Лекции по математической логике и теории алгоритмов. Часть 1

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'# F (p1 : : : pn )  A1 : : : An { ) . ( '#"   F $"' A1 : : : An " "+ p1 : : : pn   ,   $"', &" F (A1 : : : An),     '%*"  ": (1)  F (p1 : : : pn ) = pi ,  F (A1 : : : An) = Ai  (2)  F (p1 : : : pn ) = (),  F (A1 : : : An) = () (3)  F (p1 : : : pn) = :G(p1 : : : pn)  G(A1 : : : An)  ,  F (A1 : : : An) = :G(A1 : : : An) (4)  F (p1 : : : pn) = G(p1 : : : pn)  H (p1 : : : pn),   # & ^, & _, & ;!,  G(A1 : : : An)  H (A1 : : : An) '-  ,  F (A1 : : : An) = G(A1 : : : An)  H (A1 : : : An). ! 2.2.  q1 : : : qn {      ,     A1 : : : An  1 : : : n 2 B .  ,  Ai (1 : : : n) = i 2 B .  2.1.

F (A1 (1 : : : n) : : : An(1 : : : n)) = F (1 : : : n):

#  "- &#       2.1  "* '!  % $"'. &  1.  F (p1 : : : pn)  G(p1 : : : pn),  A1 : : : An { ,  F (A1 : : : An)  G(A1 : : : An). &  2.  A1 : : : An B1 : : : Bn { ,   Ai  Bi i = 1 : : : n,  F (A1 : : : An)  F (B1 : : : Bn). 2  1 &  #  " 1  +  (2.1)  (2.2)   (2.2)  (2.3),  - " 2  +  (2.4)  (2.5)   (2.7)  (2.8). 2  2  # # " 2  +  (2.5)  (2.6).

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- P (x) { " " x &   " P " { 0 "  .  "  ,  " "  " " x  "- &# & ", & -".  ",5  P (x) { "x {  ",  P (2) =  P (6) =  P (224 + 1) =  P (22 + 1) =  (0   '  . /). ,    " &# M ,  "   P (x) "- " #  $'!%  M "-  B = f g. ( $'!   #  .  "'  #   #+ "+. "/ n n-"" "  "-  M   &- P : M ;! B . ,-  M   " &#%,  ", "%*  M , { "" "". ,-  P = f< x1 : : : xn >2 M n j P (x1 : : : xn ) = g   &#%   P . "" ,   %& C  M n '* ' n-"  P , ,  P = C ,  ", P (x1 : : : xn ) =  ()< x1 : : : xn >2 C: 2.

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2.5.

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'# P (x) 1  x ^ x  3 Q(y) 2  y ^ y  3 { ,    R+. 6""  R(x y) = P (x) _ Q(y)  S (x y) = P (x) ^ Q(y).   P 0 '  P & " $  " y,   Q0 '  Q & " $  " x. ( P = P R+ (.1), Q = R+ Q (.2). ( # R = P Q  S = P \ Q &-  '+ 3  4   . "/ 2.6. '# P (x0 x1 : : : xn) { (n + 1)-"   "-  M .  ,  n-"  Q(x1 : : : xn ) '  P ""   '*   ( &*)  " x0  1' Q(x1 : : : xn) = 9x0 P (x0 x1 : : : xn) (Q(x1 : : : xn ) = 8x0P (x0 x1 : : : xn ))   %&+ u1 : : : un 2 M Q(u1 : : : un) =    # ,    ( %&) u 2 M ,  (  ) P (u u1 : : : un ) =  1.

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 Q(x1 : : : xn) = 9x0 P (x0 x1 : : : xn))  &#  Q  M n  Q     &   P  M n+1  P . /   '  &" ""  " ! ',  M = R { ". . 5. ! 2. '# P (x y) y = x2,  P  R2 # & &   XOY .  !  # OY # -# '# (.6), .. 9x(y = x2 ) = y  0,  "- ' #   '+ &- { *   y    "  *   x   # ,   !#. ! &  # OX     #%. % ',  9y(y = x2 )  . / -  , #' %&  x "-    . ! 3. 6"" +"  P (x y z) x + z = y  R+ .  &#  # #  x + z = y ,  - -#"    OXY Z (.7). ! 0    -#   c XOY &-  .8.   ,  9z (x+z = y) = y  x,     '+ &-: y  x   # ,  y ; x  0. ."",  # '*   #'  $,  " &#   !#+ .   - "  "#  " R,   &#    & '- % # x + z = y,   ! #  OZ     #% XOY , .. 9z (x + z = y)  ,  -  , #' 0" '   z  "- # y ; x.   &*  "   "  !,  "-'  " &*  '*   "   #,  %*, ,  # &   , '+ ""   &*. $ 2.7.  P (x) {  ,       x ,   ,    - !  .  "   

:8xP (x) = 9x:P (x) :9xP (x) = 8x:P (x)

(1)

  . '#   #      . ( 8xP (x) =  ,   %   &*,  0" m 2 M , ,  P (m) = . .# M { " &#  P . ( :P (m) =  ,   %   '*  , 9x:P (x) = .    #      ,  '* ' m 2 M , ,  :P (m) = . / ,  P (m) =     %   &* 8xP (x) = , .. :8xP (x) = . (" ""      17

.     "- &#      $"#. .""  "     P  :P . (     !, '" :8x:P (x) = 9xP (x) ( #, " !  &" "         !, '"     &'"+    ! 0" #  " %'  ,   #+ "+, +*+  P (" " x), $ . 6   (1)  %,    &* "- &# -    '*    &. .

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8xP (x) = :9x:P (x) 9xP (x) = :8x:P (x)

' & " &    (1) "- " #  &    ,. "" ,  " &# , .. M = fa1 a2 : : : ang,  8xP (x) = P (a1 ) ^ P (a2 ) ^    ^ P (an ) 9xP (x) = P (a1 ) _ P (a2 ) _    _ P (an )     (1) +    :(P (a1 ) ^ P (a2 ) ^    ^ P (an )) = :P (a1 ) _ :P (a2 ) _    _ :P (an ) :(P (a1 ) _ P (a2 ) _    _ P (an )) = :P (a1 ) ^ :P (a2 ) ^    ^ :P (an )   % & &   ,. !  &#   8y(x2 ; y2  1) ^ 9x(x2 + y2  1):    +*+ 3%!%    #  " x.  &#  #    &   :8y(x2 ; y2  1) = 9y:(x2 ; y2  1) = 9y(x2 ; y2 > 1): &#   x2 ; y2 > 1 1+   .9.     !  # OX #  ;1  x  1. /  # &#   8y(x2 ; y2  1). &#   9x(x2 + y2  1) # !  # OY ' x2 + y2  1 (.10), ..  ;1  y  1. ( #, '- -,  " 1, ",  &#  1  #   ;1 1] ;1 1] (.11). 4.

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(1) QxP (x) = QtP (t),    t       P  (2) :8xP (x) = 9x:P (x) :9xP (x) = 8x:P (x) (3) 9x(P (x) _ Q(x)) = 9xP (x) _ 9xP (x) 8x(P (x) ^ Q(x)) = 8xP (x) ^ 8xQ(x) (4) 9x(P (x) ^ R) = 9xP (x) ^ R 8x(P (x) _ R) = 8xP (x) _ R,  R      x (5) QxQyP (x y) = QyQxP (x y)   . 6   (1)  (5)    '%      . 6   (2)  " 2.6. -"      (3).  9x(P (x) _ Q(x)) = ,  '* ' m 2 M ,  M { " &#, ,  P (m) _ Q(m) = , . . P (m) =   Q(m) =  (  1 ,   #+ "+, +*+  P  Q $ ).  " ' 9xP (x) = ,   #  9xP (x) _ 9xP (x) = .  ' "  . '#, &, 9xP (x) _ 9xP (x) = . / ,  & 9xP (x) = , & 9xQ(x) = .  " ' '* '  m 2 M ,  '   ' % P (m) = ,   #  P (m) _ Q(m) = , .. 9x(P (x) _ Q(x)) = .  ' "  .     (3) "  " $"#   ,  "',  0 &  #  " 2.6. .""  "    (3)  P  :P ,  Q {  :Q  ""  &" " '    !. '"   

:9x(:P (x) _ :Q(x)) = :(9x:P (x) _ 9x:P (x))     '    ""     (2),     {   ,. '"    8x:(:P (x) _ :Q(x)) = :9x:P (x) ^ :9x:Q(x) "     '      ,     !,   -"'   3%!    { 22

    (2)     !, '" &'"   . -"  #      (4). '#    # ,  '* '  m 2 M ,  P (m) ^ R = , .. P (m) =   R = . ( # ,  9xP (x) = , ..     #    (4). &,     # (4),  9xP (x) =   R = . 2 #, '* '  m 2 M ,  P (m) = ,    P (m) ^ R = , #' R     m. % ',     #    (4).     (4) $"#       "',  0 &     (3)  & .    R      x,  QxR = R. ! 6   (1)  ,  " x - QxP (x)  '% - #   '""   "  .    ",  x "- &# " #  "'%,  +*'% P '* .  ", 9x(y < x)  # 9t(y < t),    # 9y(y < y).   -,  <  '  1     '" "- ,  "    ,  ",  + ' + "- +,  "%*+ &#1 0".    - &# R2   # "  : f (x t)dx # $'!, .

1

2 R  *  x,  "  f (t t)dt #  . 1

 #,    '*      3%!%,   &* {  3%!%  "   # "# "". ("  " " " ! 3.2. #"   $  : (1) 9x(P (x) ^ Q(x)) ;! 9xP (x) ^ 9xQ(x)   (2) (8xP (x) _ 8xQ(x)) ;! 8x(P (x) _ Q(x))   (3) 9x8yP (x y) ;! 8y9xP (x y) =       " ! (1) ,  '* '  m 2 M ,  P (m) ^ Q(m) = , .. P (m) =   Q(m) = , .. 9xP (x) =   9xQ(x) = . % ',   % " ! (1) . " !% (2) "- # ,  "-   $"#  " ! (1) (  ' -!).   " ! (3) ,  '* '  m 2 M ,  8yP (m y) = . (   # n 2 M P (m n) = ,     # n 2 M 9xP (x n) = ,  0  ,  8y9xP (x y) =   .

23

&  (1) { (3) " !   ,  ,   , -. "" ,  ", ,   " ! (1)  '*    m, &*%* '  P ,   Q.  "   %  '*    m, &*%* ' P   n, "-   m,  &* ' Q. '#,  ", M = N P (x) x = 2 Q(x) x = 3. (  ,  9xP (x) ^ 9xQ(x)) ;! 9x(P (x) ^ Q(x)) =  4& # -# " !, &  (2)  " M  P -,   1,  Q(x) x =6 2. (  ,  (8x(P (x) _ Q(x))) ;! (8xP (x) _ 8xQ(x)) = 

,  #   P "- # %&  -   ,    Q {  !.  , & #  "  " !, &  (3), "",   (3) ,  '* '  0" x,  &* P '  %&" y,   % {   - y '* ', &*  ,   0" x, &*%* P '. '#   M = N,  P (x y) y < x. (  , 

8y9xP (x y) ;! 9x8yP (x y) =  #' N  - 0" " &#1,   &#1 0". ("  "    " !, &  (3). 4&  "# "     .  "#     , %& n-"  "- - #   &#% ,  %* & "-  M n,  M { " &#. , "  - %   #" "- " M .  0" " A(x) 1' x 2 A   #'% *": (9x 2 A) B (8x 2 A) B

9x(x 2 A ^ B) 8x(x 2 A ;! B)

'!     #  "," "- " A, , , "  ". 24

!

   3.1   3.1     ,     - !                 . #  0 - "-   $"#,  #'   +   , -  " 3.1  - 3.2 (' -). ! 3.4.  P (x y) {  ,     x y ,   ,    - !  .  3.3.

(8x 2 A)(9y 2 B ) P (x y) = (9f 2 B A )(8x 2 A) P (x f (x))   . (&' # # " ! A ;! B  B ;! A,  A  B {        "      . " ! ;  :  -" $ " x   y ' # f (x).  #  " ! ;! "", ,    ,   -" $ " a 2 A "-  Ba = fb 2 B j P (a b) = g 6= . &"  - "-  Ba  0"' ba   " f : A ;! B , -  %& a 2 A f (a) = ba . ( ,   %&" a 2 A P (a f (a)) = , .. (8x 2 A) P (x f (x)) = .

(" &"  %  " !   ."",   #  - 3.4  # #   .  #,   ""  - 3.4 (, " ! ;!)  # " &. )"' ' " &   & &'- "-  6].

