cwcx � }Yy/�i��� B�i�#$ }��; d b.=�bc/Jd BRG a (�H�U = (1G+K�B * " /Jd b+* U)P1MN0�"%(1& d 0�K 4N= acbed-b P1( * BbH�BT0�K <+* M = 0 * "%0�G)]
=)- A�B d b 2 b P b & d C)K�(1U#K�B B /)4�acbed ( = (1K <%� (1G#K�B * (1G#6 5 b 0 d a S * BR&)(1G#6 * BbH�BT0 *)- 6%(1& d ]0�K 4N= 0�A - 0�BRM =�
� ( �� ) ⇐⇒ ¬ � ( �� )J ¬ Q� ( �� ) ⇐⇒ � ( �� ) & � ( �� )J
& Q� ( �� ) ⇐⇒ � ( �� ) ∨ � ( �� )J ∨ Q� ( �� ) ⇐⇒ � ( �� ) � ⇒ � ( �� )J ⇒ Q� ( �� ) ⇐⇒ (∃ � ) � ( �� ��� )J ∃ Q
� ( � � �� ) ⇐⇒ (∃ F ≤ � ) � ( �� � F )J ∃≤ Q� ( �� ) ⇐⇒ (∀ � ) � ( �� ��� )J ∀ Q
� ( � � �� ) ⇐⇒ (∀ F ≤ � ) � ( �� � F )J ∀≤ Q� ( �� ) ⇐⇒ � (
�1( �� ) �;:;:;:�� � � ( �� ))J b1=F*Jd acb+*)< 0 *�b 0�" Q
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�-< 1 ;=m3=���;0g��`g=j@;���� ;=m&= h��`m ;91 j-< 18>|m3<��8m �c<HjJ,@m&< ¬ � & � ∨ � ⇒ Z ;=m&= ��� �`1� , �8AMm3=��hJm j-m���g@? � ;0g�� ∃≤ � ∀≤ >c1e< 1|A ;=< >c1c; j|;91 j|7 JL(�H d a�4 =�Q 1@A=1�� �`mk,l< >@?BA�j�=8A=1��@;k. �j6g��BA Z 1�� �����g8A�g@?PA=1 <:>��`g�< j|;BA��-< 1(;=m&= ��h�mejcm��kg@? �W;0g�� ∃ � ∀ �QP@ ! 2*Y, ������Z�5 2 J�Y Q m 0�A - 0�" � ( �� ) BRE =�bed �����������������W� �� b1= D d b a�< P�( d bK�BR& d a�, 2 b1=�b1/ &)(1K d a�, 0�G =)< & * "+0�" � ( �� ) 2
� ( �� ) ⇐⇒ �( �� )↓ :
JU� Q m 0�A - 0�" � ( �� ) BRE =�bed Σ01b1= D d b a�< P1( d b�b1=�b1/ &)(1K d a�, 0�A - 0�" � ( �� ��� )
� ( �� ) ⇐⇒ (∃ � ) � ( �� ��� ) :QP@ N 2 *Y, Z / '�� [*2 +l1 g@C`. �^g@?YAM1e<:<Hjcm���<iAM1k,21 Z �-< 1 ;k76A ;@=>�E1-?H1 j@� �=j|7 � ( �� ) J > Q�� � ( �� ) g@?PA=1 < 79,B< 1|AM1���� m ,B< >E.��JL! Q�� � ( �� ) g@?PA=1 < Σ0
1
�J N Q�� � ( �� ) < >-1@A=m hJm < g@? ;k7eA < j-m��-=8A=1 ,H? 1
� ( �� ) ⇐⇒ (∃ � ) � ( �� ��� ),Eg > hJm <H1 h ���@;=m �Jg8A0? � 1@A=1�� �`mk,l< >:. � ( �� ��� ) �
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i�� 2�� �=�z'&2&'�< , ���=��6 { 5n��: { ��#9��5 O1d
j-X Z�<\#3��} [*2 J > Q � ⇒ J N Q b P1S * ( C)B 4 &I"�K b�acb1= ( =Jd a�, 6 K�(1& O�, 6 J N Q� ⇒ J�! Q* B * & d K�K -T=�b acbed JL! Q� ⇒ J > Q C -f* ( =F*�b 6
�( �� ) = � � � ( �� ��� ) �
-b* 0 d P1(1G(∃ � ) � ( �� ��� ) ⇐⇒ �
( �� )↓ : a
� A�(1G+K�B ,�/ "70�G =�b1=F*I, 0�B d�* ( a H b 0 d a S7P b & <1/ B d D�K b 0�A - 0�"+6 P1(1G?BFE =�bed "�K d b1=�b ]/ &)(1K d a�, b H#H < S1A d|b1=�b1/ &)(1K d a�, 2 * "c0�A - 0�" * BR&)K b+*Jd 0�K�(1U �
( � � � ) J N @ Q Q QP@ Q,2 *Y, Z / '�� [ J � E.���������� � ��� � � �12 � ;@= �:1 ? 1�j@� �=j|7 � ( �� ) g@?PA=1 <E1@A=1 �� �`mk,l< >:. 1|A^>c1e<@,9A6A=m6A 1|A�7 � ( �� ) >-1 < 7 ��6A 78j@. ;k7 � ¬ � ( �� ) g@?YAM1e<:>-1 <Wm <���<Jm70,l<H1@A=1�� �`mk,l< > �����j-X Z�<\#3��} [*2 B = " � ( �� ) BFE =�bed|b1=�b1/ &)(1K d a�, 2 * S * B acbed ( d
� ( �� ��� ) ⇐⇒ � ( �� ) � � ( �� ��� ) ⇐⇒ ¬ � ( �� )BFE =�bed|b1=�b1/ &)(1K d a�- 6�2 acbed - A�(1G+K�B J * B * & d K�K -T=�b�Q
� ( �� ) ⇐⇒ (∃ � ) � ( �� ��� )¬ � ( �� ) ⇐⇒ (∃ � ) � ( �� ��� ) :
; d bc* " = < H+H#" acb+* BRU#C)G = 0�"�2 b1=� ( �� ) ⇐⇒ (∃ � ) � ( �� ��� )
¬ � ( �� ) ⇐⇒ (∃ � ) � ( �� ��� )K�B b1=�b1/ &)(1K d a�- 670�A - 0�B d 6 � acbed � 2 * S * B'"c0�G =)< & * "+0�"
�( �� ) = � � [ � ( �� ��� ) ∨ � ( �� ��� )]
BFE =�bed (�H d a�, b1=�b1/ &)(1K d a�, 2 acbed� ( �� ) ⇐⇒ � ( �� � � ( �� )) : a
QP@ 9 2 *Y, Z / '�� [*2 +-m j,<iAMm �8m ; �BA 70,l<H1@A=1�� �`mk,l< >@?BA j � �=jeg��BA g@?YAM1e<6>��`g�< j|;BA�-< 1 1@A=1�� �`mk,l< > ��� 1|A ;=< >c1c;91 j|; j6g�< � Z �c<H1 ;=m3=�� �=5=gk;=< >|m3<�� � h � m`;91ejc<H1J>2m&< ��;0g��`g �j|;����
& � ∨ Z �c<H1�;=m3=�� � �`1� , �8AMm3=�� h�mejcm��kg@? �W;0g�� ∃≤ Z ∀≤ Z >c1e< �c<H1�;=mcA = h�1�� C8< 18> AhJm j-m���g@? � ;k7 ∃ � �kg8A g@?PA=1 <W>��`g�< j|;BA �-< 1#;k76A���6A 78j@7 ¬ >c1e< �c<H1#;=m6Ap>c1c5Mm �c< > AhJm j-m���g@? � ;k7 ∀ �
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OPO i 2 j 2&'�< , ���=��6�%M' X ' , ��k�� [&/ %;��l�2�����'j-X Z�<\#3��} [*2 m a H+B d 0 * S * " *�b D d b b1=�b1/ &)(1K d a�- 6 b1=F*Jd acb+*�b 0 *)< 0�B d 6cBFE =�bed P�&)(#]
O@b1=I, 6�2 acbed ( dEb`a S�H�(1G#C)( d K�B *�b 0�A�"+K b+*Jd 0�K�(�E / BFE A = (1G =%* (1G+6 G)P1S�H+( d P1(1G#6�2�C)B *Jd ]a (1U+6 d 0�A�G#& d 0�K�(1U#6 * "+6'P�&)S *�b 0�"+6 (∃ � ) � ( �� ��� ) ∨ (∃ � ) � ( �� ��� ) ⇐⇒ (∃ � )[ � ( �� � � ) ∨ � ( �� � � )]
(∃ � ) � ( �� ��� ) & (∃ � ) � ( �� ��� ) ⇐⇒ (∃ � )[ � ( �� � ( � )0) & � ( �� � ( � )1)]
(∃ � )(∃ � ) � ( �� ��� � � ) ⇐⇒ (∃ � ) � ( �� � ( � )0� ( � )1)
(∃ F ≤ � )(∃ � ) � ( �� ��� � F ) ⇐⇒ (∃ � )[( � )0 ≤ � & � ( �� � ( � )1� ( � )0)
(∀ F ≤ � )(∃ � ) � ( �� ��� � F ) ⇐⇒ (∃ � )(∀ F ≤ � ) � ( �� � ( � ) � � F ) :B P�S * " =�< H+H#"7K�BR& d < 2 * ( 0�U = (�H�( * M = "+K d b1=�b1/ &)(1K d a�4 = 0�A - 0�BTM =�/ B = BRE =�bed a H+B d ]0 * S'D d b'* " = < & = "+0�" , * ( = acb C)(�H d a S'P�(10�( / BRE A * "�2+D d b+* E b H#H d 4 6 " 5`b 0 d a�, 0�A - 0�"* BR&)K b+*Jd 0�K�(1U
�( � � � ) ⇐⇒ (∃ � ) � 1( � � � ��� )C b BFE A�B�"+K d b1=�b1/ &)(1K d a�, < & = "+0�" acbed C b ,#*�b1=�b1=�b1/ &)(1K d a�, b P1S * ( Q�@ Q 2�B =)4 / B =
BFE =�bed a_ ( ����� ������ K d b 6 K�BT& d a�, 6 0�G =)< & * "+0�"+6 � ( �� ) BFE =�bed "'0�A - 0�"'0 * (1G#6 O G#0 d a (1U#6
� �( �� � � ) ⇐⇒ �
( �� ) = ���J O9> Qacbed " BfP1S1K�B = "�2 b P�H , P1&)S *�b 0�" / E = B d 0�B'P�(�H+H - 6?P1BT& d P *)4 0�B d 6 * B * & d K�K -R= BT6 b P1(#]/ BFE C)B d 6 b1=�b1/ &)(1K d a S * " *�b 6?D d b K�BT& d a�- 670�G =�b & *I, 0�B d 6 QP@ T 2 *Y, Z / '�� [*2 � ;@=>�E1-?H1 ,Eg �@< >:.�j,=iA ��-;k78j@7 � ( �� ) g@?YAM1e<:1@A=1�� �`mk,l< >:.p1|A>-1 <@,�AcAMmcA 1@Ap;=m�� ����e70, ;k7 � � �
( �� � � )g@?PA=1 < 79,B< 1|AM1���� m ,B< >E. j � �=j@7 �
j-X Z�<\#3��} [*2 B = " � ( �� ) BRE =�bed2b1=�b1/ &)(1K d a�, K�B a M /Jd a S � 2 * S * B� �
( �� � � ) ⇐⇒ (∃ � )[ � � ( � � �� ��� ) &�
( � ) = � ] �-b* 0 d P1(1G " � �
( �� � � )BRE =�bed "�K d b1=�b1/ &)(1K d a�, acbed|b1=�( �� ) = � ⇐⇒ (∃ � ) � ( �� � ��� � )
K�B a�< P�( d b�b1=�b1/ &)(1K d a�, � ( �� � ��� � )2 * S * B
�( �� ) =
(��� � ( �� � ( � )0
� ( � )1))
0
�-b* 0 d P1(1G " � ( �� ) BFE =�bed|b1=�b1/ &)(1K d a�, a_ ( * BbH�BTG *�b E ( b P�( *)- H+BR0�K b 0 � b G * S * ( B /)<1O|d ( b P�H+(+P1( d BRE 0�"+K b1=F*Jd a�< P�(�H+H - 6acb+*�b 0 a BTG - 6 QP@ d 2 *Y, Z / '�� [ J � ����
Σ01
��EF��� ������98�12 < 1^> 65 g^70,l<H1@A=1�� �`mk,l< >:.^j � �=j@7� ( �� � � )
Z = h���;�lg�<B1|AM1���� m ,B< >E. Z ,:g �|< >E. j,=iA ��-;k78j@7 �( �� ) ;��k;=me< 1 hJm&= �-< 1 > 65 g
�� Z(∃ � ) � ( �� � � ) ⇐⇒ �
( �� )↓(∃ � ) � ( �� � � ) � ⇒ � ( �� � � ( �� )) :
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i � 2 j 2&'�< , ���=��6�%M' X ' , ��k�� [&/ %;��l�2�����' O ]
j-X Z�<\#3��} [*2 B P1S * " = G�P�S1C)BR0�"�2�G)P < &)A�B dEb1=�b1/ &)(1K d a�, 0�A - 0�" � ( �� � � ��� ) *)- ]* ( d b P�(1G� ( �� � � ) ⇐⇒ (∃ � ) � ( �� � ����� ) �acbed�* (.0�G#K�P - & b 0�K b%* (1G � , K�K b+* (16 / BRE A = B *�bed BTU a (�H b�b1= C - 0�(1G#K�B�( �� ) =
(��� � ( �� � ( � )0
� ( � )1))
0
: a
� ��� �p�Byl~:�*$e}��� Q�@ c>&2�� BRE C * B?S *Jd "cK�BR& d a�, 0�G =)< & * "+0�"
�( � � � ) = 〈 �
�(( � )0)
�;:;:;:���� � (( � ) [ e ( � )−· 1)〉BFE =�bed|b1=�b1/ &)(1K d a�, � Q�@ ! 2�� 0 * M � ( �� � � )
"+K d b1=�b1/ &)(1K d a�, 0�A - 0�" *)-b* ( d b P1(1G D d b a�< C)B �� G�P < &R]A�(1G = * (1G�H < A d 0 * ( = / U+( b & d C)K�(�E �1 6= �
2*)-f* ( d ( d P�(1G � ( �� � � 1)
acbed � ( �� � � 2)
� BFE C * B S *Jd G)P < &)A�(1G = / U+(�2+(�H d a�- 6 b1=�b1/ &)(1K d a�- 6 0�G =�b & *I, 0�B d 6 � ( �� ) � � ( �� ) *)-f* ( d BR6P1(1G D d b S�H b%*�b �� 2
� ( �� � � ( �� )) & � ( �� � � ( �� )) &�( �� ) 6= �
( �� ) :� Q�@ N ∗ 2�� 0 * M � ( �� � � )
"+K d b1=�b1/ &)(1K d a�, 0�A - 0�" *)-f* ( d b P1(1G D d b�a�< C)B �� 2G�P < &)A�B d�* (1G#H < A d 0 * ( = -R=�b 6 � *)-f* ( d (16'P1(1G � ( �� � � )
JLY Q � BRE C * B7S *Jd G�P < &)A�B d (�H d a�, b1=�b1/ &)(1K d a�, 0�G =)< & * "�0�" � ( � � �� ) 2 *)-f* ( d b P�(1G� ( �� � � ) ⇐⇒ (∃ � )[ � =
�(� � �� )] :J O ! Q
JU� Q � BRE C * B S *Jdlb1= J�BbP d P�H - ( =�Q D d b a�< C)B �� 2�G�P < &)A�(1G =.< P1B d &)( d�* ( P+H , C)(16 �*)-b* ( d ( d P�(1G � ( �� � � )
2 * S * B G)P < &)A�B d (�H d a�, 2 b1=�b1/ &)(1K d a�, � ( � � �� ) P�(1G d acb1= (+P�( d BFE* " = J O ! Q acbed BRE =�bed # 1� 1 0 * ( � & 2 / "+H b1/I, 2�D d b S�H b%*�b �� � � � �W2� 6= � � ⇒ �
( � � �� ) 6= �(� � �� ) :
� � � � �czE�cwcx(� }Yy �&z+%:zEw@}��,�Bt@{)�&� � �cx��2zQPE' c>&2�Y, ������Z�5 2 _ ( * G+A b E ( 0�U = (�H�( � ⊆ N
BRE =�bed ������������W� � � �������1�7� � ����� J , 1|AM1���� m ,B< > �1��@<Y5 ,W.Jj-< ,@m 2 b b 2 i V : g�i ` `fZ VF[ ^ VAb g XcV i Y ��[ V#2 iF V Q 2 b1=� = ∅ 2 , a�< P�( d b (�H d a�, 2 b1=�b1/ &)(1K d a�, 0�G =)< & * "+0�" � : N → N b P b & d C)K�BRE * ( � 2
� =�[N] = { � (0) � � (1) �;:;:;: } :J OPN Q
QPE' ! 2g*-, Z / '�� [*2 J > Q +B1�g@Ci. � g@?PA=1 </< j-m��-<8A=1 ,:1 �c<H1 ;=m(;@=>�E1-? m � ⊆ N�
J�Y Q +@m � g@?PA=1 </1 �H1 �
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]#8 i 2 j 2&'�< , ���=��6�%M' X ' , ��k�� [&/ %;��l�2�����'JU� Q � j � �=j@7 � ∈ � g@?YAM1e<W70,l<H1@A=1�� �`mk,l< >:. Z �k;=jc</h�m3=
� =S'_+XcY#Z b
(�) = { � | � ( � )↓}
�-< 1 >eh�me< 1 1|AM1���� m ,B< >E.�,:g �|< >E. j,=iA ��-;k78j@7�� �J :)Q +@m � g@?YAM1e< hJg�h8g �`1ejJ, �8AMm Z . = h���;�lg�<��8AM1 � h��`m�� � �8A=1 1@A=1�� �`mk,l< >:. j,=iA ����;k78j@7 �
: N � Nh�m3= 1 hJ1��|<P5 ,:g@? ;=m � Z � =
�[N]
�JL! Q � 18>|m �8m3=85@? 1
0��
1�;:;:;:1eh�1��@<Y5 ,Eg@?:;91 1 �H1 �2j,<iAMm � 1 �k;=j-<2h�m3= 7 j � �=j@7 � ∈ � A=1pg@?YAM1e<:70,l<H1@A=1�� �`mk,l< >:. �
J N Q +-m ;@=>�E1-? m � ⊆ Ng@?YAM1e<c1|AM1���� m ,B< >+A 1@Al>c1e<J,9A6A=m6AW1|ABg@?PA=1 <-hJg�h8g �`1ejJ, �8AMm. = h���;�lg�< J b G+0 * "�& <�Q 1�<@CMm&=Jjc1 Z 1@A=1�� �`mk,l< >:. j,=iA ��-;k78j@7 h�m3= 1 hJ1��|<P5 ,:g@?W;=m � Z
� = { � (0) G �(1) G �
(2) G · · · } :j-X Z�<\#3��} [*2 ; d b%* ( J N Q 2�P�& 4W*�b 2�G�P�B = C)G#K�E � (1G+K�B J�� >A@ c> ! Q S *Jd "c0�G =)< & * "+0�"
�: N → N
BRE =�bed 1�<@CMm&=Jjc1 b1=�(�) G �
(�
+ 1) (� ∈ N) �
P1(1G%0�G = BfP < D�B *�bed2b K - 0�MN6 JLK�B'BbP b D�MND ,�Q S *Jd� ≤ �
(�) D
- P�B *�bed S *Jd|b1=?* ( � b P b & d C)K�BRE *�bed|b P1S.K d b�b U�C)(1G+0 b 2 b1=�b1/ &)(1K d a�, � 2 * S * B� ∈ � ⇐⇒ (∃ � ≤ � )[ � =
�(�)] �acbed�* ( � BRE =�bedcb1=�b1/ &)(1K d a S ; d b * " =?< H#H+" acb+* BTU+C)G = 0�"�2 b1= * ( � BRE =�bedcb1=�b1/ &)(#]
K d a S acbed�< P1B d &)(�2 * S * B "�(0) = ( � � )[ � ∈ � ]�
(�
+ 1) = ( � � )[ � �(�) & � ∈ � ]BFE =�bed|b1=�b1/ &)(1K d a�, 2 b UPC)(1G#0 b acbed@b P b & d C)K�BRE * ( � J > Q m 0�G = BbP b D�MND , JLY Q� ⇒ J � Q BRE =�bed�* B * & d K�K -T= " D d b * ( � = ∅ 2 acbed^b1=
� =�[N]
2 * S * B� ∈ � ⇐⇒ (∃ F )[ � =
�( F )] :
_ ( b1=F* E 0 * &)( O ( J � Q� ⇒ JLY Q BFE =�bed BbP1E 0�"+6 * B * & d K�K -R= ( D d b * ( � = ∅ 2 acbed b1=�0 ∈ � acbed � ∈ � ⇐⇒ (∃ � ) � ( � ��� ) �* S * B * ( � b P b & d C)K�BRE *�bed|b P1S * " =pb1=�b1/ &)(1K d a�, 2�(�H d a�, 0�G =)< & * "�0�"
�( � ) =
{ �0� b1= ¬ � (( � )0
� ( � )1)�
( � )0� b1= � (( � )0
� ( � )1):
_ BTH d a�< 21" 0�G = BfP b D�MWD , J :)Q� ⇒ J�Y Q BFE =�bed1* B * & d K�K -R= "�2�"cJ�Y Q � ⇒ J :)Q BFE =�bed BfP�E ]0�"+6 * B * & d K�K -R= " D d b P�BbP�BR& b 0�K -R=�b � 2 acbedlb P1(1K -T= B dN=�b / BRE C)(1G+K�B%S *Jdlb1=c* ( �BFE =�bed < P�B d &)( acbed
� = { � (0) � � (1) �;:;:;: }
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i � 2 j 2&'�< , ���=��6�%M' X ' , ��k�� [&/ %;��l�2�����' ] >
9 O 9 > N O 9 ! N :;:;:� � �
� :&4��='Fq 2�� d b D�& b1O�, BbP b1=�b H�BRE ��BTM = D d b#a�< P�( d b#b1=�b1/ &)(1K d a�, 0�G =)< & * "�0�" � 2 * S * B * ( � b P b & d C)K�BRE *�bed BbP1E 0�"+6 b P�Sa�< P1( d b -R=�b ]�P1&)(16b] -R=�b�b1=�b1/ &)(1K d a�, 0�G =)< & * "�0�" m 5`b 0 d a�,(d /)-Mb BFE =�bed =�b # /Jd b ]D�& < ��(1G#K�B *Jd 6 BfP b1=�b H , ��B d 6 &Wb P�S * " = b P b &�E C)K�"�0�" K�B * ( � 2 a�<+*Jd P1(1G h � m��@1|A0? �( / "+D�BRE 0�B'K d b (�H d a�, 2 -R=�b ]�P1&)(16b] -R=�b 1|AM1���� m ,B< >E. b P b &�E C)K�"+0�" * (1G � ; d bc* " =pb G+0 * "�& , b P�S / B d CI" b G *I, 6 * "�6 # P�&)( O@b1= S * " *�b 6 & 2 - 0 * M
� = { � | (∀ FHG �)[�( F ) 6= �
(�)]}
* ( b1=�b1/ &)(1K d a Sc0�U = (�H�( * M = C - 0�BTM = S+P1(1G acbed = (1U+&ID d b K - H+" * (1G � b P b & d C)K�(1UI]=F*�bed-b P�S * " = � � =�[N]
D d b�a�< P�( d b b UPC)(1G#0 b � ( � ) b P�S * (%K - &)(16 J N Q 2 acbed�* (� b P b & d C)K�BRE *�bed|b P1S * " = -T=�b ]LP�&)(16b] -T=�b 0�U = C)BR0�" � ( � ) =
�(�(�))
_ ( J�! Q 0�G =)< D�B *�bed|b P�S * (.A b & b`a�* "�& d 0�K�S JU� Q * (1G J > Q aQPE' N 2g* Z , �����=' 2 � c5=g 1@A=1�� �`mk,l< > A j,<iAMm �8m g@?YAM1e<c1 �H1 � Z 1�� �� = h���;�|m&=iA/1 � 1 �j�<8A=m��e1 hJm&=���g8A�g@?PA=1 </1@A=1�� �`mk,l< > Z h � � � Z ;=m
� ′ = { � | � (( � )0 � ( � )1)} :J O Q Qj-X Z�<\#3��} [*2 B =%* ( � ′ ,#*�b1= b1=�b1/ &)(1K d a S�2 * S * B b1=�b1/ &)(1K d a�, C b ,#*�b1= acbed "
0�A - 0�" * BT&)K b+*Jd 0�K�(1U J T O Q 2�B O S10�( =�
( � � � ) ⇐⇒ 〈 � � � 〉 ∈ � ′ : am a H < 0�" * M = b b 0�G = S�H+M = - A�B d P1(�H+U P�H+(1U+0 d b / (1K , acbed - A�B d K�BTH+B * "+C)BFE
B =F*�b+*Jd a�< YI /)4 C b P1BT& d (1& d 0 * (1U#K�B 0�B.J�BTH < A d 0 *�b�Q�5`b 0 d a�< b P�( * BTH - 0�K b+*�b 2�P1(1G/ E = (1G = a�< P�( d b D�BTU+0�" * M =�d /Jd ( *I,#* M =7* "�6 QPE' Q 2�Y, ������Z�5 2'�������+7 �� * (1G'0�G = S�H+(1G � 0 * ( � 2�BFE =�bed " * G#A b E b JL(�H d a�,1Qb1=�b1/ &)(1K d a�, 0�G =)< & * "+0�" � P�(1G d acb1= (+P�( d BFE * " =�d 0�( / G =�b K�E b
� ∈ � ⇐⇒ �( � ) ∈ � :J O 9 Q
; d b / U#( � � � ⊆ N2�C -f* (1G+K�B
� ≤ � � ⇐⇒ G)P < &)A�B d|b1=�b D�MND , * (1G � 0 * ( � �� ≤1 � ⇐⇒ G)P < &)A�B d -R=�b ]�P1&)(16b] -R=�b b1=�b D�MWD , * (1G � 0 * ( � �� ≡ � ⇐⇒ G)P < &)A�B d|b1=�b D�MND , * (1G � 0 * ( � P�(1GcBRE =�bed K�B *)< C)BT0�" �
S+P1(1G " �: N�→N
BFE =�bed ,:gk; c5=g=j|7 b1= BRE =�bed b K O|d K�( = (10 , K b1=F* " b1=F*Jd 0 * ( d A�E bJ -T=�b ]LP�&)(16b] -T=�b�acbed BbP1E * (1G N Q $'&)( O@b1=)4 6�2
� ≡ � � ⇒ � ≤1 � � ⇒ � ≤ � � :
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]+! i 2 j 2&'�< , ���=��6�%M' X ' , ��k�� [&/ %;��l�2�����'QPE' 9 2g*-, Z / '�� [*2 <H1 A � 1(;91 j�<8A=m��e1 � Z � Z � Z
� ≤ � � >-1 < [ � ≤ � � & � ≤ � � ] � ⇒ � ≤ � � �>-1 < ;=m�? �J< m#<Hj � < g�<��-< 1 ;=< � < j@��=���A ;0g � g��(1|AM1�1� � ��� ≤1>c1e< ≡ � g�h`< h�����mcA Z 7j@� �=j|7 1@A=1�� �`mk,l< >|m3< < j-m ,@m���@< j8,-m3< ≡ g@?PA=1 </j,=9,|,Egk; �@< >:. Z
� ≡ � ⇐⇒ � ≡ � :j-X Z�<\#3��} [*2 ; d b'* "7K�B *�b+5 b+*Jd a S * " *�b b G *)4 =�* M = 0�A - 0�BTM = 2#P b & b+* "�&)(1U#K�B S *Jdb1= 2 b P�S * " = G�P�S1C)BR0�"
� ∈ � ⇐⇒ �( � ) ∈ � acbed � ∈ � ⇐⇒ �
( � ) ∈ � �D d b / (10�K -R= BT6 b1=�b1/ &)(1K d a�- 670�G =�b & *I, 0�B d 6�2 * S * B
� ∈ � ⇐⇒ �(�( � )) ∈ � �
-b* 0 d P1(1G "c0�U = C)BT0�" � ( � ) =�(�( � )) b1=)< D�B d�* ( � 0 * ( � a
QPE' T 2�Y, ������Z�5 2 _ ( 0�U = (�H+( � BRE =�bed ���\���N�������E@8 b1= BFE =�bed2b b acbed|a�< C)Bb b � b1=)< D�B *�bed 0 * ( � K�B -T=�b ]LP�&)(16b] -T=�b b1=�b D�MND , 2 � ≤1 � ; d b P b & <1/ B d D�K b 2 * ( � ′ P1(1G (1&�E 0 b K�B 0 * " = J O Q Q BFE =�bed P+H , &)BR6�2WBfP1B d /I, a�< C)Bb b 0�U = (�H+(.BFE =�bed�* "+67K�(1& O�, 6 � D d b a�< P1( d ( � 2 acbed
� ∈ �⇐⇒ �
�( � )↓ ⇐⇒ 〈 � � � 〉 ∈ � ′ :
^ < P�M 6 b P�H+(1U+0 * BR&)(.BFE =�bed�* ( # /Jd b D 4 =Jd ( & 0�U = (�H+(� = { � | (∃ � ) � 1(
� � � ��� )} = { � | � � ( � )↓} �J O T Q* (1G%(+P1(�E (1G " P+H+"+&)S * " *�b / B = BRE =�bed�* BTH+BFE MN6 * B * & d K�K -T= " QPE' d-2g*-, Z / '�� [*2 +@m � g@?YAM1e<B1 � 1 �Bh �|.��eg����j-X Z�<\#3��} [*2 ; d b%* ( * G#A b E ( b b 0�U = (�H�(
� = { � | � ( � )↓}JLK�B b1=�b1/ &)(1K d a�, � ( � ) Q 2�C -f* (1G+K�B
�( � ��� ) =
�( � )
acbed BfP d H - D�(1G#K�B a�< P1( d ( a M /Jd a S � * "�6 � 2 -b* 0 d P1(1G D d b (+P1( d ( /I, P1( * B � 2� ∈ � ⇐⇒ �
( � ��� )↓⇐⇒ { � }( � ��� )↓⇐⇒ { � 1
1(� � � )}( � )↓ Dacbed b1O (1U b G *I, " d 0�( / G =�b K�E b#d 0�A�U+B d D d b S�H b *�b � 2 d /Jd b E * BR& b#d 0�A�U+B d D d b * (
� =� 1
1(� � � ) 2 acbed K b 6 / E = B d 0 � b G *I,%* " = P1BT&�E P * MN0�"
� ∈ � ⇐⇒ { � 11(� � � )}( � 1
1(� � � ))↓
⇐⇒ � 11(� � � ) ∈ � �
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i � 2 j 2&'�< , ���=��6�%M' X ' , ��k�� [&/ %;��l�2�����' ] N
P1(1G b1=�b D < D�B d�* ( � 0 * (�� K�B * "c0�G =)< & * "+0�" � ( � ) =� 1
1(� � � ) a
_ ( BfP1S1K�B = (�2 5`b 0 d a S C)B 4 &I"+K b / BFE A = B d JLB = K - &)B d Q S *Jd K - A�& d 6 b1=�b1/ &)(1K d a (1Ud 0�(1K�(1& O@d 0�K�(1U.G)P < &)A�B d K�S = ( -R=�b b b P�H , &)BT6 0�U = (�H�(�2 b H+H < BFE =�bed P1(�H+U.D�B =Jd ]a S * BR&)( Bd 0�A�U#B d D d b S�H b?*�b 0�U = (�H b � � � 2�S1A d K�S = ( *�b b1=�b1/ &)(1K d a�<pb P b & d C)K�" *)<� acbed|b G * S.BRE =�bed P�(1G a�<1= B d�* " =pb P1S / B d CI" * S10�( / U#0 a (�H#" QPE' O 2�� # � ,3[ �=' J � E��������� ����� �� �������12 <H1��-<�m j�<8A=m��e1 � Z � Z
� ≤1 � & � ≤1 � � ⇒ � ≡ � :j-X Z�<\#3��} [*2 m b P�S / B d CI".BRE =�bed|acb+*�b 0 a BRG b 0 *Jd a�, B a�/ (1A ,%* "�6 b P�S / B d CI"+6 * (1Ga H b 0 d a (1U p'BRMN& , K b+* (16 � : e i u_Pw�V i ] E V i b�` dfVFZcb%0 * "70�G = (�H�(1C)BTM &�E b 2 acbed+5 b 0�E � B *�bed
0 * (.BbP�S1K�B = ( � , K�K b 2�S+P1(1G " * G+A b E b P1BfP1BT& b 0�K -T= " b`a (�H+(1G+C�E b � BTG#D 4N= = ( � 0
���0)� ( � 1
���1)�;:;:;:�� ( � � ��� � )J OPd Q
acb H�BRE *�bed ������ J P�&)(10 - D�D d 0�" d 0�(1K�(1& O|d 0�K�(1U Q D d bc*�b 0�U = (�H b � acbed � 2 b1=F 6= � � ⇒ � � 6= � � ��� � 6= � � � acbed � � ∈ � ⇐⇒ � � ∈ � ( F � � ≤ �
) :; d b a�< C)B b`a (�H�(1G#C�E b S+P1MN670 * " = J OPd Q 2�C -f* (1G+K�B
�=�
( ) = { � 0� �
1�;:;:;: � � � } � �
=�
( ) = { � 0 ��� 1 �;:;:;: ��� � } :
� 4����='�� n A ==h �� �Eg�< �8AM1 � h��`m�� � �8A=1 1|AM1���� m ,B< >E. j�=8A���@;k7Jj|7 �: N � N;��k;=me< 1 hJm&=
� ∈ � ⇐⇒ �( � ) ∈ � �;BA ;0g �-< 1�A��e1 ;91
�= { � 0
�;:;:;: � � � } � � = { � 0 �;:;:;: ��� � } acbed ���∈ � �,Bh�m��`m3< ,Eg A=1 � � m&<9,:g >eh�me< m � �∈ � Z �k;=j-</h�m3= 1@Ap7 1J>2m��im&=i5 ?H1
= ( � 0���
0)�;:;:;:�� ( � � ����� )g@?YAM1e<2>-1��|. Z g�h�? j|7 � >c1 �@. g@?PA=1 <:>c1e<W7�g�h �M>:;91ej@7
′ = ( � 0���
0)� ( � 1
���1)�;:;:;:�� ( � � ����� ) � ( � ��� ) �J O�O Q
�`7 �e1��`. � �∈ � >c1e< � ∈ � ⇐⇒ � ∈ � :j-X Z�<\#3��} [M/ �&� � 4����=' / ��5�� p -f* (1G+K�B
� 0 =�( � )
� � +1 =
{� � � b1= � � �∈ � ��( � � ) � b H#H d 4 6�2 b1= � � = � � �
acbed BfP b H#"�C)BTU+(1G#K�B / U+( 5`b 0 d a�- 6 d /Jd S * " * BR6 * "+6 b`a (�H�(1G#C�E b 6 � 0 � � 1 �;:;:;: J > Q ; d b a�< C)B F 2 � ∈ � ⇐⇒ � � ∈ �
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] Q i 2 j 2&'�< , ���=��6�%M' X ' , ��k�� [&/ %;��l�2�����'n h3A��kg�< Ci7 ; d b F = 0
2 � ∈ � ⇐⇒ �( � ) = � 0 ∈ � 2 b P�S * " = G)P1S1C)BT0�" D d b
* " = � \I P b D�MND d a�< 2 b1= � � �∈ � 2 * S * B� ∈ � ⇐⇒ � � +1 = � � ∈ �
b P1S * " = BfP b D�MWD d a�, G�P�S1C)BR0�"�2 acbed@b1= � � ∈ � 2 * S * B� ∈ � ⇐⇒ � � = � � ∈ � J b P1S * " = BfP b D�MWD d a�, G�P�S1C)BR0�" Q
⇐⇒ � � ∈ � J�BbP�B d /I, " / (10�K -T= " b`a (�H�(1G#C�E b BRE =�bed@acb H ,�Q⇐⇒ �
( � � ) = � � +1 ∈ � :JL! Q ; d b a�< C)B F 2 � � ∈ � � ⇒ (∀ � G F )[ � � 6= � � ]
n h3A��kg�< Ci7 Rm P1&)S *�b 0�" b H#"�C)BTU+B dN* B * & d K�K -T=�b\b1= � 0 =�( � ) �∈ � 2ND d b+* E 0 �b G *I, * " = P�BR&�E P * M 0�" � � =
�( � ) �∈ � D d b a�< C)B F 2 b P�S * ( = (1& d 0�K�S vI P1E 0�"+6%"
P1&)S *�b 0�" b H#"�C)BTU+B dN* B * & d K�K -R=�b D d b F = 02 acbed 2 BbP b D�MND d a�< 2 / BTA�S1K b 0 * BcS *Jd "
P1&)S *�b 0�" b H+"+C)BRU#B d D d b * ( F acbed � � +1 ∈ � $ b & b+* "+&)(1U+K�B S *Jd � � ∈ � 2 b H#H d 4 6J b P�S * ( = (1& d 0�K�S Q � � +1 = � � �∈ � < & b 2 b P1S * ( = (1& d 0�K�S�2�D d b a�< P1( d ( ��2� � = � � � � � +1 =
�( � � ) :J O ] Q
$?&)(16 <+* (+P1(�2 - 0 * M � * ( BTH < A d 0 * ( b1=F*Jd P b & <1/ B d D�K b D d b * ( � � +12 / "+H b1/I, � G
F + 1 acbed� � +1 = � � & (∀ � G �
)[ � � +1 6= ��� ] :$ b & b+* "+&)(1U+K�B'S *Jd � 6= 02�BbP�B d /I, � 0 =
�( � ) 2 � � +1 =
�( � � ) acbed BfP1(1K -T= M 6
� � +1 = � 0 � ⇒ �( � � ) =
�( � ) � ⇒ � � = � �
P1(1G.BRE =�bed�<+* (+P�(�2 b1O (1U � �∈ � B =)4 � � ∈ � < & b � =�+ 1
D d b�a�< P1( d ( � G F 2acbedcb P1S * " = BbP d H�(�D , * (1G ��2 � � ∈ � J b H+H d 4 6 � �+1 = � � acbed�* ( � +1 / B�C b ,#*�b1=
* ( BbH < A d 0 * ( b1=F*Jd P b & <1/ B d D�K b�Q acbed BbP�(1K -R= MN6�2�D d b a�< P1( d ( � 2� � = ����� � �
+1 =�( ��� ) :J ]�8 Q
� P1(�H+(�D�E � (1G+K�B � � +1 = � �
+1� ⇒ �
( � � ) =�( � � ) J b P1S *Jd 6%J O ] Q acbed J ]#8 QfQ
� ⇒ � � = ��� J " � BRE =�bed K�( = (1K�(1& O|d 0�K�S16 Q� ⇒ � � = � � J " b`a (�H�(1G#C�E b BRE =�bed@acb H ,�Q� ⇒ � � = � �
J b P1S *Jd 6%J O ] Q acbed J ]#8 QfQ :� P1B *�bedib P�S * " = BbP b D�MND d a�, G)P1S1C)BT0�" S *Jd � ≥ F 2 < & b � +1 ≥ F +1
P�(1G b1=F*Jd * E C)B *�bed0 * " = G)P1S1C)BT0�" � + 1 G F + 1
m P�&)S *�b 0�" J�! QN*)4 & b 0�G = BbP < D�B *�bed S *Jd D d b a�< P�( d ( � G �
