Nov 13, 1996 - be found in [Sco80, Section 4] and [LS86, Example 9.5 of Part II]. It is often convenient to omit the application map App , writing fx for. App Ï,Ï w.
Kripke-style models for typed lambda calculus John C. Mitchell Department of Computer Science Stanford University Eugenio Moggi Department of Computer Science University of Edinburgh November 13, 1996
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1
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We use the term set theory to mean any classical set theory, such as ZF. This should be distinguished from set theories developed in intuitionistic logic. As will become apparent, our Kripke lambda models are essentially a semantics of typed lambda calculus developed in a form of intuitionistic set theory.
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')(z]^ 1n ¼nT "¶ UbY ·r½ ¡ !# 1'85 085"!L]= &$@ !D ,.¯ P 0 #³A85!D¾ n&oF85 1&*q!D,.¯sP 0PA!D/8(. E7 0/'7 1!Dz!# a →A ? 1i!(&* JB,V!)K p%&%&*1 1L !" 0!D' 085Z 85F)7 @ 0*1! )(i )Z;D!D87 1L-Dd]q*1*n`-hF %'85F 0/* 8PAbnp587D(. 1F 1!D.* 1.#I85' 0p7$&5F¥ Ed!Dd 0!D%.*1Fr#I!D8 NF - 1> %&*1 ) 1!Do!LKM8f 0!D' 085F- 85Z)9 2 0*1! )(k BF;D!D87 1)^]^&lu %F$.wr!D,.¯sP 05f85 *1*1!)]q)(Q`
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U,V
2
Here is one example, based on the results of [Jac75]. We write M ←→ N if there is a term P with P U V =β,η M and P V U =β,η M . Then in the traditional inference system U,V
U,V
for typed lambda calculus, we have U = V ` M = N iff M ←→ . . . ←→ N . This fails when the new inference rules for empty types are added.
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2.1
Possible worlds
¤ ]^ 1nm!Dn&8eJ89 1% ? 0>Z5 $*1S PAc{eJ87 1% ? o* -,(&i!&(*q]^ 1*1* 1' 0*1/'(& 4%'85 *1*1$ >n!D87(&85L(ß !#ou %b! 7 1,.*1S]q!D85* (P`xwhF'5Z)D([!#^&)K& 1&;{4r!# *1ps=!#jBD G $%'MAeJ89 1% ? -* -,V(.!&(* ]^ 1*1*Q&)KM-9^!#c*1N 5^!# BD 9k $%'J)fBD k%b! 7 1,.*1d]q!D85* ( ßË` &85* B 1!D'5 1%S,bF]qo*1N 5 !#F$&%' σ ) ]q!D85* ( w .( w ≥ ww E∈=Z&) KM85$ a σ ¸ Y w ED95!& 0 BF)(4]^ 1n 5!Dpk/. _ / ¸ Y `zhZ.#I!D85N*1*1$A/'9 1.;iZ" 0!D!D~ps%!D8N!# D 85* B 1!D~ 1lF 1NaMAcσn EwNB'Nn.BKM85$z*1p!# σ B w ]^ 1*1*= 0!DF 1 /≤ks! ,'{*1pO!# σ 1KM85$m%b! 7 1,.*1 w ≥ w ` ¤ o]fG!LKM"#³85!D w s! %b! 7 1,.*1"#}/. /85o]q!D85* ( w A ]q!{n 1.;M 1;D&&%&%'Co]f"$zD _/ 185k!D85 *1psBA.(i(. E5 1' 0d*1N 5N$G,bP 0!Dp (&
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2.2
Applicative structures and predicate logic
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σ w