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8" > 0 9 > 0 8x(jx ; aj <  ;! jf (x) ; bj < ") 25

(4.1)

  # " +" $"' # - xlim !a f (x) 6= b, .. ! (4.1), ,     - " (2) " 3.1  -" 3.3, 0 !  1  9" > 0 8 > 0 9x:(jx ; aj <  ;! jf (x) ; bj < ") (4.2) ( #  #  1# - "    :(p ;! q)  p ^ :q  ",  : (jf (x) ; bj < ")  jf (x) ; bj  ", '" #'% $"' ' ' - xlim 6 b: !a f (x) =

9" > 0 8 > 0 9x (jx ; aj <  ^ jf (x) ; bj  ") 2 .   # (1) " a { y,  " b ; f (y), '"      $'! f  y. 2 #   # $'! f   &    1 : 8y8" > 0 9 > 0 8x(jx ; yj <  ;! jf (x) ; f (y)j < ") (4.3)  "",   "   # f   '%*" &": 8" > 0 9 > 0 8y8x(jx ; aj <  ;! jf (x) ; bj < ") (4.4)

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' (4.5), -     0    # -% 9x(9A 2 A(x 2 A) ^ y = f (x)) (4.7) 27

#'  y = f (x)  - " A   9A 2 A "-   1 &  "# ""  ""  " 9x. #' 9x(x 2 A ^ y = f (x)  y 2 f (A)  (4.7)  # -%  9A 2 A(y 2 f (A))  y 2 f (A) A2A

    (4.6). 4 . '#   f : X ;! Y . ( ' -  ",  f { %3   1  8y 2 Y 9x 2 X (y = f (x)). ' - 3.4  -  # '%*"': 9g 2 X Y 8x 2 X (y = f (g(y))).  - 8x 2 X (y = f (g(y))) ,  f  g = idX , ..,  g {   &  f &-. (" ""    "  ",  &- %3    # ,   "   &. 5 . . 1" $"#  '% "'  ",  -  ' "-  N " "#1 0": 8A(A =6 ;! 9x(x 2 A ^ 8y(y 2 A ;! x  y))) (4.8) #' (4.8)      %& "-  A,  " "- "# A  :A. "   #'"  " - "    p ;! q  :q ;! :p (4.9) . '# (4.8)  # '%*"' -%: 8A(:9x(x 2 A ^ 8y(y 2 A ;! x  y)) ;! :A =6 ) (4.10) &'" ' " ! (4.10),  #  1# '%*"   "" - "   : :(p ^ q)  q ;! :p '",  (4.10)  # 8A(8x(8y(y 2 A ;! x  y) ;! :x 2 A) ;! :A =6 ) (4.11)   #'" '%*"  " 0  ": :A =6

 A = N, :x 2 A  x 2 A  x  y  y < x,  - ""  "!, *   " 8y -  (4.9). '# '",  (4.11)  # 8A(8x(8y(y < x ;! y 2 A) ;! x 2 A) ;! A = N) (4.12) 28

,  (4.12) #   ,  ! "" '! '%* $".

" #       x   ,   , $ x,   A ,  x   A,       A.* #' " ,  (4.8)  # (4.12),  " "" ,  ' -  ",    ' "-  '#  " "#1 0",  # ! ' "" '!. ."",  0+ '-+ "   # #  !$ "-  N. (" &" 1 "  '% '%* &&*.  "",   '  "-     ' ",  -   ' "-  " "#1 0".  ,  $"#   (4.12)  (4.8)  ",  "# N   #  '  "- . (" ""

 #"  ' " "-  M     '%* ' -:

8A  (8x(8y(y < x ;! y 2 A) ;! x 2 A) ;! A = ) / ' -   ! " $ '!. 2   " 8", - "-  "- &#  ' . (" &" ! '! "- "#  #  + ' -  # # + "- ,   #  #+ &+ "- . 2'  "# ',  " 8"   0$$  {   # " &. , +   ',  "-  ' # "-  R,   #  + &   '     "-". / '*    "- " !  $ '!  + "- . 7 & &'- " 8"  '+ ' -  #+ " & "-  6]. M

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29

&#  + + ,   '+ -. 2 '  "% -,   (-)  "  ,    + +. 4&  # 0' '!% """ "" &+" # $"#    -,  '%    "* +      ,  - { $"#     - + -.  0" ,  $"#"   - - '  # 1# &   ,      0" &" - &#       .  '* ,   0" $"' '*  & &  -, " ,    '% $"'"   . -, " +#   ", "",  "" &     * &  $'!. 6'", - n;" $'! f (x1 : : : xn)   ' n + 1-"  P (x1 : : : xn xn+1) f (x1 : : : xn) = xn+1. ,   0"  "   #,    & $'!%,  0    '* " '-" $"#+ -.  ",  $,   P (x y) =  () y = f (x)   $'! f    : 8x 9yP (x y) ^ 8y8y0(P (x y) ^ P (x y0 ) ;! y = y0 )] :

- *  + &+  &# 8!yP (x y)  # "'* '   "   y, ,  P (x y). & # -   #  + "  " $'!#+ %,  "%   -",   $'!  %   '" '.  " &,  ",  " $'!#+ "  +    #'% +"  "  S (x y z )  (x y z )   . ( - 9x(x2 + px + q = 0) - &#   '%*" &": 9x8u8v8w8t((x x u) ^ (p x v) ^ S (u v w) ^ S (w q t) ;! t = 0) 2' "# ',    $"#" + $'!   %  f (x y), .. " x + y  x  y ' # +(x y)  (x y)   , . " 1 -  1  9x +(+((x x) (x p)) q) = 0] : -, , ",  ' '% #, &'" # # &*  $"  , #',   , 0     '"". 30

+"   % "-  S  + "     . "/ 5.1. ,-  S  + "     % & '%* &3 (1) C & "  x1 : : : xn : : :  & "+ (    "+ &'' ' &#  ' &' ). (2)  "-  C "   & . (3) ,-  $'!#+ "  F  $ " &-" ' : F ;! N n f0g.  0" '(f ) = n ,  f { "  $'!  n "+ (n-" $'!). (4) ,-  +  B = f g. (5) ,-  + "  P  $ " &-"  : P ;! N nf0g.  0"  # (p) = n ,  p { n-"  " . (6)  " , &    : ^ _ : ;! 8 9 = ( ) , (  =< C F P >   '    .  '  & L . ,*#% '  { jj   "*# "-  C F P . !# 1.  "  (0" "-  C ) "- " #  0-" $'!# " . ( -      "- " #  0-"  " . 2.  "",   $  ,  "-   " ,   &,  "-   "  "* ""' $ '. #'  " - $"'  L  % &    "  "   $  S ,  "*# "-  + $"'  L   + maxf@0 jjg. "/ 5.2 ($ 1.

L). (""  L  %   $  S ,  '%  '%*"  " (1) -  "  c { ". (2) - "  " xi { ". (3)  f { n-" $'!# " ,  t1 : : : tn { ",  f (t1 : : : tn) { ". (4)  " "- &# '   " 1 { 3. ("  % &    "-+ $'!,  '%  $'!, "%*+ & "-  F ,  "* .

31

'  !, -  "+     " "+. "/ 5.3 (3  1.

L ). )"'"  L  %   $  S ,  '%  '%*"  " (1)  t1 : : : tn { ",  p { n-"  " ,  p(t1 : : : tn), ti = tj  -    { $"'  L . (2)  A  B { $"'  L ,  :A, (A ^ B), (A _ B), (A ;! B), 8xA, 9xA { $"'  L (3)  $"'  L "- &# '   " 1 { 2. )"'  (1)  % 0""  "". +- " x $"'' A    ",     +-" $"''  8xB  9xB.  " '     &". )"',  "%*  &+ +- "+,   -". !. 8x(x = y + z ;! 8z(x < z ;! 8y(:x  y = z))). .#  +- x  ,   +- y z  &, #  . "/ 5.4. '#  =< C F P > {  '. & " M '     < M i >,  M {  "- ,  " " " M  i { &-,  *    -"' c 2 C 0" i(c) 2 M , -"' n-""' $'!#"' " ' f 2 F { &- i(f ) : M n ;! M , -"' n-""' "' " ' p 2 P { n-"  i(p) : M n ;! B .  F = ,  M   "#%,  P = ,  M   (' #) &. ,*#% " M   "*#  . !# 1. #1",  0     '"",  c  i(c), f  i(f ), p  i(p) - . 2. 6'", '% &'% "' "-  # "#, +  n-"+ $'!    '%*" n + 1-"" " ,  &   1.  ' + &+ " 0"  #'% -  + 1 -. &*+ - "+ 0 + &'   # #. !. 1. ';1  "- " #  & " ' < feg f g >. ( " '  1'  1):8x8y8z (x  (y  z ) = (x  y  z )) 2):8x(x  e = x ^ e  x = x) 3):8x(x  x;1 = e ^ x;1  x = e): .

32

 '&#  ' ;1 ,  # "  1  8x9y(x  y = e ^ y  x = e). !, "- '&#  ' e. (  "  1  9y8x(x  y = x ^ y  x = x),  #  8x9y8z (z  (x  y) = z ^ z  (y  x) = z ^ (x  y)  z = z ^ (y  x)  z = z ). 2.   "-  # "# '  =< fg >. "    %  -  L '%*" &": 1):8x x  x 2):8x8y(x  y ^ y  x ;! x = y) 3):8x8y8z (x  y ^ y  z ;! x  z ) 1   { 0 1 , '  %* *  ": 4):8x8y(x  y _ y  x):  '  "-  " % -  "# ' 0 =< f. "    %  -  L '%*" &" 10):8x :x < x 20):8x8y8z (x < y ^ y < z ;! x < z ) "   0" ' " : 30 ):8x8y(x < y _ y < x _ x = y): " 1)-3) 0   "" 1')-2') '%*" ".  x < y "- &# -  '%* $"'  L : x < y x  y ^ :x = y:   # " 1')-2')  # 0' $"'' "  <, , #'#  " - "   ,  #,  - 1)-3)  # -" 1')-2'),  - 1)-4)  # -" 1')-3'). &,  L  x  y    $"' x.  0" "- & #  !% {,  # ;1 , 0

0

0

33

#' 0  !     %'  .  "" " : 1):8x8y(x + y = y + x ^ x  y = y  x) 2):8x8y8z ((x + y) + z = x + (y + z ) ^ (x  y)  z = x  (y  z ) 3):8x8y8z (x  (y + z ) = x  y + x  z ) 4):8x(x + 0 = x ^ x  1 = x) 5):8x9y(x + y = 0) 6):8x(:(x = 0) ;! 9y(x  y = 1)  + p,  p {   { 0 , '  %* " AChar(p) : 1| + {z  + 1} = 0: p



  &  #  + 0 &' & #  "-  "  AChar(0)n :

: 1| + {z  + 1} = 0: n



.# n = 2 3 : : : . ( -,  , &  # & "' , &' & #  "-  "  ACln:

8a08a1 : : : 8an;19x(xn + an;1xn;1 +    + a1 x + a0 = 0)

 n = 1 2 : : :

!. 7 ,  " - &#   & ",   '%* "' "'. 4.    { 0 & " &*  ' < f0 1g f+ g fg >. " '   { 0 "  (" 1 { 6 " 3), "   (" 1 { 4 " 2  *  ",   %* 1   "  !":

8x8y8z ((x  y ;! (x + z  y + z ^ (0  z ;! x  z  y  z ))) : 2 ' +  % *  "' . *  "'"   '  , ", - 34

"    " + &  #,  - -# 0"    ". 4&  # *  "'  '-  "" '   & # " ACln  + + n  "' 8x(x > 0 ;! 9y(y = x2 )). 5.     " K { 0 ' # & ' < f0g f+g ff j  2 K g >,  f { "  ! '"-   . "    { 0 "   &  '  (".  " ,     ! -)  & "-  "  8x8y(f (f (x)) = f  (x) ^ f (x + y) = f (x) + f (y))   "-+   2 K . ,*# 0 "-  "   "*  K . &+"# '# "-  "        1" +    ' " "+, &%*+   ,  #,  "-#% "#    ". " '  '& ,   '% &&*    ',  ',  " #  "+, -  + "   " "- . /    " '-" $"' ,   '* '  '# +%. 0"' #  &&*  " % (2$"' #     0 ' {  ' -).

"/

 ,

6

'# M1 = (M1 i1), M2 = (M2 i2) {   & " ' . 7  &- ' : M1 ;! M2   "$"" &+ ",  (1)  %&  "  c 2 C '(i1 (c)) = i2(c) (2)  %& n-" "  f 2 F  %&+ m1 : : : mn 2 M1 " "    6.1.