+ 22 � � �∈ � J b1O (1U
* ( � - A�B d � + 1K - H#" Q 2 acbed�* ( � , K�K b�b H+"+C)BRU#B d K�B * " = BbP d H�(�D , � = �
� 2 ′ = �� ( � ��� ) a J � , K�K b��7Q_ (c0�G+K�K�B * & d a S � , K�K b � b P�( / E / B d D d b a�< C)B b`a (�H+(1G+C�E b acbedca�< C)B � �∈ �-R=�b � �∈ � *)-f* ( d ( P�(1G b1= " BRE =�bedEacb H , 2 * S * B acb H , BRE =�bedEacbed " BfP -0a�*�b 0�"
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i � 2 j 2&'�< , ���=��6�%M' X ' , ��k�� [&/ %;��l�2�����' ]+9 ′ = �� ( � ��� ) @ � b G *)<.*�b � , K�K b+*�b 2�".K�B *)< C)BT0�"cP�(1G.A�&)B d b%� S1K b 0 * B acb+*�b ]0 a BTG <%� B *�bed K�B /Jd b1/ (1A d a�, B O-b &)K�(�D ,c* M = B C , 6 / U+( 5 "�K <+* M = 2|C)B a-d =)4 =F*�b 6 b P�S* " = acb H , b`a (�H�(1G#C�E b
0 = (0 � � (0)) � �
0 = {0} � � 0 = { � (0)} :f g �@<Y;k;BA �B.9,:1 2
�+ 1
p -b* (1G#K�B � = min(N \ � 2� ) acbed BbP�B a�* BRE = (1G+K�B * " =
2� K�B'B O@b &)K�(�D , * (1G � , K�K b+* (16 � -f* 0 d P1(1G � ∈ � 2
�+1
� �@;=< m��B.9,:12�
+ 2 p -f* (1G+K�B � = min(N \ � 2
�+1)
acbed BbP�B a�* BRE = (1G+K�B * " =
2�+1
K�B'B O@b &)K�(�D , * (1G � , K�K b+* (16 � -f* 0 d P�(1G � ∈ �2�+2
e * ( *)- H+(16�2#" -R= MN0�" ⋃ � �� BRE =�bed D�& <1O "+K b K�B *)< C)BT0�"+6 � : N�→NP�(1G b1=)< D�B d
* ( � 0 * ( � 2� ∈ � ⇐⇒ �
( � ) ∈ � :m b1=�b1/ &)(1K d a S * " *�b * "�6 � 0�G =)< D�B *�bed@b P1S * " acb+*�b 0 a BTG ,�acbed 0�G+K�P+H+"+& 4 = B d�* " =b P1S / B d CI".S *Jd � ≡ � aQPE' ] 2 � ����������������� \B = � ≤ � � 2 acbed�* ( � BFE =�bedeb1=�b1/ &)(1K d a S�2 * S * B acbed�* (
� BRE =�bed:b1=�b1/ &)(1K d a S - P�B *�bed S *Jd2b1= � ≤ � � acbed�* ( � / B = BRE =�bed:b1=�b1/ &)(1K d a S�2* S * B acbed�* ( � / B = BRE =�bed|b1=�b1/ &)(1K d a S @ b%� E�K�B * " = P�H#"�&)S * " *�b.* (1G � 2�" b P�H , b G *I, P1&)S *�b 0�".BFE =�bed�* (cP1& 4�* (�2 5 b ]
0 d a ScBT&ID b H�BRE (.D d b * " = b P�S / B d CI"%K�"�] b1=�b1/ &)(1K d a S * " *�b 6'0�G = S�H�M = acbed 0�A - 0�BTM =�D d b+* E b1= / BRE C)(1G+K�B?S *Jd � ≤ � � K�B a�< P�( d ( � P�(1G / B = BFE =�bed|b1=�b1/ &)(1K d a S J P A 2* ( � Q 2 * S * B - P1B *�bed S *Jd-acbed�* ( � / B = BFE =�bed|b1=�b1/ &)(1K d a S QPE' c> 8 2 *Y, Z / '�� [ J�$ b & <1/ B d D�K b�Q 2 +-m j�<8A=m��im
� = { � | �6= ∅}g@?YAM1e<B1 � 1 �B1�� ���A��/<B1|AM1���� m ,B< >+A �
j-X Z�<\#3��} [*2 _ ( � BFE =�bed|b b BbP�B d /I, "c0�A - 0�"� ∈ � ⇐⇒ (∃ � )[ � ∈ � ]
BFE =�bed Σ01
; d b.=�b / BFE C)(1G#K�B?S *Jd � ≤ � � 2�C -f* (1G+K�B�( � � � ) = � � � 1( � � � ��� )
-b* 0 d P1(1G " *Jd K , � ( � � � ) BFE =�bed|b1= B C < & * " * " * (1G � 2�( � � � ) =
{� � � 1( � � � ��� ) � b1= (∃ � ) � 1( � � � ��� ) �⊥ � b H+H d 4 6 �
acbed 2�D d b a�< C)B � 2� ∈ � ⇐⇒ �
( � � � )↓ �-b* 0 d P1(1G
� ∈ � ⇐⇒ (∃ � ) � ( � � � )↓ D
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]PT i 2 j 2&'�< , ���=��6�%M' X ' , ��k�� [&/ %;��l�2�����'- P�B *�bed S *Jd|b1= ( � BFE =�bed@a M /Jd a S16 * "�6 � ( � ��� ) 2 * S * B
� ∈ � ⇐⇒ (∃ � )[{� }( � � � )↓ ]
⇐⇒ (∃ � )[{ � 11(� � � )}( � )↓ ]
⇐⇒ �11(� � �
)6= ∅
⇐⇒ � 11(� � � ) ∈ � �
< & b � ≤1 � acbed�* ( � / B = BFE =�bed|b1=�b1/ &)(1K d a S a$ b & b+* "+&)(1U+K�B'S *Jd K � b G *I, * " acb+*�b 0 a BTG , 2� ∈ � ⇐⇒ �
11(� � �
)= N
⇐⇒ * ( �11(� � �
)- A�B d�* (1G�H < A d 0 * ( = 2
K - H#"-b* 0 d P1(1G *�b 0�U = (�H b
� = { � | �
= N} � � = { � | * ( � - A�B d�* (1G�H < A d 0 * ( = 2K - H#" }
BbP1E 0�"+6 / B = BRE =�bed|b1=�b1/ &)(1K d a�< QPE' c>�>32'� ����������W� ����8?�1����4+7��1�������8 / U#(?0�G = S�H+M = � acbed � BFE =�bed�* ( * G#A b E (b1=�b1/ &)(1K d a S 0�U = (�H+( � *)-f* ( d (cP�(1G � ⊆ � acbed � ∩ � = ∅ B = � ∩ � 6= ∅ 2 * S * BI2�P�&)( O@b1=)4 6�2 / B = G�P < &)A�B dN/Jd b A�MN& d 0�K�S16 * (1G � b P�S * (
� 2 acbed@b1='* ( � BRE =�bed@b1=�b1/ &)(1K d a S acbed � ∩ ! = ∅ 2 * S * BI2�P < H d P1&)( O-b1=)4 6�2 * ( �/Jd b A�M &�E � B d�* ( = B b G * S * (1G b P1S * ( � QPE' c> ! 2 *Y, Z / '�� [ J � [\VFVAb�V Q 2 lh���;�|m&=iAW1 �H1 �cj�<8A=m��e1 �
0>-1 < �
1;��k;=m <H1ph�m3=
�0 ∩ � 1 = ∅ >-1 < �kg8A�==h �� �Eg�<E1@A=1�� �`mk,l< > A�� �8< 1B�+� �|< j8,�A��p;=m3= � 0
1eh3A ;=m �1�
j-X Z�<\#3��} [*2 p -b* (1G#K�B�
0 = { � | (∃ � )[ � 1(( � )0� � ��� ) & (∀ � ≤ � )¬ �
1(( � )1� � � � )]} �
�1 = { � | (∃ � )[ � 1(( � )1
� � � � ) & (∀ � G � )¬ �1(( � )0
� � ��� )]} �acbed / BRE A = (1G+K�B P�& 4W*�b S *Jd �0 ∩ � 1 = ∅ K�B b P b D�MWD , 0 * ( <+* (+P1( D d b+* E b1=
� ∈ � 0 ∩ � 1acbed� ∗ = � � [ � 1(( � )0
� � ��� ) & (∀ � ≤ � )¬ �1(( � )1
� � � � )]� ∗ = ��� [ � 1(( � )1
� � � � ) & (∀ � G � )¬ �1(( � )0
� � ��� )] �* S * B)2 b P�S * (1G#67(1& d 0�K�(1U+6�2
� ∗ G � ∗ � ⇒ �1(( � )0
� � ��� ∗) & (∀ � G � ∗)¬ �1(( � )0
� � ��� )� ⇒ �
1(( � )0� � ��� ∗) & ¬ �
1(( � )0� � ��� ∗) �acbed�* ( b1=)< H�(�D�( <+* (+P1(.0�G =)< D�B *�bed|b P�S * " =pb1=F* E C)B * ".G)P1S1C)BT0�"�2�S *Jd � ∗ ≤ � ∗ $ < H d P�&)(16 <+* (+P�(�2 b 6?BFE =�bed�*�b � 2 � b b *)-f* ( d b P1(1G
�0 ⊆ � � �
1 ⊆ � � �∩ � = ∅ � � ∪ � = N �
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i � 2 j 2&'�< , ���=��6�%M' X ' , ��k�� [&/ %;��l�2�����' ] d
acbed - 0 * M �= 〈 � � � 〉 G�P�(�H�(�D�E � (1G#K�B
� ∈ � � ⇒ (∃ � ) � 1( � � 〈 � � � 〉 � � ) & (∀ � )¬ �1( � � 〈 � � � 〉 ��� )BfP1B d /I, �
∩ � = ∅� ⇒ (∃ � )[ � 1( � � 〈 � � � 〉 � � ) & (∀ � G � )¬ �
1( � � 〈 � � � 〉 ��� )]� ⇒ 〈 � � � 〉 ∈ � 1
b P1S * ( = (1& d 0�K�S� ⇒ 〈 � � � 〉 ∈ � BfP1B d /I,��
1 ⊆ � �< & b � ∈ �
∩ � P1(1G?BRE =�bed�<+* (+P�( � P1B *�bed S *Jd � ∈ � 2 b1O (1U �∪ � = N
2b H+H < ( 0�G+K�K�B * & d a S16 G)P1(�H+(�D d 0�K�S1670�G =)< D�B d P < H d:b P � b G * S S *Jd � ∈ � ∩ � 2P1(1G%BRE =�bed <+* (+P1( a
� � � �p�/yl~2�#$ }��� Q�E' c>32 � BFE C * B.S *Jd G)P < &)A�B d b1=�b1/ &)(1K d a�, 2 K�BR& d a�, 0�G =)< & * "�0�" � ( � )
2 *)-b* ( d bP1(1G
�6= ∅ � ⇒ [
�( � )↓ &
�( � ) ∈ �
] :� Q�E' ! 2 ; d b a�< C)B K d b b P�S *Jd 6 B C , 6 P1&)( *)< 0�B d 6 b P�( O@b 0�E 0 * B b1= D d b * G#A b E b 2b1=�b1/ &)(1K d a�<(b P b & d C)K�" *)< 0�U = (�H b � acbed � " P1&)S *�b 0�" b H#"�C)BTU+B dW, S1A d / BRE C * B
*Jd 67C)B *Jd a�- 6 b P�( O�< 0�B d 6?0 b 6 acbed /)4 0 * B b1=F*Jd P b & b1/ BFE D�K b+*�b D d b%*Jd 6 b & = " *Jd a�- 6 JLY Q _ ( � ∩ � BFE =�bed|b b JU� Q \_ ( � ∪ � BRE =�bed|b b J :)Q \_ ( � \ � = { � ∈ � | ���∈ � } BRE =�bed|b b
� Q�E' N 2�B H#"�C)BTU+(1G = , S1A d � � 4 0 * B b P�( / BFE C)B d 6 , b1=F*Jd P b & b1/ BFE D�K b+*�b J > Q � P < &)A�B d JL(�H d a�,1Q^b1=�b1/ &)(1K d a�, 0�G =)< & * "+0�" � 1( � � � ) *)-b* ( d b P�(1G D d b S�H b*�b � � � 2
� (
� � � ) = �∪ � :
JL! Q � P < &)A�B d JL(�H d a�,1Q^b1=�b1/ &)(1K d a�, 0�G =)< & * "+0�" � 2( � � � ) *)-b* ( d b P�(1G D d b S�H b*�b � � � 2 � (
� � � ) = �∩ � :
J N Q � P < &)A�B d JL(�H d a�,1Q^b1=�b1/ &)(1K d a�, 0�G =)< & * "+0�" � 3( � � � ) *)-b* ( d b P�(1G D d b S�H b*�b � � � 2 � (
� � � ) = �\ � :
� Q�E' Q,2�� P < &)A�B d (�H d a�, 2 b1=�b1/ &)(1K d a�, 0�G =)< & * "�0�" � ( � � � ) *)-f* ( d b P1(1G D d b S�H b*�b � � � 2 �
(
� � � ) = { � + � | � ∈ � acbed � ∈ � }; ( ����������������� ����� ������������ : )
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] O i 2 j 2&'�< , ���=��6�%M' X ' , ��k�� [&/ %;��l�2�����'� Q�E' 9 2 � 0 * M �
: N → NJL(�H d a�,1Q b1=�b1/ &)(1K d a�, 0�G =)< & * "+0�"�2 � ⊆ N
2 acbed�[ � ] = { � ( � ) | � ∈ � }� −1[ � ] = { � | � ( � ) ∈ � }
" g�< >+A6A=1 acbed " 1|A ;B?Hj@; � m��e7�g�< > AcAM1 * (1G � b P1S * " = � ; d b#a�< C)B K d b�b P�S*Jd 67B C , 6 P1&)( *)< 0�B d 6 b P1( O-b 0�E 0 * B b1= b H#"�C)BTU+B d�, S1A d 2 b P�( / BFE C * B *Jd 67C)B *Jd a�- 670 b 6b P1( O�< 0�B d 6 acbed /)4 0 * B b1=F*Jd P b & b1/ BFE D�K b+*�b D d b%*Jd 6 b & = " *Jd a�- 6
JLY Q B =7* ( � BFE =�bed|b b 2�BFE =�bed@acbed�* ( � [ � ] b b �JU� Q B =?* ( � BRE =�bed2b b 2�BFE =�bed@acbed�* ( � −1[ � ] b b �J :)Q B =7* ( � BRE =�bed|b1=�b1/ &)(1K d a S�2�BFE =�bed@acbed�* ( � [ � ] b1=�b1/ &)(1K d a S��J w Q B =?* ( � BRE =�bed2b1=�b1/ &)(1K d a S�2�BFE =�bed@acbed�* ( � −1[ � ] b1=�b1/ &)(1K d a S��� Q�E' T 2�m > �ig�<Hj@;BA ;k7�;91 J : [\_#` g�i V Q � B = S16 0�G = S�H�(1G � ⊆ N
D d b K d b K�BT& d a�,0�G =)< & * "�0�" � : N � N
2�BFE =�bed�* ( BbH < A d 0 * ( 0�U = (�H+( � *)-b* ( d ( P1(1G � ⊇ � acbed* ( � BRE =�bed@a H�B d 0 * S.D d b%* " = � 2 / "+H b1/I,[ � ∈ � &
�( � )↓ ] � ⇒ �
( � ) ∈ � :JLY Q � BFE C * B%S *JdBb1=c* ( � BRE =�bedBb b acbed " � ( � ) BRE =�bedBb1=�b1/ &)(1K d a�, J�K�BR& d a�,�Q 2
* S * B acbed " a H+B d 0 * S * " *�b � * (1G � D d bc* " = � BRE =�bed|b b JU� Q � BRE C * B�S *Jd G�P < &)A�B d P1&)M * (�D�B =I, 6 b1=�b1/ &)(1K d a�, 0�G =)< & * "�0�" � ( � � � )
2 *)-b* ( d bP1(1G7D d b S�H b *�b � acbed � 2 * ( � (
� � � )BFE =�bed " a H�B d 0 * S * " *�b � * (1G � D d b * " =b1=�b1/ &)(1K d a�, K�BT& d a�, 0�G =)< & * "+0�" � � K�B a M /Jd a S �
� Q�E' d 2 � BFE C * B S *Jd`a�< C)B < P�B d &)(�2 b b 0�U = (�H+( - A�B d�< P�B d &)(�2 b1=�b1/ &)(1K d a S7G�P�(#]0�U = (�H�(
� Q�E' O 2 � BFE C * B?S *Jd�* (.0�U = (�H�(�
0 = { � | (∃ � )[ � 1(( � )0� � ��� ) & (∀ � ≤ � )¬ �
1(( � )1� � � � )]}
BFE =�bed|b b ]�P�H , &)BT6 � Q�E' ] 2 J m d /Jd S * " *�b * "�6 1|AM1�1� �c. � J i VAw g : dfZ\_�b Q D d b * " = a H < 0�" * M = b b
0�G = S�H�M = � BFE C * B S *Jd:b1= *�b 0�U = (�H b � acbed ! BFE =�bedEb b 2 * S * B G)P < &)A�(1G = b b 0�U = (�H b � 1
2 � 12 *)-b* ( d b P�(1G
� 1 ⊆ � � � 1 ⊆ � � � 1 ∪ � 1 = � ∪ � � � 1 ∩ � 1 = ∅ :� Q�E' c> 8 2 m d /Jd S * " *�b * (1G �8< 1B�+� �|< j8,-m3< J�` VB0�Y i Y�dfZ\_�b Q D d b * " = a H < 0�" * M =b b 0�G#K�P�H#"�&)MNK <+* M = � BRE C * B.S *Jd/b1=c*�b � acbed � BFE =�bed 0�G+K�P+H+"+& 4 K b+*�b(b b
0�G = S�H�M = � � acbed � � acbed � ∩ � = ∅ 2 * S * BWG�P < &)A�B d8b1=�b1/ &)(1K d a S � P�(1G /Jd b A�M &�E � B d* ( � b P�S * ( � 2 / "+H b1/I, 2
� ⊆ � � � ∩ � = ∅ :lh3A��kg�< Ci7 � I O@b &)K�S10 * B * " = P�&)(�"+D�(1U#K�B = " < 0 a "�0�"%0 *�b 0�G+K�P+H+"+& 4 K b+*�b � � acbed� �
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i � 2 * ' , '&0 . 01��6�%���< [ �=���&� , 01��6�%;6�'&�\' X ��%;��l�2�����' ]P]
�
�
�
�
� 1 = ( � \ � ) ∪ � � � 1 = ( � \ � ) ∪ �
� Q�E' c>P>&2'@ E b b P�S *Jd 6 / U#( BbP�S1K�B = BT6 P1&)( *)< 0�B d 6 b H+"+C)BRU#B d 2�B =)4 " < H+H#" / B =b H+"+C)BRU#B d � 4 0 * B b P1S / B d CI" b G *I, 6 P1(1G b H+"+C)BRU#B d acbed b1=F*Jd P b & <1/ B d D�K b#b G *I, 6P1(1G / B =�b H#"�C)BTU+B d JLY Q B = � ⊆ � acbed�*�b � � � � BFE =�bedcb b 2 * S * B�G�P < &)A�B dcb1=�b1/ &)(1K d a S%0�U = (�H+( �*)-b* ( d (cP�(1G � ⊆ � ⊆ � JU� Q B = � ⊆ � acbed�*�b � � � � BFE =�bedEb b 2 * S * B G)P < &)A�B dBb1=�b1/ &)(1K d a S 0�U = (�H+(
� *)-b* ( d (%P1(1G � ⊆ � ⊆ �
� � � u�z:wcz('*)�' }Yy �/� �et+� }�x �@w ' }Yy �&y/zB}�z+%����&� � �cx��2z@ - A�& d�*)4 & b 2 *�b K�S =�b b b 2�K�")] b1=�b1/ &)(1K d a�< 0�U = (�H b P�(1G - A�(1G+K�B 0�G =�b1=F*I, 0�B d
BFE =�bedlb b P+H , &I"�2 acbedW/ "�K d (1G#&ID�BFE *�bed " BT& 4W* "+0�" b1= a�< C)B b b 0�U = (�H+( BRE =�bedW,b1=�b1/ &)(1K d a S ,(b b P+H , &)BR6 m BbP�S1K�B = " 0�B d & < (1& d 0�K 4 =�acbed P�&)( *)< 0�BTM = J * (1G� X%Z\[ � _+` d Q / BFE A = B d S *Jd " b P+H , b G *I, B d a S =�b BRE =�bed P�(�H�U%K b`a & d < b P�S * " = P�& b D1]K b+*Jd a S * " *�b QoG c>&2�Y, ������Z�5 2 m 0�G =)< & * "+0�" � : N � N
BFE =�bed ���������(7 ��� �� ��������� �������� D d bc* ( 0�U = (�H+( � b1= BRE =�bed|b1=�b1/ &)(1K d a�, 2 -T=�b ]LP�&)(16b] -T=�b�acbed
� ⊆ � � ⇒ � ( � ) ∈ � \ �Dacbed�* (.0�U = (�H�( � BFE =�bed ���������+7 ��� ��� b1= - A�B d P b & b D�MWD d a�, 0�G =)< & * "+0�"
_ ( 0�U = (�H+( � BFE =�bed ����W�������+��� ��� b1= BFE =�bed|b b acbed�* ( 0�G+K�P+H , &)M K <%* (1G� �
= { � ∈ N | ���∈ � }BFE =�bed P b & b D�MND d a S QoG ! 2g*-, Z / '�� [*2 +@m � g@?PA=1 <��`79,B< m3= � �c< > A Z ,Eg�h�1��`1�1� �-< >E. j,=iA ��-;k78j@7
�-< 1(;=m � � ;k76Ap;91,=i;=m`;=< >:. � ( � ) = � �j-X Z�<\#3��} [*2 $'& - P�B d =�b / BRE C)(1G+K�B?S *Jd
�⊆ � � � ⇒ � ∈ � � \
� �
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> 8#8 i 2 j 2&'�< , ���=��6�%M' X ' , ��k�� [&/ %;��l�2�����'/ "+H b1/I,
(∀ � )[ � ∈ � � ⇒ � �∈ � ] � ⇒ [ � �∈ �& � �∈ � ] :
m G�P�S1C)BR0�" * "+670�G = BfP b D�MWD , 67BRE =�bed(∀ � )[{ � }( � )↓ � ⇒{ � }( � )↑]
acbed�* (.0�G#K�P - & b 0�K b b P�H <{ � }( � )↑�
BbP�B d /I,� �∈ � ⇐⇒ � �∈ � ⇐⇒ { � }( � )↑ Dacbed " G�P�S1C)BR0�"%0�G = BfP < D�B *�bed�* (c0�G+K�P - & b 0�K b 2�D d b+* E b1= { � }( � )↓ 2 * S * B�C -b* ( =F*�b 6
�= � 0 * " = G)P1S1C)BT0�" - A�(1G#K�B { � }( � )↑2�P�(1G%BFE =�bed <+* (+P1( aQoG N 2g* Z , �����=' 2 � 65 g�1 � 1 �Bh �|.��eg���j,<iAMm �8m�g@?YAM1e< �`79,B< m3= � �c< > A �B O�,�= (1G#K�B * " =pb P�S / B d CI"cD d b.< 0 a "+0�"�2 � Q1G c># _ ( b1=F* E 0 * &)( O ( b G * (1U * (1G P�(1&�E 0�K b+* (16�BbP1E 0�"+6 d 0�A�U#B d`acbed�/ E = B d�-R=�b # / (1K d a S &
A b & b`a�* "�& d 0�K�S * "�6 b b P�H#"�&)S * " *�b 6�2 b H+H < " b P�S / B d C ,?* (1G / B = BFE =�bed�* S10�( b P�H ,acbed C b * " = b1=�b+51< H+(1G+K�BcD d b * ( BbP�S1K�B = ( B /)<1O@d ( Fe * ( G�P�S�H�( d P�( b G * (1U * (1GB /�b1O E (1G acb+*�b 0 a BTG <%� (1G#K�B b b 0�U = (�H b P1(1G / B = BFE =�bed�/ "+K d (1G+&ID d a�< 2 < & b�acbedS1A d P+H , &I" QoG Q 2g*-, Z / '�� [*2 � c5=g�hJ1�� 1 �&� �c< > A�j,<iAMm �8m � �>�lg�< h8g�< � m Z 1 �H1 ��==hJm j�< �A=m��im �j-X Z�<\#3��} [*2 m d /)-Mb BRE =�bed =�b (1&�E 0�(1G+K�B * " 0�G =)< & * "�0�" � : N → N
K�B * " =b1=�b1/ &)(1K ,�(0) = � 0
� S+P�(1G �0
= ∅�( � + 1) = a�< P1( d (16 a M /Jd a S16 * (1G � ( � ) ∪ {� ( � ( � ))}
S+P1(1G " � ( � )BFE =�bed " / (10�K -R= " P b & b D�MWD d a�, 0�G =)< & * "�0�"cD d bc* ( � \B = * ( acb+*�b ]O�- &)(1G+K�B b G * S�2 * S * B?K�B'K d b BTU a (�H#"cBbP b D�MND ,./ BRE A = (1G+K�B?S *Jd D d b a�< C)B � 2
�(�) ( �
(�+1) ⊆ � �
-b* 0 d P1(1G * (.0�U = (�H�(� = �
(0) ∪ �(1) ∪ :;:;: = { � | (∃ � )[ � ∈ �
(�)}
BFE =�bed2b b 2 < P1B d &)( G�P�(10�U = (�H�( * (1G � ; d b.* ( = G�P�(�H�(�D d 0�K�S * "+6 b P bed * (1U+K�B = "�6�( � � � ) *)-b* ( d b 6'P�(1G �
( � + 1) =�(�( � ) � � ) �
C -b* (1G#K�B P�& 4W*�b� ( � ��� � � ) ⇐⇒ � ∈ �
∨ � = �
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i � 2 * ' , '&0 . 01��6�%���< [ �=���&� , 01��6�%;6�'&�\' X ��%;��l�2�����' > 8 >acbed P b & b+* "�&)(1U#K�B S *JdEb G *I, BFE =�bed "+K d b1=�b1/ &)(1K d a�, 0�A - 0�"�2 -b* 0 d P1(1G.D d b a�< P�( d (� 2
� ∈ � ∪ { � } ⇐⇒ {� }( � ��� � � )↓⇐⇒ { � 2
1(� � � ��� )}( � )↓ �P1(1G%0�"�K b E = B d S *Jd|b1= C - 0�(1G#K�B
U( � ��� ) =� 2
1(� � � ��� ) �* S * B
U(
� � ) = �∪ { � } :
_ BTH d a�< (1&�E � (1G+K�B�( ��� � ) = U( ����� ( � )) �acbed 0 * ( = (1& d 0�K�S * "+6 � 2
�( � + 1) =
�(�( � ) � � ) = U(
�( � ) ��� ( � ( � )))
-b* 0 d P1(1G �
(�+1) = �
(�) ∪ {� ( � ( � ))}S+P1MN6 b P bed * (1U+0�B " b P1S / B d CI" a
QoG 9 2�Y, ������Z�5 2 _ ( 0�U = (�H�( � BRE =�bed ������� 2 b1= BFE =�bed2b b acbed�* ( 0�G+K�P+H , ]&)M K <%* (1G � � BFE =�bed < P�B d &)( acbed�/ B = - A�B d�< P�B d &)(�2 b b G�P�(10�U = (�H�(�2 / "+H b1/I,
�∩ � = ∅ � ⇒ * ( � BFE =�bed P1BfP1BT& b 0�K -T= ( :
QoG T 2�� # � ,3[ �=' JU� X%Z\[ � _#` d Q 2 Eh �� �Eg�</1 h��&A j,<iAMm �8m �j-X Z�<\#3��} [*2 m 0�A - 0�"
� ( � ��� ) ⇐⇒ � ∈ � & � 2 �BFE =�bed "+K d b1=�b1/ &)(1K d a�, 2 < & b b P1S * ( � , K�K b Σ0
1
] I P d H�(�D , 6 QP@ d G�P < &)A�B d@b1=�b1/ &)(#]K d a�, 2�K�BT& d a�, 0�G =)< & * "+0�" � ( � ) *)-b* ( d b P�(1G
(∃ � )[ � ∈ � & � 2 � ] ⇐⇒ �( � )↓
⇐⇒ �( � )↓ &
�( � ) ∈ � &
�( � ) 2 �(:
_ ( b P bed * (1U+K�B = ( 0�U = (�H+( BRE =�bed "cB d a S =�bc* "+6 � 2� = { � ( � ) | � ( � )↓}
= { � | (∃ � )[ � ( � ) = � ]}= { � | (∃ � )[ � ( � ) = � & 2 � G � ]} �J ] > Q
S+P1(1G " * BbH�BTG *�b E b 2 5`b 0 d a�, d 0�S * " *�b 0�G =)< D�B *�bed b P1S * ( = (1& d 0�K�S * "�6 0�A - 0�"+6� ( � ��� ) J > Q _ ( � BRE =�bed "�K d b1=�b1/ &)(1K d a S�2 b P1S * ( = (1& d 0�K�S * (1G JLBfP1B d /I, * ( D�& <1O "+K b* "�6 � ( � ) BRE =�bed Σ0
1Q
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> 8+! i 2 j 2&'�< , ���=��6�%M' X ' , ��k�� [&/ %;��l�2�����'JL! Q _ (.0�G#K�P�H , &)MNK b � � BRE =�bed < P1B d &)(�2�BbP�B d /I,
� ∈ � & � ≤ 2 � � ⇒ (∃ � )[ � =�( � ) & 2 � G � ≤ 2 � ]
� ⇒ (∃ � )[ � =�( � ) & � G � ] �
P1(1G 0�G = BfP < D�B *�bed S *Jd ;=m h�m �,< � 1 h�A ;=m&= � 2 � +11��|<P5 ,-m3<�� ≤ 2 � 1@Ak.=>2m&=iA�j@;=m
� - P1B *�bed S *JdWa�< P1( d (16 � ≥ � b1=I,ia B d 0 * ( 0�G+K�P+H , &)M K b � � 2 acbed b1O (1U b G * Sd 0�A�U#B d D d b a�< C)B � 2 * ( � � BFE =�bed < P�B d &)( J N Q ; d b a�< C)B < P�B d &)( � 2 � ∩ � 6= ∅ 2�BbP�B d /I,
� < P�B d &)( � ⇒ (∃ � )[ � ∈ � & � 2 � ]� ⇒ �
( � )↓ &�( � ) ∈ �
� ⇒ �( � ) ∈ �
∩ � : aQoG d-2g* Z , �����=' 2 +l1p1 h���pj�<8A=m��e1���g8ABg@?YAM1e<-1 � 1 �-h �|.��@7 Z �� 1 ==h �� �Eg�<@1 �H1 � Z,W7 � 1@A=1�� �`mk,l< > A j�<8A=m��im hJm&=���g8A�g@?PA=1 </1 �H1 �/h �|.��eg����j-X Z�<\#3��} [*2�� =�b b P�H+S � / B = K�P�(1&)BFE =�b BFE =�bedlb1=�b1/ &)(1K d a S�2�D d b+* E * S * B * (
< P1B d &)( 0�G+K�P+H , &)M K < * (1G.C b ,#*�b1= b b A�M &�E 6 =�b *)- K = B d�* ( � acbed�/ B = K�P�(1&)BFE=�b BRE =�bed|b b P+H , &)BR6�2�D d b+* E / B = BRE =�bed / "�K d (1G#&ID d a S b P�S * "%$'&)S *�b 0�" Q1G Q� a
� � � �p�/yl~2�#$ }��� Q1G c>32 � BFE C * B S *Jd`b1= * ( � BFE =�bed�/ "+K d (1G+&ID d a S�2 * ( � BFE =�bed`b b acbed � ≤1 � 2* S * B acbed�* ( � BRE =�bed / "�K d (1G#&ID d a S � Q1G ! 2 � BFE C * B'S *Jd-b1='* ( � BFE =�bed-b P�H+S acbed�* ( � BRE =�bed@b b 2 < P�B d &)(�2 * S * B�"
* (1K , � ∩ � BRE =�bed < P1B d &I" � Q1G N ∗ 2�� BRE C * B%S *Jdlb1=%*�b � acbed � BRE =�bedBb P+H < 0�U = (�H b 2 * S * B acbed " * (1K ,
* (1G+6 � ∩ � BRE =�bed|b P�H+S 0�U = (�H+( � Q1G Q,2 ; d b *Jd 6'BfP1S1K�B = BR6 / U+( P�&)( *)< 0�B d 6�2 b P�( O@b 0�E 0 * B b1= b H+"+C)BRU#(1G =?, S1A d 2acbed /Jd acbed (�H+(�D , 0 * B * " =pb P <1=F* "+0 , 0 b 67K�B b P�S / B d CI" , b1=F*Jd P b & <1/ B d D�K bJLY Q ; d b^a�< C)B < P1B d &)( b b 0�U = (�H+( � 2�G�P < &)A�B d P+H , &I"�6�2 b1=�b1/ &)(1K d a�, 0�G =)< & * "+0�"� 2 *)-b* ( d b P�(1G%D d b a�< C)B � 2
�( � ) � acbed � ( � ) ∈ � :
JU� Q ; d b a�< C)B b b 0�U = (�H�( � K�B < P�B d &)( 0�G#K�P�H , &)MNK b 2�G)P < &)A�B d P+H , &I"�6 b1=�b ]/ &)(1K d a�, 0�G =)< & * "+0�" � 2 *)-b* ( d b P�(1G D d b a�< C)B � 2
�( � ) � acbed � ( � ) �∈ ! :
� Q1G 9 ∗ 2 J�Y Q � BRE C * B S *Jd8b1= * ( � BFE =�bed8b P�H+S�2�" � ( � ) BFE =�bed (�H d a�, 2 b1=�b1/ &)(1K d a�,acbed�-T=�b ]LP�&)(16b] -T=�b 2 acbed " b1=F* E 0 * &)( O "%B d a S =�b � −1[ � ] - A�B d�< P1B d &)(c0�G#K�P�H , &)MNK b 2* S * B * ( � −1[ � ]BFE =�bed|b P+H�S.0�U = (�H�(
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i � 2�� �KJ�� � # � ,3[ �=' j 2&'�< , ���=4�5 > 8 NJU� Q � BRE C * B.S *Jd/b1= P b & b H+BFE ��(1G+K�B%K d b b P1S *Jd 6.G)P1(1C - 0�B d 6 m��-< >E. 2 �8A=1 � h � m�� �
�8A=1 2 h8g�< � m j�= ,Bh �|.����@,21�;=m&= � −1[ � ]2 * S * B * ( 0�G#K�P - & b 0�K b * (1G7J�9 :)Q�/ B = d 0�A�U#B db P b & b E * " *�b
� � � � x!s�����$i� wet(�Wz ���cz:�cwcx �B~2�e�� b G * S * ( B /)<1O@d ( C bpb P�( / BFE C)(1G#K�B -R=�bpb P b+* "#H < b P�H+S C)B 4 &I"�K b7* (1G � [\VFVAb�V#2
P1(1G S1K�M 6 - A�B d B a P�H#" a�*Jd a�<�d 0�A�G+& - 6 acbed-b P1&)S1(+P * BR6'0�G =)- P�B d BT6?0 * " C)BTM &�E b (1& d ]0 d K�S * " *�b 6.J D�B =Jd a�<�Q acbed:b`a S1K�" acbed 0 * " 0�G = (�H�(1C)BTM &�E b I /)4 C b * ( A�&I"�0 d K�(#]P1( d , 0�(1G#K�B K�S = (?D d b K d b 2+0�"�K b1=F*Jd a�, B O-b &)K�(�D ,�* (1G P1(1G ( O BFE H+B *�bed 0 * ( = � ^�e�Z [\[acbed *�b G * E � B d�*�b b b 2�P�H , &I" acbed *�b / "�K d (1G#&ID d a�< 0�U = (�H b 2 b H+H < C b * ( 5 &)(1U#K�BP1(�H+U%A�& , 0 d K�( acbed|b &ID�S * BT& b 2�0 * ( ^ B O�< H bed ( T Q S c>32 � # � ,9[ �=' J � � � � � E.�������� �������������98 2 � [ VFVAb�V Q 2 <H1^>c5=g 1@A=1 �
� �`mk,l< >:. Z ,:g �|< >E. j�=8A���@;k7Jj|7 �( � � �� ) Z = h���;�lg�< �8A=1�� 1��|<P5 ,�A�� � ∗ ;��k;=m < m�� h�m3=�-< 1 >c5=g
�� Z� � ∗( �� ) = { � ∗}( �� ) =
�( � ∗ � �� ) :J ]#! Q
� < �8< >+A`;0g � 1 Z ==h �� �Eg�< h ���@;=m �Jg8A0? � 1|AM1���� m ,B< >E.�j,=iA ��-;k78j@7 �( � )
J P1(1GcB C b & *)< ]*�bed K�S = ( b P1S * (cK ,8a (16 � * "�6 H�E 0 *�b 6 �� = �
1�;:;:;: � � ��Q ;��k;=m <H1 hJm&=�1|A � = �
� Z;BA ;0g 7 J ]#! Q < j@��<kg�<@,:g � ∗ =�( � )Z �`7 �e1��`. �c<H1 A � 1 ;91 � � �� ) Z
� �(
�)( �� ) = �
�(�( � ) � �� ) :J ] N Q
_ (cC)B 4 &I"+K b b P�( / E / B d-b K - 0�M 6 K�BT& d a�- 6 b P�H - 6 P1&)( *)< 0�B d 6�P�(1G / BFE A = (1G = S *Jd "a M /Jd a (+P�(�E "�0�" * M =�b1=�b1/ &)(1K d a�4 = K�BR& d a�4N= 0�G =�b & *I, 0�BTM = - A�B d P1(�H#H - 6 b P�&)(10�]/ S a " * BT6%J acbed-a�< P1MN6'P�BR&�E BT&ID�BR6 Q d /Jd S * " * BT6 Q S ! 2 *Y, Z / '�� [ J�$ b & b1/ BFE D�K b+*�b�Q 2 Eh �� �@m3=8A �(=Jjc< >|m ?|1��@<Y5 ,@m-? � 1 � � 4 ;�� �;=me< me</h�m3=
