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" # $ $ $ $ $"$% & $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ' ( " & " ) $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ * $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ + , & $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ . 0 2 & $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ '/ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ '1 3 & $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ *4 & $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ *4 ! 3 & ) , 3 & $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ *4 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ *- & $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ 45 2 3 & $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ 46 & $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ // ) , 3 8 & $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ /7 ! $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ 6' 9 ! 2! $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ 64 & $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ 6+ Introducción , , ; & , , <= & & # , ! , & & , $ , , ? , & , ! & # = = <= % ! . , ! , , & < > , ! , ! " "%$ ) ; . , , & , : ! & , :2 , , . ! $ 0 ! & . . , , $ @, # &$ A , & &< & ( #!& . @, ! & & ! %$ %< & <, &$ :': , , @ & & Lenguaje Formal de la Lógica de Predicados ! ! , & = < , & , = ) ! @ A & , , , , & ! $ ! < & , & , , :# $ 2 , ! !& # ! ( 'A • ( • • . & ! , < $ < . ; , & , ! ! '% # %<, ! '% ) ! , & ! ∈ L ( , A ! '% , ∈ & ; < & ! , ∈ L , ! ' . , $ L ( ' & . & ∈ A ! & L . $? ( %$ ? , . $ ( ! ! % ( ; $ . ( % ! ! ( , A ! &% !" # & $ ? ! # ! $ & ; . ! " #, & ! ! , , = ( : , ; $ . <, & (<! , ! , & $ 2 , $ , ! $ :*: ; ( , $ A • ( ; & ! , '% ? , K∈ ( <, %? , , ∈ ( , A ? < , $ A ¬ ∧ ∨ → ↔ %? ! A ∀ ∃ ? () A) # ! )A $ , " , ( % : $ L :? ! L & $ : , , ; $ # ! 4A * " ) ! A ! , *" B . $ , ; ( < , &$ :4: . ! # ! +% # ! " ! ! " : !& :? ¬ ¬ ( &! !' ! !& :? !& ! " 'A ! " / " % % # ! " ∀ 4 ! - ! ! "* ! =" " 6A @, ! " , # → E! ! 4C , ∀ <∃ < !