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σ w,w 0
0!D'7 E55^!# 2ßä!#gu %b! 7 1,.*1-]q!D85* (7wN%'85F *1*1$>n!D87(&85)(",&$ ≤ A • N#n 1*1$ {A } !#j5 1&(&0.)(k,&$S $%'L σ .(o]q!D85* ( w ∈ ßËA • @#³ 1*1$ {ã ¼M¼ } !#2us%%.*1 )F 1!D~%b7w ã ¼ ¼ A × A → A • 1&(&0.)(",&$S%' 187!#cF$&%') σ, τ .("]f!D85* ( w ∈ ß:A S#n 1*1$ {i } !#pu 587'7 1 1!D{#}/.' 0 1!D'7w i A → A 1&(&0.)(2,&$ • F$&%') σ .("%' 187!#c]f!D85* ( w ≤ w 5/M,.¯sP 0ds!knN#I!*1*1!)]^ 1.;o 0!D.(. 1 1!D'P`a@p] n&N587'9 1F 1!D{#³85!D A s! A !,b-n- (&F 1 $ A → A E^Z- (& 1 $ A ¶Ð ¡ i &("!Dn&8^587'7 1 1!D"#I/&' 0 1!D'qs!O 0!D%b! 5R7T Ñ ¼ ¡ ¢ i *1* w ≤ w ≤ w i ◦i 5!"n&)Jn&85 EJ0.D 0Z*1$4!D&N%&% 1&;k!# 1 ! ;D 1KMi#}!D8 `ra@ * 5!85P_ / 185-Z&)^%%.*1 )F 1!DG.(o587'9 1F 1A!D2 0!DrA/&F- 12&) /87w*Q]q≤$w U ¸ Y ¡ ∀f ∈ A .∀a ∈ A . ã ¼ ¼ f, a¡5¡ ¢ ã ¼M¼ 5 i f ¡ , i a¡5¡ , i ]^ 9"$o,'g(.87]^ ã ¼ ¼ σ w
σ,τ w
σ,τ w
σ w,w 0
σ w,w 0
0
σ→τ w
σ w
σ w
τ w
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σ w
σ w,w
σ w
σ w 0 ,w 00
σ w,w 0
σ w
0
σ w,w 00
σ w
σ→τ w
τ w,w 0
σ w
σ,τ w0
Aσ→τ × Aσw0 w0
0
σ w0
σ w
σ,τ w0
σ,τ w
00
6
σ→τ w,w 0
σ w,w 0
- Aτ 0 w
6
iσ→τ × iσ
iτ
ã ¼ ¼ - A &("]^ 1*1*Q,'g(&L7 087 1,'L(G 1.#}!D85*1*1$S,b*1!)]g` E 0!D%&*1F)^Zg(&0
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0
σ w
σ,τ w
σ,τ w
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σ w
τ w
σ,τ w
σ w0
σ→τ w
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' 1F 1!D4u 1'9 (&Pw=^ BF;D!D85$ è ,$k85;M87(. 1.;Sn&g]q!D87(us7wOD=NB 1.;zu !D,.¯sP 0^#³85!D è wo&(uF#}/.' 0 1!D'w DpB 1.;+u !D85% E5Í#³85!D è w` /'Nl%&%.*1 B 1KMo7585/& 0 /85S 1 è 3N 0!L*1*1P 0 1!D!#^!D,.¯sP 05 {C } 1.(0.)(\,&$\F$&%'Lr.([4 0!*1*1P 0 1!D!#^!D85% E5O ã } 1.(0.)(|,$m%' 187 σ, τ !#- $%b)O5/. ln&) ã ¼M¼ .Dn&4(.!DN 1 { ¼M¼ &(l 0!&(.!DN 1l;D 1KMl),b!LKMM`m ![(&87 1KM"!D/8O(&0
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τ w
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'&Jn&J9p*1p!# $%' τ B]q!D85* (oñH`xw 1' 0-!D85F85O ,'4(&0
'&)(l]^&|]q2.BKMG!D854KL87 ),.*1)PAq 1O 3S 0*1B8on&)S&$mñ ñ ]^ 1*1*q.)KMo)p*1)D5nk*1N 5N!#dñ`@a&) EN%'85&%V*1)9N!D,&K 1!D/'p E≥n.) ]^ 1n+85)5%bP 0G !l 085s 16* -,V(.ln&!D87 1)BA ]q\N$&)KM4!D85@P_/')F 1!D'kB ñ ñH`f ! ? gO7 1%.*1g0%.*1MA'5/%&%b! -]fr&)KM-O 0!D'5s AV.( ¢ ∅ Aj]q *1 ≥E ,'on&S7 1.;D*1OP_/'B 1!D {λx τ.c ¢ λx τ.λy σ.y} ` ¤ g]q!D85* ([c ñvσ→σ !Dp%85!LKM c ¢ λy σ.y #³85!D E `~ I Ep EN! 5pBD7 1*1$[(&!D'5587)ZL(,&$@ 9 "85;D/&N PAqD 1vP 0 1!D®.` Ã'` ¡ < !)]qKM8PAH 1 EOBD7$s!kZ&)OB ]f!D85* (zñ ¢ {z τ } ≥ ñAj]fO.)KM E ` ñ . c ¢ λy σ.y `G M/'pnS%85!)%'85 1) !# n&d587'9 1F 1!Do#}/.' 0 1!D' i 85]q*1*>n!DF 1KBF)(S,$O%85!L%'85 1)f!# n&d%85!!# 5$&7Z°#I!D8 $%'L(* -,V(.p * 0/*1/'P` a@ ]^ 1*1*b/'^F$&%' D7 1;DN 5Dru %V! 99 1,&*1 ]f!D85* (7wN 12%85!LK& 1&;Nn&g 0!D%.*1Z&)9=n!D852` 0