� �1( � ) = � 1
� �2( � ) = � 2 + � �
3= { � 3}
�4
= {0 �;:;:;: � � 4} :j-X Z�<\#3��} [*2 ; d b.* ( � 1 2 B O@b &)K�S � (1G#K�B * ( !+( p'B 4 &I"�K b B =�b1/ &)(1K , 6 0 * " 0�G)]
=)< & * "�0�"�( � � � ) = �acbed C -f* (1G+K�B � 1 = � ∗ - P�B *�bed S *Jd
� �1( � ) =
�( � 1 � � ) = � 1 :
3 d G�P�S�H�( d P�BR6 b P�( / BFE C)B d 6 BRE =�bed P b &)S1K�( d BR6 acbed B C�E 0�(1G b P�H - 6 a
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> 8 Q i 2 j 2&'�< , ���=��6�%M' X ' , ��k�� [&/ %;��l�2�����'j-X Z�<\#3��} [M/ �&�KJo�&� � # .�, 4��=' / ��5 j 2&'�< , ���=4�5 2 m K�BT& d a�, 0�G =)< & * "+0�"
�(� 1� ( � � � ) � �� ) BRE =�bed b1=�b1/ &)(1K d a�, 2 acbed BfP1(1K -T= M 6 G�P < &)A�B d�O G#0 d a S16 N*)-b* ( d (16 P1(1G
{ � 1� ( �� � )}( �� ) = { }( � � �� ) =�(� 1� ( � � � ) � �� ) D
* ( C)B 4 &I"+K b 0�G =)< D�B *�bed|b P � b G *I,%* " = B C�E 0�M 0�" b1= C - 0�(1G#K�B� ∗ =
� 1� ( ��� ) :; d bc* " =�d 0�A�G#&)S * BR&I"cB a�/ (1A , J ] N Q 2 - 0 * M � a M /Jd a S16 *)-b* ( d (16'P�(1G
� ( � � � � �� ) = ��(� 1� ( � � � ) � �� )
-b* 0 d P1(1G%( =
� 1�+1( � � � )BFE =�bed@a M /Jd a S16 * "+6 � � ( � 1� ( � � � ) � �� ) 2 acbed " � " * (1U+K�B = ".0�G =)< & * "�0�"cBFE =�bed "
�( � ) =
� 1� ( ��� ) =� 1� ( � 1�
+1( � � � ) � � 1�+1( � � � )) : a
� 6 J P1(�H+U 0�"�K b1=F*Jd a S * BT&)( Q P b & <1/ B d D�K b * "+6 / U =�b K�"+6 * (1G !#(1GFp?BTM & , K b ]* (16 B =�b1/ &)(1K , 6�2 / BFE A = (1G#K�B * ( b1=F* E 0 * &)( O ( * (1G QoG N 2 S *Jd 2 / "#H b1/I, 2 a�< C)B / ")]K d (1G+&ID d a S.0�U = (�H�(.BFE =�bed|b b P�H , &)BT6cJ acbed@a�<+*Jd P1BT& d 0�0�S * BR&)( Q Q S N 2 � # � ,9[ �=' J � ^�e�Z [\[ Q 2 +B1 g@C`. �pg@?YAM1e<W< j-m��-<8A=1 ,:1��c<H1 ;=m ;@=>�E1-? m 1 � 1 �j�<8A=m��im � �J > Q Eh �� �Eg�</1|AM1���� m ,B< >E.�,:g �|< >E. j�=8A���@;k7Jj|7 � ( � )
;��k;=m <H1 h�m3= � ∩ � = ∅ � ⇒ [� ( � )↓ & � ( � ) ∈ � � \
�] :
JL! Q Eh �� �Eg�</m �c< >:. 1@A=1�� �`mk,l< >:. j�=8A���@;k7Jj|7 � ( � );��k;=me< 1 hJm&=
�∩ � = ∅ � ⇒ �
( � ) ∈ � � \ � :J ] Q Q
J N Q +-m � g@?PA=1 < �i70,l< m&=����-< >+A Z �i7�� 1��i.(7 J ] Q Q < j@��<kg�<l,:gp,l<H1��8A=1 � h � m�� � �8AM11|AM1���� m ,B< >E. �( � )
�J Q Q +-m � g@?PA=1 </1 �H1 �Bh��@. � g����� < �8< >+A`;0g � 1 Z ;=m ;@=>�E1-? m 1 � 1 �Bj,<iAMm �8m � g@?YAM1e<Bh �|.��eg��p1|A >-1 <@,�AcAMmcA 1|A�g@?PA=1 <�`79,B< m3= � �c< > A �j-X Z�<\#3��} [*2 J > Q ⇒ J�! Q ; d b * " / (10�K -R= "�2 b1=�b1/ &)(1K d a�, K�BT& d a�, 0�G =)< & * "+0�"
� ( � )2�G)P < &)A�B d J b P1S * ( !�_�p?B 4 &I"+K b B =�b1/ &)(1K , 6 Q�O G+0 d a S16 � *)-b* ( d (16'P�(1G
{ � 11( � � � )}( � ) = � � ( � � � ) =
{ ��(�) �%b1= � ( � 1
1( � � � ))↓ �⊥ � b H+H d 4 6 :
p -f* (1G+K�B � ( � ) = � (� 1
1( � � � ))D d � b G * S * ( � acbed P b & b+* "�&)(1U#K�B S *Jd " � ( � )
BFE =�bed(�H d a�, 0�G =)< & * "+0�"�2�BbP�B d /I,
�( � ) = � (
� 11( � � � ))↑ � ⇒ �
11( �� �
) = ∅ b P1S * ( = (1& d 0�K�S� ⇒ � (
� 11( � � � ))↓ :
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i � 2�� �KJ�� � # � ,3[ �=' j 2&'�< , ���=4�5 > 8�9I P d P�H - ( = 2�B O S10�( = � ( � )↓ 2 � 1
1( �� �
) = � 2 < & b �
∩ � = ∅ � ⇒ �( � ) = � (
� 11( � � � )) ∈ � � \ � 1
1( �� �
) = � � \ �
P1(1G%BRE =�bed�* ( � " * (1U#K�B = ( JL! Q ⇒ J N Q J B G *I, "'0�G = BfP b D�MWD ,'/ B = BbP d acb H�BRE *�bed#* (?!#( p'B 4 &I"�K b B =�b1/ &)(1K , 6�2acbed C b K�P�(1&)(1U+0�B =�b BRE A�B / (1C)BFE 0 * (.B /)<1O|d ( NPG Q; d b * " / (10�K -R= " 0�G =)< & * "�0�" � ( � )
P1(1G d acb1= (+P1( d BRE * " = J ] Q Q 2 P b & b+* "�&)(1U#K�BP1& 4�*�b S *Jd G)P < &)A�B d2b1=�b1/ &)(1K d a�, K�BR& d a�, 0�G =)< & * "�0�" � ( � ) *)-b* ( d b P1(1G
�(
�) = � ∪ { � ( � )} D
acbed K�B *)< C -b* (1G#K�B)2 b1=�b1/ &)(1K d a�< 2�(0 � � ) = �
�( F + 1 � � ) =
�(�( F � � )) �
-b* 0 d P1(1G J�BRU a (�H b 2�K�B?BfP b D�MWD , 0 * ( F Q �
(�+1� �
) = �∪ { � ( � (0 � � )) � � (� (1 � � )) �;:;:;: � � (� ( F � � ))} :
� P1B *�bed S *Jd D d b F 02
J ]#9 Q � ∩ � = ∅� ⇒ �
(�( F � � )) ∈ � � \ (
�∪ { � (� (0 � � )) � � ( � (1 � � )) �;:;:;: � � ( � ( F − 1 � � ))}) �
acbed 2�B d /Jd a S * BR& b 2 �
∩ � = ∅ � ⇒ (∀� G F )[ � (� ( F � � )) 6= �(�(� � � ))] :J ]PT Q
_ BTH d a�< 2�C -b* (1G#K�B�(0) =
�(0) �acbed D d b * ( = J b1=�b1/ &)(1K d a S Q (1& d 0�K�S * "+6 � ( � +1)
2�G�P�(�H�(�D�E � (1G#K�B�P1& 4�*�b?/Jd b1/ (1A d a�<*Jd 6 *Jd K - 6 � ( � (0 � � +1)) �;:;:;:�� � (� ( � +1 � � +1)) acbed C)BRMN&)(1U+K�B / U#( P�BR& d P *)4 0�B d 6 f g �9?Hh ; �/j|7�� FB = ( d *Jd K - 6 b G *)- 6 BFE =�bed S�H�BT6 /Jd b1O (1&)B *Jd a�- 6�2 * S * B K d b b P �b G *)- 67BRE =�bed�/Jd b1O (1&)B *Jd a�, b P1S *Jd 6 � (0) �;:;:;:�� � ( � )
2 acbed C -b* (1G#K�B�
= ( �+F ≤ ( � + 1))(∀ � ≤ � )[�(�( F � � + 1)) 6= �
( � )]�( � + 1) =
�(�(� � � + 1)) :
f g �9?Hh ; �/j|7�� �� P < &)A�(1G = F � � ≤ � + 12 F 6= ��2 *)-b* ( d b P�(1G � (� ( F � � + 1)) =�
(�(� � � + 1))
e�� b G *I, * " = P�BR&�E P * M 0�".C -b* (1G#K�B�( � + 1) = max{ � (0) �;:;:;:�� � ( � )} + 1 :
I E =�bed P�&)( O@b1=)- 6 b P1S * ( = (1& d 0�K�S%S *Jd " � ( � )BRE =�bed-b1=�b1/ &)(1K d a�,�acbed�-R=�b ]�P1&)(16b]
-R=�b 2 acbed�* (.P�M 6 BRE =�bed P b & b D�MND d a�, 0�G =)< & * "�0�".D d b.* ( � � - P�B *�bed:b K - 0�MN6 b P�S*Jd 6%J ]PT Q acbed J ]+9 Q
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> 8PT i 2 j 2&'�< , ���=��6�%M' X ' , ��k�� [&/ %;��l�2�����'J N Q ⇒ J Q Q �B = " � ( � )
BRE =�bed P b & b D�MWD d a�, 0�G =)< & * "�0�"'D d b * ( � � acbed#* ( � BFE =�bed* G+A b E ( b b 0�U = (�H�(�2 * S * B7J b P�S * (c!#(�p?B 4 &I"+K b B =�b1/ &)(1K , 6 Q G)P < &)A�B d�O G+0 d a S16� *)-b* ( d (16 P1(1G
� � ( � � � ) =
{1 � b1= � ∈ � &
�=�(� 1
1( � � � )) �⊥ � b H#H d 4 6 D
"c0�G =)< & * "�0�"�( � ) =
�(� 1
1( � � � ))BFE =�bedN-R=�b ]�P1&)(16b] -R=�b J�M 6%0�U = C)BR0�" K�( = (1K�(1& O@d 0�K 4N=�Q 2 acbedBb1=)< D�B d�* ( � 0 * ( � 2M 6?B C , 6 B =�� ∈ � 2 * S * B � 1
1( �� �
) = { � ( � 11( � � � )} = { � ( � )} �cacbed
�( � ) �∈ � � ⇒ �
11( �� �
) ∩ � = ∅� ⇒ �
(� 1
1( � � � )) ∈ � � \ � 11( �� �
)
� ⇒ �( � ) ∈ � � \ { � ( � )} �
P1(1G BRE =�bedlb1=F*Jd O@b+*Jd a S < & b � ( � ) ∈ � -B P�S * " =.< H+H#" K�BR& d < 2 b1= � �∈ � 2 * S * B �
11( �� �
) = ∅ ⊆ � � ��< & b � ( � ) =�(� 1
1( � � � )) ∈ � � a
� � � �p�/yl~2�#$ }��� Q S c>&2�� BRE C * B7S *Jd D d b a�< P1( d ( � 2 � = { � � � + 1 �;:;:;: } = { � | � ≥ � } � Q S ! 2�� BRE C * B7S *Jd D d b a�< P1( d ( � 2 � � ( � ) =
� · � � Q S N 2�� BRE C * B S *Jd D d b^a�< C)B (�H d a�, 2 b1=�b1/ &)(1K d a�, 0�G =)< & * "�0�" � ( � ) 2 , G)P < &)A�B da�< P1( d (16 � *)-b* ( d (16 P1(1G%( � ( � ) =�b BFE =�bed P�BR& d *F* S16�2 acbed D d b S�H bc*�b"� 2
� � ( � ) =�( � + � ) �
, G�P < &)A�B d@a�< P1( d (16 � *)-f* ( d (16'P�(1Gc( � ( � ) =�b BFE =�bed < & *Jd (16�2 acbed D d b S�H bc*�b"� 2� � ( � ) =
�(2 � + � + 1) :
� Q S Q 2 B H+"+C)BRU#B d', S1A d� D d b a�< C)B b1=�b1/ &)(1K d a�, 2 (�H d a�, 0�G =)< & * "+0�" �( � )G�P < &)A�B d@a�< P1( d (16 � *)-f* ( d (16'P1(1G
�( � ) = � Db P1( / BRE C * B * " =pb P <1=F* "�0 , 0 b 6
� Q S 9 2 B H+"+C)BRU#B d', S1A d� D d b a�< C)B b1=�b1/ &)(1K d a�, 2 (�H d a�, 0�G =)< & * "+0�" �( � )G�P < &)A�B d@a�< P1( d (16 � *)-f* ( d (16'P1(1G
� �( � )(
�) = � � (
�) (
� ∈ N) Db P1( / BRE C * B * " =pb P <1=F* "�0 , 0 b 6
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i � 2�� �KJ�� � # � ,3[ �=' j 2&'�< , ���=4�5 > 8 d� Q S T 2 J�Y Q � BFE C * B S *Jd D d b a�< C)B (�H d a�, 2 b1=�b1/ &)(1K d a�, 0�G =)< & * "+0�" � ( � ) 2#G�P < &R]A�B d@a�< P�( d (16 b & d C)K�S16 � *)-b* ( d (16'P�(1G
� = { � ( � )} :JU� Q � BRE C * B?S *Jd G)P < &)A�B d|a�< P�( d (16 b & d C)K�S16 � *)-b* ( d (16'P�(1G� � ( � )↓ acbed� � = { � � ( � )} :
� Q S d ∗ 2�� 0 * M b1=�b1/ &)(1K d a�, 2WK�BT& d a�, 0�G =)< & * "+0�" � ( � ) *)-f* ( d b P�(1G D d b S�H b*�b � 2 b1=� � = N ��* S * B � ( � )↓ D/ BFE C * B?S *Jd G)P < &)A�(1G =pb & d C)K�(�E � acbed ��2 *)-f* ( d ( d
� = {0 � 1 �;:;:;:�� � } acbed � ( � )↓ :VvX Z�<\#3��} [���I O-b &)K�S10 * B * (.!+( p?B 4 &I"+K b B =�b1/ &)(1K , 6?0 * "cK�BT& d a�, 0�G =)< & * "�0�"
�( � � � ) =
{1 � b1= (∀ � ≤ � )¬ �
1(� � � ��� ) �
⊥ � b H#H d 4 6 �S+P1(1G �
( � ) = {� }( � )
� Q S O ∗ 2�� 0 * M b1=�b1/ &)(1K d a�, 2WK�BT& d a�, 0�G =)< & * "+0�" � ( � ) *)-f* ( d b P�(1G D d b S�H b*�b � 2 b1=� � = ∅ � * S * B � ( � )↓ D/ BFE C * B?S *Jd G)P < &)A�B d@a�< P1( d (16 � *)-b* ( d (16 P1(1G � = { � } acbed � ( � )↓
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^ I�� B��8B�� 3 9
� ����)� n�� � �7� n� � � �P ���� � �
e�� b G * S * ( a B O�< H bed ( C b K�BTH+B *I, 0�(1G#K�B *Jd 670�A - 0�B d 670 * (1G#6 O G+0 d a (1U+6 b & d C)K�(1U#6P1(1G K�P�(1&)(1U =%=�b # acb+*�b 0 a BRG b 0 * (1U =�& C)B a-d =)4N=F*�b 6%K�B *Jd 6 b1=�b1/ &)(1K d a�- 6 0�A - 0�B d 6acbed B O-b &)K�S � ( =F*�b 6 BfP b1= B d H+"+K�K -R=�b * (1G+6 * BbH�BT0 *)- 6 * "�6 P1&)M * ( 51< C)K d b 6 H+(�D d ]a�, 6 � b 0 d a�< P1(1&�E 0�K b+*�b b G *I, 6 * "�6 K�BbH -f* "�6 BFE =�bed�*�b�a H b 0 d a�< C)BRMN& , K b+*�b* M = `WY i `sr�Z�2 tvu_�w�VF[ acbed G e g�i : e 0 *�b B /)<1O|d b 9 E 2�9 G
�+����� z:w@}��,�Bt|{@}Yyl~!}/$`wcz:w)#!�z3 d "�K d b1=�b1/ &)(1K d a�- 6cJ Σ0
1Q 0�A - 0�B d 67BRE =�bed�* "�67K�(1& O�, 6
(∃ � ) � ( �� ��� )K�B a�< P�( d b b1=�b1/ &)(1K d a�, � ( �� ��� ) 2 acbed+-b* 0 d # b P - A�(1G =�& J�0�B�P1(�H+G�P+H�( a S * " *�b�Q K�S�H d 6�8A=1 = h�1�� C8< 18> A hJm j-m���g@? � ;k7 b P�S *Jd 6 b P1( a &�E 0 d K�BR6 J b1=�b1/ &)(1K d a�- 6 Q 0�A - 0�B d 6 3BbP�S1K�B = (16.(1& d 0�K�S16cBRE =�bed A�& , 0 d K�( BT&ID b H�BRE ( D d b * " *�b C d = S1K�"�0�" P�(�H�U)P�H+( a M = 2b1=�b P1( a &�E 0 d K�M = 0�A - 0�BTM = 9 @ c>32*Y, ������Z�5 J � ���1�7������1� �� � E)������4 �� �12 3 d a H < 0�B d 6 J�0�U = (�H b�Q 0�A - ]
0�BTM = Σ0
�2Π0
�2∆0
�(1&�E � ( =F*�bed K�B * "cB C , 6 b1=�b1/ &)(1K ,
Σ01 :
( d "+K d b1=�b1/ &)(1K d a�- 670�A - 0�B d 6Π0
� = ¬Σ0
� :( d|b & =I, 0�B d 6%JL0�G+K�P+H+"+& 4 K b+*�b�Q * M = 0�A - 0�BTM = 0 * ( Σ0
�Σ0
� +1 = ∃Π0
� :( d 0�A - 0�B d 6'P�(1G d acb1= (+P1( d (1U = K d b d 0�( / G =�b K�E b� ( �� ) ⇐⇒ (∃ � ) � ( �� ��� ) � S+P1(1G " � ( �� ��� ) BRE =�bed Π0
�∆0
� = Σ0
� ∩ Π0
� :( d 0�A - 0�B d 6'P1(1G%BRE =�bed Σ0
�acbed Π0
�:
_ ( * G#A b E (%0�U = (�H�( � ⊆ NBFE =�bed 0�B K d b b P � b G *)- 6 *Jd 6 a H < 0�B d 6 Γ b1= " 0�A - 0�"� ∈ � b1=I,ia B d 0 * " = Γ
9 @ ! 2 � �������1� � �18 ������ �@81 3 d|a H < 0�B d 6 b G *)- 6 * "+6 1��@<Y5 ,W7�;=< >:. � < g �`1��;�*? 1��
J P�&)( O@b1=)4 6 Q A b & b`a�* "+&�E � ( =F*�bedeb P�S *Jd 6 B C , 6 # acb1= ( =Jd a�- 6�K�(1& O�- 6 & 2+K�B * " =�-R=)= ( d bS *Jd K d b 0�A - 0�" � ( �� ) b1=I,8a B d 0 * " =�a H < 0�" à b1= BRE =�bed-d 0�( / U =�b K�" K�B * " acb1= ( =Jd a�,
> 8P]
>P> 8 q 2 j 2&'�< , ���=4;6�'&�8� , ��������Z /�[&/ 'K�(1& O�, D d b%* " = Γ
2�D d b a�< P1( d b b1=�b1/ &)(1K d a�, 0�A - 0�" � Σ0
1 : (∃ � ) � ( �� ��� )Π0
1 : (∀ � ) � ( �� ��� )Σ0
2 : (∃ � 1)(∀ � 2) � ( �� ��� 1 ��� 2)Π0
2 : (∀ � 1)(∃ � 2) � ( �� ��� 1 ��� 2)Σ0
3 : (∃ � 1)(∀ � 2)(∃ � 3) � ( �� ��� 1 ��� 2 ��� 3) ; d b P b & <1/ B d D�K b 2 b1= "c0�A - 0�" � ( �� ) BRE =�bed Π0
2
2 * S * BI2 b P1S * (1G+6?(1& d 0�K�(1U#6�2� ( �� ) ⇐⇒ ¬ � 1( �� ) K�B � 1 ∈ Σ0
2�
⇐⇒ ¬(∃ � 1) � 3( �� ��� 1) K�B � 3 ∈ Π01�
⇐⇒ ¬(∃ � 1)¬ � 4( �� ��� 1) K�B � 4 ∈ Σ01�
⇐⇒ ¬(∃ � 1)¬(∃ � 2) � ( �� ��� 1 ��� 2) K�B � b1=�b1/ &)(1K d a�, �⇐⇒ (∀ � 1)(∃ � 2) � ( �� ��� 1 ��� 2)
9 @ N 2 � # � ,9[ �=' 2 J > Q < 1 >c5=g � ≥ 1Z m <B>��� j6g�< �
Σ0
�ZΠ0
�>-1 <
∆0
�g@?PA=1 <> �ig�<Hj@;������-< 1 1@A=1�� �`mk,l< > ����1@Ak;=< >-16;91ej@; ejeg�< � >c1e< �-< 1�;=m&= � ;0g��ig=j|;����
&Z ∨ Z ∃≤>-1 < ∀≤ � � h`< h�����mcA
• � >��� j|7 ∆0
�g@?PA=1 <:> �ig�<Hj@;k. �-< 1(;k7eA ��eAk7Jj|7 ¬ �
• � >��� j|7 Σ0
�g@?PA=1 <:>��`g�< j|;k. �c<H1 ;=mcA = h�1�� C8< 18> A hJm j-m���g@? � ;k7 ∃ �
• � >��� j|7 Π0
�g@?YAM1e<2> �ig�<Hj@;k. �-< 1(;=m6A >-165=m��-< >+A h�mejcm��kg@? �W;k7 ∀ �
JL! Q < 1 >c5=g � ≥ 1Z
Σ0
� ⊆ ∆0
� +1�J ] d Q
>-1 < g�h�mk, �8A�� � m < 1��|<P5 ,/7J;=< > ��� > ��ejeg�< � < >c1|AMmeh�me< m3<8A ;=m(g@C`. � �J< ���`1k,|,:1�j�= �,BhJg �|< �|.��6g��BA
Σ01 Σ0
2 Σ03
⊆ ⊆ ⊆ ⊆ ⊆ ⊆∆0
1 ∆02 ∆0
3 · · ·⊆ ⊆ ⊆ ⊆ ⊆ ⊆
Π01 Π0
2 Π03
j-X Z�<\#3��} [*2 $'& 4W*�b / BRE A = (1G+K�B * " = a H�B d 0 * S * " *�b S�H+M = * M =#b & d C)K�" *Jd a�4N=a H < 0�BRM = D d b(b1=�b1/ &)(1K d a�- 6 b1=F*Jd acb+*�b 0 *)< 0�B d 6�2WK�B BbP b D�MND , 0 * ( � " P�&)S *�b 0�"BFE =�bed D = MN0 *I, D d b � = 1 b P�S * " = $'&)S *�b 0�" Q�@ 912 acbed BbP b D�MND d a�< J�D d b * " =
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q � 2 � ' , ��k�� [&/ ��6�4M��# , ' , :&�z' >�>P>
P1BT&�E P * MN0�" Σ0
� +1Q G�P�(�H�(�D�E � (1G#K�B
� ( �� ) ⇐⇒ � (�1( �� ) �;:;:;: � � � ( �� ))
⇐⇒ (∃ � ) � (�1( �� ) �;:;:;:�� � � ( �� ) ��� )S+P1(1G � ∈ Π0
�2 b P�S * ( = (1& d 0�K�S
⇐⇒ (∃ � ) � ′( �� ��� )S+P1(1G � ′ ∈ Π0
�b P�S * " = BfP b D�MWD d a�, G)P1S1C)BT0�" :
_ b G�P�S�H�( d P b * (1G J > Q / BRE A = ( =F*�bed BTU a (�H b 2�K�B BfP b D�MWD , 0 * ( � acbed B O-b &)K�(�D - 6* M = K�B *�b 0�A�"+K b+*Jd 0�K 4N=?* "�6 b P�S / B d CI"+6 * "�6?$'&)S *�b 0�"+6 QP@ 9 _ ( J�! Q / BFE A = B *�bed K�B?BfP b D�MWD , 0 * ( ��2�S+P1(1G%0 * " 51< 0�"�2 b1=
� ( �� ) ⇐⇒ (∃ � ) � ( �� ��� )K�B * " = � b1=�b1/ &)(1K d a�, 2 * S * B%" � BRE =�bed 0�E D�(1G#& b Σ0
2
2 b1O (1U a�< C)B b1=�b1/ &)(1K d a�,0�A - 0�"cBRE =�bed Π0
1
2 b H#H < BRE =�bed@acbed Π02
2 b1O (1U12�P�&)( O@b1=)4 6�2� ( �� ) ⇐⇒ (∀ � )(∃ � ) � ( �� ��� )acbed "c0�A - 0�"� 1( �� � � ��� ) ⇐⇒ � ( �� ��� )BFE =�bedEb1=�b1/ &)(1K d a�, m b P1S / B d CI" 0 * ( BfP b D�MWD d a S 5+, K b BRE =�bedlb`a & d 5�4 6%E /Jd b 2 acbed
( d 0�G#K�P1BT& d H , ��B d 6 * (1G /Jd b D�& < K�K b+* (16 0�G =)< D�( =F*�bed BRU a (�H b#b P1S * " = J ] d Q�acbed* B * & d K�K -T= (1G+670�G�H+H+(�D d 0�K�(1U#6 a$ d ( B =)/Jd b1O�- &)( = BFE =�bed�* ( BbP�S1K�B = ( C)B 4 &I"+K b P1(1G /Jd acbed 4 = B d�* " = BfP1( = (1K b 0�E b# d BT& b &)A�E b%& D d bc*Jd 6 a H < 0�B d 6 Σ0
�2Π0
�
9 @ Q,2 � # � ,9[ �=' J � E.�������� �c�1�7�� ���1� ���8��;E���������4 ���8 2 � [ VFVAb�V Q 2J > Q J B P b &�E C)K�"�0�" * (1G Σ0
�Q < 1�> 65 g � ≥ 1
>c1e< >c5=g � ≥ 1Z ==h �� �Eg�<
(�
+ 1)�,Eg��@. ��j � �=j@7 �
�� � ( � � �� ) j@;k76A/> ��ej@7 Σ0
�h�m3= 1 hJ1��|<P5 ,:g@?E;=< � � � ,Eg��ig@? � Z
Σ0
�j � � �j6g�< � Z �`7 �e1��`.�� 7�;@= �:1 ? 1 � ( �� ) g@?PA=1 < Σ0
�1|A >c1e<-,9A6A=m6A 1|A��c<H1 >eh�me< m � Z
� ( �� ) ⇐⇒ ��� � ( � � �� ) :
JL! Q J B P b &�E C)K�"+0�" * (1G Π0
�Q <H1^> 65 g � ≥ 1
>c1e<`> 65 g � ≥ 1Z ==h �� �Eg�<
(�
+ 1)�,Eg��@. �\j@� �=j|7 � � � � ( � � �� ) j|;k7eA > ��ej@7 Π0
�hJm&=#1eh�1��@<Y5 ,Eg@? ;=< � � � ,:g��`g@? � Z
Π0
�j@� �=j6g�< � Z �i7�� 1��i. � 7�;@=>�E1-?H1 � ( �� ) g@?YAM1e< Π0
�1|A >c1e<-,9A6A=m6A 1|A��c<H1 > hJm < m � Z
� ( �� ) ⇐⇒ � �� � ( � � �� ) :J N Q J � BT& b &)A�E b�Q � < j�= ,BhJg �|< �|.��6g�< � ;=m&= �J<H1 ��� k,|,:16;=m�� j|;k7eA f �-A`;91ej@7 9 @ Ng@?YAM1e<HA �ig���1�=�j|;k7������ Z �`7 �e1��`.��
Σ01 Σ0
2 Σ03
( ( ( ( ( (
∆01 ∆0
2 ∆03 · · ·
( ( ( ( ( (
Π01 Π0
2 Π03
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>P> ! q 2 j 2&'�< , ���=4;6�'&�8� , ��������Z /�[&/ 'j-X Z�<\#3��} [*2 ; d b%*�b J > QWacbed JL! Q C -f* (1G+K�B b1=�b1/ &)(1K d a�<
�1� � ( � � �� ) ⇐⇒ (∃ � ) ��� ( � � �� ��� )
� �� � ( � � �� ) ⇐⇒ ¬ � �
� � ( � � �� )�� +1
� � ( � � �� ) ⇐⇒ (∃ � ) � �� �
+1( � � �� ��� ) �acbed ( d6b P1( / BRE C)B d 6 BFE =�bed BTU a (�H+BR6�2�K�B�BfP b D�MWD , 0 * ( � ; d b * ( J N Q 2�P b & b+* "�&)(1U#K�BS *Jd " # /Jd b D 4N=Jd b%& 0�A - 0�"
�� (� ) ⇐⇒ �
��1(� � � )
BFE =�bed Σ0
�acbed�/ B = K�P1(1&)BRE =�b BRE =�bed Π0
�2�D d b+* E 2 b1=%,�*�b1= B)2 * S * B D d b a�< P1( d ( � C b
BFE A b K�BI2¬ � �
�1(� � � � ) ⇐⇒ �
��1( � � � )P1(1G7BFE =�bed�<+* (+P1( D d b * ( � = �
� P�B *�bed S *Jd D d bpa�< C)B ��2�G)P < &)A�(1G = 0�A - 0�B d 6�P1(1GBFE =�bed Σ0
�b H+H < / B = BRE =�bed Π0
�2 acbed:b P � b G * S 0�G =)< D�B *�bed BTU a (�H b " b G+0 * "�&)S * " *�b
S�H�M =7* M = 0�G#K�P1BT& d H , ��BTM =7* (1G /Jd b D�& < K�K b+* (16 a9 @ 9 2 J $'H , &I"+6 Q ��� �1������������ K d b 6�0�A - 0�"+6 � ( �� ) 0 * " = b & d C)K�" *Jd a�, d BR& b &)A�E bBFE =�bed ( acb C)(1& d 0�K�S16 * "+6�#fBTH < A d 0 * "+6 &�b & d C)K�" *Jd a�, 6 a H < 0�"+6 0 * " = (+P�(�E b b1=I,8a B d
" � ( �� ) 2 / "#H b1/I, " b P�S / B d CI" P1&)S *�b 0�"�6 * "�67K�(1& O�, 6� ∈ Σ0
� \ Π0
�, � ∈ Π0
� \ Σ0
�, � ∈ ∆0
� +1 \ (Σ0
� ∪ Π0
� )D d b a�< P�( d ( � ; d b P b & <1/ B d D�K b 2�0 * ( QPE' c> 8 / BFE C b K�B?S *Jd{ � |
�6= ∅} ∈ Σ0
1 \ Π01:
m P+H , &I"�6 *�b C d = S1K�"+0�" K d b 670�A - 0�"+6 BFE =�bed 0�B?K�BR& d a�- 67P1BT& d P *)4 0�B d 67P1(�H+U / UI]0 a (�H#"�2 acbed 0�G#A =)< b & a (1U+K b 0 * B 0 * ( = G)P1(�H+(�D d 0�K�S a�< P�( d (1G�# <1= M O & < D�K b+* (16 & 2/ "+H b1/I, a�< P1( d (1G � *)-b* ( d (1G?P�(1G � ∈ Σ0
�, � ∈ Π0
� m 5 b 0 d a�, K - C)( / (16�D d b * ( =
G�P�(�H�(�D d 0�K�S�# a�<+* M O & < D�K b+* (16 & 2�S *�b1= b G * S BFE =�bed B O|d a�* S�2�BFE =�bed " b P1S / B d CI" S *Jd" / (10�K -R= " 0�A - 0�" BRE =�bed h �|.��@7 � 0�B a�< P�( d b a H < 0�" Σ0
�, Π0
�S+P1MN6'0 * (cBbP�S1K�B = (b P1( *)- H�BT0�K b
9 @ T 2 *Y, Z / '�� [*2 J > Q +-m�j,<iAMm �8m �= { � | 7 � � g@?YAM1e<Bm �c< >:. } g@?YAM1e< Π0
21 ������kg8A�g@?YAM1e<Σ0
2
�JL! Q +-m j,<iAMm �8m � Z b = { � | ;=m � g@?YAM1e<Bh8g�hJg � 1 j8, �8A=m } g@?PA=1 < Σ0
2 \ Π02
�
j-X Z�<\#3��} [*2 J > Q _ ( <1= M O & < D�K b BRE =�bed P1&)( O-b1=)- 6�2 b1O (1U� ∈ � ⇐⇒ (∀ � )(∃ � ) � 1( � � � ��� ) :
; d b =�b / BFE C)(1G#K�B JLK�B b P b D�MWD , 0�B <+* (+P1( Q S *JdW* ( � / B = BFE =�bed Σ02
2 C)BTM &)(1U#K�BK d bc* G#A b E b Π0
2
0�A - 0�"�2�P�(1G d acb1= (+P1( d BRE * " =�d 0�( / G =�b K�E b� ( � ) ⇐⇒ (∀ � )(∃ � ) � ( � � � � � )
K�B a�< P�( d b�b1=�b1/ &)(1K d a�, � ( � � � � � ) 2 acbed C -f* (1G+K�B�( � � � ) = � � � ( � � � � � ) :
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q � 2 � ' , ��k�� [&/ ��6�4M��# , ' , :&�z' >�> N
B = � BFE =�bed@a M /Jd a S16 * "�6 J b1=�b1/ &)(1K d a�, 6 Q � ( � � � � � ) 2 * S * B� ( � ) ⇐⇒ (∀ � )[
�( � � � )↓ ]
⇐⇒ (∀ � )[{ � 11( � � � )}( � )↓ ]
⇐⇒ � 11( � � � ) ∈ � D
- P�B *�bed S *Jdeb1= * (f� ,#*�b1= Σ02
2 * S * B a�< C)B Π02
0�A - 0�" C b ,#*�b1= Σ02
2+P1(1G b1=F*Jd * E C)B *�bed0 * ( p?B 4 &I"+K b � BT& b &)A�E b 679 @ Q J N Q JL! Q _ ( <1= M O & < D�K b BFE =�bed P < H d P�&)( O@b1=)- 6�2
� ∈ � Z b ⇐⇒ (∃ � )(∀ � )[ � ∈ � � ⇒ � ≤ �] :
; d bc* ( a�<+* M O & < D�K b 2 - 0 * M � ( � ) * G+A b E b Σ02
0�A - 0�"�2 -f* 0 d P�(1G� ( � ) ⇐⇒ (∃ � )(∀ � ) � ( � � � � � )
K�B a�< P�( d b�b1=�b1/ &)(1K d a�, � p -b* (1G#K�B�( � � � ) = � � (∀ F ≤ � )¬ � ( � � F � ( � ) � ) �
-b* 0 d P1(1G b1= ( � BFE =�bed@a M /Jd a S16 * "�6 � 2 * S * B(∃ � )(∀ � ) � ( � � � � � ) ⇐⇒ { � | � ( � � � )↓} BFE =�bed P�BbP�BR& b 0�K -R= (
⇐⇒ { � | {� }( � � � )↓} BFE =�bed P�BbP�BR& b 0�K -R= (⇐⇒ { � | { � 1
1(� � � )}( � )↓} BFE =�bed P�BbP�BR& b 0�K -R= ( �/ "+H b1/I,
� ( � ) ⇐⇒ � 11(� � � ) ∈ � Zcb Db H+H < b G * S%0�G = BfP < D�B *�bed S *Jd�* ( � Zcb / B = BFE =�bed Π0
2
2�D d b+* E b1=',#*�b1= 2 * S * B a�< C)B Σ020�A - 0�"cC bc,#*�b1= Π0
2
2�P1(1G%BRE =�bed <+* (+P1( a
�+��� �p�Byl~:�*$e}����9 @ c>&2 _ b C d = (1K , 0 * B'0 * " =pb & d C)K�" *Jd a�,�d BR& b &)A�E bc* (.0�U = (�H�(
� = { � | �⊆ {0 � 1}} :
��9 @ ! 2 _ b C d = (1K , 0 * B'0 * " =pb & d C)K�" *Jd a�,�d BR& b &)A�E bc* (.0�U = (�H�(� = { � | * (
� BRE =�bed P1BfP1BT& b 0�K -T= ( acbed K�")] a B = S } :��9 @ N 2 _ b C d = (1K , 0 * B'0 * " =pb & d C)K�" *Jd a�,�d BR& b &)A�E bc* (.0�U = (�H�(
� = { � | G�P < &)A�(1G = < P�B d &)( d�* (cP+H , C)(16 / E / G#K�( d P1& 4�* ( d ≥ � } �S+P1(1G.( � BRE =�bed�/ E / G#K�(16 P�& 4W* (16 b & d C)K�S16 b1= ( � acbed ( � + 2
BRE =�bed:acbed ( d�/ U#(P1& 4�* ( d
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>P> Q q 2 j 2&'�< , ���=4;6�'&�8� , ��������Z /�[&/ '��9 @ Q 2 _ b C d = (1K , 0 * B'0 * " =pb & d C)K�" *Jd a�,�d BR& b &)A�E bc* "c0�A - 0�"
� ( � � � ) ⇐⇒ ��v � �
⇐⇒ (∀ � )[ ��( � )↓ � ⇒ [ � � ( � )↓ & �
�( � ) = � � ( � )]] :
��9 @ 9 2 _ b C d = (1K , 0 * B'0 * " =pb & d C)K�" *Jd a�,�d BR& b &)A�E bc* (.0�U = (�H�(� = { � | * (
� - A�B d�* (1G�H < A d 0 * ( = � K - H#" } :��9 @ T 2 _ b C d = (1K , 0 * B 0 * " = b & d C)K�" *Jd a�, d BR& b &)A�E b * ( 0�U = (�H+( / B d a�*)4 = # <1= M
O & b D�K -T= M =�& b1=�b1/ &)(1K d a�4 = 2�K�BT& d a�4 = 0�G =�b & *I, 0�BTM = 2� = { � | D d b a�< P�( d ( � acbed S�H bc*�b"� ��� � ( � )↓ � ⇒ �
�( � ) ≤ � } :
��9 @ d-2�� 0 * M � * G#A b E (�2 b1=�b1/ &)(1K d a S%0�U = (�H�( *)-b* ( d ( P1(1G � ( N *�b C d = (#]K , 0 * B'0 * " =pb & d C)K�" *Jd a�, d BR& b &)A�E bc* ( 0�U = (�H+(!