& *" & ( &$ +A ! $ ∃ ! - !! # ! ! $ D ! % A ∀ # ! " ! !& !& ! " ! ? ∧ ∨ → ↔ , !& ! < . ! ! , & $$ & :/: . ! =" . " *A " %∀ 0 !! # : , ! ! & <, < $ , 1 ! ! -A ? # ! . $ < $ !& ! ! " , $) $) & ! ( , < ! , , , !& ∃ !→∀ !! < $ ! & !& . . A ∀ 0 F <, & <, ! ! . G :6: , ! ∃ < Otros Ordenes = ? . =. <@ & , ( , & # , , & , , <! ! . & <, ( , < &A < < @ ! ! & ! "# & , %" & %$ ) ! $ , 4A ! ! & ! , . . # ! . < ! & ! A ' & % ! ! & @ % ∃∀ ∃ % & ! +% ! A∃ ∀ ∀ " ! ! $ ( , , $ & 2 . , /? $% !& , ( A !! → "# $% , != & ! #" !" & $ " ! !! ∧ , %$ < , ! ! ! → "# < , , $ ? , , , H ∧ → H < !" . , , " ? <@ < ! !& ∀ , B " $# , $ $ ! , !→ . ( !! , &< , !& ! , ! 6 & ?, & !& , " , @ , ! %< & % , & ) , , " $ & ) ) :+: , ) , $ / Interpretación , #F % ! & !& $ ) . , !& !(< . , , $ , &< . , $ ! ! < ! ! !& , : $ ? , ( & !& < # " ! → 6 × × % , & ∈ & , →{ & , $ & ! $ :2 × ! ∈ :2 × × ! < <, :2 × & $ 1A # ! ! " < , #V% B ! , * ! & } , , & , & , &$ , , ; :-: $ < # " ( ( ) ∧ ( ) → ( ( ))) ∧ =∀ 6A ? " , ! 0; % ) = = '% != != != % != , & H >' & # != ( . , % != $ !& ; != $ != 8( A A . ; , 7 * . H C . ( , & A H ! ! 7A # ! , !+ , :? != ' , & K ,&I ( !& ' , . K @, ( ' K , $ 0 ! , " $ . #" #" != ¬( & K ' ! ! -A ! " ! -" ( ! ! + & ! , :? ( , < , . H$ !& < & ' (! = (! = !& K $ & , = $? ( @, <, ! & & , $ . <. ! :1: , B $ < :? ! (∧) != #" (! = )! = :? ! (∨) != #" (! = )! = :? ! (→) != #" (! = :? ! (↔) != #" (! = :? ! #" != #" != # , , !& " , *∈ A ! ( * !! = $ *∈ <( * ! ! , % + , <, !& $ 0& ? , , 5< . , ∃ ( ! 3% & !& !& , ={ A A , , =∀ ∃ A & !< = = < ( ( } = = ! = )− != & , , , A =. , ; $ )! ! ( * !! = ! :? ! ∀ ( )! = + + ! *+, + :7: ! &$? ! , , , #, -%$ ) ∧ ( ( ))) → ¬ ( ( < & , ! & < @ ! )) < ) ! =" ≤ " A ! =" = ) @ . = " !& )∧ ( ( , , , & ( ( ))) → ¬ ( ( ! )) =. , , 045% ? % ? < . < <, . ( ( ' ' ( , < J'$ FA , ) ∧ ( ( ))) → ¬ ( ( ' ! )) ' 4 4 ' 4 * % ) . , J' , < ( ( ' * $) ) ∧ ( ( ))) → ¬ ( ( ' J* ! )) ' 4 4 * * ' V 2 =. % # $ :'5: = A % ) 04)% ,45% ( ( ) ∧ ( ( ))) → ¬ ( ( * ' * ! )) ' 4 ' * 4 * # = % ) 046% < =. ; % ,45% ( ( ) ∧ ( ( ))) → ¬ ( ( 4 ' 4 ! )) ' 4 ' ' 4 * = !& ( , , & :'': . !