0
0
0
0
σ w,w 0
3.2
Environments and meanings of terms
¤ ~X0U âP¶ WsTU Ñ X0UbY #I!D8 OeJ89 1% ? r%&%&*1 ) 1KM-5585/& 0 /85 ã EdO%'85F *C%&% 1&; #I85!DÕK87 ),.*1)=.η("]f!D85* (s!*1N 5!# ã 5/. 9"n.B 7XU â ¡ hZ# ηxw ∈ A &( w ≥ w, n& ¢ i ηxw ¡ . ηxw P¨ 0
σ w
0
σ w,w 0
hZF/ 1 1KM*1$ Ajz&K& 185!Dp %b4K87 ),.*1 s!\6u %'85 *q*1p7w ]^ 9g$J0& 35q }!D8,'f(&0
'&)η( ¡ )5!DN]q!D85* (PAB,x/.C!DP 0L9789 1*1$J*1*]f!D85* (ηxB` 1' 0=-F$&%'$r,'=% $p)!D&q]q!D85* (.(n!D&%F$p* BF8PAM]qL( ! &)KM&K 185!DN 55/& gn.) ηxw E /..(&0
'L(r)5!DN w AL.(gZ\u ,'P 0!DpL7w (0
'&)(~){* )Z8 w ≥ w `[a@"]^ 1*1*85 /87~ !\n 3%b!D 1 OL#IZ8(&0
' 1.;{Z pB 1.;M!#cZ85OP` hn# η Ed4&K 185!DN d&( a ∈ A A']qg]^87 1F η a/xj#I!D8 n&g&K 185!DN (F *s! η !DGKL87 ),.*1)!Dn&8=n. x A.("]^ 1n η a/x ¡ xw ¢ i a #}!D8=*1* w ≥ w `a@-s ? η a/x ¡ xw s!,'-/.&(&0
'&)(o#I!D8 w !D ≥ w ` hn# η 3fkK& 185!DN #I!D8f%&%.*1 B 1KM=5585/. 0F/85 ã A&(oñ{ 3qrF$&%'D97 1;D.> pPA.]fg7$ w ·0¸ Y ¶E·}ö X · ñ ¸ Y η A']^87 1ZF wk−ño η # #}!D8=*1* x σ ∈ ñ . ηxw ∈ A ê=!DZrn&)^ # wk−ño η.( w ≥ w An& w k−ñS ηZ` '!D8j$geJ87 1% ? H!(&* ã &(rK& 185!Dp −ñS ηZA]fq(0
'&HZfpB 1.; ñ . M σ ηw !#cZ85ñ . M σ 1"K& 185!Dpwk η )^]f!D85* ( w ,&$S 1.( /. 0 1!Do!D ng7585/& 0 /85J!#jF85OP` ñ . x σ ηw ¢ ηxw ñ . M N τ ηw ¢ ã ¼M¼ 5 ñ . M σ→τ ηw ¡ 5 ñ . N σ ηw ¡ ñ . λx σ.M σ→τ ηw ¢n&-/& _/ d ∈ A 7/& oZ&)#}!D8^*1* &( w, ã ¼M¼ i d¡ a ¢±a ∈ñ ,Ax σ . Mw τ ≥η a/x w Äf!D-, 1&)s!D87H.(0&F'7 1!D.*1 1F$N;D/'87 FdZ&) 1On& ñ . λx σ.M σ→τ DMA 0& E7sc.(r 3/. _/M` Ej Ec%85!LKML(rDj 1n 0* D99 *Z 1.;'A)/'9 1.;d587'5* L> d F 1!Do 1 s!N 0!D-, 1.) !D87J Î89LÃ'A n!D87(&85L(o h ß , ≤i D^N )Z;D!D85$S 1"n&/'5/'*]q$ ` %'P 0
b *1*1$ A&n&d!D,.¯sP 05H!# Z E BF;D!D85$o85n*1N 5q!#pßä&(Sn&85 3q /& _/iu *1L9s>nZ&.>n!D8s>nP_/'*>ns!MwO8585!)] ` #³85!D w s! w ² w ≤ w `d 1' 0p BF;D!D85$4g/'5-.)KMr (& 1 1)J&(i,'O 0*1!9L({/..(&8 0!D%V! 7 1 1!DCAQ]qp*1 ` ,'gn&J (F 1 $S!D w &("(&0
'&J 0!D%V! 7 1 1!DG,&$ ¢ ` . ` ◦` .M
A
A
0
w,w 0
0
w,w
w 0 ,w 00
w,w 0
Ã
w,w 00
¬ 1KM"eJ87 1% ? J%%.*1 )F 1KMd5585/& 0 /85 ã A& 1f 3BD5$Os!9n.BBD o $&%' σ (F85 1L=N#I/&' 0s!D8 ý #³85!D h ß , ≤i !O5P`%'P 0
b *1*1$ A.]qs ? ý w¡ ¢ A ý ` ¡ ¢ i &(m/'" 0!D.(. 1 1!D' å ¶ Ðæ2&( å R9T Ñ ¼ æG 1ln2(&0