= { � | �⊆ � } :
��9 @ O 2 J�Y Q � BRE C * B?S *Jd "c0�A - 0�"� ( � ) ⇐⇒ * ( � BRE =�bed2b b P+H , &)BR6
BFE =�bedBb & d C)K�" *Jd a�, 2 acbed�* (+P1(1C)B *I, 0 * B b G *I, * " 0�A - 0�" 0�B a�< P1( d ( 0�G�D a B a & d K -T= (0�U = (�H�( Σ0
�, Π0
�2�S10�( P d (.A b K�"#H < 0 * " = b & d C)K�" *Jd a�, d BR& b &)A�E b K�P�(1&)BFE * B J @ " =
P1&)(10�P b C , 0 * B =�b / BFE C)B * B7S *Jd " *�b C d = S1K�"�0 , 0 b 6?BFE =�bed P+H , &I"�6 QJU� Q ^ <1=F* B * ( E /Jd (.D d bc* "c0�A - 0�"
�( � ) ⇐⇒ * ( � BFE =�bed / "+K d (1G+&ID d a S :
��9 @ ] 2�� BRE C * B?S *Jd�* (.D�& <1O "�K b� �
( �� � � ) ⇐⇒ �( �� ) = �
* G+A b E b 6 m �c< >:. � 0�G =)< & * "�0�"�6 � ( �� ) BFE =�bed Σ0
�b1= acbed K�S = ( = b1= BFE =�bed ∆0
�
��9 @ c> 8 2�m (�H d a�, 0�G =)< & * "+0�" � ( �� ) BFE =�bed m��@<H1J> 1|AM1���� m ,B< >E. J�[ Z\X%Z d i V : g�i ]`fZ V Q b1= G�P < &)A�B d|b1=�b1/ &)(1K d a�, 2�(�H d a�, 0�G =)< & * "�0�" � ( � � �� ) *)-b* ( d b P�(1G�( �� ) = lim� →∞
�( � � �� ) �
S+P1(1G * (.S1& d ( b`a (�H+(1G+C�E b 6 O G#0 d a�4N= (1&�E � B *�bed MN6?0�G =I, C)MN6�2lim� →∞
� � = � ⇐⇒ (∃ � )(∀ � ≥ �)[ � � = � ] :
� BFE C * B S *Jd " * G#A b E b (�H d a�, � ( �� ) BFE =�bed (1& d b`a�< b1=�b1/ &)(1K d a�,�b1=�acbed K�S = ( = b1= * (D�& <1O "+K b � � * "+6 � ( �� ) BFE =�bed ∆02
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q � 2�jY, ��k�� [&/ ��6 { 57��: { ��#3��576�'&� / � � # � ,9[ �=' / �&� ��������� >�> 9��� � � w@}��,�Bt@{@}Yy/�`��� B�`�*$e}���y/zl}^{cx ��$i� wet(�Wz {cx � ��� �����
� b 0 d a S16?0 * S1A�(167K b 670 � b G * S acbed�* ( BfP1S1K�B = ( * ( B /)<1O@d ( BFE =�bed =�b / BRE C)(1G+K�B*�b�a H b 0 d a�< C)BTM & , K b+*�b\b1=�b P1( a & d 0 d K�S * " *�b 6 * M = `WY i `sr�Z�2 tvu_Pw�VF[ acbed G e g�i : eD d bc*Jd 6 -T=)= ( d BR6 * "+6 b H , C)B d b 6 acbed�* "�6 b P1S / B d CI"�670 * " = J�(�H d a�,�Q�< H+D�B 5 & b
�= (N � 0 � 1 � + � ·) �J ] O Q
/ "+H b1/I,.* " = a H b 0 d a�, h��3�@;=m � c5 ,B< 1��8m ,W. * M =cO G+0 d a�4 = b & d C)K 4 = K�B?P1&)S10�C)BR0�"acbed P�(�H+H b P�H b 0 d b 0�K�S p b C)B a-d =I, 0�(1G#K�B K�B�K d b 0�U =F* (1K�" BfP d 0 a S+P�"+0�" * M = 0�A�B ]*Jd a�4 = (1& d 0�K 4 = b P1S * " = P�&)M * ( 5�< C)K d b H�(�D d a�, D d b (�H d a�- 6 < H+D�B 5 &)BT6
= ( � 0 � 1 � � 1 �;:;:;: � ��� )
P1(1G BRE =�bed D�B =Jd a BTU+0�B d 6 J acbed K d a & - 6�P b & b H#H b D - 6 QN* M = (1& d 0�K 4 = 0 * (cB /)<1O|d (%! @ 9 E' c>&2�� ����7���� � �x������ � ��������� �
=�(0 � 1 ��� 1 �;:;:;:���� � )) � �����#��� ���� m
� BRE =�bed K d a & , P b & b H+H b D , * "�6 � (
)P�(1G (1&�E 0 b K�B.0 * ( B /)<1O|d ( ! @ 2 b H#H < C b
BbP b1=�b H <+5 (1G+K�B%J�0�G = (+P *Jd a�<�Q * (1G+6 0�A�B *Jd a (1U#6 (1& d 0�K�(1U#67D d b =�b /Jd BTG a & d = E 0�(1G+K�BP�H , &)MN6 *Jd 6 /Jd b1O (1& - 6 \m�� - A�B d1c;=m ,B< > ��� ,Egk;91�� �@7J;���� �
0���
1�;:;:;: �
1c;=m ,B< > ����j|;9165 g ����� 0 � 1j�=8A=1��@;k7Jj-< 18> ��� j@;91c5=g ����� �1�;:;:;:���� � (
Y i Z d ^( � � ) =
� � )*�b j@70,:g@?H1 j|;B? C=g�� � � ( )
* ( 0�U+K 5 (�H�( * "+6 d 0�S * " *�b 6 =acbed�*�b j�< ,��:m � 1(;k7 ��h��3�@;=m � c5 ,B< 1�� �im �-< >E. � ¬ & ∨ → ∃ ∀$ b & b+* "+&)(1U+K�B�S *Jd " � (0 � 1 ��� 1 �;:;:;:���� � ) /Jd b1O�- &)B d6b P�S * " = � ( )M 6 B C , 6 N/ B =
- A�B d b+* (1K d a�- 6 0 *�b C)BT& - 6�2 -b* 0 d P1(1G / B = a�<1= B d acb K�E b 0�G#D a B a & d K -R= " b1=�b1O (1& <0 * ( 0�U = (�H�( / B = P b & - A�B d 0�U+K 5 (�H d 0�K�S D b * " /Jd b`a H <1/ MN0�" acbed 2 * ( a G#& d S#]* BR&)(�2�P b & - A�B d 0�G+K 5 (�H d 0�K�S D d b * BTH+BR0 *)- 6 * "�67P�&)( *�b 0 d b`a�, 6 acbed P�&)M * ( 5�< C)K d b 6H�(�D d a�, 6 3 d 0 *�b C)BR& - 6 0 acbed 1 acbed ( d b+* (1K d a�- 6 K�B *�b+5 H+" *)- 6 J�M 6 b`a (�H�(1G#C�E BR6 K , ]a (1G+6 1 Q BFE =�bed ����4�� ���� ������� 2 acbed ( d J b D = (�E 2W&I" * (�E Q ������� (1&�E � ( =F*�bedBb1=�b1/ &)(#]K d a�< 23C)B a-d =)4 =F*�b 6'K�B * (1G+6 b &)A d a (1U#6 S1&)(1G+6 acbed A�&I"+0 d K�(+P�( d 4 =F*�b 6 *�b 0�G =�b & * ")]
0 d b`a�< 0�U#K 5 (�H b 2 -b* 0 d P�(1G a�< C)B?S1&)(16'P�(1G / B = BFE =�bed|b &)A d a S16 BRE =�bed�* "+67K�(1& O�, 6� � ( � 1
�;:;:;:�� � � � ) S+P�(1G ( d � 1�;:;:;: � � � � BFE =�bed S1&)( d K d a &)S * BR&)(1G K ,ia (1G#6 7@ B * (
0�G = (+P *Jd a S * &)S+P1( P1(1G /)4 0 b K�B * ( = b1=)< H�(�D�( (1& d 0�K�S S1&)M = D d b K�BR& d a�- 6 < H+D�B ]5 &)BT670 * " = J Q�> Q 2
� :≡ � � | 0 | 1 | ��� ( � 1�;:;:;: � � ��� )J ]�] Q
�c��4�� ���� �� ������� BRE =�bed ( d H - C)B d 6 * "+67K�(1& O�, 6� 1 = � 2
����� �!�#"�$&%('*)�+!,!-!+/. �/0!"�$1 �/2 3!4/+!5/670 28�9#:�+/; +/<���,8��5!=#> "/>�2? "&@A4/+/0 2#>�2 4#0/>���0!6CB#0�3/+!,/"ED�:A4/=!- FG�H4/+�DI<�F/5!�#>�+J; � K#"('L�MJN FO:!>AF/5/.�4�PH+!9�D ?/Q!Q R D ?SLGTUM!V ' L!LYR
>P> T q 2 j 2&'�< , ���=4;6�'&�8� , ��������Z /�[&/ 'S+P1(1G ( d � 1
2 � 2BFE =�bed S1&)( d 2 acbed ( d � ������� * "�6 � (1&�E � ( =F*�bed b1=�b1/ &)(1K d a�< 2 C)Bf]a-d =)4 =F*�b 6 K�B * (1G+6 b &)A d a (1U#6 * U�P�(1G+6 acbed A�&I"�0 d K�(+P1( d 4N=F*�b 6 * (1G#6 * BTH+BR0 *)- 6 * "+6
H�(�D d a�, 6�2 -f* 0 d P1(1G a�< C)B * U�P�(16.P�(1G / B = BRE =�bed b &)A d a S16 BRE =�bed 0�B.K d b\b P�S * "+6K�(1& O�- 6
¬(�
1) (�
1) & (�
2) (�
1) ∨ (�
2) (�
1) → (�
2) ∃� � ( � 1) ∀� � ( � 1)
S+P1(1G%( d �12 �
2BFE =�bed�* U)P1( d K d a &)S * BT&)(1GcK ,ia (1G#6 e G = (+P *Jd a�<
J > 8+8 Q �:≡ � 1 = � 2 | ¬(
�1) | (
�1) & (
�2) | (
�1) ∨ (
�2) | (
�1) → (
�2)
| ∃ � � ( � 1) | ∀� � ( � 1)
3 d ����������E � ��E@8 ∃ acbed ∀ / "+K d (1G+&ID�(1U = K�BT& d a�< acbed = (1U+&ID d b O@bed = S1K�B =�b 0 * "D�H 4 0�0 b P1(1G / B = G)P < &)A�(1G = 0 * " = J�D�B =Jd a�<�Q b P+H�(1U#0 * BT&I" D�H 4 0�0 b �
(
)P1(1G
- A�B d K�S = ( S1&)(1G+6 m E� � ���1����� J�BTD�D�& b1O�,1Q K d b 6 K�B *�b+5 H+" *I, 6 � � 0 * ( = * U�P�( �BFE =�bed E)� E.� ��E���� b1= ( � BFE =�bed|b &)A d a S16�2 , J b1=�b1/ &)(1K d a�<�Q b1= ( � BRE =�bed�-T=�b 6 b P�S* (1G+6 * U)P1(1G#6�0 * " = J > 8+8 QEacbed "7BRK O�<1=Jd 0�" * "+6 � � BRE =�bed BTH+BRU#C)BR&I" 0 * ( = �
1, 0 * ( =
�22 b H#H < 2�0 * " =?* BTH+BRG *�b E BR6 / U#(%P1BT& d P *)4 0�B d 6 � 6= F BRK O-b1= E 0�B d 6 * "+6 � � P�(1G / B =BFE =�bed BbH�BTU+C)BT&)BR6 BFE =�bed ��E)�� EI�������E@8 2 acbed �����������(E�� 8 BFE =�bed ( dN* U)P1( d P�(1G / B =
- A�(1G = acb K�E b BbH�BTU+C)BT&I" BTK O�<1=Jd 0�"cK�B *�b+5 H+" *I, 6 9 E' ! 2�� ����7���� � �x��W��� ��� ������� �
=�( � 1 �;:;:;:���� � )) � ������������������ ���
� P�M 6 0 * (7B /)<1O@d ( ! @ 2 1eh�m ;B? ,W7Jj|7 0 * " =�< H#D�B 5 & b BFE =�bed " * G#A b E b 0�G =)< & * "+0�"�
: {� 0 ��� 1 �;:;:;: } → N
P1(1G b1=�b C -b* B d 0 * " a�< C)B K�B *�b+5 H#" *I, ���?-R=�b 0 * ( d A�BFE ( �( � � ) ∈ 2 acbed " ;=< ,/.
� ��� � (� � � ) * (1G S1&)(1G � 0 * " = D d b * " = b P�( * E K�"�0�" � (1&�E � B *�bed/b1=�b1/ &)(1K d a�<
K�B * ( = P1&)( O-b1=I,%* &)S+P�( � ����� (0 � � ) = 0 � ����� (1 � � ) = 1 � ����� (��� � � ) =
�(��� )
� ��� � ( � � ( � 1�;:;:;: � � � � )) =
� � ( � ��� � ( � 1)�;:;:;: ��� ��� � ( � � � ))
m BR&)K�" = BFE b%* M =?* U�P�M = acb C)(1&�E � B *�bed|b P�S * " a H b 0 d a�, 0�A - 0�" � ������������ �����98J�`fY�dfZ\` � Y : dbZ _Pb Q * (1G�`WY i ` r�Z b1=)< K�BR0 b 0�B * U)P1(1G#6�2 acbed:b P1( *Jd K , 0�B d 6 P�(1G (1&�E � B *�bedb1=�b1/ &)(1K d a�< MN67B C , 6
� � |= �1 =
�2 ⇐⇒ � ����� (
�1� � ) = � ����� (
�2� � ) � � |= ¬(
�1) ⇐⇒ � � 6|= �
1 � � |= (�
1) & (�
2) ⇐⇒ � � |= �1acbed � � |= �
2 � � |= (�
1) ∨ (�
2) ⇐⇒ � � |= �1, � � |= �
2 � � |= (�
1) → (�
2) ⇐⇒ � � 6|= �1, � � |= �
2 � � |= ∃ � � ( � 1) ⇐⇒ G�P < &)A�B d � ∈ *)-b* ( d (16'P�(1G � � {� � := � } |= �1 � � |= ∀ � � ( � 1) ⇐⇒ D d b a�< C)B � ∈ � � � {��� := � } |= �
1�
����� �!�#"�$&%('*)�+!,!-!+/. �/0!"�$1 �/2 3!4/+!5/670 28�9#:�+/; +/<���,8��5!=#> "/>�2? "&@A4/+/0 2#>�2 4#0/>���0!6CB#0�3/+!,/"ED�:A4/=!- FG�H4/+�DI<�F/5!�#>�+J; � K#"('L�MJN FO:!>AF/5/.�4�PH+!9�D ?/Q!Q R D ?SLGTUM!V ' L!L �
q � 2�jY, ��k�� [&/ ��6 { 57��: { ��#3��576�'&� / � � # � ,9[ �=' / �&� ��������� >�>Hd
S+P1(1G� { � � := � }( � � ) =
{ � � b1= � = F ��( � � ) � b H#H d 4 6�2 b1= F 6= �
BFE =�bed " E)���������7���� * "�6 b P1( * E K�"+0�"+6 � K�B * " = # b1=)< C)BR0�" & � � := � � b 0 d a�, d /Jd S * " *�b * (1G (1& d 0�K�(1U * (1G�`WY i `sr�Z BRE =�bed S *Jd " b H , C)B d b * "+6%0�A - 0�"+6
� � |= � B C b & *)<+*�bed K�S = ( b P1S *Jd 6 *Jd K - 6 * "�6 � 0 *Jd 6.BTH+BRU#C)BR&)BT6 K�B *�b+5 H+" *)- 6* (1G � 2 / "+H b1/I,
(∀ F )[ " � � BFE =�bed BbH�BTU+C)BT&I" 0 * ( = � � ⇒ �1( F ) =
�2( F )]
� ⇒ [ � �
1 |= � ⇐⇒ � �2 |= �
] :B G * S / BRE A = B *�bed BRU a (�H b K�B�BbP b D�MND , 0 * ( = * U�P�(�� 2 acbed 0�G = BfP < D�B *�bed S *Jdeb1= (��BFE =�bed P�&)S *�b 0�" JLA�MN&�E 6 BTH+BRU#C)BR&)BT6 K�B *�b+5 H+" *)- 6 Q 2 * S * B7" 0�A - 0�" � � |= � BFE =�bedb1= B C < & * " * " b P1S * " = b P�( * E K�"�0�" � 2 acbed K�P�(1&)(1U+K�B =�b * " = P b & b H�BRE ��(1G#K�B b P�S* ( 0�G+K 5 (�H d 0�K�S�2
� |= � ⇐⇒ D d b a�< P�( d b � � � � |= �
⇐⇒ D d b a�< C)B � � � � |= � :J P�&)S *�b 0�"�� Q9 E' N 2�Y, ������Z�5 2 3 * U)P1(16 � ��� ���E������ � �T(E)�� �(4 ����� � ( �� ) E%�1� 8 E �
��� ������� �@8� 1 �;:;:;:�� � � 0 * " = < H+D�B 5 & b 2 b1= " b`a (�H�(1G#C�E b � 1 �;:;:;:�� � � P�BR& d - A�B dS�H�BT6 *Jd 67BbH�BTU+C)BT&)BR6 K�B *�b+5 H#" *)- 6 * (1G � acbed D d b S�H bc*�b"�
1�;:;:;:�� � � ∈ 2
� ( � 1�;:;:;:�� � � ) ⇐⇒ � { � 1 := �
1�;:;:;: � �
�:= � � } |= � DJ > 8 > Q
* BTH d a�< 2�K d b 0�A - 0�" � ( �� ) BFE =�bed ������ J�VF[ VFXcVAb+dTY i ^ Q�, h �@;=m65�65 ,l<H1 J ��i ` d ]L_ i w�V iw�V � b�Y ��[ V Q 0 * " =?< H+D�B 5 & b b1= (1&�E � B *�bed-b P�S a�< P�( d ( * U�P�(%K�B a�< P�( d b b`a (�H+(1G)]C�E b K�B *�b+5 H+" *)4 = 2 acbed K d b 0�G =)< & * "�0�" � :
�→ BFE =�bed �����1 b1=�* ( D�& <1O "+K <
* "�6 J O9> Q BRE =�bed|b & d C)K�" *Jd a�, 0�A - 0�" 9 E' Q 2'�c�1�7�� ���1� ���@8 �(4 �)�(E#� 8 ����� ���������#����(E#� 8� I /)4 B =)/Jd b1O BT&)S1K b 0 * Bd /Jd b E * BR& b 0 * " = BT&)K�" = BRE b * "+6 P1&)M * ( 51< C)K d b 6 D�H 4 0�0 b 6 � (+ � ·) 0 * " =?< H#D�B 5 & b �* "�6 b & d C)K�" *Jd a�, 67J ] O Q 210 * " = (+P�(�E b ( (1& d 0�K�S16 * M = S1&)M = P b E & = B d1* " = B C , 6�2 b P�H ,
K�(1& O�, � :≡ � � | 0 | 1 | ( � 1) + ( � 2) | ( � 1) · ( � 2)3 d P1&)M * ( 51< C)K d BT6.0�A - 0�B d 6 acbed 0�G =�b & *I, 0�B d 6 * "�6 � acb H�(1U =F*�bed ���1�7������1� � �18 2acbed D d � b G *)- 6 C b A�&)B d b 0 * (1U#K�B J�0�A�B / S =�Q K�S = ( * ( = B C , 6 A b & b`a�* "�& d 0�K�S�2 P1(1G
/ BFE A = B *�bed BTU a (�H b�b P1S * (1G+6?(1& d 0�K�(1U#6 9 E' 9 2�� # � ,3[ �=' 2 +@m\j,<iAMm �8m#; �BA 1��@<Y5 ,W7�;=< >@?BA j@� �=j6g��BA g@?YAM1e<�;=m g���B�B< �j|;=m j�<8A=m��im j@� �=j6g��BA A j@;=m3=����(=�j-< >2m&< ��1��@<Y5 ,@m&< � ,Eg ;=< � g@C`. ��< �J< A`;k7J;0g�� J > Q�� </j � �=jeg�< �
� = �� � = 0 � � = 1 � � + � = � � � · � = �g@?YAM1e<Bj|;=m A �
����� �!�#"�$&%('*)�+!,!-!+/. �/0!"�$1 �/2 3!4/+!5/670 28�9#:�+/; +/<���,8��5!=#> "/>�2? "&@A4/+/0 2#>�2 4#0/>���0!6CB#0�3/+!,/"ED�:A4/=!- FG�H4/+�DI<�F/5!�#>�+J; � K#"('L�MJN FO:!>AF/5/.�4�PH+!9�D ?/Q!Q R D ?SLGTUM!V ' L!L�V
>P>AO q 2 j 2&'�< , ���=4;6�'&�8� , ��������Z /�[&/ 'JL! Q +@m A g@?YAM1e<:> �ig�<Hj@;BA��-< 1 1|A ;=< >c1c;91 j|;k.Jj6g�< � ,:g�;=< � h��`m �:m ����� � �� ( �� ) >-1 <;=m3=�� ;0g��`g=j@;���� ;k7 � �8m �c< >:. � Z ¬ Z & Z ∨ Z ∃ Z ∀ �� hJgk;91e<*A`;=< 7 j � �=j@7 ;k7 ��1@A�< j�A`;k7J;91��
� ≤ � ⇐⇒ (∃ � )[ � + � = � ]g@?YAM1e<@1��@<Y5 ,W7�;=< >:. Z >-1 < A`;=<:;=m j,<iAMm �8m ; �BA 1��@<Y5 ,W7�;=< >@?BA j � �=jeg��BA g@?PA=1 <6>��`g�< j|;BA�-< 1 1��|<P5 ,/7J;=< > ����1@Ak;=< >-16;91ej@; ejeg�< � Z 1��`m&<� (�1( �� ) �;:;:;: � � � ( �� ))
⇐⇒ (∃ � 1) · · · (∃ � � )[�1( �� ) = �
1 & · · · &� � ( �� ) = � � & � ( � 1
�;:;:;:�� � � )] :$ - & b1= b G * (1U�2�C b BbP d acb H�BT0 * (1U#K�B BfP�E 0�"�6 K�BR& d a�- 6 J acb+*)< * ( P+H�BRE 0 * ( = P1&)( O-b ]
= BFE 6 Q d /Jd S * " * BR6 * "+6'0�A - 0�"+6 * "�6 d acb1= (+P1(�E "+0�"+6'0 * " = � P�(1G b P1(1&)& - (1G = BTU a (�H bb P1S * ( = (1& d 0�K�S * "+6 ; d b * ( BbP�S1K�B = (�2 5 b 0 d a S�p?B 4 &I"+K b J P�(1G /Jd acbed (�H�(�D�BRE * " = ( = (1K b 0�E b # b & d CR]
K�" *Jd a�, d BT& b &)A�E b%& D d b *Jd 6 a H < 0�B d 6%0�G = S�H+M = Σ0
�2Π0
�Q 2�A�&)B d b%� S1K b 0 * B * ( B C , 6�2b P�H+S � , K�K b�b P�S * " = b & d C)K�(1C)BRMN&�E b 2�P�(1G�K b 6 / E = B d K d b J b`a S1K�" � Q a M /Jd a (+P�(�E "�0�"b`a (�H�(1G#C d 4 = 2 b G *I, * " O (1& < b & d C)K�" *Jd a�,
9 E' T 2 � 4����=' J m 0�G =)< & * "+0�"�� * (1G tvu_Pw�VF[ Q 2 � j,=iA ��-;k78j@7�( � ����� F ) =
i X( � � 1 + ( F + 1) � )g@?YAM1e< 1��|<P5 ,/7J;=< >E. Z >-1 <��-< 1�> 65 g(18>|m �8m3=85@? 1�1��@<Y5 ,l?BA �
0�;:;:;:�� � Z = h���;�|m&=iA�(=Jjc< >|m ?B1��@<Y5 ,@m-? � >-1 < � Z ;��k;=m < m </hJm&=
��� = �( � ��� � F ) ( F = 0 �;:;:;: ��� ) :
j-X Z�<\#3��} [*2 m 0�G =)< & * "+0�"�� ( � ��� � F ) BFE =�bed|b & d C)K�" *Jd a�, 2�BbP�B d /I,�( � ��� � F ) = � ⇐⇒ (∃ )[ � = (1 + ( F + 1) � ) + � & � G 1 + ( F + 1) � ] :
; d bc* ( / BRU * BR&)( d 0�A�G#& d 0�K�S * (1G � , K�K b+* (16�2 - 0 * M� = max( � 0
�;:;:;: � � ��� ) + 1� = � !
� � = 1 + ( F + 1) � = 1 + ( F + 1) � ! ( F = 0 �;:;:;:���� ) :$ b & b+* "+&)(1U+K�B S *Jd ( deb & d C)K�(�E � 0 � � 1 �;:;:;: � � BFE =�bed j �lgk;=< >|? � h �`?@;=m < 2 / "+H b1/I, / B =G�P < &)A�B d P�& 4W* (16 b & d C)K�S16%P1(1G =�b /Jd bed &)BFE / U#( b P � b G * (1U+6 D d b+* E b1= ( � BFE =�bedP1& 4�* (16 acbed@a ( d = S16 /Jd bed & -b* "+6 * M = � � acbed � � K�B F*G � ≤ � 2 * S * B ># � � 2 b H#H d 4 6 � | � !
2 acbed|b G * S BRE =�bed <+* (+P1(�2 b1O (1U � |1 + ( F + 1) � !2 acbed
! ( � /Jd bed &)BFE * " /Jd b1O (1& < (� − F ) � !