=V = V !& < " =∀ ∃ !& 7% ={ A !! ∧ , ! & A } = A A . ! / 0 0 1 1 / ) = ={ A ? , !& V # ! 58% # ! $ 55% " , $) } . < < ! = F$ , ! # ! , ; , , < . ! = V$ , " & !& # ! @ !& * @ = <, & !& . ={ !}< ! & , =. $ 5)% 56% & . !& @ ! !& ! $ :'*: , & . @ ! !& < . ( " 5A # "A ? ¬ !=F, , . A ! !& ! , $ ! V !=V , . &< @ :'4: , , < & ¬ & * . , # $ %$ ) , !& !& ! , & $ & $) . ! = V$ < -. { % } % # 0 1 != != = /. ! / + #, !=V 2 "/ , != " 3 + V V !/ 4 { , { } } % . / 5 / & # "% ∧ 5$: ? → ∧ . != ∧ ) 1? 2 ∧ & " )A ∧ != ∧ != = ∧ ∧ ∧ != A ∧ ∧ != . != → $ , & ! = F$ ! = F$ * != $ :'/: . , . , . → → = ∧ → @ ∧ ∧ , ∧ ! = V( 59A ? . & ⇔ % , # ! ≡ , !& $ , & < 6 # " != , . ! # # / / . 2 2 # → A " ≡¬ ∨ ∧( ∨ 6 1 ≡( → ↔ A " )≡ )∧ ( ) → ∨( ∧ ∧ F≡ F ∨V≡ V ∧V≡ ∨ F≡ ∧¬ ≡ F 1/ ∧ 2 7 ∨ / 7 /6 7 /, 8 ∧( ∧ ∨( ∧ ∨¬ ≡ V ≡ ≡ ∨ ∨ )≡ ( )≡ ( ¬( ∨ , * ∧ ∧ 2 )∧ ∨ )∧ ( )≡ ¬ ∧¬ ¬¬ ≡ , )≡ ! :'6: ∨( ∨ ∨ ) ∧( ∨ ¬( ∧ ≡ ≡ ∧ )≡ ( )≡ ( ∨ ∧ )≡ ¬ )∨ )∨ ( ∨¬ ∧ , # : " ¬∃ ( )≡∀ -, ! ∧ ¬ %,& % () ¬ ! ;% A ¬∃ ! ! 2 & . . AH ! A ∀ ∀ !≡∀ ∀ ! ∃ ∃ ∃ ∀ % , , ∀ ∃ ! ( !) -, ! → %,& !≡∃ ∃ &A3 ! ! ! , ; ! / ∃ ∀ ∀ ∃ ? !) ≡ ∀ H$ " < % A ! ) ≡ ∀ (¬ -, ! ∨ %,& , ( ) ¬ !! 2, ∀ ¬( -, ! ∧ ¬ %,& ( )≡ ∃ ¬∀ 4* < . !$ < "4 A % $ $ & % = ∃ < ( ( ) ∨ ( )) ≡ (∃ ( ) ∨ ∃ ( )) %0 ∃ ∀ ( ( ) ∧ ( )) ≡ (∀ ( ) ∧ ∀ ( )) , , ( ( ) ∧ ( )) ≡/ (∃ ( ) ∧ ∃ ( )) <, ( , <, A ' 5 0 4 " A , & 5 . 4 ' 5 ! 3 6 ! A∀ ( ( ) ∨ ( )) ≡/ (∀ ( ) ∨ ∀ ( )) < ( 4 ! $ $ , A 4 $ :'+: . ! = % # ∨∃ ( )≡ ∃ ( ∨ ( )) ∨∀ ( )≡ ∀ ( ∨ ( )) ∧∃ ( )≡ ∃ ( ∧ ( )) ∧∀ ( )≡ ∀ ( ∧ ( )) , % :'-: Formas Normales 53% ? # ! ! . !& " < " 0 & < ' . #∀ ∃% ! !& . L " !& / 2 & . !& # , , . . . @ !& ) 0 , ! , @$ ? $ , ) !& , !& 0 , @$ & # "% & & . ) . . 0 % !& " 6% ) !& . 0 < ! $ $ ! ! > ! ! !& !& < @$ , A :3 ! : & # ↔ %< , : , &A :2, & : ) ! $ $) & < 2 ! → (! ∧ ( → ! → ( ≡ ¬ ∨( C !& ! ↔(≡ $ ! & , ! . $ , ! !& 0 ( &A :'1: , 0 ∧ /" / ∧ / ∧L ∧ / !& "= # ! !& 57% % !& # ! ∨ /" ¬ K !& 5;% K & & / ∨ / ∨L ∨ / !& "= ! # !# % " < " & < ' " ( & ! # ! ∧ ∨ ! "= < "' '= !& 5>% $ "' " < " # , & <#' " & ! " +% ) 0 . # $ # "% ( "= . "' '= !& # ( !& < 0 , . " 9% # "% ? 0) < , ! , !