' 1 1!Dz!#-eJ87 1% ? G%&%.*1 B 1KM 5585/& 0 /85Os!45&!L]ôn.BgZ Er%[ Er#I/&' 0s!D87 *n`ka 1*1S 1r$@9Ý7 1%.*1)5 !l/'i#}/.' 0s!D872ý DG!D,.¯sP 05o!# A Z EG$ (& #}$F$&%')" 1+Z@ D ]^85 σ 6¢ τ 5$ D 0 *1*1$ Ab,/& A C¢ A &%&%'\s!o,bn7N9P`p 1' 0 ]fo]f!D/* (@!DpP 0L9789 1*1$i] ps!4 (& #}$\%&%&*1 ) 1!D\#}/.' 0 1!D'g!DZO ]q! F$&%')BAQn Eg 0!D/* ([*1BD(\s!G/&&P 0)785$4 0!D.#I/'7 1!DC` &850#}!D85MAC]f]^ 1*1*/'n F$&%'-0.%85)99 1!D'^Dn&-!D,.¯ P 05!# C ` 1' 0 BD O $%b (&Z85 1&)Hg#}/.' 0s!D8PAD]q=]^ 1*1*b/'=.B /87*587's#I!D85N) 1!D' D=n&-!D85% E5O^!# C `Hb!D8^KM85$k%' 18=!#j $&%') σ .( τ A' 0!D.(. 1F 1!D å U ¸ Yµæg 1 n-(&0
' 1 1!Do!#ceJ87 1% ? -%%.*1 )F 1K 5585/. 0F/85J7$.fZ&)n% w 7→ ã ¼ ¼ A ]^ 9+]q[5.*1*-]^87 1Zm7 1%.*1$vD ã ¼M¼ AJ E2l.)F/87*-587's#I!D85B 1!Dv#I85!D ý × ý s!ý `mþ 9 1&; ã ¼M¼ AH]fG lkZ&)KM85$[;D*1!D,'*^*1N a !#j $%b σ→τ 1.( /. 0)^.)F/87*587's#I!D85N) 1!D ν #I85!Dý s!Sý A.&N*1$ ¢ ã ¼M¼ a , · ¡ ν '!D8g0.Z'9 1!D&*H%&%.*1 B 1KM5585/. 0F/85Lo I.(@&' 0O!(&* A 1r ErBD5$\s!4 n&) #' ]q!=;D*1!D,'* *1N 5 a .( b (&F85 1fnq7N&) /¡87* 587's#I!D85N) 1!DCA n a ¢ b )-KM85$i]f!D85* ( w `oa@S*1gn&O!D85% E5Og#I85!D σ s! τ 1 C ,'S*1*.B /87*587's#I!D85N)F 1!D' ν Cý →ý 1.(/& 0L({,&$4;D*1!D,'*H*1psg!# ã !#HF$&%' σ→τ .(4*1J 0!D%b!7 1 1!Di!#!D85% 35O,'!D87(. 1&85$2 0!D%V! 7 1 1!D\!# &)F/87*Q587'5#}!D85B 1!D'q 1\ÓX0Y ` ¤ 85!D/.F 1&f * 0/* ) 1!DN5&!L]= n.) # ã EcJeJ87 1% ? q* -,V(.d!(&*nAn& E d )Z;D!D85$-]^ 1nN!D,.¯ P 0 #}!D8BD 9pF$&%'MAL&(rn&85f Ejd!D0>n!D&H 0!D8585)7%b!D.(C' 0 ,'F]q;D*1!D,'*.*1p5!#V $%b σ→τ 1 A &(!D85% E7Oj#I85!D σ s! τ 1 C ` hZND((. 1F 1!DCA) 1 EBD5$-s! 7!L][n.B C Ej 85ZL7 0*1! )(g # ã .D %85!&( /. 05 .( Z85 1.*C!D,.¯sP 0P` -85* ) 1!D'5 1%",'F]qGn&g BF;D!D87 *C 1Z85%85sB 1!D2!# Z85OPA&DH 1\& 0! ªL A.(n& NB 1.;g#I/.' 0F 1!DS]fd&)KM^;D 1KMO 3f5/.87 1)( σ
σ w σ w,w 0
σ σ
w,w 0
A
σ
σ w
τ w
A
A
σ,τ w
σ,τ
σ→τ
σ
σ,τ
τ
σ
σ,τ w
w
w
τ
w
w
A
σ
τ
hW ,≤i
A
A
A
3
3
There is a minor source of confusion in [Sco80, page 413]. In assigning an arrow of a category to an open term M , we must decide which variables to consider free in M . In particular, we may want to consider some variables “vacuously” free. Scott’s slightly informal discussion does not address this point. However, in the formalism of the present paper, we have explicit type assignments, and so we simply treat all variables in Γ as occurring free in Γ . M : σ.