2 < & b � |(� − F ) , � | � !2
acbed/b G * S BRE =�bed <+* (+P�(�2 b1O (1U � − F ≤ � G � G � acbed ( � / B = /Jd bed &)BRE * ( � !2
S+P1MN670 * ( >� \_ BTH d a�< 2 * (.H+BTD�S1K�B = ( � <YA � �8< >2m � gM? �-70,21 EhJm��im-?Hh&�BA 5 B 5 bed 4N= B dS *Jd|b1O (1U �
0 G � 0 �;:;:;: � � G � 2�G)P < &)A�B d|a�< P�( d (16 � *)-b* ( d (16'P�(1G�
0 =i X
( � � � 0) =�( � ����� 0) �;:;:;:�� � =
i X( � � � ) =
�( � ��� ��� ) : a
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q � 2�jY, ��k�� [&/ ��6 { 57��: { ��#3��576�'&� / � � # � ,9[ �=' / �&� ��������� >�> ]
9 E' d-2�� # � ,3[ �=' 2 J > Q � 65 g h ���@;=m �Jg8A0? � 1|AM1���� m ,B< >E. j�=8A���@;k7Jj|7 g@?PA=1 <c1��@<Y5 �,W7�;=< >:. �JL! Q � ;@=>�E1-?H1 j � �=j@7 g@?PA=1 <-1��|<P5 ,/7J;=< >E.^1@AE>c1e<J,�AcAMmcA/1|ABg@?PA=1 < Σ0
�Z �-< 1^> hJm < m
� Z �`7 �e1��`.A =
⋃� Σ0
� =⋃� Π0
�:
j-X Z�<\#3��} [*2 J > Q B & a BRE =�b / BFE C)(1G#K�B S *Jd ( d ��2 � � acbed � �� BFE =�bed^b & d C)K�")]*Jd a�- 6�2 acbed S *JdW* ( 0�U = (�H�( * M = b & d C)K�" *Jd a�4 = 0�G =�b & *I, 0�BTM = BRE =�bedWa H+B d 0 * S D d b0�U = C)BR0�" acbed P�&)M * (�D�B =I, b1=�b1/ &)(1K , 2 acbed-b P � b G *)< K�S = ( * ( * BbH�BTG *�b E ( / B = BFE =�bed* B * & d K�K -T= ( $ b & b+* "+&)(1U+K�B'S *Jd|b1= " � ( � � �� ) (1&�E � B *�bed K�B * " = P�&)M * (�D�B =I, b1=�b1/ &)(1K ,
�(0 � �� ) =
�( �� )�
( � + 1 � �� ) =�(�( �� �� ) ���� �� ) �
* S * B)2 b P�S * "%H�BbD�S1K�B = " # S'VAw�V r�Zcb3w b1=)< H+G+0�" * "�6 b1=�b1/ &)(1K , 6 & 2�( � � �� ) = � ⇐⇒ (∃ � 0
�;:;:;:�� � )[� ( �� ) = �0 & � = �
& (∀ F*G � )[ � ( � � � F � �� ) = � �+1]] Db G * S'BFE =�bed P�&)( O@b1=)- 6�2+K�B ��� = �
( F � �� ) D d b * " acb+* BTU+C)G = 0�" J ⇒ Q 2 acbed K�B BfP b D�MWD ,0 * ( F ≤ � D d b%* " acb+* BTU+C)G = 0�" J ⇐ Q � P�B *�bed 2 b P1S * ( � , K�K b 2�S *Jd
�( � � �� ) = � ⇐⇒ (∃ � )(∃ � )[� ( �� ) =
�( � ����� 0) & � =
�( � ������� )
(∀ F*G � )[ � ( � ( � ��� � F ) � F � �� ) =�( � ����� F + 1)]]
P1(1G?0�G = BfP < D�B *�bed6b K - 0�MN6�2 b P1S *Jd 6 d /Jd S * " * BT6 a H�B d 0 * S * " *�b 6 * (1G A S *Jd " � ( �� �� )BFE =�bed|b & d C)K�" *Jd a�, JL! Q m 0�G+K�P�BR&�E H#"��"
Σ0
� ⊆ A BRE =�bedWb P�H , 2 K�BcBfP b D�MWD , 0 * ( ��2 acbed " b1=F* E ]0 * &)( O "c0�G#K�P1BT&�E H+" ��" A ⊆ ⋃
� Σ0
�0�G =)< D�B *�bed|b P1S * ( A b & b`a�* "+& d 0�K�S.9 E' 9 a
9 E' O 2'�c�1�7�� ���1� �������� ������ @ B a�< C)B 0�U#K 5 (�H+( * "+6�P�&)M * ( 5�< C)K d b 6 D�H 4 0�]0 b 6 * "+6 b & d C)K�" *Jd a�, 6 � =
�(+ � ·) 0�G#0�A�B * E � (1G+K�B -R=�b.O G+0 d a S b & d C)K�S [ ] K�B * " =b P�H , b P b &�E C)K�"+0�"�2
0 1 + · = ¬ & ∨ → ∃ ∀ ( ) � �0�1:;:;:
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 :;:;:-b* 0 d P�(1G [∃] = 9
2[ � 0] = 14
2 a H#P acbedla M /Jd a (+P�( d (1U+K�B * (1G#6 * U)P1(1G#6cM 6 P�Bf]P1BT& b 0�K -T= BR6 b`a (�H+(1G+C�E BT6 b P�S 0�U#K 5 (�H b 27A�&I"+0 d K�(+P�( d 4 =F*�b 6 a�< P�( d b 270 *�b C)BT& , 2P1&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�, a M /Jd a (+P1(�E "+0�" b`a (�H�(1G#C d 4 = 2
[ � 0 � 1 · · · � � ] = 〈[ � 0] � [ � 1] �;:;:;: � [ � � ]〉 �P A 2
[∃ � 2( �= �
2)] = 〈[∃] � [ � 2] � [(] � [ �] � [=] � [� 2] � [)]〉 :
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> !�8 q 2 j 2&'�< , ���=4;6�'&�8� , ��������Z /�[&/ '@ B *Jd 6 K�BTC)S / (1G+6 * (1G ^ B O-b H b E (1G N 2 / B = BRE =�bed+/ U+0 a (�H�( =�b?/ BFE C)(1G#K�B S *Jd ( d�5`b 0 d ]a�- 6 ,:gk;91k,21c5J70,21c;=< > ��� 2�0�G =F*�b`a�*Jd a�- 6 0�A - 0�B d 6 * "�67D�H 4 0�0 b 6 BFE =�bed P�&)M * (�D�B =)4 6b1=�b1/ &)(1K d a�- 6�2�P A 2�( d
_ U)P1(16( � ) ⇐⇒ ( � BFE =�bed@a M /Jd a S16 * U�P�(1G
I H+BRU#C)BR&I"( � � F ) ⇐⇒ ( � BFE =�bed@a M /Jd a S16 * U�P�(1G �acbed " � � BTK O@b1= E � B *�bed BTH+BRU#C)BR&I".0 * ( = �
$?&)S *�b 0�" ( � ) ⇐⇒ _ � BRE =�bed@a M /Jd a S16'P�&)S *�b 0�"+6 �a H#P 8_ BbH d a�< C -b* (1G#K�BJ > 8�! Q ` ibg dfe
=` ibg dfe
(�
)
= { � | ( � BRE =�bed@a M /Jd a S16 P1&)S *�b 0�"�6 P1(1G b H+"+C)BRU#B d 0 * " = � } :B G * S'BFE =�bed�* (?0�U = (�H�( O G+0 d a�4 = b & d C)K 4N= P1(1G a M /Jd a (+P1( d BRE�S�H+BR6 * "+6�J P�&)M * ( 5�< CR]K d BR6 Q # b & d C)K�" *Jd a�- 6 b H , C)B d BR6 & � a�< C)B 0�"+K b1=F*Jd a�, P1&)S *�b 0�" * "�6�C)BTM &�E b 6 b & d C)K 4 =P1(1G b H#"�C)BTU+B d�- A�B d�* ( = a M /Jd a S * "�6?0 * (�` ibg dfe 9 E' ] 2 � 4����=' 2 � c5=g 1��@<Y5 ,W7�;=< >:.^j@� �=j|7 � ( �� ) 1|A ��gk;91e<cj|;=m j,<iAMm �8m ` ibg dfe Z�`7 �e1��`. = h���;�lg�< J -T=�b ]LP�&)(16b] -T=�b�Q 1|AM1���� m ,B< >E. j,=iA ��-;k78j@7 � ( �� ) ;��k;=m <H1 hJm&=
� ( �� ) ⇐⇒ �( �� ) ∈ ` ifg dbe :J > 8 N Q
j-X Z�<\#3��} [*2 ; d b a�< C)B O G+0 d a S � (1&�E � (1G#K�B * ( = S1&)( ∆( � ) K�B * " = b1=�b1/ &)(1K , 2∆(0) = 0 � ∆( � + 1) = (∆( � )) + (1) �
-b* 0 d P1(1G (∆( � ) BRE =�bed/a H+B d 0 * S16 JLA�MN&�E 6.K�B *�b+5 H+" *)- 6 Q acbed - A�B dW* " = E /Jd b *Jd K ,
D d b a�< C)B b P1( * E K�"+0�"��N2� ��� (∆( � ) � � ) = �(:
� P1B *�bed JLBTU a (�H b 2 b P1S *Jd 6�0�G = C ,ia BT6 * (1G `WY i `sr�Z Q S *Jd D d b�a�< C)B * U�P�( � K�B�BbH�BTU)]C)BR&)BT6 K�B *�b+5 H#" *)- 670 * "%H�E 0 *�b � 1 �;:;:;:�� �
� 2� � { � 1 := �
1�;:;:;: � �
�:= � � } |= �
⇐⇒ � � |= (∃ � 1) · · · (∃ ��)[ � 1 = ∆( � 1) & · · · & �
�= ∆( � � ) &
�]
⇐⇒ [(∃ � 1) · · · (∃ ��)[ � 1 = ∆( � 1) & · · · & �
�= ∆( � � ) &
�] ∈ ` ibg dfe Dacbed BbP�(1K -R= MN6�2 b1= ( * U�P�(16�� (1&�E � B d�* " 0�A - 0�" � ( �� ) 0�U#K O M =�b K�B * " = J > 8 > Q 2
* S * B'" J > 8 N Q d 0�A�U+B d K�B * ".0�G =)< & * "+0�"�( �� ) = [(∃ � 1) · · · (∃ �
�)[ � 1 = ∆( � 1) & · · · & �
�= ∆( � � ) &
�]]P1(1G%BRE =�bed 2�BTU a (�H b 2 b1=�b1/ &)(1K d a�, acbed -T=�b ]LP�&)(16b] -T=�b a
9 E' c> 8 2�� E.�������� ����� � � �� ��� +-m�j�<8A=m��im ` ibg dfe �kg8Apg@?YAM1e< 1��|<P5 ,/7J;=< >+A Z>-1 <lg�h�mk, �8A�� ��m&<i;0g 1|AM1���� m ,B< >+A �j-X Z�<\#3��} [*2 B = * ( ` ifg dbe ,#*�b1= b & d C)K�" *Jd a S�2 * S * B C b ,#*�b1= Σ0
�2�D d b a�< P�( d (
��2 acbed BbP�(1K -R= MN6�2 b P�S * ( � , K�K b 2 a�< C)B b & d C)K�" *Jd a�, 0�A - 0�" C b ,�*�b1= Σ0
�2WP1(1G
b1=F*Jd * E C)B *�bed 0 * ( p?B 4 &I"+K b � BT& b &)A�E b 679 @ Q J N Q a
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q � 2 � ';k�# .�, 4��=' / ' /|. 2���������� 6�'&� � �������� > ! >; d b * " =\d 0 * (1& d a�,#b`a &�E 5 B d b 2 P1& - P1B d'=�b P b & b+* "�& , 0�(1G#K�B B /)4 S *Jd (;`WY i ` r�Zb P -T/ B d C)B K�S = ( * ( P�& 4W* ( K - &)(16 * (1G C)BTM & , K b+* (16�2 S *JdN* ( 0�U = (�H+(�` ibg dfe / B =
BFE =�bed�b & d C)K�" *Jd a S�2 acbed'/ B = ,�*�b1=�acb1= 0�B C - 0�" =�b / BFE C)B d'* ( / BTU * BT&)( K - &)(16�2BbP�B d /I, / B = , C)BT&)B S *Jd *�b b1=�b1/ &)(1K d a�< 0�U = (�H b BRE =�bed:b & d C)K�" *Jd a�< � S+P1MN6 BfP�E 0�"�6/ B =c, C)BT&)B acbed�* " 0�A - 0�" b1=)< K�BT0 b 0�B b1=�b1/ &)(1K d a S * " *�b acbed G)P1(�H+(�D d 0 d K�S * " *�b 2/ "+H b1/I,%* ( B E * "+K b G e g�i : e1]a` g�i Zcb h�
��� ��� z �,$3) we~(�Wz|{cz {,) ������ �����y/zl} ��� ������_ b / U+( b G *)< C)BTK�BTH d b`a�<�b P�( * BTH - 0�K b+*�b - A�(1G = =�b�a�<1= (1G = K�B * " 5`b 0 d a�,
-R=)= ( d b7* "+6 ;@= hi< >:. � 1 h�A���g�< C`7 � * "�6�H+(�D d a�, 6�2+P�(1G / B = BRE =�bed 0 * ( C - K b K b 6�2 b H+H <b P�H - 6$# P b & b1O & < 0�B d 6 &W* (1G#6�BFE =�bed BRU a (�H b P1(1&�E 0�K b+*�b * "�6�C)BTM &�E b 6 b1=�b1/ &)(1K d a�4 =0�G =�b & *I, 0�BRM = acbed|b C�E � B d�=�b%*Jd 67B a C - 0�(1G+K�B?B /)4 9 G c>&2'� ������E#� ���1� � � ����� ���������� ;�B =Jd a�< 2 ;@==h`< >E.(1 h�A���g�< C`7 JL0 * " = b & d CR]
K�" *Jd a�,�Q BFE =�bed K d b b`a (�H�(1G#C�E b�b P�S * U)P1(1G#6�
0�;:;:;:�� � � �
*)-b* ( d b P�(1G a�< C)B�� � BRE =�bed 1 C1? �@,21 J * "+6 H�(�D d a�, 6 ,�* "+6 b & d C)K�" *Jd a�, 6 Q�, 0�G =)< D�B *�bedK�B a�< P1( d ( >-1@ABA6A=1 * "+6 H+(�D d a�, 6 ,?* "�6 b & d C)K�" *Jd a�, 6 b P1S a�< P�( d b � �
1
�;:;:;:�� � ��� K�B�1�;:;:;:�� � � G F 2 acbed ( * BTH+BRG *�b E (16 * U)P1(16 � � BFE =�bed P1&)S *�b 0�"�2 * ( #b0�G+K�P - & b 0�K b%&
* "�6 b P1S / B d CI"�6 n hJm���g�< >E;=< >+A\j,<Jj@;k70,21 J 0 i _1_�� ` ^�` dfVFX Q P BRE =�bed K d b b G+0 * "�& ,/Jd b+* U�P�M 0�" * M = b P b & b E * " * M = # b C d M K <+* M =�&2acbed # acb1= S = M =�& P�(1G K b 6 acb C)(1&�E � (1G =P1( d BT6 b`a (�H�(1G#C�E BR6 * U�P�M = BRE =�bed # b P�( / BFE C)B d 6 & 2 acbed1*�b 5 g�� �-.0,21c;91 * (1G P BRE =�bed�*�b0�G#K�P1BT& < 0�K b+*�b%* M = b P1( / BRE C)B 4N= * (1G12
`P� ⇐⇒ G)P < &)A�B d2b P�S / B d CI"��
0�;:;:;:�� � � 0 * ( P K�B�� � ≡ � :
� P < &)A�(1G = 2 51-f5 bed b * B * & d K�K -T=�b�b P�( / B d a�*Jd a�< 0�G#0 *I, K b+*�b 2�P A 2 b G * S S+P1(1Ga�< C)B b`a (�H+(1G+C�E b7* U�P�M = J P1(1G * BbH�B d 4 = B d K�B P�&)S *�b 0�" Q BFE =�bed6b P�S / B d CI"�2 ,7* ( < H+H+(S+P1(1G'( d b P�( / BFE C)B d 6 BRE =�bed S�H+BR6 ( d b`a (�H�(1G#C�E BR6 K ,ia (1G#6 1
2�K�B K�E b 2 b H+"+C , P�&)S *�b 0�"� ���vC d b K�BbH -f* "�6'S1K�M 6 BFE =�bed�*�b b P1( / B d a�*Jd a�< 0�G#0 *I, K b+*�b P�(1G d acb1= (+P�( d (1U ='*Jd 6B C , 6 / U+(�2 5 b 0 d a�- 6 d /Jd S * " * BR6 J > Q � ������������� J b & d C)K�" *Jd a�, BbD a G+&)S * " *�b�Q < 1 >c5=g h���A ;91 j|7 � Z 1@A `P
� Z;BA ;0g |= � 2 -b* 0 d P�(1G *�b C)BRMN& , K b+*�bc* (1G P =�b BRE =�bed S�H b b H+"+C , JL! Q � �������1���������������� ����� ��������E ��E#� 8 �� j@� �=j|7
� i _1_ �P( � ) ⇐⇒ (∃ � 0
�;:;:;:�� � � ∈ P)[ � = 〈[ � 0]�;:;:;:�� [ � � ]〉]g@?YAM1e<B1|AM1���� m ,B< >E.
m / BRU * BR&I" 0�G = C ,ia " B a�O & <%� B d J�K�B * ( B E * "+K b G e g�i : e1]a` g�i Zcb h Q * " 5`b 0 d a�,P1& b`a�*Jd a�, * M = K b CI"+K b+*Jd a�4 = 2NS *Jd ( d 0�A�G#& d 0�K�S16 * (1G _ <1/ BcS *Jd - A�B dWb P�( / BFE C)B da�< P1( d ( C)B 4 &I"+K b J * " = U)P b &HCI" b P1BRE &)M = * ( P�H , C)(16 /Jd / U+K�M = P1& 4�* M = 2ND d b P b ]& <1/ B d D�K b�Q 2�K�P1(1&)BRE =�b #bBbH�BbD�A�C)BFE & � G�P < &)A�B d D�B =Jd a�, K - C)( / (16'P1(1G b P1( O-b 0�E � B db1= b G *)< P1(1G H - B d ( _ <1/ B b P1( * BbH�(1U = b G#0 * "+& , b P1S / B d CI" * (1G d 0�A�G#& d 0�K�(1U * (1G
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> !#! q 2 j 2&'�< , ���=4;6�'&�8� , ��������Z /�[&/ ', S1A d 2�0�U+K O M =�b K�B *�b b C d 4 K b+*�b acbed�* (1G#6 acb1= S = BT6 * "+6 H+(�D d a�, 6 P�(1G - A�(1G#K�Bb P1( / BTA * BFE
� P < &)A�B d:acbed K d b.* &�E * " d /Jd S * " *�b P1(1G.C b C - H b K�B =�b - A�B d�* ( d /�b1=Jd a S b P1( / B d ]a�*Jd a S 0�U#0 * "+K bJ N Q � �1������������� <H1 > 65 g h �-A`;91ej@7 � 2
`P� , `P ¬ � :
e * ( P�& 4W* ( * &�E * ( * (1G !�8�(1G bed 4N=�b - D d =�b1= P�(�H+H - 6�P�&)(10�P < C)B d BT6 =�b 5 &)BRC)BRE -R=�bb P1( / B d a�*Jd a S 0�U#0 * "+K b D d bc* " =pb & d C)K�" *Jd a�, P1(1G =�b BRE =�bed (1&)C)S�2 b P�( a &�E 0 d K�(cD d bb P1( / BRE C)B d 6 acbed P�H , &)BT6�2�P1(1G C b a M /Jd a (+P1( d (1U#0�B'S�2 *Jd A�&)B d <%� B *�bed|b P1S * "%H+(�D d a�,D d b =�b H+U+0�(1G+K�B S�H b *�b P�&)( 5 H , K b+*�b * "�6cC)BTM &�E b 6 b & d C)K 4 = 3 d P�&)(10�P < C)B d BT6b G *)- 6 b P -b* G#A b1= 2 acbed�/ B = K�P1(1&)(1U#0 b1= =�b P�B * U#A�(1G =�9 G ! 2�� E��������� � ������������������8 ������������ � � � g8A = h���;�lg�<^1 hJm���g�< >E;=< >+Aj�<�j|;k79,:1 �-< 1 ;k76A 1��@<Y5 ,W7�;=< >:. hJm&= AM1�g@?YAM1e<Bm��@5�A Z 1eh�m8>�� ? j-< ,@m �c<H1 1eh�m��kg@? CMg�< �>-1 </h �|.��eg����j-X Z�<\#3��} [*2 B = * ( P - A�B d acbed *Jd 6 * &)B d 6 b G *)- 6 d /Jd S * " * BT6�2 * S * B D d b a�< C)B
P1&)S *�b 0�"��`P
� � ⇒ |= �
BbP�B d /I,%* (.0�U#0 * "+K b BRE =�bed|b & d C)K�" *Jd a�< - D a G+&)(�2 acbed|= � � ⇒ 6|= ¬ �
� ⇒ 6`P ¬ � JL(1&)C)S * " *�b�Q� ⇒ `P
� J P+H+"+&)S * " *�b�Q :� P1B *�bed S *Jd
` ibg dfe= {[ � ] |`P
� }= { � | $?&)S *�b 0�" ( � ) & (∃ � )[ � i _1_�� P( � ) & � =
[ Y�` d( � )]} �
acbedW-b* 0 d�* ( 0�U = (�H+( ` ifg dbe BFE =�bed Σ01
2 b P�S * " = G�P�S1C)BR0�" * "�6 b P�( a & d 0 d K�S * " *�b 6D d b�b P�( / BFE C)B d 6 * (1G P 2�P1(1G b1=F*Jd * E C)B *�bed 0 * ( p'B 4 &I"�K bc* (1GY`WY i `sr�Z�9 E' c> 8 am b H , C)B d b , S1A d�* (1G B d *I, K b+* (16 G e g�i : e1]a` g�i Zcb h / B = P b E � B d &)S�H+( 0 *Jd 6%0�G)]
D a B a & d K -T= BR6 B O-b &)K�(�D - 6 * (1G�p?BTM & , K b+* (16 B P�H#"�&)S * " *�b 6�2�BfP1B d /I, 0 * " P�& < CI"�2*�b 0�G#D a B a & d K -R=�b 0�G#0 *I, K b+*�b P1(1G A�&I"+0 d K�(+P�( d (1U =F*�bed b P�S K b CI"+K b+*Jd a (1U+6 acbed- A�(1G = K�BbH�B * "�C)BRE�BFE =�bed S�H b�b P�( a &�E 0 d K b D d b\b P�( / BFE C)B d 6 � b`a & d 5�4 6 BbP�B d /I, "0�G = C ,ia " JL! Q BRE =�bed�O G+0 d a�, ; d b.=�b /Jd b+* G)P 4 0�(1G#K�B * (�p'B 4 &I"�K b%* (1G G e g�i : e 2�BbP d 0 * & -TO (1G#K�B?0 * " P1&)M * (#]
51< C)K d b D�H 4 0�0 b � ( � 1 �;:;:;:���� � )0�B b G+C b E &)B * ( H+B C d H+S�D d ( ( � 1 �;:;:;: ��� � )
2 a B d D d b* " =7* G#A b E b P�&)S *�b 0�"�� * "�6 � ( � 1 �;:;:;: ��� � )
C -f* (1G+K�B
` � ⇐⇒ D d b a�< C)B < H#D�B 5 & b 2 |= �
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q � 2 � ';k�# .�, 4��=' / ' /|. 2���������� 6�'&� � �������� > ! NacbedJacb H+(1U+K�B * " = � �I��� ����� J Y�[\Z w Q@b1= ` � �_ ( a H b 0 d a S � gM? �@79,:1 f �|7��-A`;k7J;91��* (1G tvu_Pw�VF[ / "#H 4N= B d S *Jd
` � ⇐⇒ "�� BFE =�bed C)B 4 &I"+K bc* "�6 a H b 0 d a�, 6�2�P�&)M * ( 5�< C)K d b 6?H+(�D d a�, 6 �acbed - A�B d MN6 P1S1& d 0�K b S *Jd �-< 1 >c5=g �ig@C8< �3A �-< m �
1�;:;:;: ��� � Z ;=m j�<8A=m��im ; �BA J a M�]
/Jd a�4 = * M =�Q ���J>"= ���BA h � m`; ejeg��BA�;k7 � � ( � 1 �;:;:;: ��� � )g@?YAM1e< 79,B< 1|AM1���� m ,B< >+A JLK�B
* " = P�&)( O@b1=I, a M /Jd a (+P�(�E "�0�" Q \B P1S * " = < H#H+"cK�BT& d <9 G N 2�� E��������� ����� ����� ���� < 1(> hJm <H1 �
( � 1 �;:;:;: ��� � )Z ;=m h���A ��|79,:11|Ap7�;@=>�E1-?H1 h �-A`;91ej@7 � g@?PA=1 < ���J>"= �@7�g@?YAM1e<B1|A9g�h�? ��=8;=m �
VvX Z�<\#3��} [*2�� 0 * M�
= (�0� �
1�;:;:;:�� � � )
* G+A b E (�2�P�&)M * (�D�B =)- 6 b1=�b1/ &)(1K d a ScP�&)S�D�& b K�K b 0�U#K O M =�b K�B * ( = (1& d 0�K�S > E' c> T � B = BRE =�bed�/ U#0 a (�H+( =�b acb+*�b 0 a BTG < 0�(1G+K�B K d b P�&)S *�b 0�"
� �=�
1 & · · · &� �
0 * " = � ( � 1 �;:;:;: ��� � )P1(1G�#fB a�O & <%� B d�* G�P d a�<%& * (1G+6 (1& d 0�K�(1U#6'0 * ( � b1= 2�P A 2 * (0�U#K 5 (�H+(%P1&)( 5 (�H , 6 � 3
2
BTK O@b1= E � B *�bed 0 * ( � 2 * S * B a�< P�( d b�b P�S *Jd 6 � � BFE =�bed "(∀� 1)(∀ � 2)(∀� 3)[ � 3
2(�1���
2: �
3) = �2]�
acbed|b1= " � (1&�E � B *�bed 0 * ( � K�B * "%P�&)M * (�D�B =I, b1=�b1/ &)(1K ,�(0) = 5�
( � + 1) =�(�( � ) ��� ) �
* S * B a�< P�( d b�b P�S *Jd 6 � � BRE =�bed "� (0) = ∆(5) & (∀ � 1))[ � ( � (� 1)) = � ( � ( � 1) ��� 1)] :@ B b G * S * ( = (1& d 0�K�S�2 - P�B *�bed A�MN&�E 6 K�BbD < H+" / G+0 a (�H1E b S *Jd D d b�a�< C)B 0�G =)< & * "+0�"
� � P1(1G%(1&�E � B *�bed 0 * ( � acbed S�H+(1G+6 * (1G+6 �� = �1�;:;:;:�� � � � � ∈ N
2� � ( �� ) = � ⇐⇒ ` �
�→ � � (∆( � 1)
�;:;:;:�� ∆( � � )) = ∆( � ) :_ BTH d a�< 27B O@b &)K�S � (1G#K�B b G * S * ( � , K�K b 0 * " = P�BR&�E P * M 0�" P�(1G " � � BRE =�bed "A b & b`a�* "�& d 0 *Jd a�, 0�G =)< & * "+0�" * "+6 0�A - 0�"+6 �
1(� � � ��� ) acbed�/ BFE A = (1G#K�B JLBTU a (�H b
P d b�Q S *Jd " b P�( a & d 0 d K�S * " *�bc* "+670�A - 0�"�6?BTD a G#&)S * " *�b 67( / "+D�BRE 0�B <+* (+P�( a
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^ I�� B��8B�� 3 T
� ���� �R�W� � � � � �� � ��� � �P� � � � �7�� �K�� � � � � � ���������v� �
e�� b G * S * ( a B O�< H bed ( C b K�BbH�B *I, 0�(1G+K�B D�B =Jd a BTG+K -T=�b P�&)(�D�& < K�K b+*�b 2 # b1=�bed *Jd (#]a & b+*Jd a�<%& acbed K�B #fB C)M * BR& d a�- 6 & JLBbH�BTU+C)BT&)BR6 Q 0�G =�b & * "�0 d b`a�- 6 K�B *�b+5 H+" *)- 6�2 acbedC b /Jd BR&)BTG =I, 0�(1G#K�B *Jd 6.B O@b &)K�(�D - 6 * (1G+6 J a G+&�E MN6 Q 0 * " C)BRMN&�E b #bG)P1(�H+(�D�E 0 d K�M =0�G =�b & * "�0 d b`a�4 =�& _ b P1BT& d 0�0�S * BR& b b P1( * BbH - 0�K b+*�b P�(1G C b / BFE C)(1G#K�B d 0�A�U+(1G =D d b S�H�BT6 *Jd 67K�BR& d a�- 6 < H#D�B 5 &)BR6�2 b H#H <%*�b O@bed = S1K�B =�b P1(1G%K b 67B =)/Jd b1O�- &)(1G = BTK1]O@b1= E � ( =F*�bed P�H , &)MN6 0 * " = a H b 0 d a�, < H#D�B 5 & b �
0 = (N � 0 � 1 � � � �� ) acbed 0 *Jd 6BbP�B a�*)< 0�B d 6 * "�6�2�K�B *Jd 67(+P�(�E BR6 acbed C b�b 0�A�(�H+"+C)(1U+K�B
�+������cz:�cwcx � }Yy �&�O� �cz:we{et:� }�z:y �T @ c>32*Y, ������Z�5 2��*E)�1� �(EI�������� ������������W� ��� ��������������� * "�6 �
0BFE =�bed
* ( * G#A b E ( 0�U+0 * "�K b b1=�b1/ &)(1K d a�4 = B C d 0 4 0�BRM =( � 0)
�0( �� 0) =
�0 (= � 0[
�0�;:;:;: � �
�� �
1�;:;:;:�� � � ])
( � � )�� ( ��� ) =
�� (= � � [
�0�;:;:;: � �
�� �
1�;:;:;: � � � ])
J � Q
S+P1(1G ( d 0�G =�b & * "�0 d b`a�- 6 K�B *�b+5 H#" *)- 6 �0�;:;:;: � �
�� �
1�;:;:;:�� � � BRE =�bedW/Jd b1O (1&)B *Jd a�- 6K�B *�b C)U * (1G+6.J�S+P1MN6 0 * ( 5 b 0 d a S (1& d 0�K�S 0 * ( B /)<1O|d ( ! E Q 2 b H+H < * ( � P b & - A�B d
(1& d 0�K�(1U#6 K�S = ( D d b *Jd 6 E)�(7���E��1� � �18 J / BR0�K�BRG#K -R= BT6 Q K�B *�b+5 H#" *)- 6 �0�;:;:;:�� �
�2
B =)4 BbP d * & - P�B dBacbed�* " = BTK O�<1=Jd 0�" * M = E��7���E)�1� � ��� J�BTH+BRU#C)BR&)M =�Q K�B *�b+5 H+" *)4 =�1�;:;:;:�� � � 0 * (1G#6 b D = (1U+6.S1&)(1G+6 � 1
�;:;:;: � ��2 S+P1MN6.G�P�( / BFE C b K�B.K�B * ( = J P�&)(#]
0�MN& d = S Q 0�G+K 5 (�H d 0�K�S � � [ �0�;:;:;:�� �
�� �
1�;:;:;:�� � � ] � =�b *)-f* ( d ( P�&)S�D�& b K�K b BR&R]
K�" = BTU+B *�bed O G#0 d a�< 0�B * G+A b E BT67BbP�B a�*)< 0�B d 6(�
0���
1�;:;:;:���� � ) = (N � 0 � 1 � � � �� ��� 1
�;:;:;:���� � )* "�6 �
02 b1= C)BTM & , 0�(1G#K�B / "#H b1/I, *Jd 6%K�B *�b+5 H#" *)- 6 �
1�;:;:;:�� � � M 6 0 *�b C)BT& - 6 P1(1G( = (1K <%� (1G = *Jd 6�K�BT& d a�- 6�0�G =�b & *I, 0�B d 6 �
1�;:;:;: ��� � 2�S+P�M 6 ( d�� 1 �;:;:;: ��� � ( = (1K <%� (1G =
*Jd 6 / (10�K -T= BR6�K�BR& d a�- 6 0�G =�b & *I, 0�B d 6 �1�;:;:;:�� � � K d b 6�K�BR& d a�, 6 < H#D�B 5 & b 6 N@ B * (> !#9
> !�T x 2 j 2&'�< , ���=��6�%M����2&' ,3/�[ ���z'�6�%;6�'&� � X ������0P��� /|{ 5 X�, %�}3#9��50�G#K 5 (�H d 0�K�S * "�6%J�9 Q Q acbed K d b P1&)( O-b1=I, P b & b H+H b D , * (1G�2�C -b* (1G#K�BJ > 8 Q Q � �
( �� ��� 1�;:;:;: ��� � ) = �
⇐⇒ (�
0���
1�;:;:;:���� � ) � � ` �
0( �� ) = � K�B �1 := �
1�;:;:;: � � � := � �
⇐⇒ � � �� := �� ` �0( �� ) = ���
S+P1(1G12�D�B =Jd a S * BR& b 2�D d b a�< C)B?BT0�M * BR& d a�, K�B *�b+5 H#" *I, � * (1G � 2J > 8�9 Q � � �� := �� ` �
( �� ) = �
⇐⇒ (�
0���
1�;:;:;:���� � ) � � ` �
( �� ) = � K�B �1 := �
1�;:;:;:�� � � := � � :
I /)4 (%0�G#K 5 (�H d 0�K�S16 P b & b H�BRE P1B d�* " = b1=�b1O (1& < 0 * " =7< H+D�B 5 & b �0B O S10�( = b G *I,
BFE =�bed " K�S = " < H#D�B 5 & b P�(1G.C b A�&I"�0 d K�(+P1( d , 0�(1G#K�B70 � b G * S * ( a B O�< H bed (�2 b H+H </ BFE A = B d�* " = ������� � ����� �
1 := �1�;:;:;: � � � := � � * M = B C)M * BR& d a�4N= K�B *�b+5 H+" *)4 =�b G *I, BRE =�bed A�& , 0 d K�"�2NBbP�B d /I, P�(�H+H - 6 O (1& - 6cC b BT&)K�" = BTU+0�(1G+K�B * ( * ( E /Jd ( P�&)S#]
D�& b K�K b K�B /Jd b1O (1&)B *Jd a�- 6 � S�H�BT6 *Jd 6 / G =�b+*)- 6 � b P�( *Jd K , 0�B d 6 * M = B C)M * BT& d a�4 =K�B *�b+5 H+" *)4 =7* (1G _ b K�")] D�B =Jd a BRG#K -R=�b P1&)(�D�& < K�K b+*�b JLA�MN&�E 6'B C)M * BT& d a�- 6?0�G =�b & * "+0 d b`a�- 6'K�B *�b ]
5 H#" *)- 6 Q acb H�(1U =F*�bed ��������������� T @ ! 2��?�������#����������� �� m K�BR& d a�, 0�G =)< & * "�0�" � ( �� � �� ) 0 * " = J > 8 Q Q BFE =�bed P b ]& <1/ B d D�K b ���������#��������������� JL0 * ( N Q 2 / "#H b1/I, K�BT& d a�, 6 0�G =)< & * "�0�"�6 P�(1G /)- A�B *�bedM 6 *Jd K - 6cB d 0�S / (1G O G#0 d a (1U#6 b & d C)K�(1U#6 acbed P+H�B d (1K�BTH+BFE 6.K�BT& d a�- 6c0�G =�b & *I, 0�B d 6�2acbed J�S *�b1= 0�G#D a H�E = B d Q b P1( / E / B d�*Jd K , 0 * ( N
\I P d P�&)S10�C)B *�b P b & b1/ BFE D�K b+*�b�
1( �� ) =�( �� ) (
S+P�(1G �: N
�� N) �
�2(� � ) = � (0) (� : N � N � � : N2 � N) �V Y#[ �
( �� ��� ) = � ( �� ) :_ (��
1/Jd BTG a & d = E � B d S *Jd�/ B = BFE =�bed-b P b & b E * " * " " B C < & * "+0�" B = S16'0�G =�b & * "+0 d b`a (1Ub P1S K�BR& d a�- 6c0�G =�b & *I, 0�B d 6�2 -f* 0 d P�(1G >c5=g ,Eg �@< >:. j�=8A���@;k7Jj|7�g@?YAM1e< j�=8A=1��@;k7 �j-< 18> A acbed+* (��
2BRE =�bed P b & <1/ B d D�K b 0�G =�b & * "�0 d b`a (1U?K�B BRE 0�( / ( A�MN&�E 6 O G#0 d a (1U#6b & d C)K�(1U+6�� K�S = ( K�BT& d a�- 6�0�G =�b & *I, 0�B d 6 $ b & b+* "�&)(1U#K�B�S *Jd D d bpa�< C)B /Jd K�BTH , K�B ]
& d a�, 0�G =)< & * "+0�" 2�K�B � ( � ) = � + 1 acbed � * " = # a B =I,�& K�BT& d a�, 0�G =)< & * "+0�"�2�
2(� � ) =
�(0) = 1 � �
2(� � ) = � (0) = ⊥ :
_ ( V Y#[ � BRE =�bed E 0�MN6 * ( b P+H�(1U#0 * BT&)( K�"�] * B * & d K�K -T= ( 0�G =�b & * "�0 d b`a S * "+6 J ��]K�BTH+(1U+6 Q � �1����98 2�P A 2
V Y�[ 1( � � � ) =�( � ) = � + 1 � V Y�[ 3( � ��� � � � � 3
2 ) = � 32 ( � ���� � ) = ��:
T @ N 2*Y, ������Z�5 2 _ ( * G+A b E ( 0�G =�b & * "�0 d b`a S � ( �� � �� ) BRE =�bed ������������W� ��� 2 b1=G�P�(�H�(�D�E � B *�bed@b P1S a�< P�( d ( b1=�b1/ &)(1K d a S P1&)S�D�& b K�K b � K�B B C)M * BT& d a�- 6?K�B *�b+5 H+")]*)- 6��
1�;:;:;:�� � � 2 / "#H b1/I, � =
� � 0 * " = J > 8 Q Q 2 , 2 d 0�( / U =�b K b 2�( �� ��� ) = � ⇐⇒ � � �� := �� ` �
( �� ) = �
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< H+H+M = 2�P A 2�(1& d 0�K�(1U#6 * "�6?K�(1& O�, 6�( � � � ������ � � � ) =
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C)(1G+K�B BRU a (�H b � 0�G =I, C)M 6cK�B b P+H , BbP d 0 a S+P+"�0�" � * " = b1=�b1/ &)(1K d a S * " *�b P1(�H)]H 4N= 0�G =�b & * "+0 d b`a�4N= ; d b P b & <1/ B d D�K b 2 b1=�* ( � ( � � � � � ) BFE =�bed b1=�b1/ &)(1K d a S�2 * S * Bb1=�b1/ &)(1K d a S BRE =�bed@acbed�* (
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�1(� ��� ��� � � ) =
�( � ���� � ) JL&I" *)<�Q
�2(� ��� ��� � � ) = =
�( � ) = � + 1
J�&I" *)<�Q�
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JL&I" *)<�Q�
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V Y�[ 2( � 1(� ������ ) � � 2(
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3(� ��� ��� � � ) =
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�1(� ������ � � ) = 0) * S * B�� 2(
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e B�P�(�H+H - 6�P1BT& d P *)4 0�B d 67S1K�MN6�2�(cBRG a (�H�S * BT&)(16 * &)S+P1(16 =�bc/ BFE C)(1G#K�B'S *Jd-a�< P�( d (0�G =�b & * "�0 d b`a S7BFE =�bed`b1=�b1/ &)(1K d a S BRE =�bed b P�S * ( = (1& d 0�K�S � =�b D�& < ��(1G+K�B a�< P�( d (P1&)S�D�& b K�K b P1(1G * ( G)P1(�H+(�D�E � B d
�+��� �p�Byl~:�*$e}����T @ c>&2�� BRE C * B * ( K - &)(16 JU� Q * (1G � , K�K b+* (16gT @ Q�
��� � � �czB}P{@}�x2y/wcz2{@}Yyl~�z(�czE�cwcx(�B~e * ( = (1& d 0�K�S ! E' Q b1O "�&I"+K -R= M = K�"�A b1=)4N= BfP d * & - � b K�B b1=�bed *Jd ( a & b+*Jd a�- 6 0�A - ]
0�B d 6 K�B *�b+5�< 0�BTM = 2 -f* 0 d P1(1G *�b BbP d A�B d & , K b+*�b G�P - & * (1G B d *I, K b+* (16 G e g�i : e1]` g�i Z b h 0 * ( B /)<1O|d ( N�E =�b BRE =�bed S10�( * ( / G =�b+* S = d 0�A�G+&)S * BT& b 2 b H#H < K - A�& d�*)4 & b/ B =.- A�(1G#K�B K�BTH+B *I, 0�B dBb1=�bed *Jd ( a & b+*Jd a�<(b1=�b1/ &)(1K d a�< P�&)(�D�& < K�K b+*�b - A�(1G = acbedb G *)<%*Jd 67B O@b &)K�(�D - 6 * (1G+6 T E' c>&2�Y, ������Z�5 2'������� �1�����������1� ��� �������������� ���c��������������� BRE =�bed#* ( * GI]
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�0( �� 0) = � 0
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J � Q
S+P1MN6 0 * ( = 3'& d 0�K�SnT @ c> 2 S+P1(1G *)4 & b BfP d * & - P1(1G#K�B.P�BR& d 0�0�S * BT&)(1G+6 b P1S -R=�b1=(1& d 0�K�ScD d b%*Jd 67BT0�M * BR& d a�- 6 0�G =�b & * "+0 d b`a�- 67K�B *�b+5 H#" *)- 6 � &I"�0 d K�(+P1( d (1U#K�B * ( =E /Jd ( 0�G+K 5 (�H d 0�K�S D d b *)-f* ( d b 2ND�B =Jd a BTG+K -T=�b P�&)(�D�& < K�K b+*�b 2 ( d g=j-�@;0g �@< > ��� acbedg@C1�@;0g �@< > ��� K�B *�b+5 H#" *)- 6 * (1G#6 (1&�E � ( =F*�bed S+P�M 6 P�& d = 2 acbed ( d >c1c;91 j|; j6g�< � 2'( d= h�m �8m �c<HjJ,@m-? acbed ( d ;0g �J,21c;=< >2m-?�= h�m �8m �c<HjJ,@m-? b1=�bed *Jd ( a & b+*Jd a�4N= P�&)(�D�& b K�K <+* M =(1&�E � ( =F*�bed`b`a & d 5�4 6 S+P1MN6 acbed D d b'*�b bed *Jd ( a & b+*Jd a�< P�&)(�D�& < K�K b+*�b 0 * (?B /)<1O@d (7! EK�S = ( P�(1G *)4 & b BbP d * & - P�(1G+K�B * " = U)P b &HCI" P1(�H#H 4N= G�P�(�H�(�D d 0�K 4 = K�B * " = E /Jd bBFE 0�( / ( � b & b`a�* "+& d 0 *Jd a S b G *)4N= * M = P�&)(�D�& b K�K <+* M = BRE =�bed S *Jd�/ B =�acb C)(1&�E � (1G =b P b & b E * " *�b K d b K�BR& d a�, 0�G =)< & * "+0�" � D d b a�< C)B?BT0�M * BR& d a�, K�B *�b+5 H+" *I, �
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$ b & b+* "+&)(1U+K�B�S *Jd-a�< C)B P�&)S�D�& b K�K b K�P1(1&)BRE =�b C)BRMN&I"�C)BRE acbed MN6 b1=�bed *Jd ( a & b ]*Jd a S�2�B O S10�( = (c(1& d 0�K�S16 / B = b P bed * BRE * " = U�P b &HCI" P1(�H#H 4N= (1& d 0�K 4N= D d b a�< P�( d bK�B *�b+5 H+" *I, � b P+H <%* (1G+6?BbP d * & - P�B d @ BR& d a�< P b & b1/ BRE D�K b+*�b 2�P�& d = (1&�E 0�(1G+K�B b G+0 * "�& < * " 0�"+K b 0 d (�H+(�D�E b D d b *�bb1=�bed *Jd ( a & b+*Jd a�< P1&)(�D�& < K�K b+*�bT E' ! 2 _ (cP�&)S�D�& b K�K b
�( � ) = 0
J �1Q
�( � ) = 1
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(0)�
B P�S * " = < H+H#"cK�BR& d < 2 * (%P1&)S�D�& b K�K b�( � ) = �J �
2Q
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� ( � ) = 0�( � ) = � (
�( � ))
P1&)( O-b1=)4 6 G)P1(�H+(�D�E � B d * " = �( � ) = 0
2 b1= acbed 2�P < H d 2 / B =\b1=F*Jd 0 * ( d A�BRE acb K d <K�BR& d a�, 0�G =)< & * "+0�" � 0 * ".K�B *�b+5 H#" *I, � ; d b a�< C)B � 2 * ( �
2- A�B d�/ U+( G)P1(�H+(�D d ]
0�K�(1U#6 * "�6 *Jd K , 6 �( � ) 2�K�B * (1G#6'G)P1(�H+(�D d 0�K�(1U#6 P1(1G / BFE A = ( =F*�bed 0�A�"+K b+*Jd a�< 0 * (e A , K b T
B a S1K�" P d ( B =)/Jd b1O�- &)( = BRE =�bed�* (cP1&)S�D�& b K�K b� ( � ) = �J �
3Q
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D d b'* (7(+P1(�E ( P < H d �( � ) = 0 b H#H <'*)4 & b7- A�B d1< P�B d &)(1G#6 * ('P�H , C)(16 G�P�(�H�(�D d 0�K�(1U+6D d b a�< C)B'BFE 0�( / (�2�P1(1G / BRE A = ( =F*�bed 0�A�"�K b+*Jd a�< 0 * ( e A , K b d1
_ ( * BbH�BTG *�b E ( P b & <1/ B d D�K b BRE =�bed`b1=�bed *Jd ( a & b+*Jd a (1U�P1&)(�D�& < K�K b+* (16 K�BNK d b B C)M�]* BR& d a�, K�B *�b+5 H#" *I, (
�)P1(1G%G�P�(�H�(�D�E � B d 0�G =�b & * "�0 d b`a S
�0(� ) =
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� ( � ) =�(1)
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@ B * ( = (1& d 0�K�S J > 8 Q Q * (1G 0�G =�b & * "+0 d b`a (1U P1(1G G�P�(�H�(�D�E � B *�bed b P1S -T=�b P�&)S#]D�& b K�K b J b1= / BTA * (1U+K�BNS *Jdib G * S16 d 0�A�U#B dJacbed D d b b1=�bed *Jd ( a & b+*Jd a�< P1&)(�D�& < K�K b+*�b�Q 2* ( �
4K�B * " =pb P�( * E K�"�0�" �
:= � G)P1(�H+(�D�E � B d�* (.0�G =�b & * "�0 d b`a S�( � ��� ) =
{0 � b1= � (0)↓ ∨ � (1)↓ �⊥ �^b H#H d 4 6 :
J > 8 O Q@ B *)< b P1S b G *)< *�b P�&)( acb+*�b & a�*Jd a�< 2 (1&�E � (1G+K�B *Jd 0�"�K b E = B d D d b -T=�b 0�G =�b &R]
* "�0 d b`a S =�b #bG)P1(�H+(�D�E � B *�bed & b P1S a�< P�( d ( b1=�bed *Jd ( a & b+*Jd a S�2 b1=�b1/ &)(1K d a S P�&)S#]D�& b K�K bT E' N 2�Y, ������Z�5 2'������� �1�����������1� � � �������������� ���@8c E)�1� ���@8%���������#����(E#� 8
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0� � ` �
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K�B?BRE 0�( / ( �: �� acbed�* BR&)K b+*Jd a�, acb+*)< 0 *�b 0�" : � :@ B < H#H b H�S�D d b 2 * ( � G)P1(�H+(�D�E � B d�* " = � ( �� ) b1= 2�D d b a�< C)B �� J > Q � P < &)A�B d1* (1G�H < A d 0 * ( ='-T=�b 6 * BR&)K b+*Jd a S16�G)P1(�H+(�D d 0�K�S16�K�B�BFE 0�( / ( �
: �� acbed* BR&)K b+*Jd a�, acb+*)< 0 *�b 0�" :�( �� ) J�! Q a�< C)B * BR&)K b+*Jd a S16 G)P1(�H+(�D d 0�K�S16 * (1G � 0 * " = BFE 0�( / ( �
: �� - A�B d�* BR&)K b+*Jd a�,acb+*)< 0 *�b 0�" :�( �� ) $ b & b+* "+&)(1U+K�B S *Jd (.(1& d 0�K�S16?BbP d * & - P�B d�* " = U)P b &HCI"cG�P�(�H�(�D d 0�K 4 = P�(1G b P1( a H�E ]
= (1G = J�K�B�# < P�B d &)( K ,8a (16 &bQ 2�S+P1MN670 * ( �3P d (cP <1= M
3 (1& d 0�K�S16%BbP�B a�* BRE = B *�bed BTU a (�H b 0�B%0�G =�b & * "�0 d b`a�< 0 * " =�b P�H , P1BT&�E P * MN0�"K�B K d b K�S = ( B C)M * BT& d a�, K�B *�b+5 H+" *I, D d b K�BT& d a�- 6�0�G =�b & *I, 0�B d 6�2 * ( * G#A b E (70�G =�b &R]* "�0 d b`a S�� ( �� ��� ) BRE =�bed 1|AM1e<P;=< mJ>��`1c;=< > 1@A=1�� �`mk,l< > A b1= G)P < &)A�B deb1=�bed *Jd ( a & b+*Jd a SP1&)S�D�& b K�K b � K�BNK d b B C)M * BR& d a�, K�B *�b+5 H#" *I, � acbed8a�< P�( d b BT0�M * BR& d a�, K�B *�b+5 H+" *I,
� 2 -f* 0 d P�(1G%D d b S�H b%*�b �� ��� � � 2�( �� ��� ) = � ⇐⇒ � � � := � ` �
( �� ) = � :p -f* (1G+K�B
R ��� = * (.0�U = (�H�( * M = b1=�bed *Jd ( a & b+*Jd a�<�b1=�b1/ &)(1K d a�4 = 0�G =�b & * "+0 d b`a�4N=3:T E' Q 2 � 4����=' 2 � ;@= �:1 ? 1 ,Eg �@< >:.�j,=iA ��-;k78j@7 �
( �� ) g@?PA=1 < 1|AM1���� m ,B< >E. 1|A>-1 <@,�AcAMmcA 1@A�g@?YAM1e<B1|AM1e<P;=< mJ>��`1c;=< > 1|AM1���� m ,B< >E.��j-X Z�<\#3��} [*2 m a M /Jd a (+P1(�E "+0�" 0�G#K 5 S�H+M = 2 P1&)(�D�& b K�K <+* M = 2 acb+*�b 0 *)< 0�BRM = 2
G�P�(�H�(�D d 0�K 4 = 2 acbed D�B =Jd a�< S�H#"�6 * "�6 C)BRMN&�E b 6 b1=�b1/ &)(1K d a�4N= P1&)(�D�& b K�K <+* M = 0 * (B /)<1O@d ( N�@ BfP1B a�* BFE = B *�bed�* B * & d K�K -T=�b D d b *�b b1=�bed *Jd ( a & b+*Jd a�< P�&)(�D�& < K�K b+*�b 2 acbedK < H d 0 *�b K�B%K�S = ( K d b /Jd b1O (1& < 0 *Jd 6%H�BfP * (1K - &)B d BT6 0 * ( = (1& d 0�K�S * "+6.0�A - 0�"+6$?&)(�D�&
( � ) b P+H < P b & b H�BRE P1(1G#K�B * " = * BTH+BRG *�b E b 0�G = C ,8a " P�(1G b P b D�(1&)BRU#B d P1(�H)]H b P+H�(1U#67(1& d 0�K�(1U+6 * "�6 E /Jd b 670�G =�b & * "+0 d b`a�, 67K�B *�b+5 H+" *I, 6 8B G * S BbP d * & - P�B d�* " =U�P b &HCI"�P�(�H+H 4 = G)P1(�H+(�D d 0�K 4N= D d b * " = E /Jd b BFE 0�( / (�2 / "#H b1/I, K�P1(1&)BRE =�b G�P < &)A�(1G =�1���
2*)-f* ( d b P1(1G
�1 6= �
2 & � � ( � � �� ��� 1) & � � ( � � �� ��� 2) �b H+H < / B =^b H+H <%� B d�* ( 5`b 0 d a S 0�G+K�P - & b 0�K bBb1=^d 0�A�U#B d "%J > 81] Q D d b a�< P�( d b � acbeda�< P1( d (%P1&)S�D�& b K�K b � A�M &�E 67B C)M * BT& d a�- 67K�B *�b+5 H+" *)- 6 acbed K�B a M /Jd a S � 2 * S * B�( �� ) = � ⇐⇒ (∃ � )[ � � ( � � �� ��� ) &
�( � ) = � ] �
-b* 0 d P1(1G * (cD�& <1O "+K bc* "+6 � ( �� ) BFE =�bed Σ01acbed " � ( �� ) BFE =�bed|b1=�b1/ &)(1K d a�, a; d b 0�G =�b & * "�0 d b`a�< 2�S1K�MN6�2 " b1=�bed *Jd ( a & b+*Jd a�, b1=�b1/ &)(1K , BRE =�bed D =I, 0 d b BbP - ]a�*�b 0�" * "+6 bed *Jd ( a & b+*Jd a�, 6 b1=�b1/ &)(1K , 6�2 acbed K < H d 0 *�b * ( P b & <1/ B d D�K b J > 8 O Q?/ B =
BFE =�bed J bed *Jd ( a & b+*Jd a�<�Q/b1=�b1/ &)(1K d a S m b P�S / B d CI" 0 * "+&�E � B *�bed 0 * " = * BTH+BRG *�b E b b P�S*Jd 67B C , 6 * &)B d 6 5`b 0 d a�- 6 d /Jd S * " * BR670�G =�b & * "�0 d b`a�4 =
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> N ! x 2 j 2&'�< , ���=��6�%M����2&' ,3/�[ ���z'�6�%;6�'&� � X ������0P��� /|{ 5 X�, %�}3#9��5T E' 9 2�Y, ������Z�5 2 _ ( 0�G =�b & * "+0 d b`a S � ( �� ��� ) BRE =�bed�• ������������1� ��� J�X%_Pb�_�db_�b�Z :7, X%_Pb�_�db_�b�V Q b1= D d b S�H�BT6 *Jd 6 K�BT& d a�- 6 0�G =�b &R]*I, 0�B d 6 � 2 �12 acbed S�H b%*�b �� 2 � 2
[�( �� ��� ) = � & � v �
] � ⇒ �( �� � � ) = � D
• ������E@4 �@8 J : _Pb+dbZ b g _ g ` Q b1= D d b a�< C)B � acbed S�H bc*�b �� 2 � 2�( �� ��� ) = ��� ⇒ (∃ )[ v � &
�( �� � ) = � &
" BRE =�bed P1BfP1BT& b 0�K -T= " ] �S+P1(1G K d b K�BR& d a�, 0�G =)< & * "�0�" BRE =�bed h8g�hJg � 1 j8, �8Ak7 b1= - A�B d P1BfP1BT& b 0�K -T= (P1B / E ( 0�U#D a H d 0�"+6 acbed
• ��� �1�����������1� ��� J w�VRdbV i XcZ b�Z\` dbZ :)QWb1= D d b a�< C)B � acbed S�H b%*�b �� 2 � 2�( �� ��� ) = ��� ⇒ (∃! v � )[
�( ���� ) = � & (∀ ′ v )[
�( �� � ′)↓ � ⇒ ′ = ]] :3 d (1& d 0�K�(�E�BFE =�bed P b &)S1K�( d ( d D d b 0�G =�b & * "�0 d b`a�< � ( �� � �� ) K�B P�BR& d 0�0�S * BT&)BR6?K�B *�b ]5 H#" *)- 6�2�K�S = (.H�E D�(cP d ( / U+0 a (�H�( d�=�b /Jd b+* G)P1MNC)(1U = P�H , &)MN6
; d b P b & <1/ B d D�K b 2 * (�(� ) =
{0 � b1= � (0)↓ �1 � b H+H d 4 6 �
/ B = BRE =�bed K�( = ( * ( =Jd a S * (�(� ) =
{0 � b1= " � BRE =�bed (�H d a�,⊥ � b H#H d 4 6
/ B = BRE =�bed 0�G = BRA - 6 acbedI* ('0�G =�b & * "+0 d b`a S J > 8 O Q / B = BFE =�bed8bed *Jd ( a & b+*Jd a S�2 � 0 a "+0�"��T E' ! T E' T 2�� # � ,3[ �=' 2 � c5=g 1|AM1e<P;=< mJ>��`1c;=< > 1|AM1���� m ,B< >+A j�=8A=1��@;k7Jj-< 18> A � g@? �A=1 < ,@m6A=m`;=mcAk< > Ap>c1e<2j�=8A0g>� ��� Z >c1e<@g�h`< h�����mcA Z > 65 g J bed *Jd ( a & b+*Jd a�<�Q 1|AM1���� m ,B< >+Aj�=8A=1��@;k7Jj-< 18> A g@?YAM1e<B1e<P;=< mJ>��`1c;=< >+A �� hJgk;91e< A`;=<�= h���;�|m&=iA 1@A=1 <Y;=< m8>�� 16;=< >\1@A=1�� �`mk,l< >�j,=iAM1��-;k78jc<H1J> �h�m3= ��g8Ag@?YAM1e<B1|AM1���� m ,B< > Z h � � � Z 1,=i;BA hJm&= m��9? � gk;91e<B1eh3A(;k7eA J > 8 O Q �j-X Z�<\#3��} [*2 ; d b * " K�( = ( * ( =Jd a S * " *�b 27G�P�(1C -b* (1G#K�B S *Jd D d b a�< P�( d ( J E 0�M 6
b1=�bed *Jd ( a & b+*Jd a S Q P�&)S�D�& b K�K b � K�B a U#& d ( 0�U+K 5 (�H�( �0acbed B C)M * BR& d a�, K�B *�b ]
5 H#" *I, �
�( �� ��� ) = � ⇐⇒ � � � := � ` �
0( �� ) = ���S *Jd@a�< P�( d (167G)P1(�H+(�D d 0�K�S16
�0 : �� → �
1 :�
1 → · · · → � � :� � →: �J >�> 8 Q
* (1G � K�B * " =�b P1( * E K�"+0�" � := ��b P1( / E / B d�* " =�*Jd K , � ( �� ��� ) = � 2 acbed S *Jd � v � acbed P b & b+* "+&)(1U+K�B7S *Jd " E /Jd b b`a (�H+(1G+C�E b�acb+*�b 0 *)< 0�BTM = BRE =�bed G)P1(�H+(�D d 0�K�S16 * (1G� D d bc* " =pb P1( * E K�"+0�" �
:=�12�BbP�B d /I, ( d K�B *�b+5�< 0�B d 6
� ∗ �: �� � ∗ → � ∗ : � ( �� ) � ∗
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x � 2 j 2&'&� / ����6 , ' / ��6�4 '&2&'�< , ���=4 >ANPN
P1(1G acb H�(1U =?* " = � / B = BFE =�bed�< D�( = BT6 J b H#H d 4 6'(cG�P�(�H�(�D d 0�K�S16'C b 0 *�b K b+* (1U#0�B Q 2< & b � ( �� )↓ 2 acbed BbP�(1K -R= MN6 � ( �� ) = � ( �� ) " E /Jd b K�B *)<+5`b 0�" d 0�A�U+B dWacbed D d b * ( =G�P�(�H�(�D d 0�K�S.K�B * " =pb P1( * E K�"+0�" �
:=�
@ B * ( = E /Jd ( * &)S+P1(�2�( G�P�(�H�(�D d 0�K�S16 J >P> 8 Q D d b * " = b P1( * E K�"+0�" �:= � BFE =�bed
BbP1E 0�"+6 G)P1(�H+(�D d 0�K�S16?D d b.* " =�b P1( * E K�"+0�" �:= 2 S+P�(1Gc" ( �� ) 0�G#D a H�E = B d K�S = (D d b *Jd 6 J P1BfP1BT& b 0�K -T= BR6 * ( P+H , C)(16 QW*Jd K - 6 * "+6 � P1(1G acb H�(1U =F*�bed 0 * ( = J >P> 8 Q 2 acbed
-b* 0 d�* ( � ( �� ��� ) BFE =�bed 0�G = BRA - 6 _ BTH d a�< 2 b1= * ( P1&)S�D�& b K�K b � BRE =�bed bed *Jd ( a & b+*Jd a S�2 * S * B.K�S = ( -R=�b 6 G�P�(�H�(#]D d 0�K�S16 J >P> 8 Q G�P�(�H�(�D�E � B d�* " *Jd K , �
0( �� ) = � D d b * " = b P�( * E K�"�0�" �:= � 2 acbed
" J P1BfP1BT& b 0�K -T= " Q K�BR& d a�, 0�G =)< & * "�0�" 0 * " = b P�S / B d CI" * "+6 0�G =)- A�B d b 6.BFE =�bed "BTH < A d 0 * " v � K�BR& d a�, 0�G =)< & * "�0�" *)-f* ( d b P�(1G � ( �� � )↓ 2 b H+H d 4 6c( J >�> 8 Q C b* BR&)K <+*Jd � B # P d (cD�& , D�(1& b%& a; d b *�b 0�G =�b & * "+0 d b`a�< 0 * " = a H < 0�" R ���
G�P < &)A�B d K d b b P�H ,pacbed A�& , 0 d K�"�# acb ]= ( =Jd a�, K�(1& O�, & 2�P�(1G%A�&I"�0 d K�(+P1( d BRE * " = B C , 6 a M /Jd a (+P�(�E "�0�" T E' d-2 � 7��1� �������� ����� ��EF��E������������7�� ���������#����(E@7�� ���������������,7��� ; d ba�< C)B � ∈ N
2�C -b* (1G#K�B
� ( � � F ) =
{( � ) � −· 1 � b1= FHG [ e
( � ) & ( � ) � 0 �⊥ � b H#H d 4 6
� � ( F ) = � ( � � F ) �� � = { F | � � ( F )↓} :
m K�BR& d a�, 0�G =)< & * "�0�" � ( � � F ) BRE =�bed J P�&)M * (�D�B =)4 6 Q b1=�b1/ &)(1K d a�, " b`a (�H+(1G+C�E b� 0� � 1
�;:;:;:b P b & d C)K�BRE�S�H�BT6 *Jd 6 P�BbP�BR& b 0�K -R= BT6�2�K�BR& d a�- 6?0�G =�b & *I, 0�B d 6'K d b 6?K�B *�b+5 H#" *I, 6 acbed" b`a (�H�(1G#C�E b
�0� �
1�;:;:;:
b P b & d C)K�BRE S�H b%*�b P�BbP�BR& b 0�K -R=�b 0�U = (�H b.-b* 0 d P1(1GF ∈ � � ⇐⇒ FHG [ e
( � ) & ( � ) � 0 �
acbed 2�B d /Jd a S * BR& b 2�"c0�A - 0�" F ∈ � � =�b BRE =�bed P1&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�, T E' O 2�� # � ,3[ �=' J � �������1� �� � ������ ����� ��� R ���
�12 +@mp;@= �:1 ? mpj,=iAM1��-;k7 �j-< 18> A �( �� ��� ) g@?PA=1 <|1@A=1 <Y;=< m8>�� 16;=< > 1@A=1�� �`mk,l< > A Z 1|AB>c1e<i,9A6A=m6A 1|A �c<H1 >eh�me< 11|AM1���� m ,B< >E. j � �=j@7 � ( �� � � � � ��� ) Z
�( �� ��� ) = � ⇐⇒ (∃ � )[ � � v � & (∃ � ) � ( �� � � � � ��� )] :J >�>P> Q
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> N Q x 2 j 2&'�< , ���=��6�%M����2&' ,3/�[ ���z'�6�%;6�'&� � X ������0P��� /|{ 5 X�, %�}3#9��5j-X Z�<\#3��} [*2 ; d b%* "cK d b acb+* BTU+C)G = 0�"�2�K�B �� = ( � 1
�;:;:;: � � � ) 2�C -b* (1G#K�B� ∗� ( � � �� � � � F ��� ) ⇐⇒ ( � BFE =�bed@a M /Jd a S16 a�< P�( d (1G J E 0�M 6 b1=�bed *Jd ( a & b+*Jd a (1U Q
b1=�b1/ &)(1K d a (1U P1&)(�D�& < K�K b+* (16 �acbed K�S = (c" � 1� BRE =�bed B C)M * BR& d a�, 0 * ( �acbed ( � BRE =�bed@a M /Jd a S16 * BT&)K b+*Jd a (1UG�P�(�H�(�D d 0�K�(1U * (1G � 0 * " = BRE 0�( / ( � �
0 : ��D d bc* " =pb P1( * E K�"+0�" � 1� := � � :m 0�A - 0�" b G *I, BRE =�bed P�&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�, K�B *Jd 6%K�BRC)S / (1G#6 * (1G N @ 2 acbed 2BRU a (�H b 2 b1=7* ( � BFE =�bed�*)-b* ( d (%P1(1G
�( �� ��� ) = � ⇐⇒ � � � 1� := � ` �
�0 ( �� ) = ���
* S * B�( �� ��� ) = � ⇐⇒ (∃ � )[ � � v � & (∃ � )[ � ∗� ( � � �� � � � F ��� ) &
�( � ) = � ]] :
; d b * " =.< H#H+" acb+* BTU+C)G = 0�"�2�G)P1(1C -f* (1G+K�B S *Jd * ( � ( �� ��� ) d acb1= (+P1( d BRE * " = d 0�(#]/ G =�b K�E b J >�>P> Q 2�C -b* (1G#K�B�( � � �� ) =
(� � � ( �� � ( � )0 � � � ( � )1)
)0
�acbed P b & b+* "�&)(1U#K�B?S *Jd�* (.0�G =�b & * "�0 d b`a S�( � � �� ��� ) = b1= (∀ F*G [\e
( � ))[ � � ( F )↓ � ⇒ � ( F ) = � � ( F )] * S * B � ( � � �� ) b H#H d 4 6 ⊥BFE =�bedlb1=�b1/ &)(1K d a S�2 b P�S *Jd 6 d /Jd S * " * BR6 a H�B d 0 * S * " * BR6 * M = b1=�b1/ &)(1K d a�4 = 0�G =�b &R]* "�0 d b`a�4 = T @ Q� � 0 * M � P�&)S�D�& b K�K b P�(1GcG)P1(�H+(�D�E � B d�* ( � ( � � �� ��� ) -b* 0 d P1(1G
�( � � �� ��� ) = � ⇐⇒ � � � := � ` �
( � � �� ) = ���acbed�- 0 * M � ′ * ( P1&)S�D�& b K�K b P�(1G / "�K d (1G#&ID�BFE *�bed K�B * " = P�&)S10�C)BT0�" * M = B C , 6B C d 0 4 0�BRM = 2�K�B acbed = (1U+&ID d BT6 K�B *�b+5 H#" *)- 6
� (�) = 0
� (�) =
�( � (
�))
�( �� ) =
�( � (0) � �� ) :
3 d�* BR&)K b+*Jd a (�E G�P�(�H�(�D d 0�K�(�E * (1G � ′ P�(1GvC)B a-d = (1U = K�B �: �� BRE =�bed�* "�67K�(1& O�, 6
�: ��
�� : 0 ��
�: �|�� : �
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x � 2 j 2&'&� / ����6 , ' / ��6�4 '&2&'�< , ���=4 >AN 9acbed BfP1B d /I,�b P1S * " 0 *Jd D�K , P1(1G G)P1(�H+(�D�E � B *�bed " *Jd K , � / B = B = BT&ID�(+P1( d (1U =F*�bed P d b( d K�B *�b+51< 0�B d 6 P1(1G P1&)(10�C - 0 b K�B?0 * ( � 2 - P�B *�bed S *Jd
� =�( �� ) =
�( � � �� ��� ) �
-b* 0 d P1(1G%K�B * (1G+6?(1& d 0�K�(1U#6�2� � v � & (∃ � ) � ( �� � ��� � ��� ) �J >�> ! Q
/ "+H b1/I, � ( �� ��� ) = � �B P1S * " =.< H+H#" K�BT& d < 2 b1= � ( �� ��� ) = � 2 * S * B G)P < &)A�B d �*)-b* ( d ( P1(1G d 0�A�U+B d " J >P> ! Q 2 acbed K�B *Jd 6cK�B *�b+51< 0�B d 6 * "�6 � (�)0 * ( � ′ 2WG)P < &)A�B dG�P�(�H�(�D d 0�K�S16 * (1G � ′ P�(1G O�*)<1= B d 0 * (.0�"+K�BFE (
�: �|�� D
D d b P b & <1/ B d D�K b 2 b1= � = 22�(.G�P�(�H�(�D d 0�K�S16 O�*)<1= B d 0 *Jd 6 acb+*�b 0 *)< 0�B d 6
�: ��
�� : 0 ��
� �� : 0 ��
� � �� : 0 ��
� � �: 0 ��
� �: 1 ��
�: 2 ��(:
B P�S / M acbed K�B *)< ( G)P1(�H+(�D d 0�K�S16 0�G = BRA�E � B d K�B *Jd 6 K�B *�b+5�< 0�B d 6 * (1G � acbed* BTH d a�<�b P1( / E / B d�* "c0�M 0 *I, *Jd K ,
�( � � �� ) =
�( � � �� ��� ) =
�( �� ��� ) : a
T E' ] 2���� �� ��� �������(7 ��������������� gm * G+A b E b (�H d a�, 2�K�( = (1K�BbH , 6 0�G =)< & * "+0�"�
: N → N1@A� �Jgk;91 <E1|AM1���� m ,B< > 0�B K d b1=?< H#H+" � 2 b1= G�P < &)A�B d6a�< P1( d ( b1=�b1/ &)(#]
K d a S 0�G =�b & * "+0 d b`a S � (� ��� ) *)-f* ( d (cP�(1G
�(�) =
�(� � � ) (
� ∈ N) :J >�> N Qp -f* (1G+K�B
� ≤ � � ⇐⇒ " � b1=)< D�B *�bed@b1=�b1/ &)(1K d a�< 0 * " �� ≡ � � ⇐⇒ � ≤ � � &
� ≤ � � :3 / BFE a�* "+6 � 0 * (?0�G#K 5 (�H d 0�K�S b1=�b1O�- &)B *�bed 0 * ( = ` g�i Zcb h 2 acbed "'0�A - 0�" ≤ � a ( d =)<acb H�BRE *�bed � � KMFHD`_ 1|AM1�1� �c. acbed 0�G =I, C)MN6cB O-b &)K�S � B *�bed 0�B 0�U = (�H b 2WK - 0�M * M =A b & b`a�* "�& d 0 *Jd a�4 =7* (1G#670�G =�b & *I, 0�BRM = 2
� ≤ � � ⇐⇒ � � ≤ � ���
� ≡ � � ⇐⇒ � ≤ � � & � ≤ � � :
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> N T x 2 j 2&'�< , ���=��6�%M����2&' ,3/�[ ���z'�6�%;6�'&� � X ������0P��� /|{ 5 X�, %�}3#9��53 dia H < 0�B d 6 d 0�( / G =�b K�E b 6 * "+6 ≡ � 0 *�b 0�U = (�H b acb H�(1U =F*�bed J ` g�i Zcb h Q �:1c5 ,@m-?|1@A=1 �hJmJ>��@<Hjc< ,�A ;k7�;91�� J w�V h#i VFVF` _ � g b�`f_#[ Y ��Z [\Z d ^ Q
w�V h� ( � ) = { � | � ≡ � � } �acbed " # P�&)( /Jd <+*�b CI" & ≤ � (1&�E � B d 0 * (1G#6 5 b C)K�(1U#6 * "cK�BR& d a�,c/Jd <+*�b CI"
� ≤ � � ⇐⇒ (∃ � ∈ � � � ∈ �)[ � ≤ � � ]
⇐⇒ (∀ � ∈ � � � ∈ �)[ � ≤ � � ] :
I P1E 0�"+67C -f* (1G+K�B0 =
w�V h(∅) (=
w�V h( � )
D d b a�< C)B b1=�b1/ &)(1K d a S � )
0′ =
w�V h( � ) �
S+P1(1G � BRE =�bed�* ( acb1= ( =Jd a S�2�P+H , &)BR6 b1=�b1/ &)(1K d a�<�b P b & d C)K�" * S 0�U = (�H+( J O T Q m C)BRMN&�E b.* "+67K�BR& d a�, 6 /Jd <+*�b CI"�6 ≤ � 0 * ( 0�U = (�H+( D * M =75`b C)K 4 =pb1=�b P�( a & d ]
0 d K�S * " *�b 6'BFE =�bed J b`a S1K�" Q -R=�b 6 b P�S * (1G#6 P+H - ( = / & b 0 *I, & d (1G+6 a H <1/ (1G#6 - &)BRG =�b 60 * " C)BTM &�E b b1=�b1/ &)(1K , 6�2 * ( = (+P�(�E ( S1K�MN6 / B C b acb H�U ��(1G#K�B B /)4 p b P�BR& d (1& d ]0 * (1U#K�B 0�B K�BR& d a�- 6 b P+H - 6�2�0�A�B *Jd a�- 6 b 0 a�, 0�B d 6 acbed 0 * " /Jd b+* U)P1MN0�" * & d 4N= b P�S*�b%5 b 0 d a�< C)BTM & , K b+*)<%* (1G P�(1G%G)P1( / BRE A = (1G = K d b b P1S *Jd 6 a U+& d BT6 acb+* BTG+C)U = 0�B d 6* (1G T E' c> 8 2 � # � ,9[ �=' 2 J > Q < 1 > 65 g�j,<iAMm �8m � ⊆ N
==h �� �Eg�<^>eh�me< m � ,:g,Eg�� 1 �,<i;0g �`m �:1c5 ,9A 1@A=1 hJmJ>��@<Hjc< ,�A ;k7�;91�� Z w�V h( � ) G � w�V h ( � )
�JL! Q � [\VFVAb�VT] � _+` d lh���;�|m&=iA j,<iAMm � 1 � � � ⊆ N
hJm&=���g8A j�= �8>�� ?PA=m6Ak;91 < � �h��`m�� ;k7 ≤ � Z �i7�� 1��i. � 6≤ � � >-1 < � 6≤ � � �J N Q � i Z VAw�� V ifh ] � g : e3b�Z r Eh �� �@m3=8A 1 � 1 � j�<8A=m��e1 � � � ⊆ Nh�m3=���g8A j�= �
�8>�� ?PA=m6Ak;91 <H� ��h � m�� ;k76A ≤ � Z �i7�� 1��i. � 6≤ � � >-1 < � 6≤ � � Z ��`1 >c1e<0 G � w�V h ( � ) G � 0
′
>-1 < ;=m ? �J< m �-< 1(;=m � �m 5 BTH * E M 0�" J N QN* (1G.J�! Q /Jd b+* G�P 4 = B *�bed C)BRA�MN& d 0 *)< BfP1B d /I, ,#*�b1= b1= ( d a�* S P�&)S#]