& 0) ! / @ 0 $ . ' 0 ( % !& & !& 2? " & <2' ?I , & . $ & I !& ! !& ! , # 4 ! . & ( " < " $ !& 0 , , 0 !& )5% . !& C " < " 0 !& 0 ) ! ) ! < !& % . . # ! 0 ) $ "' %$ )8% !& 0 ) # 0 )%< ) @$ # ! 2 @ ∨ ∧ ! $ :'7: & , , ! @ " & ? "/ '$: ? ' , ! *$: ? &<! 4$: ? ! , ! B " , ! !& 4< ∀ ∀ ∃ ∀ ! . 0? !& 4 / $ ! @ < & ! ! ! $ , <. !$ !! $ ∀ ∃ ! JH' ∃ ∀ ! H ! J H! ! . !& ! . ! ∀ , $ . /< ! & A H7 " J ∀ ∃ ! , ∃ & = @ !! $ ?I ! $) ! ! . < " , , ∀ & !& . ! ! %< , < , ) @ # 0 )%$ ) @ ; , ( ! , < $ ? < !& $ ∃ ! ! %$ 0 , $ '$ . # "% C 0 , ? , # ! ! & A∀ ∀ ! 3% & ? , #. ! %$ K !& . '$ ∀ ∃ #% , B , $ @ , /$: ? @ A ∀ ∀ ! @ # @ $ & , $ !$ ? A . . ! 7 " $ & , 7 " $ :*5: . # ! !& ))% ! ( " % ) :: , >% $ & ! ! J@ < = !& ) (() ∀ (¬() 5 8" ! ! : J@ :+ J@ , ( A !∨ ! ) → (∃ ! !& ( ! ) ∧ ∀ (( ( !∧ )→ < ! ))) , A A !∨ ! ) → (∃ ( !∧ ! ) ∧ ∀ (( ( ) → ! ) ∨ (∃ ( !∧ ! ) ∧ ∀ (¬( ( ) ∨ !∨ *$: ! ))) ! ))) $ ((¬) ∀ ∃ ∀ ! ) ∨ (∃ !∧¬ 4$: ) ! ((¬) !∧¬ /$: ) ( !∧ ! ) ∧ ∀ (¬( ( ) ∨ , , !& ! ) ∨ (( . $? ! ))) 0)$ ! ) ∧ (¬( ( ) ∨ !∧ 0 <, ∀ ∃ ∀; 1 & , → '$: ∀ A ! J@ '$: 3 ∀ ! 9 $$ :7 "J@ K (() % , ;$ A :- : 2#@<%J @ , ∀ 0? " , , ! ))) $? 0 )$ ¬: ∨ ∧ ¬: ∨ " ∧ ¬: ∨¬+ ; ∨ 7 " ; ∧ ¬- ∨ ∧ ¬- ∨ " ∧ ¬- ∨¬+ ; ∨ 7 " ; 0?< , $ :*': ( 6$: K ! @ $? ¬: ∨ ∧ ¬: ∨ " ∧ ¬: ∨¬+ ; ∨ 7 " ; ∧ ¬- ∨ ∧ ¬- ∨ " ∧ ¬- ∨¬+ ; ∨ 7 " ; ∀ ∀; +$: ? , < ! ( < $ ¬) !∨ !! ¬) !∨ !! ¬) ¬ !∨ !! ¬ !∨ !! ¬ # ! , 0? L )6% < ! A ! ∨ ¬( ( ) ∨ ! ! ∨ ¬( ( ) ∨ ! . ← = , $ ? = '< A L H← H H B( , ! ( H # ! $) , ❏ )+% ? ! &< ! :**: . & Algoritmo de Resolución Introducción 3 , , & ( %$ ? '7+6$ ? # , ! ( , . & , & , , $ , < $2$ < ! , , , 8 , , , ! , , , < , $ , , , & < , , ! , & & $ ; , A ! 5 $) < # 5 / Resolución Proposicional - 2 " B $ $ )5A ? B (H$ ? , AH < , & 6+% ¬& ∈ < A & !=( " " %*- -, $ C- ) . A H" ) A ∨ ∨ & ! . { } = & ∨& ∨& ∨ ∨& # "% ? @ . &∈ , − {¬&}) $ ? $ B (H$ ∨ . " 3& C- A ( → ∨ ≡ ¬( ∨ ∨ # ! − {&}) ∪ ( - , H H? , A (∨ ∨ ! ! " $ * ( '% = ¬& ∨ & ) ∨& ∨ ∨& $ , & :*4: ! =& ∨& ∨ & ∨& ∨& ∨ ∨& ) ∧ ( → & ∨ *< , { . ) } $ ? ,& ( . ) @ '' ∨M ∨ ∧¬ ∨ ' *' . < < !& ∨M ∨ → * '' < ∨ ' *' ∨ * , , < $ # "% ? ; <, < !