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1On&^#}!*1*1!L]^ 1.;Z!D852`cê=!DF n.)]^ 1nO%85!(/. 0F$&%')BAM$ñ . M τ 3BD9 1*1$ 587's#}!D85N)(~ 1s!@9 *1*1$P_/ 1K*1 SëC σ . M τ ]^ 1n|!D*1$m!D"#³85 KL87 ),.*1M`d n 1%.*1$k85%&* D 0rng 0!*1*1P 0 1!DG!#jK87 ),.*1)= 12ñ,$"7 1.;D*1gK87 ),.*1 !#jn&g%&%85!)%87 )Z-%85!&( /. 0F$&%'M` ¡ ÿ È3ó ã ¶E·2¸ à W ¶ ¼bá X2Ê ¸ÑSÒ Ð ¸[Ñ TPÐ X0Ê ¶ Y}º ¼ W5TPлRY ·4¸ UVÐzYZXW Ѷ U ¸ Ê YIÏ ¼ X JY}ºXUlY}º.X ¶ UbYZX0W ¼ W7X0Y ¸ Y ¶ TUlTsópë σ . M τ ¶ U C ¸L· Ð X ö UX5Ð ¶ U nÓR7T ¶E· Y}º.XdU ¸ YI»&W ¸ ÊbYIW ¸ U · óTW Ѹ Y ¶ TUdóW5T Ñ ý YnT-ý ¶ UVлR9X5Ð Ò ÏY}º.X ¹ Ê1T Ò5¸ ÊQXÊ X Ñ XUbY w 7→ λx σ.M σ→τ ∅ w cº.X0W5X ∅ ¶3· Y}º.XpX Ñ ¼ YIÏkXU âP¶ W5TU Ñ XUbY ½
850#I!D85gGP_ /') 1!Dk&!* ( 1 ã ²\ 1^!* ( 1 C `hF=5&!D/* (o,bg%b!D 1Z)(2!D/& n&) Zp#}/.' 0s!D8^#³85!D C s!2ÓXY Ab%% 1.; τ s!Gý AV E=#n 1n.#I/*.(2%850> 985KML%85!(/& 05PA&,/.^ 1^$S!D^,'J#I/*1*Q!D8^N$o!D^%85)85KMJ#I/&' 0 1!DG5%'D 0)B`
85BD7!D@ EJn.BgGeJ87 1% ? * -,V(.G!(&* E^¯ /'5-k
'875-!D87(&8g5585/. 0 /85N 1 s!L%b! o!#r%85)7B)KMLP`: &850#}!D85MAZi 1 F85%85 )F 1!DCAý !# &)( !D^,'pý →ý ç&0&F'7 1!D.*1 1F$o!D*1$S85P_/ 185)n.)=ý ,'g@us5/M,.#Iσ→τ /&' 0s!D87wN!# ý →ý ` a@!)]° ? i"Nn&!(i#I!D8-(0
' 1.;GGeJ87 1% ? * -,(&2!(&*j#I85!D&$ 5N*1*q 85ZL7 z 0*1!9L( )Z;D!D85$ `G4!D85o5%'P 0
b *1*1$A ]qoD5/.p]qS85S;D 1K 6D5! 0 ) 1!D!#r $%bi 0!D'5s 5os!z!D,.¯sP 05k.(|F85ø 0!D'5s 5Ss!~8585!)]= !#^{5*1* 85Z)9 m 0*1!9L([ BF;D!D85$ D `2/& [zD5! 0 ) 1!D[(F85 1Lr 1 Z85%85s)F 1!DO!# $%'L(* -,V(.g * 0/*1/'H 1 D A 1On&=9'!#& 0! ªLZ`a@=]^ 1*1* 5!L]Z&)Z85q0& 355deJ87 1% ? q* -,V(. !&(& * 69B Es#I$ 1.;^n&7pfP_ /') 1!D' D D `hFon&-5%'P 0 *Q DZ&)!D/8 )Z;D!D89 *Q 1 Z85%85s)F 1!D"!# $%b)("* -,V(. * 0/*1/' Eqn 1 F87.*Q* &;D/';D!# I .D®) A&n E^ 0!D'5585/. 0F 1!Do;D 1KM)q/' eJ87 1% ? -* -,V(.!&(* ô]^ k 3DP_ / 1K*1 ^s! ¡ D } 1Gn&J/'5/'*C )Z;D!D87 * 9' ¡ A&,/&^!D^&P 0)787 1*1$o E5!D!D85% D`
85g85Jn85-5F%bq#³85!D5N*1* Ä^Ä^Äs!NeJ89 1% ? J* -,V(.!(&*n` &
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'L A &({0.Z'7 1!D.*1 1 $ Ab]^ {85*1*j
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'&S(¡ fo*1N s87 1*1$P_/ 1KL*1f%&%&*1 ) 1KM=5585/. 0F/85= 1{ÓX0Y A ]^85 W E=%b!9^]^ 1n2*1)D5^*1pP`
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'875G5F%v/')kn&{Á!DL(&z-,'L((. 1&; !# 1 s!~nis!)%V! k!# %85)5&B)KM)H!)KM8 D `c Ef%85!(/& 0Lfp BF;D!D87 *V 1YZ85%85sDB 1!Dk 1Sn& s!L%b! f!# %85)5&B)KM)d!)K 8 D ADJ5%'*1*1L(i!D/&J 1l& 0! ªLZ`-'!D8 1'5s' 0MAV # d En&p 1Z8s> %85 )F 1!DN!#bn&q,'D9fF$&%' σ 1 D ALn&]fq/' Y d¡ DZq 1 F85%85 )F 1!DN!# σ 1Gn-s!L%b! =!#%85)7B)KMLPA.(G7 1 1* 85*1$o#I!D8qZ- 1Z85%85sB 1!D'^!#j 0!D'7 sB` ® 0
A
σ
τ
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A
hW ,≤i
τ
σ→τ
σ
σ
τ
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D op
D op
W
D