5 H#"�K b J * ( #f$'&)S 5 H#"�K b'* (1G � _+` d &fQlb P�S * ( > ] Q�Q K - A�& d�* "'H�U#0�" * "+6 * ( > ]+9�T�2 acbed" #bK - C)( / (16 P�&)( * BR& bed S * " *�b 6 & J 0 i Z _ i Z d ^ XcVRdfe�_ow Q K�B * " = (+P�(�E b * BTH d a�< H�U#CI" a B- A�B d B C)BTH d A�C)BFE 0�B K d b b P1S *Jd 6 5`b 0 d a�- 6?K�BRC)S / (1G#6 * "�6?C)BTM &�E b 6?G�P�(�H�(�D d 0 d K�S * "�]*�b 6
��� � �p�/yl~2�#$ }����T E' c>32�B P�S *�b B C , 6?0�G =�b & * "�0 d b`a�< 2�P�( d b BFE =�bed|b1=�b1/ &)(1K d a�< acbed P1( d b BFE =�bedb1=�bed *Jd ( a & b+*Jd a�<�b1=�b1/ &)(1K d a�< � � d acbed (�H+(�D , 0 * B *Jd 6 b P b1=F*I, 0�B d 670 b 6
�(� ) = b1= (∃ � )[� ( � ) = 1] * S * B 1 b H+H d 4 6 ⊥�(� ) = b1= (∃ � )[� ( � ) = 1 & (∀ F*G � )� ( F ) = 0] * S * B 1 b H+H d 4 6 ⊥
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x � 2 j 2&'&� / ����6 , ' / ��6�4 '&2&'�< , ���=4 >AN1d
��T E' ! 2 � BFE C * B K�B b1=F*Jd P b & b1/ BFE D�K b+*�b S *Jd acb K d <�b P�S *Jd 6 0�G = C ,8a BR6 0 * ( =(1& d 0�K�S�T E' 9 / B = 0�G =)< D�B *�bed|b P�S *Jd 6 < H#H�BT6 / U#(
��T E' N 2 � BFE C * B%S *Jd D d b (�H d a�- 6�2NK�( = (1K�BbH�BRE 6c0�G =�b & *I, 0�B d 6 � � � 2 b1= G)P < &)A�B db1=�bed *Jd ( a & b+*Jd a�<�b1=�b1/ &)(1K d a S 0�G =�b & * "+0 d b`a S � (� ��� ) -b* 0 d P1(1G
�(�) =
�(� � � ) �
* S * B � ≤ � � VvX Z�<\#3��} [��KI P d acb H�BT0 * BRE * B * (�p'B 4 &I"�K b ^ b1= ( =Jd a�, 6 @ (1&R]O�, 6 T E' O�
��T E' Q,2 � BFE C * B?S *Jd � ≤ � � &� ≤ � � � ⇒ � ≤ � �
��T E' 9 2 � BFE C * B?S *Jd@b1= " � BRE =�bed|b1=�b1/ &)(1K d a�, acbed " � b1=)< D�B *�bed 0 * " � 2 * S * Bacbed " � BFE =�bed|b1=�b1/ &)(1K d a�, ��T E' T 2 � BFE C * B�S *Jdcb1= � ≤ � � acbed�* ( D�& <1O "+K b * "+6 � BFE =�bed Σ � 2 * S * B acbed�* (D�& <1O "+K bc* "+6 � BFE =�bed Σ � ��T E' d 2 � BFE C * B?S *Jd
0 G � 0′
T E' c>�>32*Y, ������Z�5 J � �������������#��� ������� �1�����������1� �� ������������� �12g_ ( b1=�bed ]*Jd ( a & b+*Jd a ScP1&)S�D�& b K�K b
�( � ) = 0
�( � ) = 1 �
/ B = G�P�(�H�(�D�E � B d K�BR& d a�, 0�G =)< & * "+0�"�2 BfP1B d /I, 2 P�&)( O@b1=)4 6�2 D d b a�< C)B � - A�B d�/ U#(G�P�(�H�(�D d 0�K�(1U+6
�: � →∗ : 0
�: � →∗ : 1 �acbed�-b* 0 dW/ B = acb C)(1&�E � B d K d b 0�G#D a B a & d K -R= " *Jd K , �
( � ) I E =�bed S1K�M 6 O G+0 d a S =�bC)BRMN& , 0�(1G+K�B M 6 # *Jd K , & P�(1G G)P1(�H+(�D�E � B *�bedcb P1S * ( � * (%0�U = (�H+( {0 � 1} * M =7/ U#(*Jd K 4 = 0 acbed 1 2 acbed ( BbP�S1K�B = (16�(1& d 0�K�S16 a�<1= B deb G+0 * "�& , b G *I,'* " O G+0 d a�,7-R=)= ( d b ; d b a�< C)B70�G =�b & * "�0 d b`a�, K�B *�b+5 H#" *I, � 0�B / (10�K -R= ( b1=�bed *Jd ( a & b+*Jd a S b1=�b1/ &)(#]K d a ScP�&)S�D�& b K�K b � J�A�M &�E 6?B C)M * BR& d a�- 6 K�B *�b+5 H+" *)- 6 Q 2�C -b* (1G#K�B
�( �� ) = { � | � ∈ N &
G)P < &)A�B d G)P1(�H+(�D d 0�K�S16 �: �� → · · · → : � * (1G �
, � = ⊥ acbed G)P < &)A�B d2b P�( a H1E = M = G�P�(�H�(�D d 0�K�S16 * (1G ��
: �� → �1 :�
1 → · · · } :; d b P b & <1/ B d D�K b 2�" # *Jd K , & * (1G P1&)(�D�& < K�K b+* (16
�( � ) = 0
�( � ) = 1
�( � ) =
�( � )
BFE =�bed �( � ) = {0 � 1 � ⊥}
��T E' O 2 � 4 0 * B P b & b1/ BFE D�K b+*�b#b1=�bed *Jd ( a & b+*Jd a�4 = P�&)(�D�& b K�K <+* M = K�B a U+& d (0�U#K 5 (�H+( � D d bc*�b (+P1(�E b
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> N�O x 2 j 2&'�< , ���=��6�%M����2&' ,3/�[ ���z'�6�%;6�'&� � X ������0P��� /|{ 5 X�, %�}3#9��5J > Q �
( � ) = N ∪ {⊥} J�! Q �( � ) = {0 � 1 �;:;:;: � � }
��T E' ] 2 � 0 * M �+P1&)S�D�& b K�K b P�(1G G�P�(�H�(�D�E � B d�* " = P�&)S10�C)BT0�"�2 acbed � "
b1=�bed *Jd ( a & b+*Jd a�, BfP -0a�*�b 0�" * (1G �+K�B * (1G+6?(1& d 0�K�(1U#6
�( � ) = 1�( � ) =
�( � ) +
�( � ) :
$?( d (.BRE =�bed�* ( 0�U = (�H+( �(0)�
��T E' c> 8 2�� 0 * M �2P1&)S�D�& b K�K b P1(1G G�P�(�H�(�D�E � B d�* " 0�G =)< & * "+0�" 2 � 2 acbed � "
b1=�bed *Jd ( a & b+*Jd a�, BfP -0a�*�b 0�" * (1G �2K�B * (1G+6?(1& d 0�K�(1U#6
�( � ) = 1�( � ) = 2
�( � ) :
$?( d (.BRE =�bed�* ( 0�U = (�H+( �(0)�
��T E' c>P> ∗ 2�� BRE C * B S *Jd8b1= D d b^a�< P�( d ( b1=�bed *Jd ( a & b+*Jd a S P�&)S�D�& b K�K b � K�B a U+& d (0�U#K 5 (�H+( � * ( 0�U = (�H+( �
( � ) BRE =�bed�< P�B d &)(�2 * S * B ⊥ ∈ �( � ) 2 / "#H b1/I, G)P < &)A�B db P1( a H�E = M = G)P1(�H+(�D d 0�K�S16 * (1G � P1(1G b &)A�E � B d K�B * " = acb+*)< 0 *�b 0�" �
: � J B G *I," < 0 a "+0�"cA�&)B d <%� B *�bed -R=�b�b P�( *)- H+BR0�K b�b P�S * "c0�G = (�H�(1C)BTM &�E b Q��� ��� x �-x ��$i� wet(�Wz ���cz:�cwcx �B~2�_ ( C)B 4 &I"+K b 0 � b G * S * ( B /)<1O|d ( BRE =�bed C)BRK�BbH d b`a S D d b * " C)BTM &�E b b1=�b1/ &)(1K , 6
; d bc* " =pb P1S / B d C ,%* (1G12 / BFE A = (1G#K�B P1& 4�*�b -R=�b b P�H+S � , K�K b T G c>&2 � 4����=' 2 <H1^>c5=g 1@A=1 <Y;=< m8>�� 16;=< >p1|AM1���� m ,B< >+A j�=8A=1��@;k7Jj-< 18> A �
( �� � �� ) Z7�,Eg �@< >:. j,=iA ��-;k78j@7�( �� � � 1
�;:;:;: � � � ) =�( �� ���
�1
�;:;:;: ��� � � )g@?YAM1e<B1|AM1���� m ,B< >E.��j-X Z�<\#3��} [*2 ; d b * " = P�BR&�E P * M 0�" K�B K d b K�S = ( 0�G =�b & * "+0 d b`a�, K�B *�b+5 H+" *I, 2
S+P1(1G ( 0�G#K 5 (�H d 0�K�S16.BRE =�bed b P+H�(1U#0 * BT&)(16�2 - A�(1G+K�B b P1S * (7p?B 4 &I"+K b ^ b1= ( =Jd ]a�, 6 @ (1& O�, 6 T E' O K d b�d 0�( / G =�b K�E b�( �� ��� ) = � ⇐⇒ ∃ � )[ � � v � & (∃ � ) � ( �� � � � � ��� )] �
S+P1(1G " � ( �� � ��� � ��� ) BRE =�bed "�K d b1=�b1/ &)(1K d a�, 0�A - 0�" < & b�( �� � � ) = � ⇐⇒ (∃ � )[ � � v �
�& (∃ � ) � ( �� � � � � ��� )]
⇐⇒ (∃ � )[(∀ FHG [ e( � )[ � � ( F )↓ � ⇒ �
�( F ) = � � ( F )]
& (∃ � ) � ( �� � � � � ��� )] �acbed " � ( �� � � )BRE =�bed|b1=�b1/ &)(1K d a�, BfP1B d /I,%* (.D�& <1O "+K <c* "+67BRE =�bed Σ1
a
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x�� 2 � �)(�� � # � ,9[ �=' j 2&'�< , ���=4�5 >AN ]
T G ! 2�� # � ,3[ �=' J � � �1� � E��������� � ���������� �8 �12 <H1 > 65 gW,@m6A=m`;=mcAk< > A>-1 <Wj,=iA9g>� ��� j�=8A=1��@;k7Jj-< 18> A �( � 1
�;:;:;:�� � � ��� ) A hJm&= 7 � g@?PA=1 < � � ,Eg��@. � j,=iAM1����;k78jc<H1J>E. ,:gk;91 ��|7�;k. Z = h���;�lg�<Eg�����/< j|;k7��,<Jj@7 � ;k7 � 1@A=1�� �`mk,l< >:. � g@C&? j-�/j@7 �� ( �� ) =
�( �� ��� ) �
hJm&= �E1�� 1J>E;k7�� ?��9gk;91 <W1 h�A ;=< � j,=iA 58.=>@g��J > Q �c<H1 >c5=g �� Z � ( �� ) =
�( �� � � ) Z >-1 <
JL! Q �c<H1 >c5=g � : N�� N
Z1|A
(∀ �� � � )[�( �� � � ) = ��� ⇒ �
( �� ) = � ] � ;BA`;0g � v � :� hi<Hh � ��m6A Z 1@Ap;=m �
( �� ��� ) g@?PA=1 <B1@A=1 <Y;=< m8>�� 16;=< > 1@A=1�� �`mk,l< > A Z ;BA ;0g 7 � g@?PA=1 <1|AM1���� m ,B< >E.��j-X Z�<\#3��} [*2 $ b & b+* "�&)(1U#K�B P1& 4�*�b S *Jd " ,@m6A=1��8< >+A`;k7J;91 * "�67K�BR& d a�, 6 0�G =)< &R]
* "�0�"�6 � P�(1G d acb1= (+P�( d BFE *�b J > Q acbed JL! Q%* (1G C)BRMN& , K b+* (16 BRE =�bed * B * & d K�K -T= " BbP�B d /I, b1= " � - A�B d2b G *)- 6 *Jd 6 / U+( d /Jd S * " * BT6 acbed " � ′ *Jd 6 b1=F* E 0 * ( d A�BR6(1)′ � ′( �� ) =
�( �� � � ′) 2
(2)′D d b a�< C)B � : N � N
2
(∀ �� � � )[�( �� � � ) = ��� ⇒ �
( �� ) = � ] � ⇒ � ′ v � �
* S * B � v � ′ b P�S * " = (1)′ acbed�* " = (2)K�B �
= � ′ 2 acbed � ′ v � b P1S * " = (1) acbed�* " =(2)′
K�B��= � � * 0 dcb P1(1K -T= B d�=�b%/ BFE C)(1G#K�B * " = U�P b &HCI" K�BR& d a�, 6'0�G =)< & * "+0�"+6 �
P1(1G d acb1= (+P�( d BFE *Jd 6 J > QWacbed J�! Q 2 acbed�* " = b1=�b1/ &)(1K d a S * " *)< * "�6�2�0 * " = P1BT&�E P * MN0�"P1(1G * ( � ( �� ��� ) BFE =�bed|b1=�bed *Jd ( a & b+*Jd a�<�b1=�b1/ &)(1K d a S p -f* (1G+K�B
� 0( �� ) = ⊥ �acbed 2 b1=�b1/ &)(1K d a�< 2
��+1( �� ) =
�( �� � �
�) :
� .9,2,21 � 0 v � 1 v � 2 v · · · 2�k;=j-</h�m3< �c<H1 > hJm <H1�,:g �|< >E. j,=iA ��-;k78j@7 � : N � N
2
� ( �� ) = � ⇐⇒ (∃ � )[��( �� ) = � ] :J >�>AQ Q
n h3A��kg�< Ci7�;=m3= � .0,|,:16;=m���� � BFE A = (1G#K�B�K�B�BbP b D�MND , 0 * (�� S *Jd � � v ��+1 m
51< 0�"�2 � 0 v � 1 BFE =�bed P�&)( O@b1=I, 6�2�BfP1B d /I, � 0 v �12�D d b a�< C)B � ; d bc* (.BbP b D�MND d a S
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>AQ 8 x 2 j 2&'�< , ���=��6�%M����2&' ,3/�[ ���z'�6�%;6�'&� � X ������0P��� /|{ 5 X�, %�}3#9��55�, K b
��+1( �� ) = � � ⇒ �
( �� � ��) = �
b P1S * ( = (1& d 0�K�S� ⇒ �
( �� � ��+1) = �
b P1S * " = BfP b D�MWD d a�, G�P�S1C)BR0�" � � v ��+1
acbed�* "cK�( = ( * ( =Jd a S * " *�b%* (1G�� ( �� ��� )� ⇒ �
�+2( �� ) = �b P1S * ( = (1& d 0�K�S :
n h3A��kg�< Ci7�;=m3= J > Q $'& - P�B d =�b / BRE C)(1G+K�B?S *Jd D d b a�< C)B �� acbed � 2� ( �� ) = � ⇐⇒ �
( �� � � ) = � �acbed C b BfP b H#"�C)BTU+0�(1G+K�B C)BRA�MN& d 0 *)<c*Jd 6 / U#( 0�G = BfP b D�MWD - 6 ; d bc* " = � ( �� ) = ��� ⇒ �
( �� � � ) = � P�& 4W*�b 2�G)P1(�H+(�D�E � (1G+K�B � ( �� ) = � � ⇒ (∃ � )[�
�+1( �� ) = � ] b P�S * ( = (1& d 0�K�S
� ⇒ (∃ � )[ � ( �� � ��)) = � ] b P�S * ( = (1& d 0�K�S
� ⇒ �( �� � � ) = � K�( = ( * ( =Jd a S * " *�bc* (1G � (� ) :
; d b?* " =^b1=F* E 0 * &)( O " acb+* BRU#C)G = 0�"�2 - 0 * M S *Jd � ( �� � � ) = � B P�S * " 0�G =)- A�B d b7* (1G�( �� ��� ) 2 - P�B *�bed S *Jd G�P < &)A�B d:a�< P�( d b P1BfP1BT& b 0�K -T= "�2�K�BR& d a�, 0�G =)< & * "�0�" � ∗ v � 2
K�B P�B / E ( 0�U#D a H d 0�"+6{ �� 0
�;:;:;: � �� � −1} = { �� | � ∗( �� )↓} �*)-b* ( d b P�(1G
�( �� ��� ∗) = ��:J >�> 9 Q
B P�S * ( = (1& d 0�K�S * "�6 � 2�D d b a�< C)B F*G ��2�G�P < &)A�B d@a�< P�( d (16 � � 2 *)-b* ( d (16 P1(1G� ∗( �� � ) = � ( �� � ) = �
���( �� � ) ( FHG �
) �acbed|b1= � = max{ � 0�;:;:;: � �
� −1} + 12 * S * B
� ∗( �� � ) = ��( �� � ) ( FHG �
) �/ "+H b1/I, � ∗ v �
� \m K�( = ( * ( =Jd a S * " *�b%* (1G � ( �� ��� ) 0�G = BbP < D�B *�bed�*)4 & b S *Jd�( �� ��� ∗) = � � ⇒ �
( �� � ��) = �
� ⇒ ��+1( �� ) = �
P1(1G%K�B * " = J >P> 9 Q 0�G+K�P+H+"+& 4 = B d�* " =pb P�S / B d CI" * (1G J > Q n h3A��kg�< Ci7�;=m3= J�! Q � 0 * M S *Jd D d b a�< P1( d b � : N � N
(∀ �� � � )[�( �� � � ) = ��� ⇒ �
( �� ) = � ] :_ ( � " * (1U+K�B = ( � v � b`a (�H�(1G#C)BFE b P�S * " =
��v �
(� ∈ N)
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x � 2&VvX ������0P��� /|{ 5 X�, %�}9#3��5 > Q�>
P1(1G d 0�A�U+B d P1&)( O-b1=)4 6?D d b � = 0 \I P b D�MND d a�< 2
��+1( �� ) = � � ⇒ �
( �� � ��) = � b P�S * ( = (1& d 0�K�S
� ⇒ �( �� � � ) = � b P�S * " = BbP b D�MND d a�, G)P1S1C)BT0�"
acbed�* ".K�( = ( * ( =Jd a S * " *�bc* (1G �� ⇒ �
( �� ) = � b P�S * " = G�P�S1C)BR0�"cD d bc* " = � :_ BTH d a�< 2 b1= * ( � ( �� ��� ) BRE =�bed�b1=�bed *Jd ( a & b+*Jd a�< b1=�b1/ &)(1K d a S�2 * S * B " K�BT& d a�,
0�G =)< & * "�0�"�( � � �� ) =
�( �� ���
�)BFE =�bed|b1=�b1/ &)(1K d a�, b P1S * ( � , K�K b T G c> 2 -b* 0 d P�(1G%D d b a�< P1( d ( =pb & d C)K�S � 2
{ � 11( � � � )}( �� ) =
�( �� ���
�) :
� P1B *�bed S *Jd2b1= ( � 0BFE =�bed2a�< P�( d (16 a M /Jd a S16 * "�6 # a B =I, 6 & K�BT& d a�, 6 0�G =)< & * "+0�"+6
� 0( �� ) = ⊥ acbed C - 0�(1G#K�B)2 b1=�b1/ &)(1K d a�< 2�(0) = � 0
�(�
+ 1) =� 1
1( � � � ( � )) �* S * B)2�D d b a�< C)B �W2�( � ( � )
BRE =�bed@a M /Jd a S16 * "�6 � � < & b� ( �� ) = � ⇐⇒ (∃ � )[{� ( � )}( �� ) = � ] �acbed " � BFE =�bed|b1=�b1/ &)(1K d a�, 2�B O S10�( = - A�B d Σ1
D�& <1O "�K b a
��� � � %Ex��2x+' }��B{c�i� %Ew)�(.,$e}��e�� b G * S acbed�* ( BbP�S1K�B = ( B /)<1O@d ( C)BRMN&)(1U+K�B 0�G =�b & * "+0 d b`a�< � ( �� � �� ) 2�S+P1(1G12S1K�M 6�2�( d K�B *�b+5 H+" *)- 6 �
1�;:;:;: ��� � a G#K b E = ( =F*�bed 0 *Jd 6 1@A=1�� �`mk,l< > ��� J acbed S1A d@b G)]C b E &)B * BT6 Q K�BR& d a�- 6 0�G =�b & *I, 0�B d 6�2 -b* 0 d P1(1G * ( � K�P1(1&)BRE =�b # acb H - 0�B d & *Jd 67K�B *�b ]
5 H#" *)- 6 * (1G K�B * (1G+6 a M /Jd a (1U#6 * (1G+6 � / "+H b1/I, # acb+* � S = (1K b%& 0 * " = (1&)(�H+(�D�E b* M = D�H+M 0�0 4N= P�&)(�D�& b K�K b+*Jd 0�K�(1U 8e G = "�C�E � B *�bed-b G *)< *�b 0�G =�b & * "�0 d b`a�<%=�bpacb ]H�(1U =F*�bed # P1& < C)B d 6 & J _&0�V i Y)dbZ _Pb�` Q T�S c>32*Y, ������Z�5 2 � ��� ��� JL0 *Jd 6 b1=�b1/ &)(1K d a�- 6'K�BT& d a�- 6 0�G =�b & *I, 0�B d 6 Q BRE =�bed "
* G+A b E b K�BR& d a�, 0�G =)< & * "+0�"�
: N�× P � 1 × · · · × P
���� � N �
S+P1(1GP��BFE =�bed * ( 0�U = (�H�( S�H�M = * M = b1=�b1/ &)(1K d a�4 = K�BT& d a�4 = 0�G =�b & *I, 0�BRM = �
K�B *�b+5 H+" *)4 = 2P�� = { � �
�| � ∈ N} Dacbed "�P�& < CI"�� BRE =�bed �1������������� �� J V�� V : dfZ V _&0�V i Y)dbZ _Pb Q2b1= "'K�BT& d a�, 0�G =)< & * "+0�"
�( �� � � 1
�;:;:;: � � � ) =�( �� ���
�1
�;:;:;:���� ��
)J >�> T QBFE =�bed|b1=�b1/ &)(1K d a�,
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>AQ ! x 2 j 2&'�< , ���=��6�%M����2&' ,3/�[ ���z'�6�%;6�'&� � X ������0P��� /|{ 5 X�, %�}3#9��5$ b & b+* "+&)(1U+K�B S *Jd " K�BR& d a�, 0�G =)< & * "�0�" � P�(1G G�P�(�H�(�D�E � B d�* " P�& < CI" � d acb ]
= (+P1( d BRE * " = B C , 6 ������������ ������������ 7�����8 J Zcb Y i Z\Y�b : V�0 i _10 V i d ^ Q ��1
= � �1
�;:;:;:���� ��
= � ��
� ⇒ �( �� � � 1
�;:;:;:�� � � ) =�( �� � � 1 �;:;:;:�� � � ) DJ >�>Hd Q
acbed J P1&)( O-b1=)4 6 Q-a�< C)B b1=�b1/ &)(1K d a�, K�BT& d a�, 0�G =)< & * "+0�" � P�(1G d acb1= (+P1( d BRE * " = J >�>Hd QG�P�(�H�(�D�E � B d�* " = P�& < CI".0 *Jd 6 b1=�b1/ &)(1K d a�- 67K�BT& d a�- 670�G =�b & *I, 0�B d 6
�( �� ���
�1
�;:;:;:���� ��
) =�( �� � � 1
�;:;:;: � � � ) :_ ( 5`b 0 d a S C)B 4 &I"�K b b G * (1U * (1G B /�b1O E (1G BFE =�bed S *Jd ( d G)P1(�H+(�D d 0 *)- 6 P1& < C)B d 6
JL(1G#0 d b 0 *Jd a�<�Q 0�G+K�P1E P * (1G = K�B *�b b1=�bed *Jd ( a & b+*Jd a�< b1=�b1/ &)(1K d a�< 0�G =�b & * "�0 d b`a�< 2acbed " K d b acb+* BRU#C)G = 0�"c0�G =)< D�B *�bed@b K - 0�MN6 b P1S * ( � , K�K b T G c>� \_ ( a H�B d / E�D d b* " =pb1=F* E 0 * &)( O " acb+* BTU+C)G = 0�".BRE =�bed�* ( B C , 6T�S ! 2�� 4����=' 2 � 65 g = h�m �8m �c<Hj@;k. h�� Ci7�g@?YAM1e<-,@m6A=m`;=mcAk< >:.p>c1e<Bj�=8A0g>�W. ���j-X Z�<\#3��} [*2 p'BRMN&)(1U+K�B'K�S = (cP�& < C)B d 6 � : P
�1 � N
2�B O S10�( = " b P1S / B d CI".0 * "D�B =Jd a�, P�BR&�E P * M 0�".BFE =�bed ".E /Jd b�� K�S = (cP d (cP1BT&�E P�H+( a ".0 * (.0�G#K 5 (�H d 0�K�S ; d bc* " =pb P1S / B d CI" * "+67K�( = ( * ( =Jd a S * " *�b 6�2 - 0 * M � v �12�S+P�(1G
� = �� acbed � = � � �
acbed � ( a M /Jd a S16 * "+6'K�BR& d a�, 6?0�G =)< & * "+0�"+6�P�(1G G)P1(�H+(�D�E � B d�* " = � 2 / "#H b1/I, 2�D d ba�< C)B � 2�( � � ) = { � }( � ) :J >�> O Q
� P1(1C -f* (1G+K�B'BbP1E 0�"+67S *Jd�( ��) = � �
acbed P�& - P�B d�=�b./ BFE C)(1G#K�B?S *Jd � ( � � ) = � m 0�A - 0�"
� ( � � � � � ) ⇐⇒ ��( � ) = � , [{ � }( � ) = � & � � ( � ) = � ]
BFE =�bed "+K d b1=�b1/ &)(1K d a�, "cG)P1S1C)BT0�" � � v � � 0�G = BbP < D�B *�bed S *Jd� ( � � � � � ) � ⇒ � � ( � ) = �(D
< & b " � ( � � � � � ) BFE =�bed D�& <1O "+K b�a�< P�( d b 6 K�BR& d a�, 6�2 b1=�b1/ &)(1K d a�, 6 0�G =)< & * "+0�"+6�( � � � ) acbed BfP1(1K -T= M 6�2 * ( !+(vp?B 4 &I"+K b B =�b1/ &)(1K , 6�0�G = BbP < D�B *�bed S *Jd � � ∗( � ) =�( � ∗ � � ) D d b a�< P1( d ( =pb & d C)K�S � ∗ 2 -f* 0 d P1(1G
� � ∗( � ) = � ⇐⇒ ��( � ) = � , [{ � }( � ∗) = � & � � ( � ) = � ] :J >�> ] Q
$ b & b+* "+&)(1U+K�B *)4 & bc*�b B C , 6 J > Y Q � ( � � ∗) = { � }( � ∗) = � I P�B d /I, 0 * " =�b1=F* E C)B * " P�BR&�E P * M 0�"�2 b P�S
* " = J >P> ] Q 2 � � ∗ = �� 2 < & b � ( � � ∗) =
�( ��) = �
J > � Q � � ∗ = � � 2 acb+* BRG#C)BFE b1=pb P1S * " = G)P1S1C)BT0�" � � v � � acbed�* ( J > Y Q � P1B *�bed S *Jd � ( � � ) =
�( � � ∗) = �
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x � 2&VvX ������0P��� /|{ 5 X�, %�}9#3��5 > Q1N
m acb+*�b 0 a BRG , D d b'* " = b P1S / B d CI" * "+6�0�G =)- A�B d b 6�BRE =�bed K d a & , P b & b H+H b D , b G *I, 6P1(1GcA�&I"�0 d K�(+P1( d , 0 b K�B'D d b.* " K�( = ( * ( =Jd a S * " *�b $'& 4W*�b.5 &�E 0 a (1G#K�B b P1S * ( !#(p?B 4 &I"+K b B =�b1/ &)(1K , 6 a�< P1( d ( � ∗ *)-b* ( d (cP�(1G
� � ∗( � ) = � ⇐⇒ (∀ � ≤ � )¬[ � 1( � � � ∗ � � ) &�
( � ) = � ] & ��( � ) = � �
J > !#8 Q
acbed P b & b+* "�&)(1U#K�B JL!#Y Q � ( � � ∗) = � \I P�B d /I, 0 * " =pb1=F* E C)B * "%P�BR&�E P * M 0�"�2
(∀ � )¬[ � 1( � � � ∗ � � ) &�
( � ) = � ] �< & b 2�D d b a�< C)B � 2
(∀ � ≤ � )¬[ � 1( � � � ∗ � � ) &�
( � ) = � ] �acbed BfP1(1K -T= M 6�2 b P1S * " = J > !�8 Q 2 � � ∗ = �
� acbed � ( � � ∗) =�( ��) = �
JL!!� Q � � ∗ v �� 2 acb+* BTG+C)BRE b1=�b P�S * " = J > !#8 Q
JL! :)Q m K�BR& d a�, 0�G =)< & * "�0�" � � ∗BRE =�bed P1BfP1BT& b 0�K -T= "�2 BfP1B d /I, 0�G�D a H1E = B d K�S = ( =b1=
� G ( ��� )[ � 1( � � � ∗ � � ) &�
( � ) = � ] :J > ! > Q aT�S N 2 � # � ,9[ �=' J � E.�������� �� ����� ����� ��� � �� � � � � �12 � ;@= �:1 ? 1ph ��-C`7 �g@?YAM1e< ==hJm��im �-< j|;k. 1@A >-1 < ,�AcAMmcA 1|A g@?PA=1 < m�hJg �|< m��@<HjJ,9A�� j|;=< � 1|AM1���� m ,B< > ���,Eg �@< > ��� j�=8A=1��@;k.Jj6g�< � >eh�me< m3= 1@A=1 <Y;=< m8>�� 16;=< >(1@A=1�� �`mk,l< >|m3< j,=iAM1��-;k78jc<H1J>2m&< Z
�`7 �e1��`. 1|A��c<H1 > hJm < m � ∈ R ���Z
�( �� ���
�1
�;:;:;: ��� ��
) =�( �� ���
�1
�;:;:;:���� ��
) :j-X Z�<\#3��} [*2 m K d b\acb+* BRU#C)G = 0�" b P1( / BRE A * " a B 0 * ( � , K�K b T G c>� ; d b * " =
< H+H#"�2�C)BRMN&)(1U+K�B K�S = ( P�& < C)B d 6�� (� )K�B K d b 2�K�( = (1K�BbH , K�B *�b+5 H+" *I, 2 acbed P b & b ]
* "�&)(1U#K�B S *Jd6b P�S * (Yp?B 4 &I"+K b ^ b1= ( =Jd a�, 6 @ (1& O�, 6 T E' O b & a BFE =�bpb P1( / BRE C)(1G+K�B* ( B C , 6 G�P < &)A�B d "+K d b1=�b1/ &)(1K d a�, 0�A - 0�" � ( � � � )
2 *)-b* ( d b P�(1G�(� ) = � ⇐⇒ (∃ � )[ � � v � & � ( � � � )] :J > !+! Q
B = ( � BFE =�bed:a M /Jd a S16 * "+6 K�BR& d a�, 6 0�G =)< & * "�0�"�6 � ( � � F ) P1(1G b P b & d C)K�BFE�S�H�BT6*Jd 6'P�BbP�BR& b 0�K -R= BT6�2�K�BR& d a�- 6 0�G =�b & *I, 0�B d 6�2 * S * B� � ( F ) = {� }( � � F ) = { � 1
1(� � � )}( F ) �acbed BbP�(1K -R= MN6�2 b1= "7K�BT& d a�, 2 b1=�b1/ &)(1K d a�, 0�G =)< & * "�0�" � ( � )
G)P1(�H+(�D�E � B d+* " = � (� )2
* S * B'"c0�A - 0�"
� ( � � � ) ⇐⇒ �( � � ) = � ⇐⇒ �
(� 1
1(� � � )) = �BFE =�bed "+K d b1=�b1/ &)(1K d a�, 2 acbed " J > !#! Q - P�B *�bed|b K - 0�M 6 b P�S * ( � , K�K b T�S ! a
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>AQ�Q x 2 j 2&'�< , ���=��6�%M����2&' ,3/�[ ���z'�6�%;6�'&� � X ������0P��� /|{ 5 X�, %�}3#9��5T�S Q,2 * Z , �����=' J ��Z : VT] � e�Y@0�Z i _ Q 2 � �
( � )g@?YAM1e<@,:g �|< >E. Z 1|AM1���� m ,B< >E. Z >-1 <
� = � � ⇒ �( � ) =
�( � ) �;BA ;0g >-1 <@,�AcAMmcA 1@A = h���;�lg�<W70,l<H1@A=1�� �`mk,l< >:. j@� �=j|7 � ( � � � )
;��k;=m <H1 h�m3=�( � ) = � ⇐⇒ (∃ � )[ � � ⊆ �
& � ( � � � )] �S+P1(1G�" b P b &�E C)K�"+0�" �
0� �
1�;:;:;: * M = P1BfP1BT& b 0�K -T= M = 0�G = S�H�M = (1&�E 0 * " a BW0 * (�T E' d �
B O�,�= (1G#K�B * " =pb P�S / B d CI"cD d b.< 0 a "+0�"�2 ��T S T
��� � �p�/yl~2�#$ }����T S c>&2�� 0 * M �
( � ) b1=�b1/ &)(1K d a�, K�BT& d a�, 0�G =)< & * "+0�" *)-b* ( d b P1(1G[�( � )↓ & � = � ] � ⇒ �
( � )↓ D/ BFE C * B?S *Jd D d b a�< C)B � 2
�( � )↓ � ⇒ G�P < &)A�B d P�BbP�BR& b 0�K -R= ( � 2 *)-b* ( d (%P1(1G � ⊆ � acbed � ( � )↓ :
��T S ! 2�� 0 * M �( � ) b1=�b1/ &)(1K d a�, K�BR& d a�, 0�G =)< & * "+0�"�2 *)-b* ( d b P�(1G � ( � ) ≤ 5D d b(a�< C)B a M /Jd a S � (�H d a�, 6�2 K�( = (1K�BbH�(1U#6.0�G =)< & * "�0�"�6 �
� RB H+"+C)BRU#B dN, S1A d "B C , 6 P�&)S *�b 0�" G)P < &)A�B dEa�< P1( d (16 � *)-b* ( d (16 P1(1G." � � / B = BFE =�bed (�H d a�, 2 b H+H <�( � )↓ acbed � ( � ) ≤ 5
\B P�( / BFE C * B * " =pb P <1=F* "�0 , 0 b 6 ��T S N 2�� 0 * M �
( � ) b1=�b1/ &)(1K d a�, 2�K�BR& d a�, 0�G =)< & * "�0�" *)-b* ( d b P�(1G �
= ∅ � ⇒ �( � )↓ :
JLY Q � BFE C * B?S *Jd D d b a�< P1( d ( �6= ∅ 2 � ( � )↓
JU� Q �� BRE C * B7S *Jd D d b a�< C)B � 2�G�P < &)A�B d@a�< P1( d (16 � *)-b* ( d (16'P�(1G � = � acbed � ( � )↓ :
��T S Q 2�� BRE C * B�S *Jd D d bpa�< C)B�G)P1(�H+(�D d 0 *I, P�& < CI"�� ( � ��� ) 2�" b1=�b1/ &)(1K d a�, B C�E ]0�MN0�"� ( � ) =
�( � ��� )
- A�B d2b1=�b1/ &)(1K d a�, J�K�BR& d a�,�Q H+U+0�" ��T S 9 2�� BRE C * B�S *Jd D d bpa�< C)B�G)P1(�H+(�D d 0 *I, P�& < CI"�� ( � ��� ) 2�" b1=�b1/ &)(1K d a�, B C�E ]0�MN0�"
� ( � ) =�( � ��� )
- A�B d BbH < A d 0 * "%H�U#0�"�2�P�(1GcBRE =�bed|b1=�b1/ &)(1K d a�, ��T S T 2 B P1( / BRE C * B * (�p?B 4 &I"+K b ��Z : VT] � e�Y@0�Z i _�T�S Q�
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x�� 2�� � � � �� ��� ��� �������6�'&�� �� ������ ��� > Q 9��T S d J � E.�������� ������ � � � �12 � BFE C * B S *Jd6b1= " (�H d a�, b1=�b1/ &)(1K d a�, 0�G =)< &R]
* "�0�" � ( � ) d acb1= (+P�( d BFE * "c0�G = C ,ia " b1=�b H+H+(�E M 0�"�6 � = � � ⇒ �
( � ) =�( � ) �
* S * B'" � ( � )BFE =�bed 0 *�b C)BR& ,
��T S O ∗ 2 JLY Q � BFE C * BcS *Jd G�P < &)A�B d/b1=�b1/ &)(1K d a�, K�BT& d a�, 0�G =)< & * "+0�" � ( � ) *)- ]* ( d b P�(1G D d b a�< C)B � 2�( � )↓ ⇐⇒ (∃ � )[ �
�( � )↓ ] �J > ! N Q
(∃ � )[ ��( � )↓ ] � ⇒ �
�(�( � ))↓ :J > ! Q Q
JU� Q � BRE C * B S *Jd�/ B = G)P < &)A�B d^b1=�b1/ &)(1K d a�, K�BT& d a�, 0�G =)< & * "+0�" � ( � )P1(1G =�bd acb1= (+P1( d BRE *Jd 670�G = C ,ia BT6%J > ! N Q 2�J > ! Q Q 2 acbed BfP d P+H - ( = * " =
��
= � � � ⇒ �( � ) =
�( � ) :J > !+9 Q
��� ��� z��,$�) we~+�Wz2{cz�� � � � � � � ����� ����� � ����� � � ��� ��� \y/zB}������ �� �� � � �_ ( 5`b 0 d a S K ,�= G#K b * (1G T S N BRE =�bed S *Jd�/ B = G)P < &)A�B d�* &)S+P�(16 =�b A�&I"�0 d K�(+P1( d , ]