& , & < ∧ A * →❒ < ∧ $) , ( & <, ! ❒ . . $ & =. < ( ! . $ <, $? ! $ < > , ( $ " ( % 2 7 ! 7 % ∈ '$: 8 *$: ? 6$: ? " ! & /$: ? 4$: ? . @ . , . &∈ ¬& ∈ A 4$: , ∨M ∨ * ! @ & . . E & < !& $ * " 7% ? < , = A D ) . , > !=❒ & <2> & ( . 2@ C ! 7 B A 2@ C ))% ? 7 . 7 7 ! ' . ! 7 7 ! { ( , &$) :*/: ¬ ∨ A ¬ ¬ ∨¬ ∨ }< #¬ % :? :? = :? = ) # ¬ ∨ ¬ ∨ %< #¬ ∨ % ! , ! 74O$ 2 <@ N?, < $ , , & , , $? , & , 0 ! & , N8 :2 ; , ! ! = )6% ? " ❏ <7 & $ A¬ , . 2 ¬ ∨¬ , & 7'O ( 7 ( ! ! = { ¬ ∨ ¬ ¬ ∨ ¬ ∨ } <, ( . !& < ∧(¬ ∨ ) ∧(¬ ) ∧ (¬ ∨ ¬ ∨ ) $ , A . < ! $ ¬ F ¬ F ¬ ∨ ! ¬ F ¬ ∨¬ ∨ ! F ¬ ! " 5% ? ( , ! . #, $ , # "% ) <* ! $ ( , . ( . $ # "% ) " +% ! ! ! . = < ! ! 7 " 6% , # ! =( ! < ! < " )% * ! 7 ! ! , & , . 7 69% ? < , &% $2 < < ! ! $ ! < . & < @ $ . ! ! < ; =( $ :*6: ! " 9% , =( $ <, & ! ! = , < $ ! # "% . < . . A " ¬ ! =F ! =V ' F :) . %< ! !& ' < 9< , = ¬ ∨ ,# 9 ∨ ,# 9 0 = , , . ' . %$ , ' 0 A ! = ,# 9 ' ∨ ,# 9 0 0 <, ' ,# 9 < ' $ % ' < ,# 9 ' $) " 0 < < ' 0 ! = ,# 9 ! , 0 <, { # %# %$ " , , 9 0 0 ' ! . ¬ & &! % ) ,# 9 ; ' ' :) , . 0 F %! ¬ ∨ ¬ ¬ ∨ . }( ! ! ! < , ( A ( , < <( ¬ ¬ ! F ¬ F ¬ ∨ ! ¬ F ¬ ∨ ! F ¬ ! ! ! < + *$ ! - :*+: #¬ % . ! ! /, ' ∨ ,# 9 0 " ;& " @ " <, ! '% ? &< C ( $ # "% ! # , ! ( ! '%< < @ =( # ( , ,. > . ( , $ #, <@ ! ! 6% # /% > ( $ ? < . ! % < <$$$< , ! . , Resolución General = < & , , & & , , & , , < ! . $ ! & , . $ & , ! ! , , <, & . ! & , $ 2 & , . < , ( ! @, , @, <= ! !& $ &< . @, $ # ! {- : - : 63A σ } - : ( < -" - - L - -" $ ? σ( σ( )= $ < 0 ( & & " ) )+A ! " " 6;A ? σ = {- : - : - : ', σ < , σ ( )< , & -" # ! " 67A 0 ! $ * # ! ! } σ( ? !! $ = !! :*-: @, & < @, & . ', " ) ! #" = K ' * σ ={ : : %$ !} σ ={ : 6>% ? < # ! : " { :σ ( ) :σ ( ) } ' ∉{ σ ={ : !"- . σ ={ : σ ! : : }7 , , " & σ$ :< : #'} , & ? '= K < ( ") . :σ : , & ? = σ != ! " }/ !! < σ σ } σ : σ (σ !! ! A σ oσ σ σ @, . . )7A σ oε = ε oσ = σ , ={ : : @, $ }$ A ! +5A . , & $ "" )3A ? σ = { : σ ={ : ! : : ( " )$ σ σ σ , % =σ ( , < " } ( )) = σ σ ( ) , - 7" , , ε$ , " # ! ! :# } &A :# σ oσ : & , : σ (σ σ ={ : σ ={ :# % σ o σ oσ ! = σ oσ ! oσ , :@ 1 } σ & ) , ' " ! +8A # ! .