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g )Z;D!D89 *C 1 F85%85 )F 1!DG 1GZ-s!L%b! ^!#c%85)7B)KMLf!LKM8 ;D 1K )^/' S%%.*1 )F 1K f5585/. 0F/85 ã 1n&=7ps!L%b! P`%'P 0
b *1*1$ A ã EZ=D%&%.*1 B 1KM 5585/& 0 /85r]^ 1Z{n&NF$&%' σ 1 F85%85Z)(@DJS#}/.' 0s!D8Jý #I85!D D s!4ÓXYA.( ã ¼M¼ DnkK*1/'B 1!Dz!D85% 35 eval #I85!Dø ný →ý ¡ × ý s!ý ` I F/85 ã 7B Es
'LnOL.∼ 1!DOO→ K ¡ AH S ¡ .(\¡ n0.Z'9 1!D&*1 1 $4 0!D.(. 1F 1!D å X5ëDYµæ A D 0!D87(. 1&;s!n&geJ87 1% ? 0"> !M!)$ *NF q!#j#}!D85g/* D 1\ÓQX0Y I9S .D®) ¡ ` a@g!L]+&)KMJ* -,V(.!&(* 1@ÓX0Y A.,/.=!D^&P 0)787 1*1$" à W ¶ X * -,V(.N!(&*9 1' 0 D $O!Dq,'Ap%b!9P`c &dn 187(k5F%o!# ZJ 0!D'5585¼b/.á > F 1!Di/'9Ln$ # D 0!DL7 0/\ 0!)KM8P`J &p;D&87*c 0!D'5585/& 0 1!D4;D 1K { 1\Æc%&*1 ¨ ` .(NÄf!D85!*1* 85$ ¿ ` ¿ !# !M!Dbª%85!&( /. 0)j %V! .(r^#I/.' 0s!D8qÐ 5/& Gn.B=&$o%&%.*1 B 1KM-5585/. 0F/85 ã 1@ÓX0Y EW=*1p 87 1*1$kP_/ 1KLW*1→D ^ ! 2%%.*1 )F 1KM5585/. 0F/85% ± 1\ÓXY !D,s 1L(",$o 0!D%b! 9 1&;)D 9k#I/&' 0s!D8^.( &)F/87*f587's#I!D85N)F 1!Dz 1 ã ]^ 1nËÐ W →D `\hF%'85 0/* 8PA ã &&( Þ7)s> Es#I$n&27NoP_/'B 1!D',' ]f~ $%b)(m* -,V(.@F85OP`l KL*1- ! C .( C EJ E5!DD!D85% C s! C ` 1&*!D,.¯ P 0k!# D `± !z%85!&( /. 0 C E5!D!D85% {s! C A ]f{ ? W 1Z (0
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'L(kD 1[ !D!DbªLZA.0.Z.(&)(os! ⊥ ,&$O%% 1.;NnJ% $ 9P_/' 0=s!rn& F85 1&*!D,.¯sP 0 !# D `jê !DZn.BZ85 E0D 0n*1$!D&=]$Ns! 0&F&(\Ðo !N!D85% E5Of#³85!D ⊥ s!$ w ∈ W A ,bP P/' Eqn& 1 1 *!D,.¯ P 0 1 D `o2!D85O0.%.*1 0 1n*1$ AHÐ\ 3-n&#}/.' 0s!D8J#³85!D W ! D 5/. 9@Z&)J#}!D8g&$ 9P_/' 0 f , . . . , f !#c 0!D%b! 7),.*1-N%Vg }!D,.¯ P 0^!# W ¡ A&]q-*1 ÐQ f , . . . , f ¡ ¢ ( n & g# 0n!(.¢v!Dª 12!# 1 A!DZ85]^ EM` f D hn# ZJ9P_/' 0 w E w #I!*1*1!)]q)(o,$ f , . . . , f A&n&\Ðo%bqn/. _ /J8585!L] D
D
op
σ
σ,τ
Φσ ,Φτ
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σ
τ
σ
τ
τ
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D op
op
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W
op
σ→τ
σ
τ
B
A
A
B
B
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1
2
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2
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nJ!D*1$k!D85% E5¦#³85!D s!4ÐQ A. # ¢ QÐ ` ¡ ¢ nJ (& 1 $S!DÐQ w ¡ A # n ¢6ª w ¡ w ⊥ n- 0!D%V! 7 1 1!DG!# f , . . . , f A.!Dn&85]^ E9M` 1' 0NZN!(.
'L(4#I/.' 0s!D8Ð →D 5 1*1*j9B Es
'L n& 0!D&(. 1 1!D' !#^Äf!D85!L*> * 85$ ¿ ` ¿ 1y !D!DbªLZAZ2!&(. W
')* ( ¾85N 1'S*1p 87 1*1$~P_/ 1KL*1Os! ã ` 2!D85!LKM8PAn&g;D*1!D,'* *1N !#Hý 1@ÓXY 85r 12.B /87*C 0!D8585)7%b!D.(' 0 ]^ 1nn&O;D*1!D,'**1N 5-!#dý 1lÓX0Y `khZ@#nD 0PA n Er 0!D'7585/& 0 1!D@&Dgn #}!*1*1!L]^ 1&;%85!L%'85 1)P ÓXY , ý ¡ ∼¢ ý ⊥¡ ,&$SZJÁ!D&)(.QA ý ⊥¡ ¢Ëý ¡ ,'P P/'rý ¢+ý ◦ Ð".([ÐQ ⊥¡ ¢ A ÓX0Y , ý ¡ ∼¢ ý ¡ ;M 1G,$onJÁ!D&)(.Q`
