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['R��6E7MPR?] ['OWOFG4$UB<"�� ��h2!G!dKR���!�2
|= AX6VHp^�
M |= AGPDWR�� �?DFE � ['Z6MPZ�E7DWU�E7R)q}�?DFE Q6R�G DWR>U�E7M�EHG�GPDFE7M�E7R)� J�S~O ��LNX/E7R��
Mgr |= AZ�E7D�c~Urq���[DcA['Q�G �6E7R w�LNMP{�E7OWR DWR
Mgr
]�LNOWS~U�3:Q6RT['X6Z�E'R6S~DWS�kmLNR E7DWR�E7{ �mLNR6�#M�E7U�E7RIX6VHp^�
I, βq
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� v R6R>U�E^['Q�jIZ u EHG�E7UPVrE�]��6MC ['U�E7R�GPUPMPQ6�#UPQ6M�E7R:E7R>UPZT['OWU�E7R)q�V�� �,�
∀x∀s first(prefix (x, s)) = x�
{S|~}(���Z�3|F�Z���Z�,�5�q��| �z�q���5��|��8���~�(�L�L�q�Y�8�q�Z|T� ��}5�5�������q|L������}5|~�����T���5��|
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Ek[AV�h2!�+FE�V
⇒ A"�� ��hU!G!dK ���
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'�H�
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⇒ AB�ES(�H2+ V�hU!o+FE�V
�
EGEk[AV�h2!�+FE�VXW
|= A[⇔ ∅ |= A
� 'Zt�'c!BA � DWD �∅ |= A ⇔
⇔
⇔ AS~�6OWUPDWS�� 2
2n$%&(p �/1 u Ya"�EA ∈ For
$UBQ'M ⊆ For �
�#��B�B V�E !o+M |= A
V2"�BQ��$b' ��B�B�
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]��6M}['OWOFEI, β : I, β |= M ⇒ I, β 6|= ¬A
���dM�LN�/L~GPDWUPDFLNR � � � D �P�S�E7RT['Q:��['R6R�p E7R6RM ∪ {¬A}
DWG�U�Q6R�E7M�]4�6OWOWXT['Mr�2
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]��6M¬A
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I, β['OFG ��E!OFE7So��[�]4�6M;�!q�G�LNRTGPU DFGPUKEHGKS~�6OWUPDWS �4p E7R6R �6DWEsJ�E7M�GPQ�jIZ�E
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� � ���/�-� � �� ���� �D���
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P = P0 = {p0, p1, p2, ...} � Q6R�� � = � DFGPU k�E7MPX/LNU�E7R)��edG UPM�E7U�E7R�DWR�w�LNM�{�E!OWR['OFG�L*�mE7DWR�E�cxE7MP{�E ['Q?]�q#�6E7{�RT[NjIZ�['Q�jIZ �mE7DWR�E�J ['MPDt['X6OFE7R)�PEG%�+ �6E7{�RT[NjIZ��?DFE�h:E7R6S�E,�6E7M�['U�LN{l['M�E7R
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I |= AS�E7RT['Q:��['R6R�p E7R6R
I, β |= A�
-&� e���S~DWX6U^EHG,�mE7DWR�E�J�['MPDt['X6OFE7R)q)��E7OFE7S~Q6R6S�E7R)q∀xq Q6R��
F = ∅ EHG,S~DWX6U ['OFG�LKR>Q6M�
P0,→,¬�a�iR>U�E7M�g
�6M�E7UI['UPDFLNR�ZT['X/E7R:�6E7{�RT[NjIZ:�?DFE^w�LNMP{I : P → {T, F}�YDWMd� v R�R�E7Ra['OWOFE.� R>U�E7MP�6M�E7UI['UPDFLNR�E7R �4X6VHp^�~DWZ6M�E7RlM�E7OFE7kN['R>U�E7RacxE7DWO � {�DWUPU�E7OFG��f�DH&��H;"�EF+ %�+,� ��"�!��?Q6M�jIZ?g
�6M�LNX6DFE7M�E7R��S�E7RT['Q ��['R6R)q p�E7R6R ['OWOFE�e�DWR>UPM�E'S�E�E7DWR�E7M^wTLNMP{�E7OTGPDWR���q/DFG�U �?DFE�wTLNMP{�E7O�S~�6OWUPDWS�� u ��O g
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A ≡ p → (q → p)
B ≡ (¬p → ¬q) → (q → p)����LNR>UPMI['�/L~GPDWUPDFLNR �
C ≡ (¬p → ¬q) → (p → q)
p q q → p A ¬p → ¬q B p → q C
T T T T T T T T
T F T T T T F F
F T F T F T T T
F F T T T T T T
AQ6R��
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2
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� 'c!BA5w,!#'*+#'-� � �CDWR���E7DFGP�6DFE7O7;?� � DFGPU�E7M�]��6OWOWXT['Mr�
;?��� E7D#k>DFE7OFE7R&b�U�LN{�E7R �n��DFGPUy�?DFEHGxJdE7M�] ['Z6M�E7R&GIE7Z6M ['Q?]4p E7R��?DWS��
2n k/E7DWOFE7R ��'E7k#UPO � � S�E7VHDFE7OWU�E��#Q�jIZ�E � qV�� �,���� E7DW{ b}R>UPM;['S�['Q?] ��[NG,JdLNM��6D_��OWLN{ GPU�E7Z�E7R E7DWR p E7D�c�EHGrq�E7DWR M�LNU�EHG,Q6R�� E7DWR X6Ot['Q�EHG wTLNMP{*Q6Ot['M VHQ6M