0�(1G#K�B -T=�b P1&)S�D�& b K�K b � J d 0�( / U =�b K b * ( = a M /Jd a S * (1G � Q P1(1G G�P�(�H�(�D�E � B dK d b%* G#A b E b b1=�b1/ &)(1K d a�, K�BT& d a�, 0�G =)< & * "�0�" � 0 * ( = G)P1(�H+(�D d 0�K�S d /Jd ( *I,#* M =7* "+6� < H+H+(16 b P1S * ( = P1&)( O-b1=I, 0 * " P�(1&)BFE b * (1G G�P�(�H�(�D d 0�K�(1U K�P�(1&)(1U+K�B =�b A�&I"�0 d ]K�(+P1( d , 0�(1G#K�B * ( � D d b7=�b G)P1(�H+(�D�E 0�(1G#K�B S+P1( d b'*Jd K , � ( � ) * "+6 � A�&)B d b%� S1K b 0 * B _ ( b1=)< H�(�D�(%P1&)S 5 H+"+K b D d b G)P1(�H+(�D d 0 *)- 6?P�& < C)B d 670�B m��-< > ����1|AM1���� m ,B< > ����j�= �A=1��@;k.Jj6g�< � BRE =�bed / G+0 a (�H�S * BT&)( acbed - A�B d P d (.B =)/Jd b1O�- &)(1G#0 b�b P <1=F* "�0�" ; d b�a�< C)B � = 1 � 2 �;:;:;: 2 F � �
BFE =�bed�* (?0�U = (�H+(?S�H+M = * M = b1=�b1/ &)(1K d a�4N= JL(�H d a�4 =�Q0�G =�b & *I, 0�BRM = � K�B *�b+5 H#" *)4N= 2
F�� = { � �
�| (∀ �� )(∃ � ) � � ( � � �� ��� )} �
acbed 2?B d /Jd a S * BR& b 2 F�1
BRE =�bed ( a H b 0 d a S16 �:? �`m���� VJFHK9I 2 * ( 0�U = (�H+( S�H�M = * M =18>|m �8m3=85�< ?BA b P�S O G+0 d a (1U+6 b & d C)K�(1U+6 T�� c>32*Y, ������Z�5 2 � ��� ��� J�0 *Jd 6 (�H d a�- 6 b1=�b1/ &)(1K d a�- 6c0�G =�b & *I, 0�B d 6 Q BRE =�bed "
* G+A b E b K�BR& d a�, 0�G =)< & * "+0�"�
: N�× F
�� 1 × · · · × F
�� � � N D
acbed " P�& < CI" � BFE =�bed ==hJm��im �-< j|;k. b1= G�P < &)A�B dla�< P�( d b(b1=�b1/ &)(1K d a�, K�BT& d a�, 0�G)]=)< & * "�0�" � ( �� � � 1
�;:;:;:�� � � ) *)-f* ( d b P1(1G��1
�;:;:;: ��� ��∈ F
� � ⇒ �( �� ���
�1
�;:;:;:���� ��
) =�( �� � � 1
�;:;:;: � � � ) �J > !PT QS+P1(1G
F�
=⋃� F
��
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>AQ T x 2 j 2&'�< , ���=��6�%M����2&' ,3/�[ ���z'�6�%;6�'&� � X ������0P��� /|{ 5 X�, %�}3#9��5I /)4 " K�BR& d a�, 0�G =)< & * "�0�" � P1(1G G)P1(�H+(�D�E � B d�* " P1& < CI" � d acb1= (+P1( d BRE * " j,=iA �58.=>E7 1@A=1�� �8m ? �/j|7 � J Zcb Y i Z\Y�b : V�0 i _10 V i d ^ Q
J > ! d Q ��1
= � �1
�;:;:;:���� ��
= � ��∈ F
�
� ⇒ �( �� � � 1
�;:;:;:�� � � ) =�( �� � � 1 �;:;:;:�� � � )
P1(1G BRE =�bed 0�"+K b1=F*Jd a�<�b 0�C)B =)- 0 * BT&I" b P�S * " = b1=)< H�(�D�" 0�G = C ,ia "7J >�>Hd Q D d b P1& < C)B d 60 *Jd 6 b1=�b1/ &)(1K d a�- 67K�BT& d a�- 670�G =�b & *I, 0�B d 6 D d b P b & <1/ B d D�K b 2�"
�( � ) =
{1 � b1= (∀ F ≤ � )[ �
�( F ) = 0] �
⊥ � b H#H d 4 6d acb1= (+P1( d BRE * " = J > !�T Q J acbed G)P1(�H+(�D�E � B d 2 a�< P�M 6 b1O U#0 d acb 2 * " = P�& < CI"
�(� ) = 10 * ( A 4 &)( E Y#Z i V F
�1Q 2 B =)4 2�P�&)( O@b1=)4 6�2 / B = d acb1= (+P1( d BRE * " = J >�>Hd Q acbed�/ B = G�P�(#]
H�(�D�E � B d P1& < CI".0 * (.A 4 &)( P�
3 (1& d 0�K�S16 / "+K d (1G+&ID�BRE *Jd 6 B C , 6 / U+( BR&)M *I, 0�B d 6 � D d b * " = b P�H+(1U+0 * BR&I"cP�Bf]&�E P * MN0�" P1& < C)BTM = � : F
�1 : � N
T�� ! 2�5������������ � � 3 hJm�� m&<9,:g�A=1 ��`m3< ,Eg Z �-< 1\> 65 g = h�m �8m �c<Hj@;k. h ��-C`7�
: F�1 � N
Z >eh�me< 1(1@A=1�� �`mk,l< >:. Z ,Eg �@< >:. j,=iA ��-;k78j@7 h�m3= AM1 ;k76A ==hJm��im ��?��9g�<j�< , ���BAM1 ,Eg(;k7eA J > !�T Q Z >-1 <^h�m3=\A=1\< >-1@A=m hJm < g@? ;k7eA <Hj � = �@. j�=8Ak5J. >:7(1@A=1�� ��im-? �/j@7 � J >�>Hd Q��� 0�( / U =�b K b � S+P1MN67C b./ (1U+K�B T�� N 2�5������������ � � Eh �� �Eg�< Z �-< 1(>c5=g�= h�m �8m �c<Hj@;k. h ��-C`7 �
: F�1 � N
Z1|AM1���� m ,B< >+A J �=j@; �&1@A=1 <Y;=< m8>�� 16;=< > A Q j�=8A=1��@;k7Jj-< 18> A � ∗ : P1 � N;��k;=m < m h�m3=
� ∈ F�1� ⇒ �
(�
) =� ∗(
�);
_ ( 5`b 0 d a S P1BT& d BRA�S1K�B = ( * (1G-p?BTM & , K b+* (16 �'i VFZ `fVF[ ]���Y : _+X���VT] � e�_1VAb � VF[ w acbed* (1G b1=F*Jd P b & b1/ BRE D�K b+* (16 * (1G � i Z\VAw���V ibh P�(1G�C b'/ BRE C)(1G+K�BW0 � b G * S * ( B /)<1O@d ('BFE =�bedS *Jd " b P <1=F* "�0�" BRE =�bed 5=gk;=< >:. acbed 0 *Jd 6 / U#( b G *)- 6 BR&)M *I, 0�B d 6 �-< 1 ==hJm��im �-< j|;����m �c< > ��� h ��-C=g�< � �
: F�1 → N
2�D d b%*Jd 67(+P1(�E BT6� ∈ F
�1� ⇒ �
(�
)↓ �b H+H < J�D�B =Jd a�<�Q 1��6A 7J;=< >E. D d b P�& < C)B d 6 P1(1G b P1( a H�E = (1G = D d b a�< P1( d BT6 *Jd K - 6 * M =K�B *�b+5 H+" *)4 =7* (1G#6 T�� Q,2�� 4����=' 2 � j|; � �
(�
)==hJm��im �-< j|;k.�h ��-C`7�j@;=m �:? �`m
F�1
>c1e< �( � )1|AM1���� m ,B< >E.�,:g �|< >E. j,=iA ��-;k78j@7 h�m3= ;k76A ==hJm��im ��?��9g�< Z
��∈ F
�1� ⇒ �
( ��) =
�( � ) �>-1 <��=j|; � �
= ��∈ F
�1
1|AM1���� m ,B< >E. 18>|m �8m3=85@? 1\;��k;=me< 1�hJm&= �(�
) =�( � ) =
� ∈ N� < 1 >c5=g � ∈ N
Z ==h �� �Eg�</1J>2m��im&=i5 ?H1��� : N → N
;��k;=m <H1 h�m3= �J > Q � ≤ � � ⇒ �
� (�) =
�(�)�
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x�� 2�� � � � �� ��� ��� �������6�'&�� �� ������ ��� > Qod
J�! Q � (�� ) =
�(�
) = � �J N Q � �
�g@?YAM1e< ;0g��-< > ,/7 �kg8A�< >E. Z �i7�� 1��i. = h���;�lg�< > hJm < m�� � ;��k;=me< m��(h�m3=
� � � ⇒ �� (�) = 0
�
j-X Z�<\#3��} [*2 B P1S * ( !#(np?B 4 &I"+K b B =�b1/ &)(1K , 6 Q S c> 2ND d b(a�< C)B ��2 G)P < &)A�B da M /Jd a S16 � b1=�b1/ &)(1K d a�, 6�2�K�BT& d a�, 670�G =)< & * "+0�"+6�2 *)-f* ( d (16'P1(1G� � (
�) =
{ ��(�) � b1= � ≤ � , (∀ � ≤ �
)¬[ � 1( � � � � � ) &�
( � ) = � ] �0 � b H#H d 4 6 �J > ! O Q
S+P1(1G%( � BFE =�bed@a M /Jd a S16 * "�6 � 2 / "#H b1/I, � ( � ) = �� ( � )
p -f* (1G+K�B�� (�) = � � (
�)
acbed BfP b H#"�C)BTU+(1G#K�B /Jd b1/ (1A d a�<c*Jd 6 b P bed * (1U+K�B = BR6 d /Jd S * " * BT6 J > Q m � � (
�)BRE =�bed (�H d a�, 2 b1=�b1/ &)(1K d a�, 0�G =)< & * "+0�" acbed 2?D d b�a�< C)B � ≤ ��2
� � (�) = �
�(�)2 acb+* BRG#C)BFE b1=pb P1S * " = J > ! O Q
JL! Q (∃ � )[ � 1( � � � � � ) &�
( � ) = � ]2 / "+H b1/I, � ( � � ) =
�( � ) = � B = S1A d 2 * S * B
(∀ � )¬[ � 1( � � � � � ) &�
( � ) = � ] Db P � b G * S - P1B *�bed S *Jd 2 D d b a�< C)B � 2 (∀ � ≤ �
)¬[ � 1( � � � � � ) &�
( � ) = � ] < & b� � = �
� 2 b P1S * " = J > ! O Q 2 acbed � ( � ) =�( � ) = � 2 b P�S * " = G�P�S1C)BR0�" S *Jd " � BFE =�bed
F�1
] b1=�b H#H�(�E M * " J N Q ; d bpa�< C)B � ( ��� )[ � 1( � � � � � ) &
�( � ) = � ]
2 � � ( � ) = 02 acb+* BRG#C)BFE b1= b P�S
* " = J > ! O Q a_ ( � , K�K b75 B 5 bed 4N= B d S *Jd ( d ;0g��-< > ,/7 �kg8A�< > ��� J acbed BfP1(1K -T= M 6 b1=�b1/ &)(1K d a�- 6 Qb`a (�H�(1G#C�E BR6?BRK O-b1= E � ( =F*�bed # P�G a�=)<%& 0�B a�< C)B'0�U = (�H�(
�� = { � ∈ F
�1 | � (
�) = � } ( � ∈ N) �
/ "+H b1/I, 2�D d b a�< C)B � ∈ ��2WG)P < &)A�(1G =c* BTH d a�< K�" / B =Jd a�- 6 b`a (�H+(1G+C�E BT6%0 * ( �
�P1(1G�#f0�G#K O M = (1U =�& K�B * " = � 0�B (10�( =)/I, P1( * B K�BTD < H b 2 b &)A d a�< * K , K b+*�b 3 d* BTH d a�< K�" / B =Jd a�- 6 b`a (�H+(1G+C�E BT6 a M /Jd a (+P�( d (1U =F*�bed b P+H < 2�K�B * " = E /Jd b J 5 b 0 d a�<�Qa M /Jd a (+P�(�E "�0�"7P�(1G7A�&I"�0 d K�(+P1( d , 0 b K�B D d b *Jd 6�P�BbP�BR& b 0�K -R= BT6�2�K�BR& d a�- 6 0�G =�b & *I, ]0�B d 6 T�� 9 2 � 7��1� �������� ����� ��E)��� � � �����E)�1� � ��� ���������#����(E17��� � 0 * M
( � � F ) =
{( � ) � � b1= F*G [ e
( � ) �0 � b H#H d 4 6 �
� ( F ) = ( � � F ) :� P1B *�bed S *Jd "c0�G =)< & * "�0�" ( � � F ) BFE =�bed|b1=�b1/ &)(1K d a�, 2 acbed " b`a (�H�(1G#C�E b
0��
1�;:;:;:
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>AQPO x 2 j 2&'�< , ���=��6�%M����2&' ,3/�[ ���z'�6�%;6�'&� � X ������0P��� /|{ 5 X�, %�}3#9��5b P b & d C)K�BRE S�H�BT6 *Jd 6 * BbH d a�< K�" / B =Jd a�- 6 b`a (�H�(1G#C�E BR6�2 -b* 0 d P1(1G
� ( F ) 6= 0 � ⇒ F*G [\e( � ) :
I P1E 0�"+6�2�G)P < &)A�B d P�&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�, 0�G =)< & * "�0�"� ( � ) =
� 11( �� � ) (
S+P1(1G �� (� � � ) = ( � � � )) �J > !P] Q
*)-b* ( d b P�(1G � ( � ) = ( � � � ) = ���
(�)(�) :
T�� T 2 � # � ,9[ �=' J �?��� �14 E#��� �1������������� ����� ����� ��E17�� � ���F�1
�12 <H1^>c5=g= h�m �8m �c<Hj@;k. h ��-C`7 �
: F�1 � N
>-1 <B> 65 g � ∈ F�1
Z 1|A �(�
)↓ Z ;BA ;0g==h �� �Eg�<> hJm < m�� � ∈ NZ ;��k;=me< m���hJm&= �c<H1 > 65 g � ∈ F
�1
Z[�(�
)↓ & (∀ � ≤ �)[�
(�) =
�(�)]] � ⇒ �
(�
) =�(�
) :I d /Jd a S * BT& b 2 b1= " � BRE =�bed (�H d a�, 2 -f* 0 d P�(1G�2 D d b�a�< C)B � ∈ F
�1
2 �(�
)↓ 2 * (0�G#K�P - & b 0�K b P b E & = B d�* " =pb P�H+(1U+0 * BR&I"cK�(1& O�, 2(∀ � ≤ �
)[�
(�) =
�(�)] � ⇒ �
(�
) =�(�
) :j-X Z�<\#3��} [*2�� 0 * M �
( � ) b1=�b1/ &)(1K d a�, 2�K�BR& d a�, 0�G =)< & * "+0�" P1(1G G�P�(�H�(�D�E � B d* " = � (
�)2 -f* 0 d P�(1G
��
= � � ∈ F�1� ⇒ �
( � ) =�( � ) :
m d /)-=bc* "+6 acb+*�b 0 a BRG , 6?BFE =�bed =�bc5 &)(1U+K�B a�< P1( d ( � *)-b* ( d (%P1(1G� � (
�) =
{ �(�) � b1= 0�B ≤ � # 5+, K b+*�b%&�/ B = 0�G#D a H�E = B d " � ( � ) �
� ( � ) �%b H+H d 4 6S+P1(1G7" * BTH d a�< K�" / B =Jd a�, � BfP d H - D�B *�bed JL0�G+K 5`b+*)< K�B * " = P1& 4�* " P1BT&�E P * MN0�"�2 b1=G�P < &)A�B d Q�-b* 0 d P1(1G
�( � )↓ &
�( � ) 6= �
(�
) :B =c* ( acb+*�b1O�- &)(1G#K�B b G * S�2 * S * B C b - A�(1G#K�B � ( � ) =
�(�
)2�K�B * " = ( d a BRE b P d bb P1S / B d CI"�2 acbed|b P � b G * S C b 0�G+K�P�BR& <1= (1G#K�B?S *Jd@b1=
�=( # b & d C)K�S16 5 "�K <+* M =�& 0 * ( = (+P1(�E (.0�G�D a H1E = B d " � ( � ) �
* S * B �kg8A = h���;�lg�< � *)-b* ( d (cP�(1G�( � )↓ &
�( � ) 6= �
(�
) & (∀ � ≤ �)[ � ( � ) =
�(�)] D
b P1S * (?(+P�(�E (�2+K�B * ( � , K�K b T � Q 2+C b 0�G#K�P1BT& <1= (1G+K�BWP�BR& bed *)- &)M S *Jd ��g8A ==h �� �Eg�<1|AM1���� m ,B< >E. 1J>2m��im&=i5 ?H1 � ;��k;=m <H1 h�m3=(∀ � ≤ �
)[�
(�) =
�(�)] &
�(�
)↓ &�(�
) 6= �(�
) �P1(1G%BRE =�bed�* ( � " * (1U#K�B = ( @ B H�BfP * (1K - &)B d BT6
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x�� 2�� � � � �� ��� ��� �������6�'&�� �� ������ ��� > Q ]
B P�S * ( !#(Rp'B 4 &I"�K b B =�b1/ &)(1K , 6�2 D d b a�< C)B � 2�G�P < &)A�B d:a M /Jd a S16 � 2 *)-b* ( d (16P1(1G
� � (�) =
��(�) � b1= � ( � )↓ & (∀ � ≤ �
)¬[ � 1( � � � � � ) &�
( � ) =�( � )] �
�( �� �
)(�) �%b1= � ( � )↓ & (∃ � ≤ �
)[ � 1( � � � � � ) &�
( � ) =�( � )] �
⊥ � b H+H d 4 6�2 / "+H b1/I, b1= � ( � )↑�
J >AN 8 Q
S+P1(1G: _+X 0 ( � � � ) = � � � 1( � � � ��� )BFE =�bed�* (.K ,ia (16 * (1G%G�P�(�H�(�D d 0�K�(1U * "�6 � ( � ) J b1= � ( � )↓ Q acbed
J >AN3> Q �( � � � ) = ( � � )[ � ( � )↓ &
�( � ( � ))↓ &
�( � ) 6= �
( � ( � ))& (∀ � ≤ : _+X 0 ( � � � ))[ � ( � ) = �
�(�)]] :
I /)4 " � ( � ) BFE =�bedcb P1S * " = J > !�] Q 2 -b* 0 d P�(1G ��� ( � ) = �pacbed 2 b P1S * ( � , K�K b Q�@ dΣ1] I P d H+(�D , 6�2�" � ( � � � )
0�G#D a H�E = B d-b`a & d 5�4 6 b1= G)P < &)A�B dca�< P1( d (16 �.*)-b* ( d (16 P1(1G�( � )↓ &
�( � ( � ))↓ &
�( � ) 6= �
( � ( � )) & (∀ � ≤ : _+X 0 ( � � � ))[ � ( � ) = ��(�)] �
acbed 2�S *�b1= 0�G#D a H�E = B d 2�BfP d H - D�B d|a�< P1( d ( � =�( � � � )
K�B b G *)- 6 *Jd 6 d /Jd S * " * BT6 � P1(1C -f* (1G+K�B *)4 & b S *Jd � = �
�∈ F
�1acbed � (
�) =
�( � )↓ J > Q � ( � )↓ >-1 < � ( � ) =
�( � ) b H#H d 4 6 � � = �
�∈ F
�1acbed � ( � ) =
�( � )
2�P1(1GBFE =�bed <+* (+P�( p -f* (1G+K�B
�= : _#X 0 ( � � � ) :J >AN ! Q
JL! Q � g8A = h���;�lg�<W;0g��-< > p,W7 ��g8Ak< >:. j�=8A���@;k7Jj|7 � Z ;��k;=me< 1 hJm&=�( � )↓ &
�( � ( � )) =
�( � ) 6= �
( � ) & (∀ � ≤ : _+X 0 ( � � � ))[ � ( � ) = ��(�)] :
; d b+* E b1= G)P , &)A�B)2 * S * B � ( � � � )↓ acbed " � ( � � � ) - A�B d@b G *I, * " =pd /Jd S * " *�b 2 -b* 0 d P�(1G�2b P1S * " = acb+*�b 0 a BRG , 2 � � = �( �� �
)acbed � ( � ) =
�( � (�( � � � ))) =
�( � ( � �
�)) 6=�
( � )2�P�(1G b1=F*Jd * E C)B *�bed 0 * ( J > Q
J N Q < 1 >c5=g � ∈ F�1
2
[�(�
)↓ & (∀ � ≤ �)[�
(�) =
�(�)]] � ⇒ �
(�
) =�(�
) :B G * S - P�B *�bed *)4 & b b P1S * ( � , K�K b T � Q D d b+* E b1= � (
�) = � 6= �
( � )2 * S * B
G�P < &)A�B d@a�< P�( d bc* BbH d a�< K�" / B =Jd a�," � P�(1G%0�G#K O M = BFE�K�B * " = � J acbed 2�BfP1(1K -T= M 6�2acbed K�B * " = � Q D d b � ≤ ��2 acbed D d b?* " = (+P�(�E b � ( � ) = � 6= �( � )
2#P�(1G b1=F*Jd * E C)B *�bed0 * ( JL! Q aB G * S * ( C)B 4 &I"�K b BRE =�bed " 5 b 0 d a�, b1=�b`a�< H�G ��" acbed 2 P�(�H+H - 6 O (1& - 6�2 acb H+BFE *�bed
* ( # p?B 4 &I"+K b �'i VFZ `fVF[ ]���Y : _+X���VT] � e�_1VAb � VF[ w & 2 , b P1( / E / B *�bed 0 * ( =�� 4 0�( K b CI")]K b+*Jd a S��G VFZ dbZ b P1(1G * ( b P -R/ B d C)B b1= B C < & * " *�b v_ ( S = (1K b 2�S1K�MN6�2 *�bed & d <%� B d P d (P1(�H+U%0 * (.BbP�S1K�B = ( d 0�A�G+&)S * BT&)( b P�( *)- H+BR0�K b
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> 9�8 x 2 j 2&'�< , ���=��6�%M����2&' ,3/�[ ���z'�6�%;6�'&� � X ������0P��� /|{ 5 X�, %�}3#9��5T�� d 2 � # � ,9[ �=' J �'i VFZ `fVF[ ]���Y : _+X���VT] � e�_1VAb � VF[ w 2 �G VFZ dbZ b Q 2� <H1 >c5=g ==hJm��im �
�-< j|;k. h�� Ci7 �: F
�1 � N
==h �� �Eg�< 1|AM1���� m ,B< >+A(j,=iAM1��-;k78jc<H1J>+A � ∗ : P1 � NZ;��k;=me< m hJm&=
[� ∈ F
�1 &
�(�
)↓ ] � ⇒ �(�
) =�(�
) DJ > N�N Q< �J<H1-?Y;0g �`1 Z 1|Ap7 �
: F�1 → N
g@?YAM1e</m��-< >E. Z ;BA ;0g� ∈ F
�1� ⇒ �
(�
) =� ∗(
�) �
acbed ( d BR&)M *I, 0�B d 6gT � ! acbed T�� N - A�(1G = C)B *Jd a�- 6 b P b1=F*I, 0�B d 6'D d b (�H d a�- 6?G�P�(�H�(#]D d 0 *)- 6'P1& < C)B d 6 j-X Z�<\#3��} [*2 � P1MN6�P <1=F*�b 2 - 0 * M � a M /Jd a S16 a�< P1( d b 6 b1=�b1/ &)(1K d a�, 6�2�K�BT& d a�, 60�G =)< & * "�0�"�6 P1(1G%G)P1(�H+(�D�E � B d�* ( � 2 / "#H b1/I,
��∈ F
�1� ⇒ �
( ��) =
�( � ) = �
� ( � ) :
m a &�E 0 d K�" P b & b+*I, &I"+0�"�BRE =�bed S *Jd " b P1S / B d CI" * (1G=p'BRMN& , K b+* (16 e G =)- A�B d b 6\T � TBFE =�bed >c1c;91 j >-g3=J1 j|;=< >E. 2�0�G#D a B a & d K -R=�b S *Jd ( b & d C)K�S16 � 0 * " = J > N ! Q BRE =�bed " *Jd K ,K d b 6 b1=�b1/ &)(1K d a�, 6�2NK�BT& d a�, 6c0�G =)< & * "�0�"�6 � ( � )
2�P1(1G 0�G�D a H1E = B d S *�b1= ( � BFE =�beda M /Jd a S16 b1=�b1/ &)(1K d a�, 6 b`a (�H+(1G+C�E b 6 � � *)-f* ( d b 6�P�(1G � ( ��) =
�( � )↓ $'& < D�K b+*Jd 2" K�BT& d a�, 0�G =)< & * "+0�" � ( � � � )
BRE =�bedWb1=�b1/ &)(1K d a�, 2 M 6.0�G =)< & * "+0�" / U+( K�B *�b+5 H+")]*)4 = 2�K�B * ( = (1& d 0�K�S J >AN3> Q b P�S * " = B a�/ (1A , J ] N Q * (1G !#(1G p'BRMN& , K b+* (16 B =�b ]/ &)(1K , 6 K�B7P b & < K�B * &)(�2�G�P < &)A�B d:a�< P1( d b P1&)M * (�D�B =)4 6 b1=�b1/ &)(1K d a�, � ( � ) *)-b* ( d bP1(1G " J >AN 8 Q d 0�A�U+B d|b1= C - 0�(1G+K�B
� =�( � ) D
acbed 2 * BbH d a�< 2�( b & d C)K�S16 �.P�(1G%A�&)B d b%� S1K b 0 * B?G)P1(�H+(�D�E � B *�bed|b P1S * " =�
= � ( � ) = : _#X 0 ( � � � ( � )) :J >AN�Q QI C)B *)<%� (1G+K�B *)4 & b P < H d1* " =�b P1S / B d CI" * (1G T � T12 =�b / (1U#K�B *Jd K b 6 / E = B d �(� �9? �;k76A ==h�A65 g=j@7 A`;=< � � ∈ F
�1
� 4����=' < 1 >c5=g � Z ,:g � =�( � )Z 1|A
�( � )↓ &
�( � ) =
�( � ) & (∀ � ≤ � ( � )) �
�(�)↓ �;BA ;0g Z �c<H1 > 65 g � 2
[� ∈ F
�1 &
�(�
)↓ & (∀ � ≤ � ( � ))�
(�) = �
�(�)] � ⇒ �
(�
) =�( � ) :n h3A��kg�< Ci7 $'&)(16 <+* (+P1(�2 / BTA�S1K b 0 * B * " = G)P1S1C)BT0�"�D d b�* ( � acbed S *Jd D d b�a�< P�( d b� ∈ F
�1
2�(�
)↓ & (∀ � ≤ � ( � ))[�
(�) = �
�(�) &
�(�
) 6= �( � )] :
B P�S * ( � , K�K b T � Q *)4 & b 2�G�P < &)A�B d@a�< P1( d b%* BTH d a�< K�" / B =Jd a�, � *)-f* ( d b P1(1G�( � ) =
�(�
) 6= �( � ) & (∀ � ≤ � ( � )) � ( � ) = �
�(�) D
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x�� 2�� � � � �� ��� ��� �������6�'&�� �� ������ ��� > 9 >acbed6b P1S * ( = (1& d 0�K�S * "�6 � ( � � � ) acbed1* " = J >AN 8 Q 21" � ( � � � ) - A�B d6b G *I,?* " = d /Jd S * " *�bacbed � � = �
(
� � � ) ∈ F�1 < & b � ( � ) =
�(�( � � � )) 6= �
( � )2�P1(1G b1=F*Jd * E C)B *�bed 0 * " =
G�P�S1C)BR0�" _ ( 0�U = (�H+(
� = { � | � ( � )↓ &�( � ) =
�(�( � )) & (∀ � ≤ � ( � )) �
�(�)↓} �
* M =�b & d C)K 4N= � P1(1G d acb1= (+P�( d (1U =�* " = G)P1S1C)BT0�" * (1G � , K�K b+* (16�BRE =�bed "+K d b1=�b1/ &)(#]K d a S�2 < & b� ∈ � ⇐⇒ (∃ � ) � ( � � � )
D d b a�< P�( d b J P1&)M * (�D�B =)4 6 Q b1=�b1/ &)(1K d a�, 0�A - 0�" � ( � � � ) \_ (.0�G =�b & * "�0 d b`a S
�(� ) = ( � � )[(∀ � ≤ � )� (
�)↓ & � (( � )0
� ( � )1)]BFE =�bed J�BRU a (�H b�Q b1=�b1/ &)(1K d a S � P1B *�bed S *Jd@b1=�b1/ &)(1K d a S BRE =�bed@acbed�* (
�(� ) =
�((�(� ))0)
�
acbed / B = BRE =�bed / U+0 a (�H�(�2 b P1S * ( � , K�K b 2 =�b / BFE C)(1G#K�B *)4 & b S *Jd[� ∈ F
�1 &
�(�
)↓ ] � ⇒ �(�
) =�(�
) �P1(1G%BRE =�bed�* ( � " * (1U#K�B = ( am a�< P�M 6?P�&)(10�B a�*Jd a�, /Jd b+* U)P1MN0�" * (1G p?BTM & , K b+* (16 BFE =�bed2b P b & b E * " * "�2 B C bed ]
* E b 6 * (1G%B C , 6 b1=F*Jd P b & b1/ BFE D�K b+* (16 T�� O 2 � # � ,9[ �=' J � i Z\VAw���V ibh Q 2 lh���;�lg�< ==hJm��im �-< j|;k. h�� Ci7 � : F
�1 � N;��k;=me< 1 hJm&=��
J > Q � (�
0) = 1Z Aeh�m3= Z �
0(�) = 0
Z �c<H1 > 65 g � �JL! Q < 1 >c5=g � Z = h���;�lg�<2> hJm <H1 �
� ∈ F�1
;��k;=me< 1 hJm&=(∀ � ≤ �
)�� (�) = 0
>c1e< �(�� )↑:
� hJgk;91e<*A`;=< 7 h�� Ci7 � �kg8A�g@?YAM1e<Bm hJg �|< m��@<HjJ,9A���j|;=mF�1
>eh�me< m3= J - 0 * M b1=�bed ]*Jd ( a & b+*Jd a�<�Q 1|AM1���� m ,B< >2m&< j,=iAM1��-;k78jc<H1J>2m&<��j-X Z�<\#3��} [*2 m / BRU * BR&I".P1&)S *�b 0�" 0�G =)< D�B *�bedEb P�S * " = P1& 4�* "�2�BbP�B d /I, 2 b1= * (
� ,�*�b1= (cP1BT& d (1& d 0�K�S1670 * ( F�1a�< P1( d (1G b1=�b1/ &)(1K d a (1U�� 2 * S * B�� (
�0) = 1 < & b( G�P�(�H�(�D d 0�K�S16 * "+6 b1=�b1/ &)(1K d a�, 6 K�"+A b1=I, 6 P�(1G G)P1(�H+(�D�E � B d�* " = *Jd K , � (
�0)* BR&)K b+* E � B d@acbed 2�K - A�& d =�bc* BT&)K b+* E 0�B d 2 - A�B d # acb H - 0�B d & P1BfP1BT& b 0�K -T= BR6 * (%P�H , C)(16
*Jd K - 6 * "�6 �0 acbed b1= � BRE =�bed ( K - D d 0 * (16 b & d C)K�S16 D d b * ( = (+P1(�E ( ( G�P�(�H�(#]
D d 0�K�S16 A�&I"�0 d K�(+P1(�E "+0�B * " =%*Jd K , �0(�)2 * S * BI2�P�&)( O@b1=)4 6�2�( G)P1(�H+(�D d 0�K�S16 C b
* BR&)K b+* E 0�B d acbed C b /)4 0�B d�* " =�*Jd K , 1D d b a�< C)B � *)-b* ( d b P�(1G (∀ � ≤ �
)�
(�) = 0
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> 9#! x 2 j 2&'�< , ���=��6�%M����2&' ,3/�[ ���z'�6�%;6�'&� � X ������0P��� /|{ 5 X�, %�}3#9��5; d b =�b#acb+*�b 0 a BTG < 0�(1G+K�B * ( � acbed�=�b�b P�( / BFE C)(1G#K�B * " = P1& 4�* " P�&)S *�b 0�"�2
C -b* (1G#K�B
� ∈ � ⇐⇒ (∀ � ≤ � )[ ��(�) = 0] �J >AN 9 Q
� ∈ � ⇐⇒ (∃ � ∈ � )(∃ � )(∀ � ≤ �)[ �
�(�) = � � (
�) = 0
J >AN T Q& �
�(�
+ 1) = � � (�
+ 1)↓& �
�(�
+ 1) 6= 0]�( � ) = 1 ⇐⇒ � ∈ � ∪ � :J >AN1d Q
m K�BR& d a�, 0�G =)< & * "�0�" � BRE =�bed P1&)( O-b1=)4 6 b1=�b1/ &)(1K d a�, � 4����=' � � g@?YAM1e<
F�1
� 1@A=1�� �8m ? �@;k7 Z �i7�� 1��i.��
= � � ∈ F�1� ⇒ �
( � ) =�( � ) :
n h3A��kg�< Ci7 � 0 * M ��
= � � ∈ F�1
2 acbed � ( � ) = 1 $?& - P1B dW=�b / BFE C)(1G#K�B S *Jd
�( � ) = 1
* # , � X&/|. � [ ( 2 �
�= � � =
0 e�� b G *I, * " = P1BT&�E P * MN0�" � ∈ � 2 < & b�
( � ) = 1
* # , � X&/|. � [ J 2 � � = � � 6= 0acbed � ∈ � m J > N T Q/d 0�A�U#B d K�B � = � 2 � = �acbed � 0 * "cC - 0�" * (1G � 2 < & b � ∈ � acbed � ( � ) = 1
* # , � X&/|. � [ U 2 �
�= � � 6=
0acbed � ∈ � $?& - P1B d *)4 & b =�b G�P < &)A�(1G =
#bK < & * G#&)BR6 & � ∈ � acbed � P1(1G d acb1= (+P1( d (1U = * " = J >AN T Q acbed�5 B 5 bed 4N= (1G = S *Jd� ∈ � b`a & d 5�4 6?( d E /Jd ( d K < & * G#&)BR6 d acb1= (+P1( d (1U =7* " = J > N T Q K�B � 0 * "cC - 0�" * (1G� 2 < & b � ∈ � acbed � ( � ) = 1
m G)P1(�H+(�D d 0 *I, P1& < CI" � : F
�1 � N
P�(1G G�P�(�H�(�D�E � B *�bedlb P�S * " = � P�&)( O@b1=)4 6- A�B d�* " =�d /Jd S * " *�b J > Q 0 * (�p'B 4 &I"�K b ; d b =�b / BFE C)(1G#K�B * " = J�! Q 2�D d bc* ( * G#A b E (��2�C -f* (1G+K�B
� = { � ∈ � | � ≤ �& (∀ � ≤ �
) ��(�) = 0 & �
�(�
+ 1)↓ & ��(�
+ 1) 6= 0} :_ ( 0�U = (�H+( � BFE =�bed P1BfP1BT& b 0�K -T= (�2 acbed J b1= * ( � BFE =�bed b & a B *)< K�BbD < H�( Q S1A da B = S�2 - 0 * M
� = { � 1�;:;:;: � � � } D
C -b* (1G#K�B
�� (�) =
{0 � b1= � ≤ � , � �
+ 1 �max{ �
�1(�
+ 1) �;:;:;: ����� (�
+ 1)} + 1 � b1= � =�
+ 1 �-b* 0 d P1(1G12 b 0 O@b H 4 6�2 (∀ � ≤ �
)�� (�) = 0
2 acbed:b & a BRE =�b ( / "#D�"�C)(1U#K�B 0�B <+* (+P�(b P1S * " = G)P1S1C)BT0�"cS *Jd G)P < &)A�B d � *)-b* ( d (16 P1(1G��
=�� & � ∈ � ∪ � :
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x�� 2�� � � � �� ��� ��� �������6�'&�� �� ������ ��� > 9 N* # , � X&/|. � [ ( 2 � � =
��acbed � ∈ � \B P�S * ( = (1& d 0�K�S * (1G � 2
� ∈ � � ⇒ (∀ � ≤ � )[ ��(�) = 0]
� ⇒ � ≤ � b1O (1U � � ( � + 1) =�� (�
+ 1) 6= 0� ⇒ � ∈ �acbed�* ( * BTH+BRG *�b E (.BRE =�bed�<+* (+P1(�2�BbP�B d /I, �
� (�
+ 1) 6= ��(�
+ 1)D d b a�< C)B � ∈ �
* # , � X&/|. � [ J 2 ��
=��acbed � ∈ � �_ 4 & b G�P < &)A�(1G = K < & * G#&)BR6 � ∈ �acbed � ′ P�(1G 5 B 5`bed 4 = (1G = S *Jd � ∈ � 2 acbed � ′ =
�+ 1
2 b1O (1U (∀ � ≤ �)�� (�) = 0acbed �
� (�
+ 1) 6= 0 8I P1E 0�"+6�2 � ∈ � 2 b1O (1U � ∈ � acbed�� � (
�+ 1) 6= 0
B G *)<S1K�M 6�( / "+D�(1U = 0�B <+* (+P1(�21S+P�M 6 acbed 0 * " = P�& 4W* "?P�BR&�E P * M 0�"�2 b1O (1U �
� (�+1) 6=
� � (�
+ 1)2�D d b a�< C)B � ∈ � a
��� � �p�Byl~:�*$e}����T � c>&2 B 67BRE =�bed " � ( � ) b1=�b1/ &)(1K d a�, K�BR& d a�, 0�G =)< & * "+0�"�2 *)-f* ( d b P1(1G
(∀ � )[ ��( � ) = 0] � ⇒ �
( � ) = 3 D/ BFE C * B?S *Jd D d b a�< C)B ��2�G�P < &)A�B d@a�< P1( d (16 � 2 *)-f* ( d (16'P1(1GJ > Q (∀ � )[ � � ( � )↓ ]
J�! Q (∀ � ≤ �
)[ � � ( � ) = 0]
J N Q (∃ � )[ � � ( � ) 6= 0]
J Q Q � ( � ) = 3
��T � ! 2 @ H+"+C)BRU#B d�, S1A d "�B C , 6WP1&)S *�b 0�" B 6NBRE =�bed " � ( � ) b1=�b1/ &)(1K d a�, K�BT& d a�,0�G =)< & * "�0�"�2 *)-b* ( d b P�(1G(∀ � )[ �
�( � ) = 0] � ⇒ �
( � ) ↓ DD d b a�< C)B ��2�G)P < &)A�B d|a�< P�( d (16 � 2 *)-b* ( d (16'P�(1GJ > Q (∀ � )[ � � ( � ) ↓] J�! Q (∀ � ≤ �
)[ � � ( � ) = 0]
J N Q (∃ � )[ � � ( � ) 6= 0]
J Q Q � ( � ) ↓ J B P1( / BRE C * B * " =pb P <1=F* "�0 , 0 b 6 Q
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