850#I!D85rn&r;D*1!D,'**1p5=!#$" $%bý 1 ã 85r 1{!D0>ns!>n!D&r 0!D8585)s> %b!D.(' 0]^ 1n"n&d;D*1!D,'**1N 5q!# n& 0!D8585L5%b!D.(. 1.;p $&%'-ý 1+ `H 1' 0 nJ ]f! )Z;D!D87 1L C &( C &)KMn-7NJ9q!#!D,.¯sP 05PA ]qJN$k 0!D' 0*1/'(& n&) C &( C 85{ E5!D!D85% D`Ë%'P 0
b *1*1$ Afn&4!D85% E7O#³85!D σ s! τ 14B 1Z8 BF;D!D85$"85p 12!D&0>n!Dg 0!D8585L5%b!D.(&' 0r]^ 1n2Zp;D*1!D,'*C*1p5=!# ý Aq&(m]q ? !L]Õn.Bn&2 0!D8585)7%b!D.(. 1&;[5!#;D*1!D,'*^*1N 5O85G 1 !D0>ns!>n!D&J 0!D8585L5%b!D.(&' 0MA.,$on&g85;D/&N =),b!LKMM`
J ]f! 85ZL7 2 0*1! )(" )Z;D!D87 1L C &( D 85JP_/ 1KL*1qK& pn- 0!D8s> 85)5%V!D&(&' 0n&)qN%bqk!D,.¯ P 0 σ !# C s!pZ 1 F85%85 )F 1!D@ σ !# σ 1 D ` hZ= 3^BD5$ks!S9-Z&)dý ¢ Y 5 σ ¡ A'9 1' 0gn E E= 0*1B85*1$"585/J#I!D8=,'D9- $&%') &({%85)85KM)(\)d 1;D&8JF$&%'Ld,bP P/'ý ¢ôý →ý ¢ Y 5 σ→τ ¡ `gÎq$ (0
' 1 1!D[!# C .(zý ¢ý →ý A ]fS&)KM C σ, τ ¡ ∼¢ ÓXY ný , ý ¡ `
850#I!D85 C σ, τ ¡ 3^ E5!D!D85% Js! D 5 σ , τ ¡ ,&$ ¢ ÓX0Y ný , ý ¡ ¢:ÓQX0Y Y 5 σ ¡ , Y 5 τ ¡5¡ ∼¢ D 5 σ , τ ¡ . C σ, τ ¡ ∼
M/'gn&S 85F)7 m 0*1! )([ )Z;D!D87 1L &( 85OP_/ 1KL*1P`" 1' 0 E E5!D!D85% -s! C Abn E 0!D%&*1F)dn&gC%85! !#n.D)=n&r!D87 1;D 1.* )Z;D!D85$ DC &( ng )Z;D!D85$ C (&Z85 1&)(",&$oeJ87 1% ? -* -,(.!(&* °85rP_ / 1K*1 P` î . Ü£ØNÛ7Új&Ú ÚVy &' 0 s& H&7 , - = Ú/ þ 7 1.;OZg85* ) 1!D'5 1%b,' ]f{eJ87 1% ? -* -,V(.S!(&* =.(2 85ZL7 4 0*1! )( BF;D!D87 1)d(L7 087 1,'L(" 1"n-* D7P 0 1!DCA n&g5!D/..(.L9q&(" 0!D%.*1Z&)n0> !D85O^#I!D8=eJ87 1% ? g* -,(&S!(&* ^$",'r(&87 1KML("#³85!D]f*1*> ? !L]^Gn!D85O W
w,w 0
0
n
1
op
B σ
W
B σ
D op
A σ
W
B σ
B σ
A σ
D op
B σ
A σ
A σ
A σ
A σ
A
A
B σ
B
B
σ→τ
A
A
D
σ
σ→τ
A
σ→τ
σ
σ
A
τ
τ
D
D op
σ
τ
A
D op
A
σ
τ
D op
A
B
B
D
D
A
B ,V!D/&r* -,V(.2 * 0/*1/'g.([ 85ZL7 0*1!9L(@ BF;D!D87 1)" I94 j.D®.A Çc85rhZ ¡ ` U 9CX ÑÑO¸ ?æ 9 X0Y ã Ò X ¸ à W ¶ ¼bá XÊ ¸ÑSÒ Ð ¸GÑ TPÐ XÊ ¸ UÐ ¸ UvXU âB¶ W5TU Ñ XUbY ·0¸ Y ¶E· óÏ ¶ U ¹ ñ ¸ Y w A½ @ TW ¸ U'>Ï fX0Ê3ʵÉFYIÏ ¼ X5Ð[YZXW ÑN· ñ . N σ ¸ UVηÐ ñ , x σ . M τ BfXJº ¸Dâ X ñ . N/x M τ ηw ¢° ñ , x σ . M τ F η 1 ñ . N σ ηw/x ¡ w hZ^ E=!)]6BD5$os!%85!)KM-5!D/..(.L9q,&$S 1.( /. 0F 1!Dk!DGP_/'B 1!D&*%85!!#IP` 1 2 43456 DC å ÓTD»UVÐDUX ·9· 4 æ 9CX0Y Ò X ¸~· X0YOTsEó fXÊ}Ê É YIÏ X5ÐGX F0» ¸ Y ¶ TU ·½ È3ó ñ . M ¢ N σ Y}º.XU[X â X0WÏ Ñ TBÐ EXÊ ·0¸ Y ¶E· óÏ ¶ U ¹ E ¸ Ê · T ·¸ ¼ Y ¶E·}ö X · ñ . M ¢ N Eσ ½ ` '!D8=eJ87 1% ? * -,(.!(&* PA&]q-%85!)KMJ(&)(/& 0 1KMJ 0!D%.*1Z&)9=,$S7!L]^ 1.; ng7585!D&;D8^%85!L%'85 $oZ&)KM85$Sn&!D85$k&D^!