'(i1 (f )(m1 : : : mn)) = i2 (f )('(m1 ) : : : '(mn )) (3)  %& n-" "  p 2 P  %&+ m1 : : : mn 2 M1 "

"   

i1 (p)(m1 : : : mn ) = i2(p)('(m1 ) : : : '(mn )):

 ,    "$"  ' , !, ,  - &- &  ' + "-  % " '"   . 35

"/

'# M = (M i) { & " ' . "-  M1  M  " "'",   -  0" i(c) 8c 2 C  "' # +  !  i(f ) 8f 2 F . ."' "-  M1  M   &'% "' M1 = (M1 i1),  8m1 : : : mn 2 M1 8f 2 F 8p 2 P 6.2.

i1(f )(m1 : : : mn ) = i(f )(m1 : : : mn ) i1(p)(m1 : : : mn ) = i(p)(m1 : : : mn) : / "   " " M.   ", +* " t +  x1 : : :xn ,  &'" # t = t(x1 : : :xn ). "/ 6.3 (/ 21 ). '# M = (M i) { & " ' . -"' "' t = t(x1 : : : xn)      &- i(t) : M n ;! M  '%*"  ": (1)  t = c,  i(t) : M n ;! M {  &-: i(t)(m1 : : : mn )  c (2)  t = xi ,  i(t) = i : M n ;! M ,  i (m1 : : : mn) = mi {    i-% " ' (3)  i(t1) : : : i(tn)  ,  t = f (t1 : : : tn),  i(t) = f (i(t1 ) : : : i(tn)) { '  ! &-. !.   ""  " -  ,  # # "+    x1 : : : xn,  ' $'!    #+ "+ $ .  ""   "   ' $'!    #  " ",     "-  "+, -*  ",  # +* ". ,  "  - "+ ,     "+, ' ' "- ,   !  "   ,       "+ ' "-  x1 : : : xn,   ! 0 "   $'!  x1 : : : xn , -     . !. 1. '# K {  !   ""'  #! C = fc j  2 K g F = f+ g, " i(c) = ,   ! +    '%  -  '"- #! K . (  !" "  %  "- " 0" #!.  0"  " "'   #     - ".  ", i ((+(x y) +(x y))) = i (+(+((x x) (2 (x y))) (y y))) ( $"' #  ?). 36

 ""  " #! K "-  #  $'!%,     !  " ' '. 2. 2' < f g f: ^ _ ;!g >. (" ' ' { 0  "- ) ,  +  ! { 0   '%* "  , .. $'!  f gn f g (".   1.1 { 1.3). 3. 6"" '' < fc j  2 Rg f  = exp ln sin arcsing > :  i(c) = ,   $'!# "   '%    '%* $'!,  $'!,   %  !"  "-+ " 0 ',  % 0"" $'!". 2  , 0 "  +    !, #' # " %   %'   $'!. 4& %#    ! "  %'   $'!, '-  # &   " +" $'!"  '#  + &#   i(t),  t = f (t1 : : :tn)  &   $'! i(f )  " i(t1) : : : i(tn) '-   (.!)   1,  $"' A "  & +- # "+    x1 : : : xn,  &'" # A(x1 : : : xn). "/ 6.4 (/ 21  ). '# M = (M i) { & " ' . - $"' A(x1 : : : xn) (n  0)  L "-  #     i(A) : M n ;! f g  '%*"  ". (1)  A "  t1 = t2  p(t1 : : : tk ),  8m1 : : : mn 2 M i(t1 = t2 )(m1 : : : mn) =  () () i(t1)(m1 : : : mn) = i(t2)(m1 : : : mn ) i(p(t1 : : : tk ))(m1 : : : mn ) =  () () i(p)(i(t1 )(m1 : : : mn ) : : : i(tk )(m1 : : : mn )) = : (2) i() =  i() = : (3)  i(A)  i(B) '-  , 

i(:A) = :i(A) i(A ^ B) = i(A) ^ i(B) i(A _ B) = i(A) _ i(B) i(A ;! B) = i(A) ;! i(B) i(9xA) = 9x i(A) i(8xA) = 8x i(A): 37

  + m1 : : : mn 2 M      i(A)(m1 : : : mn ) = ,  &'" # M j= A(m1 : : : mn). -   - M  "   +-& "+. "/ 6.5. )"' A(x1 : : : xn)  B(x1 : : : xn)  # " M,    %    - , ..  8m1 : : : mn 2 M i(A)(m1 : : : mn ) = i(B)(m1 : : : mn ):

&: A M B. )"' A(x1 : : : xn )  B(x1 : : : xn)  #  &+ " K,  8M 2 K A M B (A K B). !, $"' A  B   # (A  B),    #  + &+ " ' . )"' A    &*" (` A),  A  . ! 6.6. A  B ()` A ! B ! 6.7. #"        ( .. 1 2 Q = 8 9): (1) QxA(x x1 : : : xn )  QyA(y x1 : : : xn ),  y     A(x x1 : : : xn),  A(y x1 : : : xn)     !    x  A(x x1 : : : xn)  y. (2) QxA  A,  x   !     A. (3) :8xA  9x:A :9xA  8x:A: (4) 8x(A ^ B)  8xA ^ 8xB: (5) 8x(A _ B)  8xA _ B,  x   !     B. (6) 9x(A _ B)  9xA _ 9xB: (7) 9x(A ^ B)  9xA ^ B,  x   !     B. / -        '%*+  %& " (". "' 3.1). "/ 6.8. & " M1  M2 '   % 0" 0  ",   %& - A '    ' 

M1 j= A () M2 j= A 38

!

1.  Q  R   % 0" 0  " #',  ", - 8y9x(y = x3 )  R,  - Q. 2.  R  C -   % 0" 0  " { - 8y9x(y = x2 )  C ,  - R. 3. &  '  < Z + >  < Q + >   % 0" 0  " (   "   '%* -). 4. ,- #,  &  '  < R + >  < Q + > 0" 0  . .

!

"!     !    

M1 = (M1 i1), M2 = (M2 i2) %   %    .    - %& '  & < a1 : : : an > & a.  m1 : : : mn 2 M1,  ' : M1 ;! M2 ,  & < '(m1 ) : : : '(mn ) > & '(m ). '# ' : M1 ;! M2 { "$". , -" - & &*, " &', '%* ' -.  %& $"' F (x)  L  %& & m 2 M1n M1 j= F (m ) () M2 j= F ('(m )): (6.1)  0  -"  ' -  " . '# t(x) { ",  m 2 M1n. ( 6.9.

.

'(i1 (t)(m )) = i2 (t)('(m )):

(6.2)

/ ' -    '    "$" ',  "   &  " , #' 0" '  " %& " #   '%* ,  "  &  "+. '# t = xj . ( i1 (t)(m ) = mi i2(t)('((m)) = '(mi )  " ",  0" ' ' - -    . '#  " t1 : : : tk ' - ,  t = f (t1 : : : tk ),  f {k-" $'!# " . ( i1 (t)(m ) = i1 (f )(i1 (t1)(m ) : : : i1(tk )(m )):

  % "$" '(i1 (t)(m )) = i2(f )('(i1 (t1 )(m )) : : : '(i1(tk )(m ))):

  -% '!  -   i2(f )(i2 (t1 )('(m )) : : : i2(tk )('(m ))) = i2(t)('(m )) 39

    (6.2) -"  (6.1)  "+ $"'.  F (x) "  t1(x) = t2(x),  m 2 M1n,  M1 j= F (m ) ,  i1(t1 )(m ) = i1(t2 )(m ), M2 j= F ('(m )) ,  i2(t1 )('(m )) = i2(t2 )('(m )).  (6.2)  # ',  (6.1) 0" '  # ' -% i1(t1 )(m ) = i1(t2 )(m ) () '(i1 (t1 )(m )) = '(i1(t2 )(m ))

   3  '. '#  # F (x) "  p(t(x)),  t { & " ,  p {  " . ( M1 j= F (m ) ,  i1(p)(i1 (t)(m )) = ,  M2 j= F ('(m )) ,  i2(p)(i2 (t)('(m ))) = . ' (6.1)      # "',  i2(p)('(i1 (t)(m ))) = , .. (6.1) 0" '  # ' -% i1(p)(i1 (t)(m )) =  () i2(p)('(i1 (t)(m ))) = 

     ,  ' { "$". '#  $"' F  G ' - (6.1) . ( M1 j= F (m ) ^ G(m ) () M1 j= F (m )  M2 j= G(m ) () M2 j= F ('(m ))  M2 j= G('(m )) () M2 j= F ('(m )) ^ G('(m )):   !: M1 j= :F (m ) ()    M1 j= F (m ) ()    M2 j= F ('(m )) () M2 j= :F ('(m )):  #+ +   #  "-   # #'  -%  "  . '#  $"' F (y x) ' - (6.1) . 6"" $"'' 9yF (y x).  -",  M1 j= 9yF (y m ). / ,  '* '  m0 2 M1 ,  M1 j= F (m0 m ). (   -% '! M2 j= F ('(m0 ) '(m )) ,  #,

M2 j= 9yF (y '(m )) (6.3) . '#, &,   (6.3). ( '* ' n0 2 M2, ,  M2 j= F (n0 '(m )). #' &- ' %3  '* '  40

m0 2 M1 ,  n0 = '(m0 ), .. M2 j= F ('(m0 ) '(m )). ( # ' ,  (6.1)    F "" M1 j= F (m0 m ), .. M1 j= 9yF (y m ). (" "" (6.1)    $"' 9yF .    &* #  "-   #, #'  -    '*    !. !#. 1.  -   &'  ' -   "2    ' "$ '   %". 2. ,    + &+  # #  - 6.9, & " # +' &   " "  $"'. "/ 6.10. ,-  + -, + & " M    " M  & Th(M). ,-  -, +  + "+   K &+ " ' ,     K  & Th(K).  - 6.9. ',   " M     + ", "$+ M.

 , x !  1   1    4   "/

   $"' "  '% ( $7

2.

.

7.1.

'%) "#'% $"' ( ..$.),   " 

Q1 x1 : : : Qs xs F (x1 : : : xs y1 : : : yt )

 Qi = 8  9,  F { &  $"' (s  0). $ 7.2. & "!  A(x1 : : : xn) 

 L          B      .    - $"' ..$. B ,   # $"' A     "# $" $"' A.  A { 0" $"',   &  ,  #, "     ..$..  $"' B { ..$. $"' A,  QxB { ..$. $"' QxA,  Q = 8 9. -,  Q = 8,  Q0 = 9  &.  Q1 x1 : : : Qsxs F { ..$.  A,  Q01 x1 : : : Q0s xs :F { ..$.  :A. '# R1 y1 : : :R1 yt G,  - Ri # & 8, & 9, # ..$.  B . "",    "+ xi  yi "', &*  , #.  , & &-# 0 &" " z1 : : : zt,   .

41

+  F ,  G  &"  G(z ) $"'',  ' " - +- yi G  zi   . ( ' - 6.7(1) R1 y1 : : : R1 yt G  R1 z1 : : : R1 zt G(z ). ( #,  #' #  - 6.7(5,7), '"

A _ B  Q1 x1 : : : Qs xsF _ R1 z1 : : : R1 zt G(z )   R1 z1 : : : R1 zt Q1 x1 : : :Qs xs (F _ G(z )): (" ""  ..$.  A _ B  !   #  0$$  {  '* ', "   ",   %& $"'    #'%  $"'' ..$.    1 . .

!

.

1:

8xP (x) _ 8xQ(x)  8xP (x) _ 8yQ(y)  8x(P (x) _ 8yQ(y))  8x8y(P (x) _ Q(y))

9x (P (x y) ;! 8yQ(y z ))  9x (:P (x y) _ 8yQ(y z ))   9x (:P (x y) _ 8uQ(u z ))  9x8u (P (x y) ;! Q(u z ))

2:

"/

7'"  #,   Th(M) " M =< M i > '   ' %   ,   %& $"' A(x1 : : : xn)  L '* ' &  $"' B (x1 : : : xn )  L , ,  7.3.

A(x1 : : : xn) M B (x1 : : : xn):

(7.1)

 (7.1) " "  %& " M    K,   ,    K  ' %   .  '* ' ",   %& $"' A(x1 : : : xn)     1 & '% $"'' B (x1 : : : xn ), '  %*'% (7.1)  %& " M   K,   ,    K  ' 0$$  %   . !  &!     # #  ", #' "  $"'     ".  '   0$$  %    .  ,     " &' $"' ,   7.3  #% ". .

42

!

    M (  K)    " 

 ,      9xB (x x1 : : : xn),  B {  '" $ 

           $ ,      M ( K)   !    .    ' " 7.2   #  - #  $"' ..$. #   " '!  ' n   ..$. A.  n = 0 $"' A { &   ' - .  -",  %& ..$., -* " n   ,  # &  $"'. '# 7.4.

.

A = Qn xn : : :Q2 x2 Q1 x1 C (x1 : : : xn y1 : : : ym ):  -" ,  Q1 = 9  "" $"'' 9x1 C (x1 : : : xn y1 : : : ym ).  "     ) &  $"' C   # $"' 

_k

j=1

D1j ^    ^ Ds j

(7.2)

j

 Dij # & &  $"', &  !. 2 #

0k 1 _ @ 9x1 C  9x1 D1j ^    ^ Ds j A  

j

j=1 _k

j=1

9x1 (D1j ^    ^ Ds j ): j

 ' % - $"' Wkj=1 9x1 (D1j ^    ^ Ds j )  # K  &  $"' E (x2 : : : ym ). ( # A K Qn xn : : : Q2 x2 E (x2 : : : ym )  *    $"'  ' %     -% '!.  Q1 = 8,  "" 8x1C  :9x1 :C .  0  " :C  ' (7.2)   -,  1 %"   9x1 .  0 A   -  # ..$., -* # n ; 1     # "- "#  - '!. 