& ( )} { :σ } : !) = != =σ ! ! < ( ) !! !! < =σ . ( )$ . @ ' ' ! ! ! , ! : $ :*1: %! ! $ ? < !$ ! # ! { , : +)A ? : , } " . @, & " { ( ' , , }⊆ ' :*7: . , & ' ! . Unificación , 3 @, ) , < # ! ! & , < '7+6<. , &$ < . NP $? , . < = ( . ! @, ! < A ω( ω $ ={ < . );A ? ! ? < ! ! ( , ! ω {% : -} . A ω =ω , ( , ω( ) = ω( { ( < ! , ) = ω( } , )= = ω( @, >A ? ω ω ! : ! 1!!} } ! , . $ ( {- : %}< , ! @, . ! < , : %:< < $ :45: , ω′ @ ! , ! - !!}< ( ! : ! ! ! ! ω ω , ) ω ={ : A ω ={ : $? < , ; , ω =ω ={ $ , , . . . : }, , . ω ′ = ω oσ $ ! -:% : } , {- : %} . $ " '745< < , , ! $ , , ( , )>A ? ={ ! % !! ω ={ : ! %:- : } ! ω′ = { : ! $) ( , ! ω , & σ = { : }$ , ! & !}< )= ω, ++A ? . ! ! "* # % , & σ . , ω ′ = ω oσ $ ! ! , ! &< = 17O$ +6A & ω$ ? &< ( # ! , ! > , 3 ! " . L @, 7< . ω {% : -}< , 2 ( & , @, . ! ! " @, @, @, ! < , $ ! $ ? ( = < ! < , &< < ! , , ! $ ? ,& , . = = ! ; , $ , @, $ ! , (. ! . . ! . @, @, @, @, $ ! , < $ & $ ) , # ! , & ! . ( $ ( +9A @, ={ , , # ( ! :? !=∅ A :? @, ? ! '! ={ ' != 68A ? ={ ' ' } ! A !} ! != '! , 9 . $ ! ! ! < ! < ≠∅ ( , ! ' ? '5? H'5< H ={ " @, ! A < :? }< ( )< @, &% $? ! , ! ! !} <, ! ( ' . , A < . ; :4': ! , # < ≥' ={ !} <. ={ ! < }< @, != <. 9. @ , . " '! , , ={ ≠∅ ! ( $ <. < } * <, < ! ={ != < ! } ! & A 0-= ' <σ ' -= ε '$: *$: ? σ 0 !& ! & 4$: ? A 0 0 -= ! , σ 0 Q% σ0 . A? A #QL (σ 0 ( )) @ $ {- : } σ 0+ = σ 0 0 =0+ B , A ? . ! . , ! , & ; @ . ! *$ ( & , , , ! $ & $? = , $ , @, & :4*: . , 4 , . < . $ 4< < & " 65% 2 ={ ! !} !! 5% 0 = ' ( σ ' = ε )% ' ={ ! } , = -= 6% 2 !={ )% σ ! σ ={ : ! !! !}( 0 = !!}( ! ={ . 6% < " ={ 6)% 2 !!! !} ! 0" ! $ !!} ! 5% 0 = ' ( σ ' = ε )% ' ={ } 6% 2 4/( 4 σ ={ : !={ )% σ }( 0 = !!! % 40( 4B&,' σ = { : 6% 2 ! ={ )% σ ! 6% 2 !!! 4,( 4 & ' ! ={ )% σ : ! σ ={ : ={ !!} ( ! : !} ?4) !} ={ !! !!!} !! !} !} : 2 !} ( ?46 σ $ , * " , 4 ! $ 2 & =. ( , &< =. H =. $ :44: . H# = I%$ " ={ 66% 2 ) 'A 0 = ' <σ ' = ε $ ) *A ) 4A ? ) *A σ ! ={ ) 4A ) . ' ={ ! . < } 4 "4 σ = { : 3 , !!}$ ! , ( 6+A ? ={ !!} < ! , & & , ( }<0 = < ={ ( !} 0" ! . $ " ( ! @, $ ! & , & , ! ! & @, $ , $ " ! . ! ! ' ' :4/: ! ! − − !