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w → ¬b�S�iR � LNMPU�E7R��6� E7M�E7DWR�p E7D�c�EHG}wTLNMP{*Q6Ot['M}E7DWR6M�E7DFjIZ>Urq ��['M�] ��E7DWR�X6Ot['Q�EHG}E7DWR6M�E7DFjIZ�E7R)� �
X �w ∨ rj �r → b ∧ w
b}R6S�E7R�LN{�{�E7Rr = T
q6�T['R6Rb}OFG�L
r = Fq6��['{�DWU�j �
Xp E7S�E7Rp E7S�E7Re�MPS~DWX6U��?DFE E7DWR6VHDWS�E E7M�]��6OWOFE7R��6E4� R>U�E7MP�6M�E7UI['UPDFLNR)��iR:�6E7M�b}Q�GIGI['S�E7R6OFLNS~DW�KZ�E7D�c~U �
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p q r q → rOWDWR6�#G
p ∧ qM�EHjIZ>U�G
� � �T T F F F T F
T F T T T F T� � �� � �
F T F F T F T� � �C,[��?DFE*�>�T['OWU�E7R�VHQ�OWDWR6�?G�Q6R�� M�EHjIZ>U�G�S~OFE7DFjIZ�GPDWR���q?]�LNOWS~U}�?DFE � �mQ6DWk~['OFE7R6V��
2
A ≡ p → (q → p)['Q�G���E7DFGP�6DFE7OD;?� � DFGPU�R#Q6M�E7DWR6VrE7OWR�E�wTLNMP{�E7O, � RT['UP�6M�O_DFjIZ � S~DWOWUl['Q�jIZ
¬p →
(q ∧ r → ¬p)�6b�OWOWS�E7{�E7DWR��
2n$%&(py>n1�G Ya"�EB ∈ For
V�hU!o+FE�V��fE +<`�+,��� "�Bp0, p1, ..., pn
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B �
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'�$2� (�HAi
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B′V�h2!�+FE�V
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� 'Zt�'c!BA � � E7MPU�E7R p�DWMB′
Q6R>U�E7M*E7DWR�E7M�� R>U�E7MP�6M�E7UI['UPDFLNRI ′['Q�Grq)X/EHGPUPDW{�{�E7Rzp�DWMrq�LNX
I ′ |= Ai
L>�6E7MR6DFjIZ>Ur�
I ′ |= B′Z�E'R6S~U�R6DFjIZ>U�kmLNR
Ai
q�G�LNR��6E7MPR R>Q6M�kmLNR �?DFEHG�E7Roe�MPS�E7X6R6DFG�G�E7RY['X)q��?DFE {�DWU→
Q6R��¬�mLN{*X6DWR6DFE7MPU�p E7M��6E7R)�/C�� Z)�
I ′ |= B′S�E7RT['Q ��['R6R)q p E7R6R
I |= BqTp LNX/E7D
pIi = T
S�E7RT['Qs��['R6Rsp E7R6RI ′ |= Ai
�C,[
I |= BDW{�{�E7M}S~DWOWUrq6DFGPU
B′S~��O_U�D_S��
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p ∈ P0 �?Q6M�jIZ A ∈ For�
� 'c!BA5w,!#'*+�� edM�G�E7UPVrEp�?Q6M�jIZ
p0 → p1DWR
(p → p0) → ¬p�
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G�E7DB′
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D �B ≡ q ∈ P
�
[��q ≡ p : B′ :≡
X �q 6≡ p : B′ :≡
DWD �B ≡ ¬B1 : B′ :≡ ¬(B′
1)
DWDWD �B ≡ B1 → B2 : B′ :≡
�#E7DI ′X/E7OWDFE7X6DWS�E#� R>U�E7MP�6M�E7UI['UPDFLNR��p�DWM}�6E���R6DFE7M�E7R
Iq?DWR��6E7{�p�DWM�]��6M}['OWOFE
q ∈ PG�E7UPVrE7R��
qI :=
qI′ p E7R6Rq 6≡ p
Tp E7R6R
q ≡ pQ6R��
I ′ |= A
FG�LNR�GPU
� E7Z)� �6wy�6M�['OWOFEB ∈ For
S~DWOWU�I |= B ⇔ I ′ |= B′
� E7p E7DFG �6�?Q�M�jIZ�GPUPMPQ6�#UPQ6M�E7OWOFE4� R��?Q6�>UPDFLNR��6X/E7MB
D � [��B ≡ p : I |= B ⇔ pI = T ⇐⇒ ������� I � . . . ⇐⇒ ��������� ������������������ I ′ |= B′
X �B ≡ q 6≡ p : I |= B ⇔ qI = T ⇔ . . . ⇔ I ′ |= B′
DWD �B ≡ ¬B1 : I |= B ⇔ I 6|= B1 . . . ⇔ I ′ |= ¬(B′
1) ⇔ I ′ |= B′
DWDWD � ['RT['OFLNS�]��6M→
8�E7UPVHU�GIE7Z�E7R p�DWM�?wy�6M�['OWOFEI ′S~DWOWU
I ′ |= B′S�E7RT['Q ��['R6R)q?p E7R6R ]4�6M�['OWOWE
I ′S~DWOWU
I |= B�;� GPU�['OFG�L
BS~�6OWUPDWS�qTG�L�['Q�jIZB′�
2
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�A → (B → A)
b}RT['OFLNS VHQ ;?�<=�]�LNOWS~U�['QTGB1 =||= B2
['Q�jIZB′
1 =||= B′2
�yn�[NjIZ���E7DFGP�6DFE7O ;?�<;:DFGPU�['OFG�L�]4�6M�['OWOFEA, B, C ∈ For
�A → (B → C) =||= A ∧ B → C
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; b.>,�6�#b u edn�`/- u � � �@
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T F T T T
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2
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2
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