    " "" % Th(Q) " Q '   < f     !  "  <. j

43

$



7.5.

  Th(Q)   %     "   -

  . - " :x < y &'" # y  x (". " 2 ! 5).  '* - ',  " 

%#    $"'  9y(x1 < y ^    ^ xk < y ^ u1  y ^    ^ ul  y ^ y < v1 ^    ^ y < vs^ ^y  w1 ^    ^ y  wt ) k  0 l  0   0 t  0 (7.3) 2 "+ xi ui vi wi "' &#   %*. -",  "- # '",    & ", +* $"'' (7.3), . - ,  xi    xj ,  xi < y ^ xj < y  xi < y.   u v w.  xi (ui )    " vj ,  $"' (7.3) -, #' 0" '   $"'' 9y(v < y ^ y < v) ( - $"'' 9y(v  y ^ y < v)   ). &+ ' $"' (7.3)  # &  $"' .  "  ',  xi    " wj . '# u    " w . ( $"' (7.3) - $"''  u  y ^ y  u,   # $"' y = u. 0" '  +- y "- "#  u,    (7.3)    # &  $"'.  xi    " uj ,  $"'' y  uj "- '&#  (7.3) #' 3%! xi < y ^ xi  y  # $"' xi < y. ,  vi    " wj . , # "# ',    &+ "+ $"' (7.3)   %*+.

',  y  (7.3) #    (..  k = l = 0, & s = t = 0) $"' (7.3)  # &  $"' , #' ' " "-  Q   &#1,  "#1 0".  -  "   y   &+ ,  $"' (7.3)  # $"'

^k ^s

(x < v ) ^

=1 =1 ^l ^s

^

=1 =1

^k ^t

(x < w )^

=1 =1 ^l ^t

(u < w ) ^

=1 =1

(u  w )

(7.4)

"" , # (7.4)    (7.3)    1 <  . 44

'#, &, $"' (7.4) . /  #  '%* "   maxfx g < minfv g maxfx g < minfw g (7.5) maxfu g < minfv g maxfu g  minfw g .# "-   '. 1.     " (7.5) - . 0" '  ,  maxfx u g < minfv u g  %& y, '  %*   ' maxfx u g < y < minfv u g &* ' $"'' (7.3). ( y '* ', #' '  "-  Q , .. "-' %&"   '" " 0"" "-  # . 2.     " (7.5) #,  ""    .

0" '  #,  y = maxfu g = minfw g &* $"'' (7.3) '.  ! "",    #  0$$ , .. 3  ", *  %& $"',  #'%  Q & '% $"''. 2 #  Th(Q)  ' 0$$  %   (". "   % 7.3).  #  " 7.5 " # # 1# ",   ' " "-  Q  '%* -: 1): 8x(9y(x < y) ^ 9y(y < x)) 2): 8x8y(x < y ;! 9z (x < z ^ z < y)):  '  "- , "  - 1)  2),  " " ". &"  + +  ' + "-  DL. (" &" DL #  " ' , +  - 1') { 3') " 2 ! 5  - 1) { 2) '* &!. (" &" "  ! 7.6.    DL   %     "   .  "-    ' ',  %& &  - 0 ' # $"'   ,  .

45

-* "+. %&  $"'  # & , & . (" &",     K 0 '  ' %   ,  - -  # K & , & . /   '%* ! 7.7.           ,       K   %       "   ,     K %   %   

2   - 7.6  7.7, '"  & . "!               %   %    .    , %   %    Q R. ( "   "    $,  '* ' ",   %&"' -%       ", -  0   . #' '* ' ",   %& $"'            + + "+  -,    '%* ! 7.8.    T        %     "      ,    "! !       "     ,      %   ,    T  ( .

' + - 7.7 - &  - #    , .. ",  %*  %&"' & "' -% -     ,  " '* '. 2 #,  ' + - 7.7  Th(K)  ' 0$$  %   ,   1". & .   Th(DL)  ( .

 , x #

!   3.

8

.

6"" ''  =< f0 +1 ;1g f+g f  "' Z+ =< Z i > ' ,  Z,  , "-  !+ ,  i {    ! +, $'!#+   "  .     0", - "  i %'  ',  " - %   "" " $'!"  ". (, " 1| + {z  + 1} (|;1) + {z  + (;1)} x| + {z  + x} k



k



46

k



&'" # k ;k k  x   .

" ,   &  % &   ,  "+ & " $"'" ' . #'# ! #%  ""' #% - Z,  #,  - $'!,  " " "" ' , "  k1  x1 +    + kn  xn + b ki b 2 Z:

(8.1)

2&            , "- -'% "'% $"''  t1 = t2    ' k1  x1 +    + kn  xn + b = 0

(8.2)

 -'% "'% $"''  t1 > t2 {  ' k1  x1 + : : : kn  xn + b > 0:

(8.3)

&"  '% # $"' (8.2)  (8.3)  t. ( ! $"' (8.2)  # $"' t > 0 _ t < 0,  ! $"' (8.3) { $"' t  0. /   t> 0  t;1  0 t< 0  t+1  0 t= 0  t 0^t 0  %,  '% "'% $"'' "- "# 3%! +   ,   ! { 3%! +   .   #   & '% $"''  ),  # 0 ",  " # &, #'# &' #% 3%! # 3%!,  0 $"' -  # Z+ $"'  _n m^ (ji) k1  x1 +    + kt(ji)  xt + b(ji)  0 n

j=1 i=1

(" &" "  '%* ! 8.1. )!    ,  !          !    !'    ,       $   (            $  % $   . 47

!

  Th(Z+)     "   .    6"" $"'' 9y(x = 2  y),     # ,  x { ,  -",     #  &  $"'. "" ,  & 0 $"' &  # & ,   - 8.1   &,  &#     &  &3 "- , -  + # "-  !+ 1 "    k1  x + b1  0 8.2. .

: : : : : : : : :: : : ks  x + bs  0:

 ,   "- , & , & -  ! -# (!#) ,   . ,  0" -   " &   &3 + "- ,   & "-  +    " +   %& $"'  9y(x = m  y),  m  2. / $"'    x  0( mod m). 61"  # ''   ' 0 , &     "-  "+ + "  Dm (x) m  2,  &'"   # 1 " Z+ '%*" &": i(Dm )(x) =  () x  0(mod m):  '&  - " -,  ' ' ' "  D1 (x),   '  -  . ( " Z+  '  ! + "  Dm   $" &'  & APr. " " $ 8.3. *   !   %     "     #  " " &  ' -   ,  " $"''"  "".  1. #   ax  b(mod m)  (        ,  b    d = (a m).  %    %       c,  (         : x  c(mod m)  x  c + m (mod m)  : : :  x  c + (d ; 1)m (mod m): d

d

48

  

1     



     m1 : : : mk        .       x  c1 (mod m1 ) 2 (

).

: : :: : : : : : : : : x  ck (mod mk )

    (   mod m1  : : :  mk .  %  m1 : : : mk %       $    d1 : : : dk ,   "!  c1 : : : ck (            x  d1 c1 + : : : dk ck (mod m1  : : :  mk )

/   ' - &  ' " &   ". 2'%* ' -   -" &&*"  " & +  ',  "' m1 : : : mk   %   " ".   # -    ,   # %   .  3. C     x  c1 (mod m1 ) : : :: : : : : : : : : x  ck (mod mk )

 (        ,  8i < j  k ci  cj (mod (mi mj )).  %  (      mod m1 : : : mk ].  %  m1 : : : mk %       $    d1 : : : dk ,   "!  c1 : : : ck (            x  d1 c1 + : : : dk ck (mod (m1 : : : mk ]))     8.3.  '*+ " ',   '# %#    $"' :

9x (k1 x + t1 ^ : : : ksx + ts ^ Dm1 (l1 x + u1 ) ^ : : : (8.4) ^ Dm (lt x + ut ) ^ :Dn1 (h1 x + v1) ^    ^ :Dn (hr x + vr )  ki li mi ni hi 2 Z  ti ui vi { ",  -* " x.   1,   D1 (x) "- #,  t > 0, ..   "    $"' (8.5) ' '. t

r

49

." -'% $"''  :Dn (t)   #'%  $"'' Dn (t ; 1) _    _ Dn (t ; n + 1) (:t  0(n) () t  1(n) _    _ t  n ; 1(n)),   &     9  3%!%, +"  "',   '# %#   #  $"'  (8.4)  r = 0, ..  -*+ $"'  :Dn (t). " 0, " ""' 1, "- #,  l1 =    = lt = 1.  0, " ""' 2  ""' 3, .. " "'      , 0   , +"  "',   '# %#   #  $"'  9x (k1 x + t1 ^ : : : ksx + ts ^ Dm (x + u) (8.5.)  $"' (8.5)  ki  ,  0 $"' , #'  %&+ +  &+ "+, -*  "   , '  %   &#1  &%  -#  !# x,    +    x,  '   +*"' 0' $"''  %.  " ' $"' (8.5)    '

9x(a1 x  p ^    ^ af x  pf ^ q1  b1 x ^    ^ qq  bg x ^ Dm (x + u)):

(8.6)

)"' (8.6)    '

_f _g pi pi g ^ qj = minf qj g^ = max f i ai i bj ai bj

i0 =1 j0 =1 ^ 9x(pi0

.#

pi0 ai0

0

0

0

0

 ai0 x ^ bj0 x  qj0 ^ Dm (xi0 + u))

p g # *  $"' = max f a i i i

^f i=1

aipi0  ai0 pi

q g. f   $"' qb 00 = min b i , # "# ' $"' j

j

j

j

9x(p  ax ^ bx  qj0 ^ Dm (xi0 + u))

(8.7)

 #,  0 $"'  # &  $"' bp = aq ^ p  0(a) ^ q  0(b) ^ p + au  0(ma) _ :bp = aq ^ aq  bp + abm (8.8) 50

"" , '# $"' (8.7) ,  bp  abx  bq. "-   ': ) bp = abx = aq  &) 6= bp = aq: ' ) bp +p abu  0(qmab), p.. p + au  0(q ma)  $"' (8.8)  . ' &) a  x  b , .. p q u u a + u  x + u  b + u,  .. x + u = mt,  ma + m  t  mb + m . ,-'  '" * " "    "-  # !  "  # " '  ;   1. " ' 0  #     $"' (8.8). &'% ' "      '*+ " ,  - &  " - ' 0  # "'  -  k > 0 k = 0 k  0( mod m),  k 2 Z,   ,  '* ' ",  %*  %&"' & "' ""' -%,    "  -". ( #,  #' - 7.8, "  "  " 8.3 & . *   !  (   ""  &'  1929 ' 1"#  - '#+ , ..  & N+ =< N i > ' 00 =< f1g f+g >     !   $'!# "  .  #,  0     APr '%*" ". "/ 8.4. '# M1 =< M1 i1 >  M2 =< M2 i2 > { & " ' 1  2   . 7'"  #,   Th(M2 )  '  Th(M1 ),   %&"' -% A  ' 2 "- #  - A0 ' 1,  M2 j= A () M1 j= A0 :  '* ' ",    #"' -% A  &'" - A0,   ,  Th(M2 )  ' Th(M1 ) 0$$ . ( #   '%* ! 8.5.    Th(M2) %        Th(M1) Th(M1 )  ( ,   (  Th(M2 ). +!,   %    Th(M2 )  ( ,   (  Th(M1). ! 8.6.   Th(N+ ) %        APr.     , & # A0  A '- # "# +- -   Qx A  +-   '%*    Qx  0 (". !% 3). .

51

 #      &, .. APr  ' Th(N+ ) "   8.4 (. ). 6'# &' &, - ""',  " '#"  1"  -# $"  $" "" , " "", &  " '* " 1"  ! "" #& $"! "" (". !% 2).  " ' " -, 0 ""  '" " &3". , '- $"  -"  '"-", ..  & N ' < f1g f+ g >,    1". #'  Dm (x) " Z+  " $"'" ' ,   +  ,  "+ $"'" '  Z+,    "  ,  "+ $"'" '  0 Z+. /   ' '%*  . "-  L  Zn  " ,  '* '%   a1 : : : ak b 2 Zn,  L = ft1  a1 +    + t1  a1 + b j ti 2 N i = 1 : : : kg  &3 + "- Z  "  "- ". " " '%* $ 8.7.   M  Zn      F (x1 : : : xn) c   , .. M = f< m1 : : : mn > j Z+ j= F (m1 : : : mn )g       ,  M   .

 ,

9

x $1 $  

0"  " "" % '   * + . /   R '  =< f0 +1 ;1g f+ g f.  "",  " 4 ! 5 ' ' +   %  "  -1. .#  % '' %#  &- '& . /     ( ' '%* ", -* .("'. $ $     Th(R)          %     "   .  ", "-  # # 0' "' " &3", " ""   '. -  "", 4.

.

9.1 (

).