}$ Algoritmo de Resolución General & ! & ! ) " , , # ! & $ 2 ! . & , . . , @, & . $ +3A ¬& ∈ , & , ! ω$ ? , ! * < . & ∈ . , & , & A ! = {ω & & " ! − ω & !} {ω 69A = !! ∨ ! !∨¬ & = ω ={ : ! : =¬ ! ! 63A ? " ? < !¬ !} , ! ω ={ : !∨¬ A } & !! ∨ = =¬ A & !! ! . < ( ! ! !$ . , # $ < $ ∀ !& ! , ! { !! ¬ !∧¬ . !$ ( , $ /*%< , , ( ; ( , < ! 67% ? ! ( ! , & " !! ! =❏ ) . ;$ < , . !! ! ! } != & & !! ∨ !! & = !! < { ! − ω ¬& !} !} $ :46: < = < < $? ! ( $ = =¬ !! , ! =. . $? ! < < A - = - !! =¬ ! ω ={ : ) ! !}<. . < < < ∀ . < A !! ∧ ∀ ¬ ( 4 < A { ( ( - A !! - ! =❏ { ( !} < ! < < < ''$ ! 5 # !} $ ! & !! ! % # ) ! # ; # # 3/ 6 - @ " ( % 2 7 ! 7 % ∈ '$: 8 *$: ? A 4$: > ! !=❒ /$: ? 6$: ? 4$: ? ( , $ , <2> ! , !& . 2@ C 7 B A 2@ C '' 7 ! ' . ! ∀ 7 7 ! < & !! ∨ ¬ :4+: ! =. A !! . $ !! ∨ ¬ !$ 6;A 2 " ¬ 2 & !∨ !∨ !! ¬ !¬ ! , !∨2 !∨ ! ¬2 ( !∨ !∨¬ !! ! ¬2 !∨¬ ':# 1 # ; ¬ !∨ ¬ * !∨ !∨ 2 !! ! / ! 6 ¬ !∨2 + ¬2 !∨¬ ! - ¬2 !∨¬ ! 1 ¬ 7 !∨ ! ! !! { : } −< { : } −) ε !! '5 !∨ '' { !! '4 2 '/ ¬ '6 : =−( } −) ε !! '* !! !! =− { : !} >− { : !} ?− ε ❏ '− ) . ( , , ! 4 . ! # , '*0 & !! !∨ 4 , $2 %( $ ' ) ! ( '6% , A . & . , . , ! , $ :4-: '*$ $ < , , '% ? 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U ω ′ σ . σ σ & ω ′ (σ ∪ σ (σ ∪σ ){& ∪σ & ! & ! σ R & & ( σ( σ ω σ oω ! & + ! + ! =ω <@ ! )% & @ )=σ + σ {& ; σ & & + ! ! σ ∪σ & σ & + }= & ! σ ∪σ ! ω ′ (σ ∪ σ ? ) σ ∪σ & !σ & ? @ <, ω ′ (σ ∪ σ N*O !! & & + + !} + } , ! . A !} = , ω′ $ {& < ω , . <, & σ . & & ! . & + N4O , ! ω< A )= !∪ω ∪ + # ω !∪ω < !! = != # , :/5: N*O% }< ω ′ (σ ∪ σ )( ∪ ω ′ (σ ∪ σ )( ) ∪ ω ′ (σ #) )J . ! σ ∪σ # !=σ J ω ′(σ # , )( ∪σ ! ( σ ∪σ ND ;< <σ ! !% !=σ )) ∪ ω ′(σ ( )= ! & σ & ! %< , A )) = E: O% ! J ′ ) " σ( )= = 6>A ? A σ ={ : ′< < ′ !∨ ! ! 3! ∨ : !} ! σ ={ ′: :/': =¬ ′: }$ ? ′! ′ !! ∨ ¬ ′ . ′! A = σ = l : ! : !∨ ! : ! ′= ! !∨ q σ = !! ∨ ! ′! =¬ ! m′ : ! !! ∨ ¬ ′! ∨ ¬ ′ ε ′= ? ω ={ : ! : = ′! : !∨ &<@ ′! ′ : }. ! !∨ ! !! ∨ ¬ ! . A ′! =¬ ω σ σ = ′=σ ′ !! ∨ ¬ σ= ! ′= ′! m′ : r !! ∨ ¬ ′ =σ " 5) & " ! < $ , <, ! ! & . , , & :/*: %A ? & ′ ′! r ′: ′ =¬ !! ′! ∨ ¬ A ( ′! ! A ? 7 . ( ! ! 7 #, @ L ( ! # , @ & , , & @ < ? !