52

 - $'!  Rn,  " "" ' , #  " p(x1 : : : xn)  !" 0$$!" (". " 1   % 6.3). . " $"' '  "' &#   p(x1 : : : xn) = 0  p(x1 : : : xn) > 0.     -" 7.4  '# %#    $"'  9xB (x),  B { 3%! "+ $"'  + !. "" :p = 0  p > 0 _ p < 0, :p > 0  p  0  ;p > 0 _ p = 0. ."  "* 0+ 0    ! "+ $"',   &     9  3%!%, +"  "',   '# %#    $"' 

9x(p1 (x) = 0 ^    ^ pn(x) = 0 ^ q1 (x) > 0 ^    ^ qm (x) > 0)

(9.1)

 pi  qj # "  " x, 0$$! + "  % ""  !" 0$$!"  '+ "+, & !" ". #' R p1 (x) = 0 ^    ^ pn(x) = 0  p21 (x) +    + p2n(x) = 0

 # "" $"'  (9.1), -*+ #  '  , "-, #   . , # "" # $"'  9x(p(x) = 0): (9.2)

',   # " p  " x   + 2, %    $"' (9.2) '*  '%*" &":

9x(ax2 + bx + c = 0) R a = 0 ^ b = 0 ^ c = 0_ _ a = 0 ^ b 6= 0 _ a 6= 0 ^ b2 ; 4ac  0: .# a b c { "  !" 0$$!"  '+ "+, & ! .

&*" ' %    $"'  (9.2) '*   "* " 9'". 4& " #  0 ,  &-# 1" "+ , "" $"''

9x(x4 + bx + c = 0)

(9.3)

,   1, b c { "  !" 0$$!"  '+ "+, & ! .  0" "- #,  b 6= 0, ..  " ' $"' (9.3)  # $"' c  0. 53

6"" " f (x) = x4 + bx + c.  " D(f ) = 256c3 ; 27b4.  D(f ) 6= 0,  (f f 0 ) = 1. 0" ' " 9'"  f " : f0 (x) = f (x) f1 (x) = f 0 (x) = 4x3 + b f2 (x) = ;3bx ; 4c f3 (x) = Db(3f ) :

&"  V (1)  "   # f0 (1) f1 (1) f2 (1) f3 (1),  fi (1) {  " fi (x)  x ! 1,   " &"    ' 1 0$$! " fi . .# "- 4 ' : a). b > 0 D(f ) > 0 : V (;1) = 2 V (+1) = 2 b). b > 0 D(f ) < 0 : V (;1) = 2 V (+1) = 1 c). b < 0 D(f ) > 0 : V (;1) = 1 V (+1) = 1 d). b < 0 D(f ) < 0 : V (;1) = 2 V (+1) = 1:

' " 9'"  * +  " f   V (;1) ; V (+1), ..  '* '% '+ b)  d).  D(f ) = 0,  (f f 0 ) = f2 , .. g = (fff ) { " #  ,    "   "  *  #,    " " f ( "",  g "  - ,   f ,    + g  #%,   !). # $"' (9.3)  # $"' 0

b = 0 ^ c  0 _ :b = 0 ^ (D(f ) = 0 _ D(f ) < 0):

(" &" " (  '* '   & &&* " 9'"  ' " "#+   .    1 &*  "+ $"' Th(R) ',  - &  " - '   # R & -% n > 0, & -% n  0   n 2 Z. % ,  '* ' ",  %*  %&"' & "' ""' -%    "  -". ( #, " - 7.8, '" &  2.          Th(R)  ( . "  ' -  &  .("  " & "A decision method for elementary algebra and geometry",   1 ! 40-+  (   1951 ').  # #  ( & # -" { & " &  !.   0 #  & ' * .&", 0" 54

 '"  ". 2 #  #  " (,    + 0 "-   6]. 6'# (  '*   "  # $"  $" 0" "".  '* ' & - " 1  0" &,  ",   "  # " &+ '     . #'#  " ""  "-  &#1  - 0" " $"' #  -   Th(R).  ", - 8x1 y1 x2 y2((x1 =6 x2 _ y1 =6 y2 ) ;! 9A B C (Ax1 + By1 + C = 0^ ^ Ax2 + By2 + C = 0 ^ 8A1 B1 C1 (A1x1 + B1 y1 + C1 = 0^ ^ A1x2 + B1 y2 + C1 = 0 ;! 9t(A1 = tA ^ B1 = tB ^ C1 = tC ))) - "'% "'  ",         +   ". " &" &#1  " 1# " "- &#     Th(R). /    "", + $''% *# ', &3" 1  . ., #' "- #,  &   '  *  ,   '    % 1".  " "-  # 0 ' -, -" ,  &  -. 6"" '' 1 ,  '   & "  " $'!# " . / "  " &'" &# sin    # R  $'!% sin(x). ,-  + - ' 1 , + R &'" &# Th1 (R). ! 9.2.   Th1(R).     "     ( .    , &'" # # " $",   & N ' 2 < f0g f+ g > 1". / &'  1" ' .  - 7.8 ',   &  Th(N)  ' 0$$  %   ,   & & 1". .    ' 0$$  %   .  ""  "- #,  0  &*   ' %   . -",   Th(N) 0$$   '  Th1 (R) (".   8.4).  0 1 ' - &'  #  - 8.5. 2 -",  ' 1 "-  # "  Nat(x), ,  R Nat(x) =  () x ; ;'#  .

55

  ",  '#  # 1  # !#   "#1"' -#"' '.  "  P (t) 8x(sin(x + t) = sin x) ^ t  0: ,  P (t) =  () t { !#  $'! sin(x). "#1 -#    $"' MP (t) P (t) ^ t > 0 ^ 8s(P (s) ^ s > 0 ;! t  s ( #  ,  Nat(x)  9t9s(P (t) ^ MP (s) ^ t = x  s: #  1 -      '%* ' - (".   8.4  - 8.5).  %& $"' F (x1 : : : xn ) ' 2 "- # '% $"'' F 0 (x1 : : : xn) ' 1 ,   %&+ k1 : : : kn 2 N N j= F (k1 : : : kn ) () R j= F 0 (k1 : : : kn ) / ' -    '!  % $"' F  " $"' F ' 2   F 0 = F .   $"' F  G &'" $"' F 0  G0 '- ,   , : (F ^ G)0 = F 0 ^ G0  (F _ G)0 = F 0 _ G0  (:F )0 = :F 0  (8xF )0 = 8x(Nat(x) ;! F 0 ) (9xF )0 = 9x(Nat(x) ^ F 0 ):     ,  #  {    1"   Th2 (R),  '    # " $'!# "  ' 1   ',   0 ' (  0" '  '  &# exp). / &"   * 50-+ + # # '.   1   + ,   "    ' '*   { ,  &  "+  ,  "+ $"'" 0 , - 1#   + , ..  % &  &3    "-+  (+, "'+, '+, '+  . .). "",     1" 0          &" & &  "  e  ,   ", & #,   '* ' " 56

p(x y) 2 Zx y] '  %* ' % p( e) = 0. "" ,  &  " '*  ,   & &  "  Th2 (R),   & 1"# , ..,  "# '- &    (    1" . (" &" #  1" Th2 (R)   &  # '   & &  " e  .

 ,

10

'" & & &   ,  "+ $"'" '   (.  "  '* ! ',  - &  $"' '   # 3%!   3%! $"'  p(x1 : : : xn) > 0  q(x1 : : : xn )  0,  p q 2 Zx1 : : : xn ]. &#  -  3%! # 1 " "#+   . / &-    '%*"'  %. "/ 10.1. "-  M  Rn   %   ,     "- " 1 " "#+    p1 (x1 : : : xn) > 0 : : : : : : : : :: : : : : : : : : pk (x1 : : : xn) > 0 q1 (x1 : : : xn )  0 : : : : : : : : :: : : : : : : : : ql (x1 : : : xn)  0

(10.1)

 pi  qj { "   0$$!".  &3 0"+ '&+ "-      "- ". (" &" &#   Rn,  " %& &  $"' '    & '& "- .  '&+ "- % &  &+ "& ("- 1 " &+ ' ).    '&+ "- "' # &' +  !     . 2'%* ",   '  " 9.1,  ,  0  "'  # !. 57

$

$   0 

 $  %   !      "!      !    .    . '# A  Rn+k { '& "-  c1 : : : ck 2 R. #'  %& " f (x1 : : : xn y1 : : : yk ) $'! f (x1 : : : xn c1 : : : ck )   & "  "+ x1 : : : xn,  "-  fhx1 : : : xn i jhx1 : : : xn c1 : : : ck i 2 Ag -   '&". '#  # A { "-  1 " (10.1). -",  ! A #  x1   '&" "- ". '# c1 : : : ck {  & * + 0$$! " , +*+ "' (10.1). &" " 6"" " pi (x1 : : : xn y1 : : : yk ) qj (x1 : : : xn y1 : : : yk ) 2 Zx1 : : : xn y1 : : : yk ] i = 1 : : : k j = 1 : : : l, ,  pi (x1 : : : xn c1 : : : ck ) = pi (x1 : : : xn) qj (x1 : : : xn c1 : : : ck ) = qj (x1 : : : xn). ' " 8.1 $"' F (x2 : : : xn y1 : : : yk ),   10.2 (

{

).

.

9x1 (p1 (x1 : : : xn y1 : : : yk ) ^    ^ p2 (x1 : : : xn y1 : : : yk )^ ^ q1(x1 : : : xn y1 : : : yk ) ^    ^ q1 (x1 : : : xn y1 : : : yk ))  # Th(R)  &  $"', ..  &#  # '& "-  B . ( # ! "-  A #  x1   fhx2 : : : xni jhx2 : : : xn c1 : : : ck i 2 B g ..   '&" "- " ' ",   " &! #  .

!

.

 & xy = 1 { & "& ,  # { 0" '& "-  .  ! #  OY   & '& "- , fx 2 R j x 6= 0g,    &3"  '+ 0"+ '&+ "- fx 2 R j x > 0g fx 2 R j x < 0g. 0   . #,  0 "-     0"" '&". 6""  # '' 0 ,  ' & "    "-+ + "   c  2 R. 7'" #,  - "  c  ' R  .  " 10.2 "  '%* ", '% " - &'"  # " ( { .&. 58

$

  A  Rn  ! !    ,    !        0       ,  A { !    . (" ( { .& " - "   & ". %'" "-#  "  #+ "+. &- "- Rn  "   ,   $ # '& "- . ! 10.4.  A  Rn, B  Rk f : A ;! B { !   ! ,  A = domf rngf { !   !  f { !    .     " 10.3 ',  %& "-  Rm ,  " $"' ' 0 # '& "- . '# $"' F (x y)   $ &- f . ( A = fx 2 n R j 9yF ( x y)g,  rngf = fy 2 Rk j 9xF (x y)g, .. & 0 "-   " $"'" ' 0 . C . )! !      !   !  !    .  "",  "   x 2 Rn  "-  A  Rn    dist(x A) = inf fjx ; yj j y 2 Ag. n { !    ,  ! 10.5.  A  R  $  f : Rn ;! R  ,  f (x) = dist(x A)  !    $ .    '# G(x) { $"' ' 0 ,  %* "-  A.  1" $"'' F (x y),  %*'% $ $'! f . (" "" - &' .  #,  0 $"' " '%* : 10.3.

.

.

y  0 ^ 8y(G(y ) ;! y2 > (x1 ; y1 )2 +    + (xn ; yn )2 ^ ^ 8b > 09y(G(y ) ^ (x1 ; y1 )2 + : : : (xn ; yn )2 > y2 ; b) 

!

    !    $   !    $      -  #  " ( { .&   -, -    '  +  +, "-  : 10.6.

.

59

.:".  + $$!#+    "  ". (.2. $$!#    " 0$$!". ,.:,. 1986 (&  ).

* " '     + 1  (. -   #   ' # '#. 2"  # " &* +. '# "  # & " M = hM ii '  = hC F Pi. &"  Sn (M ) { "  + n-"+ S n   M ,  "+ $"'" ' , S (M ) = S (M ). n2N -    + &"  "- #,  Sn (M ) # "  "- M n  - n 2 N.  #,  0 "  &% '%*"   ". S1 Sn (M ) { &'  & "- M n S2 ij (M ) = fx 2 M n j xi = xj g 2 Sn (M ) 1  i  j  n S3 A 2 Sn (M ) =) M A A M 2 Sn (M ) S4 A 2 Sn+1 (M ) =) (A) 2 Sn (M ),  (x1 : : : xn xn+1) = (x1 : : : xn ) S5 $ $'! i(f ) f 2 F ,  - &   i(P ) P 2 P - S (M ).  "    M "  "- T = S T n ,  n2N n M T  2 , '  %*+   " S1 { S4. (  ,  S (M ) { "#1 ''  M , &%*   " S5. (" ( - .&  ,  '& "-  &'% '''  R. / '' #  S (R),  0 { ', $''%* " 10.3. ."",  #' x  y  9z (y = x + z 2),  '& "-  &'% "#1'% ''', -*'% $ $'! x + y = z  x  y = z .   -,  S1 (R)   "- , "%*+   "   , ..  %*+ &  &3   +  . / "    '%*"'  %. "/ 10.7. 2'' T  R   -""# (order1 minimal),   - $ $'! x + y = z  x  y = z  T   "- , "%*+   "   . (" &" '& "-  &'% "#1'% ""#'% '''  R. - x = (x1 : : : xn),  ex = (ex1 : : : ex ) . ,-   fx 2 Rn j P ( x ex ) = 0g,  P { "  * " 0$$!"  2n "+,  " % ,  & 0 !#+ "-  &-   : Rn+k ! Rn  " %   "- ". n

0

0

n

-

60

$

)  #!%   $     ! " -

10.8(

).

  "    R #' "'# '&0 !#+ "- # &3, ,  +       ,   #  " #  # 1#,     '&0 !# "-  # '&0 !# "- . /  &  #    '&    Th(R) ' 00 ,  '  0 & " " $'!# "  exp    !. (" 10.8  ,  '' S (R)    '' '&0 !#+ "- . (" &" "- ' -#,  - $"'  ' 00  # R $"'  9yF (x y),  F (x y) { &  $"'. ,  "" # "-  # "' #,  - '%*'% "',    & &*   * " -""# ''. $ 10.9 ()   * ,   n+).k )!n  Rn  n = 0 1 2 : : :  !      : R ! R     00

f(x y) 2 R(n+k) j P (x y "x ey ) = 0 f (x y) = 0g  P {      % $    2(n + k)  ,  f {    $ ,           a1 b1]    an+k bn+k ]     "   , ! "   "  .  0 "  ',    '  ( & # " $'!# "  f   #   $'!%,  '% sin x   #" "    '%,  0  ,  "-  R,  " ' , "% 1#   "   . /  ,    # 0 $'!# "   $'!%,  '% sin x %'  R, #'  0   " "-  N (". #  - 9.2).   -""#" ''"    ",  "- , +* 0-""# '' &% "" +1"  "   " {  ", -  "-  " 1#   "    . 2 ' ,     ",  + "-  1.  %!  " ' #" 0$$'% "'  $'!+, +*+ ""# ''. 61

$

   $ f ,       (a b),     - -    .   "!     r   "      a = a0 < a1 <    < aN = b,      $ f     (ai ai+1)  r        !  ,  !    . 7 &'% $"!% & -""#+ ''+    *'%  " '' "-   . .: . ,. ,.: ), 1997.

% " *  - ". "#   '* # #  " ( (" 9.1)  ,   + %&" *  "'"  (". " 4 ! 5).  #  " 9'" *  "'"  -  + '&+  & 7..  

  2.. 2 #, " " ! 10.11.            %     "   .  #,   *  "'  " +' #, .. -   , "$ Q. 2 #   -* "+ " '  "%     + *  "'+ +, #' 0  ' '#   "  ! -  '"-  " 0,1,-1,   0  + #! Z  Q. 0 ,  %& " &  -   -  + *  "'+ +  ". (" "" 0        #+ & + -. ( #  - 7.8 "   & . i).           ( . ii). "!         %   %    . %,  ", ',  %&" *  "'" , %&    '& &- "'  & "   -'% '.  0,  #' $"''  $  &- f '  # - ' , ' -%*,   f     &- "'  &   "   -'% '. / -, &'' " R,   '*"'  %  %&" '" *  "'" . "" *  "' ,   R,    * + &+ .  ,  %& '   10.10.

62

   "  *  "' . "#1     * " "" + .

 , #)#    /

     x    /

 11

III.

1.

.

, +"  '  $"# # .  #  $"' "+   "   ( ).      -  L ,   % ""     %   ,  "  - %  " '+. /    %  "  . 2  "$"#" # ,       % # " -   $  S      +   -. )"#" ""   % -,  '%  "   ""  #" ""    . (" ""    - { "   . / $"# ,  *  ,    +   % "      "  "' 1 -#"' +',  "'    ( *    ). "  %   &*"+ -  L        ,  -,  '%   &*"+ -, "  %  &*"". % ',   "  -  &*". .# &  # $  ",    &*" -  % ""  . / $    " "   . /  #   " + " - -  # ,   &  - . / - ' # . - ,  '  -  ", "  & + "  .  0  ,   &  ' . "   "  .

"/

11.1 (

#    1 /



).

I. "   . '# F (p1 : : : pn ) { ) , " ` F (p1 : : : pn),  A1 : : : An { -  L . ( F (A1 : : : An) { "  . "    % ""   . 63

II. "   . '# c  d { ",  -* "+ ( , ). ( '%* -  % ""  ,   % ""   : (1) c=c (2) c = d ;! (A(c) ! A(d)),  - A(d) ' " - A(c) +   +- " c  d. III. " " ". '# A { -  L ,  " x0  + A. &"  A0 { -,  '  A " + +- " x  x0 . ( A ! A0 { "  ,    " " ". IV. "    . '# A(x) { $"'  L c    & " x, B { -  L ,  -* " x. ( '%* -  % ""  ,   % ""    . (1) 8xA(x) ;! A(t),  t { ",  -* "+. (2) :8xA(x) ! 9x:A(x). (3) :9xA(x) ! 8x:A(x). (4) 8x(B ;! A(x)) ! (B ;! 8xA(x)). ! 11.2.     ,   !   .   .  &*"# "   I  II  - " 1   IV         ! $"' &+ "+ (  6.4).  &*"# "   III  #+ "   IV '   + -   . "" . '# M =< M i > { .. ' . ( " 4     ',  '#"



8x(i(B ) ;! i(A)(x)) ! (i(B ) ;! 8xi(A)(x))  ,    , *+        0  ,  . ." " !% &+ +      3%!%  !   #  1# "    (4) " 3.1, '" &'"  -        :

A1 : : : An : (11.1) B /  # ,  - B '  "'  '  - A1 : : : An. 64

"/

  "       {   % (   modus ponens): A A ;! B 11.3.

   &&*:

B

A(c) ;! B 9xA(x) ;! B

 - B  -  c  " x. ! 11.4.    A           !     ,       !  .       modus ponens ' -  . 4& #     &&* ""  #'% .. M =< M i > ' . '# M j= 9xA(x), .. '* ' m 2 M , ,  M j= A(m). 6"" .. M0 =< M i0 >,  i0(c) = m,   + #+ &3+ ' i0    i. ( M0 j= A(c),  #',  ' %, A(c) ;! B  &*",  M0 j= B .  i0(B ) = i(B ), .. M j= B . / ,  M j= 9xA(x) ;! B ,  #' M &  #,  - 9xA(x) ;! B  &*"  "/ 11.5.  ## A1 : : : An -  L   # " - An ,  - Ai (i  n) # & "  , & '  +-&'# 1 '%*+ - 0  #  "'     . - A   "  (& ` A),  '* ' + &  #  0 -.  - 11.2  11.4 "  '%* ! 11.6.    ,   !    !# 1.  ,  - #  #  "   # ", .. - -, +* # ,   "  . 2.   ## A1 : : : An -  L  ,  - Ai (i  n) # & "  , & '  +-&'# 1 '%*+ - 0  #  "'     ,   - 0  #   "  . 4& # #  0 ", '-  -"' -% Ai 0  #,  %*"' ",  #    # . .

.

65

3.  #,    #    0$$ ", .. ',  '  0$$  ( ",   ), '* ' ",   %&   # -       ",    0  ## # ". / '  ,  "         0$$ , .. '* ' ",  %*  %&"' -

-%    1  #    0 - "  ,  - ",  %*  %&"' &' (  &, " +) -,  # '    - 0 &  '+   "  .  # +  + -   #" " " (  1" ' &'  " 1"# &"    "). 4& #  &# " #  "#+    . -  # (11.1) ,   A1 : : : An { "  ,   B { "  . ! 11.7. ,"  "     : (1)   $  $ . A1 ;! A2 A2 ;! A3 : : : An;1 ;! An A1 ;! An (2)   $ %    . A1 ! A2 A2 ! A3 : : : An;1 ! An A1 ! An (3)    '" $ . A  B  A ;! C B ;! C A _ B A _ B (A _ B ) ;! C (4)    '" $ . A ^ B  A ^ B  A ;! B A ;! C  A B A B A ;! B ^ C A^B (5)   %      . A1 ! B1 A2 ! B2 : : : An ! Bn F (A1 : : : An) ! F (B1 : : : Bn)  F (p1 : : : pn) {    . 66

(6)   %     !  .  F G {   ,  ` F (p1 : : : pn) ! G(p1 : : : pn),  A1 : : : An {      

 L ,  F (A1 : : : An) G(A1 : : : An) !#. 1.     ,   ""  "-  # -  &    .  0"   '% "  . 2.  ## - A1 : : : An, " - Ai &   "  , & '  1 '%*+  "'   +   "#+    ,  " &&*" # ".  ,   -, +* &&* #   % ""  . 3.        3%!   #' 

A ! B  A ! B A ;! B B ;! A

  .  #    (1) "   '" n = 3. &* ' "  . " ,  A1 ;! A2  A2 ;! A3 { "  . " &&* #  - A1 ;! A3 . 1:(A1 ;! A2 ) ;! ((A2 ;! A3 ) ;! (A1 ;! A3)) ( ) 2:A1 ;! A2 ("   ' %) 3:(A2 ;! A3 ) ;! (A1 ;! A3) ('  1.  2.  modus ponens) 4:A2 ;! A3 ("   ' %) 5:(A1 ;! A3 ) ('  3.  4.  modus ponens)   (2)   . 6""      3%!: A ;! B A ;! C  A ;! B ^ C '# A ;! B  A ;! C { "  . ( '%*  ##   &&*" # " - A ;! B ^ C . 1:(A ;! B ) ;! ((A ;! C ) ;! (A ;! B ^ C )) ( ) 2:A ;! B ("   ' %) 3:(A ;! C ) ;! (A ;! B ^ C ) ('  1.  2.  modus ponens) 4:A ;! C ("   ' %) 5:(A ;! B ^ C ) ('  3.  4.  modus ponens) 67

#    3%!     3%!  % . -"   0   ".  #,  '%* ) (A1 ! B1 ) ;! ((A2 ! B2 ) ;! : : : ;! ((An ! Bn ) ;! ;! (F (A1 : : : An) ! F (B1 : : : Bn ) : : : )

(11.2)

"" , " -  &  ( - " !  3%!%  !,  -   ,) +"  0   (11.2) $"' (A1 ! B1 ) ^ (A2 ! B2 ) ^    ^ (An ! Bn ) ;! ;! (F (A1 : : : An) ! F (B1 : : : Bn )

(11.3)

-",  (11.3) {  . "" ,  + &   0   (Ai ! Bi )  -    " ! (11.3),   # " " ! .  -  0 0   ,  Ai  Bi "%      %&" i,   # F (A1 : : : An)  F (B1 : : : Bn ) - "%   .

0" '    % " ! (11.3) , ..  "  . ( #, "  (11.2) n    modus ponens, '" &'".  " - #    0  + & . 1:F (A1 2:F (A1 3:F (A1 4:G(A1

::: ::: ::: :::

An) ! G(A1 : : : An) (   ' %) An) ;! G(A1 : : : An) (   3a., ". " 3) An) ("   ' %) An) (2,3 - modus ponens)

 "  #  " #  ""      . .# " &'" # # "&' #% " !:

` (p ;! (q ;! r)) ! ((p ;! q) ;! (p ;! r)) 68

(11.4)

 1"  # #  - c = d ;! d = c. 1:c = d ;! (c = c ! d = c) ("    2) 2:(c = c ! d = c) ;! (c = c ;! d = c) ( ) 3:c = d ;! (c = c ;! d = c) (!  " !) 4:(c = d ;! c = c) ;! (c = d ;! d = c) (11.4    0 . &.) 5:c = c ("    1) 6:c = c ;! (c = d ;! c = c) ( ) 7:(c = d ;! c = c) (5,6 modus ponens) 8:(c = d ;! d = c) (4,7 modus ponens) -"  #  #   . .# "  #'" 0  #% ` (p ;! (q ;! r)) ! (p ^ q ;! r) (11.5) ( # #  " : 1:a = b ;! (a = c ! b = c) ("    2) 2:(a = c ! b = c) ;! (b = c ;! a = c) ( ) 3:a = b ;! (b = c ! a = c) (!  " !) 4:a = b ^ b = c ;! a = c (11.5    0 . &.) -"  # &&*  "   :  %& $"' F (x1 : : : xn)  L ` a1 = b1 ^    ^ an = bn ;! (F (a1 : : : an) ! F (b1 : : : bn)) (11.6) #   "  n = 2 #' +  n  n + 1  -   1  2. " '% "'   , '" ` a = c ;! (F (a b) ! F (c b)) ` b = d ;! (F (c b) ! F (c d)) ( #,  #'  % ` p ;! q ;! ((r ;! s) ;! (p ^ r ;! q ^ s))  "      MP, '" ` a = c ^ b = d ;! (F (a b) ! F (c b)) ^ (F (c b) ! F (c d)): ( #, "  % (F (a b) ! F (c b)) ^ (F (c b) ! F (c d)) ;! (F (a b) ! F (c d))  ! ' " !, '" ` a = c ^ b = d ;! (F (a b) ! F (c d))   & #. 69