¬ !!} % $ 7 &% ( ; 7 . ! ( & @A % 7 # { < A , $ 7 & <, , , &< , ;$ < ! , < < . :/4: ! < <, !! ( , < ! ( A Estrategias de resolución 1 0 $ 5 # 3 * ; ; % 5 # / # 3 / , - @A 0 B/ # 0 5 8 % # 5 3! / 4 % 5 $ # 3/ . # $ ; $ / 6 % # $ / Estrategias de Borrado 4 # # $ 5 # 1 /. 0 ; # 3 # ; / # # ! !. +7% ( . @ % ; , ;. < , % &$) , $ & " ={ ( <, +8% < ! . $ . , & . , & * # ! !¬ !∨ < , !! ¬ !∨ !¬ +;% . . ∨¬ ∨ ∨¬ +5% $ $8$ , ( ! ( , . $ !} <, $ ! " < ! ( , > $ ! ! ://: ! . < & ! ( $ , <, , . " , < ( ;. ! . & , $ 0 , . " ! 1 !∨ ( ! & { : < , , $ ( { ( ! !} !¬ ! ! $ ! # ! 3 @ . +6% !∨¬ , , , $ {¬ ( +)% , " ( @ , , $ 7 +>% $? " & σ . σ !∨¬ ! ! ∨ ¬ -! ∨ 1! ¬ ! ∨ ¬ -! ∨ 1! , . : -} , < < < !⊆ $ ! !} $ ! $ ? , < . ( ( 3 , ( $ 3 $ ! & ! . < ! <, < , . =. , ( & &< , $8 ( &$ C # ! , 98% # & %$ , & & $ " '$: *$: 4$: /$: 6$: +$: " ={ ∨ <, $ ++% ? & ¬ ∨ < ∨ ¬ ∨ ¬ ∨ -$: 1$: % 7$: '5$: ¬ ¬ ¬ )! )! & . <! ?! < ?! ( $) , , . ∨ , >! . & = , ¬ }$ 2 , ¬ ∨ ( , < . , ' *, $? . :/6: $ , & , < < ! & . . $" < ! ! , ;. $ = $ ! < ={ ∨ < , < , ∨¬ ¬ ∨¬ } , . ¬ ∨ ) < , ! $ @ % ! $ & ) ! ( , < < < ( & $ ! , L . # ( < L , & Resolución de Entrada # ! , 95% ( ={ ∨ ¬ ∨ <, < ( +9% ? & " . $ , $ ¬ ∨ ¬ }$ 2 & & & , , A , '$: ∨ *$: ¬ ∨ 4$: ¬ ∨ /$: ¬ 6$: ∨ -$: ¬ 1$: 7$: )! <! ) =! ! +$: ∨ ? , , < ! . & . ! & , $ . L <, $ , & ( , < , $ Resolución Lineal 8 0 ! 5 # % / :/+: , , , 9)% ? # ! ( ( 7 2 ! #'% = ' "+ $ * . , 4@" /" "A/A " # " %< #*% , " # % , "$ & , $ & , ( < , ( /% * ! , . . , ! # ( ={ ∨ < , , ∨ ¬ ∨ . ∨¬ ¬ ∨¬ ¬ ∨ }$ " $ ∨¬ ¬ ∨¬ ¬ ❒ & = ; , , ( ? , . & , $ $2 < ( ∪ ! & . lq ! < $) ( ! , ( < < . , $ Resolución Ordenada *, 8 * #,&, "- # % 5 /8 # / 8 # # 3 ! # 3 !/ :/-: ={ ∨ # +3% ? & %A " ! ¬ ∨ = ¬ }$ 2 ¬ ∨ & , , '$: ∨ 6$: ∨ ! *$: ¬ ∨ 4$: ¬ ∨ +$: >! /$: ¬ -$: ) <! 6 ; . , & , 6< /, ' , * 4, $ , & , ? L & %$ < = # ! . ( , < < & & ' 4 , $ $) , , $ , , ; ) #2 / , ! /$ , 4 /$ +< , , , &< & ( & > . & & # 7 L & %. $ , < & ? ! :/1: &$ Prueba de Teoremas por Resolución , '4 " ! =. , & ! . , @ . $ , ! , ( ( !& , , , @ < . 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