 ,

12

6""  #   "#+    , %%*+  . ! 12.1.  x {       A(c) B (c),  c {       .  "  "     : A(c)  (2): A(c) ! B (c)  (3): A(c) ! B (c) (1): 8xA (x) 8xA(x) ! 8xB (x) 9xA(x) ! 9xB (x)    (1).  -",  A(c) { ".  1" #  8xA(x). 1:A(c) ("  ' %) 2:A(c) ;! (:A(c) ;! ) ( ) 3::A(c) ;!  (1,2 { modus ponens) 4:9x:A(x) ;!  (3 {   &&*) 5:(9x:A(x) ;! ) ;! :9x:A(x) ( ) 6::8xA(x) ! :9x:A(x) ("    ) 7:(:8xA(x) ! :9x:A(x)) ;! (:9x:A(x) ;! 8xA(x)) (  (:p ! q) ;! (:q ;! p)) 8::9x:A(x) ;! 8xA(x) (6,7 { modus ponens) 9:(9x:A(x) ;! ) ;! 8xA(x) (5,8 { !  " !) 10:8xA(x) (4,9 { modus ponens) -"   (2). 1:8xA(x) ;! A(c) ("    ) 2:A(c) ! B (c) ("  ' %) 3:8xA(x) ;! B (c) (1,2 { !  " !) 4:8y(8xA(x) ;! B (y)) (3 {   (1) 0 -) 5:8y(8xA(x) ;! B (y)) ;! (8xA(x) ;! 8yB (y)) ("    ) 6:8xA(x) ;! 8yB (y) (4,5 { modus ponens) 7:8yB (y) ;! 8xB (x) (" " ") 8:8xA(x) ;! 8xB (x) (6,7 { !  " !) 9:8xB (x) ;! 8xA(x) (#  ) 10:8xA(x) ! 8xB (x) (8,9 {   3%! { ". . 11.7) .

70

-"   (3). 1:A(c) ! B (c) ("  ' %) 2::A(c) ! :B (c) (   0  + & ) 3:8x:A(x) ! 8x:B (x) (1 {   (2) 0 -) 4:8x:A(x) ! :9xA(x) ("    ) 5:8x:B (x) ! :9xB (x) ("    ) 6::9xA(x) ! :9xB (x) (3,4,5 { !  0  ) 7:9xA(x) ! 9xB (x) (   0  + & )

-"  #,     -    (" 3.1)  % ""  .

!

   D   !      x.  "    "  , : 12.2.

(1): ` (9x(A(x) _ B (x)) ! (9xA(x) _ 9xB (x)) (2): ` 8x(A(x) ^ B (x)) ! (8xA(x) ^ 8xB (x)) (3): ` 9x(A(x) ^ D) ! (9xA(x) ^ D) (4): ` 8x(A(x) _ D) ! (8xA(x) _ D) (5): ` QxQyA(x y) = QyQxA(x y)

 Q   ! 8,  ! 9.   . #  (4):

1:(A(c) _ D) ! (:D ;! A(c)) ( ) 2:8x(A(x) _ D) ! 8x(:D ;! A(x)) (1 {   3 - 12.1) 3:8x(:D ;! A(x)) ! (:D ;! 8xA(x)) ("    ) 4:(:D ;! 8xA(x)) ! (8xA(x) _ D) ( ) 5:8x(A(x) _ D) ! (8xA(x) _ D) (2,3,4,5 { !  0  ) 71

#  (3): 1::9x(A(x) ^ D) ! 8x:(A(x) ^ D) ("    ) 2::(A(c) ^ D) ! (:A(c) _ :D) ( ) 3:8x:(A(x) ^ D) ! 8x(:A(x) _ :D) (2,   2 - 12.1) 4:8x(:A(x) _ :D) ! (8x:A(x) _ :D) ("  (4) * -) 5:8x:A(x) ! :9xA(x) ("    ) 6:(8x:A(x) _ :D) ! (:9xA(x) _ :D) (   0   ") 7:(:9xA(x) _ :D) ! :(9xA(x) ^ D) ( ) 8::9x(A(x) ^ D) ! :(9xA(x) ^ D) (1 { 7 { !  0  ) 9:9x(A(x) ^ D) ! (9xA(x) ^ D)(   0  + & )

#  (2): 1:8x(A(x) ^ B (x)) ;! A(c) ^ B (c) ("    ) 2:A(c) ^ B (c) ;! A(c) ( ) 3:8x(A(x) ^ B (x)) ;! A(c) (1,2 { !  " !) 4:8y(8x(A(x) ^ B (x)) ;! A(y)) (1 {   1 - 12.1) 5:8y(8x(A(x) ^ B (x)) ;! A(y)) ! (8x(A(x) ^ B (x)) ;! 8yA(y)) ("    ) 6:8yA(y) ! 8xA(x) (" " ") 7:8x(A(x) ^ B (x)) ;! 8xA(x) (1 { 6 { !  " !) 8:8x(A(x) ^ B (x)) ;! 8xB (x) (#  ) 9:8x(A(x) ^ B (x)) ;! 8xA(x) ^ 8xB (x) (7,8 {    3%!) 10:8xA(x) ^ 8xB (x) ;! 8xA(x) ( ) 11:8xA(x) ;! A(c) ("    ) 12:8xA(x) ^ 8xB (x) ;! A(c) (10,11 { !  " !) 13:8xA(x) ^ 8xB (x) ;! B (c) (#  ) 14:8xA(x) ^ 8xB (x) ;! A(c) ^ B (c) (12,13 {    3%!) 72

15:8y(8xA(x) ^ 8xB (x) ;! A(y) ^ B (y)) (14 {   1 - 12.1) 16:8y(8xA(x) ^ 8xB (x) ;! A(y) ^ B (y)) ! (8xA(x) ^ 8xB (x) ;! ;! 8y(A(y) ^ B (y))) ("    ) 17:8xA(x) ^ 8xB (x) ;! 8y(A(y) ^ B (y)) (15,16 { modus ponens) 18:8y(A(y) ^ B (y)) ! 8x(A(x) ^ B (x)) (" " ") 19:8xA(x) ^ 8xB (x) ;! 8x(A(x) ^ B (x)) (!  " !) 20:8x(A(x) ^ B (x)) ! (8xA(x) ^ 8xB (x)) (9,19 {    3%!)

#  " (1)  (5)  % %   ' -.  ! 6" 1  "#       #  "   %,    "  "# -  & ,  #'%*    -    ,     -    ,   # ' "  . &' # '%* "  : .

.

1: ` (8xA(x) _ 8xB (x)) ;! 8x(A(x) _ B (x)) 2: ` 9x(A(x) ^ B (x)) ;! (9xA(x) ^ 9xB (x)) 3: ` 9x8yA(x y) ;! 8y9xA(x)

x 3   "/ '# ; { "-  -  L .  ,  - A    ;  1' ; ` A,  '* '   "-  - A1 : : : Ak 2 ;,  ` A1 ^    ^ Ak ;! A. !  ` A,   "!   ;    ;`A    '# B 2 ;  2.

.

12.3.

12.4.

.

` A ;! (B ;! A) ( ) ` A (  ' %) ` B ;! A modus ponens  "     &'"  73

! 0

 ; ` A.

12.5.

 ; ` A  "! B 2 ;   ;0 ` B ,

  . '# B1 : : : Bk 2 ;  ` B1 ^    ^ Bk ;! A. '#, (1) (k) (k) (i) (i) 0 - A(1) 1 : : : As1 : : : A1 : : : As 2 ;  ,  ` A1 : : : As ;! Bi i = 1 : : :k. "" k

i

(k ) (i) (k ) (i) (1) ` A(1) 1 ^    ^ As1 ^    ^ A1 ^    ^ As ;! A1 ^    ^ As i = 1 : : : k #' 0  . 2 # (k) (1) (k) ` A(1) 1 ^    ^ As1 ^    ^ A1 ^    ^ As ;! Bi i = 1 : : : k   ' !  " !. ( #, " #     3%!, '" (k) (1) (k) ` A(1) 1 ^    ^ As1 ^    ^ A1 ^    ^ As ;! B1 ^    ^ Bk , !, (k) (1) (k) ` A(1) 1 ^    ^ As1 ^    ^ A1 ^    ^ As ;! A   ' !  " !,     &'"  &  ; ` A1 : : : ; ` Ak () ; ` A1 ^    ^ Ak  "/ ,-  T -  L   $"# ,   "' #  ", ..,  8A 2 L"  T ` A =) A 2 T . k

i

k

k

k

12.6.

12.7.

.

& 



12.8.

A1 ^    ^ Ak 2 T

!

 T {    ,  A1 : : : Ak 2 T ()

 ; {       ,  T (;) = fA j ; ` Ag,  T (;) {   (    ,  12.9.

;.

  .  ,  #,  T (;) { $"# . '# T (;) ` B . #',   % T (;) %& - C 2 T (;)  "  ;,  ' - 12.5 ; ` B , .. B 2 T (;)  "/ 12.10. ( T      ,   '* '  - A,  T ` A  T ` :A.  " '  T     . ,-  - ;      ",       T (;). 74

!

  T       ,      "!   A 

 L . #      ;       ,  "!   A 

 L    ;.    ,   T -  -,     . '#, &,  T     - A  L  ,  T ` A  T ` :A, ..   % 12.6 A ^ :A 2 T  %& - B 2 L ` A ^ :A ;! B , .. 0  . ( #     " ',  B ` T , .. B 2 T  ! 12.12.    T   , A 2= T :A 62 T ,      T fAg T f:Ag    . 12.11.

.

.

  . -"    # T fAg   . '# T fAg ` B  T fAg ` :B . ( '* '%  - A1 : : : Ak 2 T ,  ` A1 ^    ^ Ak ^ A ;! B   - B1 : : : Bl 2 T ,  ` B1 ^    ^ Bl ^ A ;! :B . '   12.6 C = A1 ^    ^ Ak 2 T , D = B1 ^    ^ Bl 2 T  F = C ^ D 2 T . #' ` (C ^ A ;! B ) ;! (C ^ D ^ A ;! B ),  F ^ A ;! B 2 T ,  F ^ A ;! :B 2 T .  ` (F ^ A ;! B ) ;! (F ;! (A ;! B ), .. F ;! (A ;! B ) 2 T ,  .. F 2 T ,  A ;! B 2 T .  A ;! :B 2 T .  ` (A ;! B ) ^ (A ;! :B ) ;! :A (     %), .. :A 2 T .  .    # T f:Ag

   

"/

1.      T   ""#,   %& - A 2= T "-  $"' T fAg   . 2.      T   ,   %& - A & A 2 T , & :A 2 T . !. ,-  - ; ,    T (;), ..   %& - A & ; ` A, & ; ` :A. ! 12.14.   T        ,     .   . '#  T ""#  A 2= T .   0" :A 2= T ,   -% 12.12 "-  $"' T fAg    ,    ""# T . '#, &,  T .  -",    ""#. ( '* '  - B 2= T ,  "-  T 0 = T fB g 12.13.

75

   . '  T :B 2 T , .. :B 2 T 0 ,    % B 2 T 0 , .. T 0   .  $ 12.15 (   ).        T         .     "   "  " "" 8. &"  T "-  +    +   L . / "-   '  1" %   ""# 0" #    "-  .  " "" 8  #,  %& ! # S  T " +%% #.  0,  % #,  #,  S T {     .  -",  T0 {  T0 = T 2S  . ( '* '% - B  - A1 : : : Ak 2 T0 , ,  ` A1 ^    ^ Ak ;! B ^ :B ,  0 ,  "-  - fA1 : : : Ak g {   . #' 0 "-  - T0 ,  '* '%      T1 : : : Tk 2 S , ,  A1 2 T1 : : : Ak 2 Tk .    '  S ',    T1 : : : Tk " &#1  %%,  ", Tk . (, fA1 : : : Ak g  Tk , .. Tk {   .  0"" S  %       ! ,  1#,  T0 {     "-  $"'. /   #  "" &'". , " '-" - #,  T0 { $"# . /   %   ' -. !. '# K {  &+ " ' . (   K { Th(K) # $"# . "" .  Th(K) ` A,  '* '%  - A1 : : : Ak 2 Th(K),  ` A1 ^    ^ Ak ;! A. 6""  #'% .. M 2 K. #' M j= A1 ^    ^ Ak  M j= A1 ^    ^ Ak ;! A (  &*" $"'),  M j= A, .. A 2 Th(K).    ,  Th(K)  Th(M).   %& .. M  Th(M) , #'  %& - A & M j= A, & M j= :A. .

.

 , x $ 1  /    1 13

3.

.

'# T { $"# .  ,  .. M   "#%  T ,  8A 2 T M j= A, .. '"  ", T  Th(M). C   M { "# "-  - ;,  M { "#  T (;).  0   ,   $"#  " "#,   76

    ( #'  %& "      ). & ' -     & -+   + " "" . $ 13.1 ($ 1  /).           .

#  0 " " "  -",  '   - $'!#+ "  #', -  +,  $'!#+ "  '   1#  "' +"' '-% # ,  &' +  + ,  - +,  &3# ! 5,  $'!#+ "   "- #,  & #+ "- $"# ,    # " & ", ,  -,  "   #  " . , '# T {      ' . - $"'    & " A(x)  L  "    '% ' cA ,  +*'% '' , & "   ',    T & "  -  9xA(x) ;! A(cA ). &" ''% %  T .  13.2.   T   .    -" ,       T 0 , ' & "  T #  - 9xA(x) ;! A(cA ). -",  T 0   . ( '* '  - U 2 T ,  ` U ^ (9xA(x) ;! A(cA ) ;! B ^ :B   - B  1 '.   ' 0  + &  ( - 11.7(6)) ` U ;! ((9xA(x) ;! A(cA )) ;! B ^ :B ) ( #,  #'  % ` ((9xA(x) ;! A(cA)) ;! B ^ :B ) ;! :(9xA(x) ;! A(cA ))    !  " !, '" ` U ;! :(9xA(x) ;! A(cA )): '# " y,  +  -. (  -% 12.1(1) ` 8y(U ;! :(9xA(x) ;! A(y))): .

77

"  "' -% -  &    (". ."  #  - 13.2), '" ` U ;! (9xA(x) ^ :9yA(y)) .. 9xA(x) ^ :9yA(y) 2 T ,   "-, #' T    . ( #,   T   ,  '* '  "-  - B1 : : : Bn,  - Bi "  9xAi (x) ;! Ai(cA ),     - U 2 T   - B ` U ^ B1 ^ : : : Bn ;! B ^ :B , .. '- "-  $"' T fB1 : : : Bn g   .  0  "-, #'  "' 1 "-  T fB1 g    ,  # "-  T fB1 g fB2 g    ,  ..  "  #  ##  T0 = T Tn+1 = Tn n = 1 S 0 1 : : :  -" Te = Tn . (  #,  Te {     n=0 . "" ,  0    ,  9A1 : : : An 2 Te, ,  ` A1 ^  ^ An ;! B ^:B   - B ,    % '* '  m,  A1 : : : An 2 Tm , .. '-  Tm   ,   "- ' "" 13.2. &"  M { "-  +   '  Te. .""  #1,  i

jMj  maxf@0 jjg:

(13.1)

"" ,  i i 2 N { '   Ti ,  "*# "-  -  L   maxf@0 jjg (". ." 2   % 5.1), ..  i + 1-" 1  ' &   &, " maxf@0 ji jg . / ,    1- 1, jL j = maxf@0 jjg, ..   "%  '% &'% "*#. . '% - "*# "  &3 + 0+ "- , ..  "-  + +    + 0 "*. '# S {  , -* T (". ""' &'").  "  "-  M 1 , - i

i

c  c0

c = c0 2 S:

-",   { 1 0  . #' c = c { "   ,  c = c 2 S , .. 1  { $ . - c = d ;! d = c { "  (". ! ! 11), ..  c = d 2 S ,  S ` d = c,    d = c 2 S . (" ""  ""# 78

1 .    # " "  a = b ^ b = c ;! a = c    # . '#  # M = M= . / "-   &'  # " "   T .  ! i      " &":  c 2 M,  i(c) #  0" c  1% .  P { n-"  " ,  -" i(P )(i(c1 ) : : : i(cn)) =  () P (c1 : : : cn) 2 S: -" - ,  0   . '# c1  d1 : : : cn  dn  i(P )(i(c1 ) : : : i(cn)) = , .. c1 = d1 : : : cn = dn P (c1 : : : cn) 2 S .  #' (11.6),  # '%*'% "'  : ` c1 = d1 ^    ^ cn = dn ;! (P (c1 : : : cn) ;! P (d1 : : : dn)): % " ',  (P (c1 : : : cn) ;! P (d1 : : : dn )) 2 S ,   #  P (d1 : : : dn)) 2 S , .. i(P )(i(d1 ) : : : i(dn)) = .  i(P )(i(c1 ) : : : i(cn)) = , ,   %, P (c1 : : : cn) 2= S: ( '  S :P (c1 : : : cn) 2 S: ( #, '- -,  '*" &!, '",  :P (d1 : : : dn)) 2 S , .. P (d1 : : : dn)) 2= S , ..

i(P )(i(d1 ) : : : i(dn)) = .  , & #,  .. M = hM ii   "#%  S ,   #   T ,    # '%* ' -.  %& $"' F (x1 : : : xn )   %&+ c1 : : : cn 2 M

M j= F (i(c1 ) : : : i(cn)) () F (c1 : : : cn) 2 S: (13.2) / ' -,  &, &'"  # '!  % $"' F . ."",   "+ $"'     '   ! + "  . -", ,  (13.2)   F1   F2 ,     F1 ^ F2 .

"" , M j= F1 ^ F2 0   "',  M j= F1  M j= F2 .   -%, 0 0   "',  F1 2 S  F2 2 S .   # ' % F1 ^ F2 2 S (  12.6). -"  #, ,  13.2   F ,    -  :F .  M j= :F ,  F - M , ' (13.2), F 2= S , .., '  S , :F 2 S .  :F - M,  M j= F ,  #'  F 0  # (13.2)  ,  F 2 S , .. :F 2= S . '#  # F "  9xG(x x1 : : : xn)   G ' - (13.2) '- . 6""  #  c1 : : : cn 2 M.  -",  M j= 9xG(x i(c1 ) : : : i(cn)).   %  $"' 79

", 0 ,    c 2 M M j= G(i(c) i(c1) : : : i(cn)), .., ' (13.2), G(c c1 : : : cn) 2 S .   #,   %& $"' A(x) ` A(c) ;! 9xA(x). "" ,  #'   -    (". "  #  - 12.2), '",  A(c) ;! 9xA(x)  8x:A(x) ;! :A(c)  -, *   , #   "    . (" "" 9xG(x c1 : : : cn) 2 S . '#, &, ,  9xG(x c1 : : : cn) 2 S .  %   n,   - - '  Tn ,   '* ' ,  c,  9xG(x c1 : : : cn) ;! G(c c1 : : : cn) 2 S , ..  G(c c1 : : : cn) 2 S '   ` p ^ (p ;! q) ;! q. (#   -% '! M j= G(i(c) i(c1 ) : : : i(cn)),   M j= 9xG(x i(c1 ) : : : i(cn)).  , " #+ +      &* "-  '#, #'  -%  '- ".  - (13.2),   "  "   , . C  13.3 ($  { &  ).    T       ,     ,       maxf@0 jjg:            (13.1). &  13.4.   !      

      , .    '# - A &*",     "  . ( "-  f:Ag    . "" ,  ` :A ;! B ^ :B , ,  #'  % ` (:A ;! B ^ :B ) ;! A    MP, " '",  ` A.   # M { "#  :A,  ' &*" A,  "   M j= A  M j= :A,   "-  &  13.5 ($ /   #.. 2 ).            ;  ,   ;  .     ;  " ",  ;   . 2 # '* '%  - A1 : : : An 2 ;,  ` A1 ^    ^ An ;! B ^ :B   - B .   '- "-  fA1 : : : Ang   ,   "- #'  " "#   "-  G  .

.

80

 #  " "  ..,#!  " " "  # ,   &,    #'  #      &+ ". "/ 13.6. '# K   &+ c" ' , "' # " (.. -* "  - "   "). 2   P &+ "  K   #",   ,    - " " M 2 K &%   " P , ',   " " M & 0"   ".  ",  K {  ' ,     ""'    #", #',  -  - '  '  G ""' ,   %&+ a b 2 G ' , - fa bg ""' , ..  %&+ a b 2 G      a  b = b  a. 1" ""  #         ' &#  -", .. -  - '  %& '   -   %. 2    -    #", #' '* '% & ' , ' +   - '   ( '   % # "). ""  '  '- '  + +   "-  N,   %   -"   N, "   (-!). *  "  #    { 0    ' &# !": -  - '    '   Q { ! (-!),  " '  Q   !   . ..,#! &  ",   "-# " "

    #  # +   &+ ".  &,  '&  1941 ' +  +    . ',    # "& " &*" " ' #+ "  ' ". / & -    "" ! ,  -   &  , {  ".

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.. '   1   0" Zm  ,   .

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-  "-  H  ; " "#. "" .   " 1#  "-   fcg1 : : : cg g, +*+ -  H. 6"" ' ' H '  G, -'% 0"" g1 : : : gn.  ' % 0 '  '  ".     ' -'% ' cg  0" gi, "   '",  H "#  H. n

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82

( #  " "  ,#!    "-  ; " "# M = hM ii. #' M j= LOAG,  M {  '  &  ' .  " &- ' : G ! M , - '(g) = i(cg ).  g1 = 6 g2 ,  M j= :cg1 = cg2 , .. i(cg1 ) 6= i(cg2 ). /  ,  ' 3 . ,  g1 + g2 = g3,  M j= cg1 + cg2 = cg3 , .. i(cg1 ) + i(cg2 ) = i(cg3 ). /  ,  '(g1 + g2 ) = '(g1 ) + '(g2). (" &" ' { 3  ""$", .. '(G) "$ G,  '(G) ' ,  '  '  '  M . 2 #  G '  "  0   1.    " " # ' &#1  "*,   "  & ". 2.  -&'# -     + # ' &#1  +,     "  + #.

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   ,  "- #  "  " { 2" { (  ",    & "  "- &#   " "- " -   #%  "$" "" ,  M = hM ii { & " ' ,  T = Th(M),   "  " { 2" { (   # "* k > maxfjj jM jg  "# M0 = hM 0 i0i  T , "*#    k. ' ,   T , Th(M0 ) = Th(M),  0 "  "' &# "$", #'  "%  "*. 2'! ",  1#    ,  ' " + &  "+   1 "#    0" "".  0"   # 0 " '   ' "    & ". (     "  . 4%   "-# "# $"' # $"#         !,  " " " "".    ' +  1#    ,  " "- $"' # "'  + . 7'" &#  "  " " &' ". #' "  - %  "- " ,  "-  #  + +  -  " . ( $"' t = sup X  1 '%*" &":

8x(X (x) ;! x  t) ^ 8s(8x(X (x) ;! x  s) ;! s  t): ( # "  +   1 :

8X (9s(8x(X (x) ;! x  s)) ;! 9t(t = sup X )):  ' ""   ,  %& '  , '  %* "  + , "$ % * + . 0   . . # "     $,  &#   X # "-  '#+ .

 *#      ( 1" ' ' #  ), "",  "  " { 2" { (    '!,    "  , "*# + $ , "$. "/ 14.2.  ,   T   "* k,    " "* k "$. 84

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)    T  

         ).

    k  maxfjj @0g,  T  .     T  ,  '*c '  - A  L ,  A 2= T  :A 2= T . 0" '  -% 12.12 "-  ; = T fAg  ;0 = T f:Ag    .  "  " { 2" { ( '* ' "# M  "-  ;, "*#    k  "# M0 "-  ;0 , "*#  -   k. #' & 0 "  % ""  T ,   ' % 1 "  "$. (, ' - 6.9, 0 " 0" 0  ,   "-, .. M j= A,  M0 j= :A  !. 1 .   '%* $ 5  . "!                 

#  0 " "- ,  ",  9].   ',   Th(DL)  + + .'. "- , " ! 7,   "* ,  #, .  '* ' 0 $ & "  ! 7  "* %   , #'   $,  %&   "   0" 0   (". - 7.7    )   '  . 2 .     $ 62 .  !                      . % ',   & "'+   %&  "* ,  #, . " 0  (". " 3 ! 5) # "   " &  " ACln  %& n 2 N  & " AChar(p)    p, &  " AChar(0)n, n 2 N. "",  '!,  1 0+  '+ "+,   &*.  "  '%* $ .       

  !                 ,        "!     . "",   & "'+   "*  . "" ,  &+  { & "  Q  "$ &"' "%,  ",  Q(e), ' "  Q  e, #'  "  0+ .

85

  0" &,   " { . 2 '  { & 0  .

  

1. /.,#.  ""'% '.    " . { ,.: ', 1971. 2. .. "#.  % ' + $'!. { ,.: ,, 1983. 3. .7', 6.-$$. "#  . { ,.: ,, 1994. 4. /./. ,"" 0" "". { ,.: ,, 1987. 5. .0. ( "-  ''"- . { ,.: ,, 1969. 6. .. , .,.  . ,*# &+ "- . & &. { -   , -  , 1998. 7. .. , .,.  . /" ""    "- . 4# 2. , &. { -   , -  , 1988. 8. .. , .,.  . /" ""    "- . 4# III. , &. { -   , -  , 1990.

86