&(*n` ÿ H5I J å è T Ñ ¼ Ê1X0YZX0UX ·9· Kæ 9CX0Y E Ò X ¸ U'Ï · X0YTsóJLX F0» ¸ Y ¶ TDU · R0Ê1T · X5л&UVÐ XW ` ½ 7 ºXW5X ¶E·p¸ à W ¶ ¼bá XrÊ ¸ÑSÒ Ð ¸kÑ TPÐ X0Ê ã ¶ Y}º ã k−ñ . M ¢ N σ ¶ M ñ . M ¢ N ½ σ∈E N 6OQPRSTUVXW &^ 0!D%.*1Z&)9n!D85° 3%85!)K )(N,$N 0!D'5585/. 0F 1.;-JZ85 !(&* ã ¢ h ß , ≤, {A }, {ã ¼M¼ }, {i }i 1Gn&J#I!L*1*1!L]^ 1.;]q$ ` ß ErnS%b! g!#q
' 1ZS $&%'OD99 1;Dpsrñ|!D87(&85)(,$i 1' 0*1/'7 1!DC`khZ • ]^&)#}!*1*1!L]=PA]f-]^ 1*1*Q]^87 1Frñ\#I!D8=285, 158785$o*1N ^!#ß:` E^Z-^!#j*1*f ñ . M σZA.]^85gñ . M σ 3^]f*1*>n $&%')(QA..( • A ñ . M σ¢ {ñ . N σ | E ` ñ . M ¢ N σ} E^n&-P_ / 1K*1' 0- 0* D9!#ñ . M σ ]^ 1nG85)5%bP 0^ ! E ` ã 5 ñ . M σ→τ , ñ . N σ ¡ ¢± ñ . M N τ • ¼M¼ é 0
σ w
σ Γ
σ,τ Γ
0
σ w,w 0
σ,τ w
σ w,w 0
5 ñ . M σ ¡ ¢± ñ . M σQ#I!D8=ñ ⊆ ñ hZp EgBD7$is!4 &P ? n.BgZk(0
' 1 1!D[ ? S'9MA&([n&)g]qO.BKM;D*1!D,'* *1ps K .( S B=*1*C%&%85!L%87 BF-F$&%'LP`c'!D80.%.*1MA ¢± λx σ.λy τ.x K
G%85!!#!#J0&F'7 1!D.*1 1F$z E\*1 1ZZ*12!D85G 1 F85)5 1.;'`/%%V! 9GZ&)4 ñ . f.(| ñ H.BK NZO7N#I/&' 0 1!D.*j,b.BK& 1!D8PAQ n` M`AC#}!D8g*1* M σ→τ ñ ≥ ñ[&(Gñ . P. σNA']fσ→τ J.BKM ñ . M P τ ¢± ñ . N P τ
CA. 1G%'85 0/* 8PA#}!D8=ñ ≡ ñ , x σ ]^ 1n x !D^ 12ñA.]q-.BKM ñ , x σ . M x τ ¢° ñ , x σ . N x τ &(z5!4,&$@85/*1i ξ ¡ .(L. 1!Dü η ¡ Ac]qS.BKM4 ñ . M σ→τ ¢¾ ñ . N σ→τ Z`
M/' A 3=eJ87 1% ? J* -,V(.!&(&*n` hZH85N 1'H !p5!L]ln.B 7) Es
')%85P 0 E9*1$Nn&=P_/'B 1!D','*1!D.;D 1.;gs! ` a@r,';D 1{,$"85* )F 1.;On&g 1AZ85%85sB 1!D4!#OF85¥s!S 15 P_ / 1K*1' 0r 0* D9B`qEhZ# ñ[ E^&$S $%bgD97 1;DN PA&]q-N$k(&0
'-G&K 185!DN η ,$ ñ ¢ ñ . x σ !D # n&x 85σ]^∈ E ñ ⊆ ñ ηx undef ined ¤ 7587 1;Ds#I!D85]q87(\ 1.(/& 0 1!D@!D[Z85ON5&!)]=gn&)-#I!D8r&$\ñ ⊇ ñ ⊇ ñAj]f &)KM ñ . M σ η ñ ¢± ñ . M σ hZ|%'85 0/* 8PA]^&&KM8 A 7) Es
')S|P_ /') 1!D~ñ . M ¢ N σ Aq]qG&)KM ñ k−ñf η ,&$o 0!D'5585/. 0 1!Dk!# η A&(G5! ñ . M σ¢° ñ . N σ 1' 0n E-85BD7!D 1&;G%&%.*1 1)Js!GKM85$4ñACKM85$4P_/'B 1!D\7B Es
'L(i,$ A g/'5 ,'g%85!LKL),.*1 #³85!D E ` a 1*1p 1= E^%V! 99 1,&*1r !O7!L]+n.B A 7B Es
'L^KM85$kP_ /') 1!Dk 1 E (. 185P 0n*1$ A ,$7 1 1* 885BD7!D 1&;'AD 085s 1O 0!D%.*1 B 1!D'c$p,'=)KM!D (&)(N,&$r85L5587 0 1&;-!D/8 BFZ 1!D2s!S 0*1! )(2Z85OP` &85g E=!o*1! 9=!#c;D&87*1 1 $o 14(.!D 1.;S7!'A'7 1' 0g 1 E^BD5$os!%85!)KMJn&g 0*1! )(kP_/'B 1!D ¢ ∅ . λx σ . . . . λx σ .M λx σ . . . . λx σ .N ¨ª 0
• iσΓ,Γ0
0
0
0
0
0
0
0
0
0
00
0
1
1
00
k
k
00
1
1
k
k
0
#I85!DÕZ-P_/'B 1!D
¢ N ,'F]q\!)%b@Z85OPAC.(iK 0KM879`pb!D8J&${ 0*1!9L({P_/')F 1!D ∅ . M ¢ N A]fJ.BK τ ∈E ñ ¢ E ` .M N τ #}!D8&$zñAH,$z85/*1 å ¸ ÐMÐ â)¸ WFæ ½ &850#I!D85MAc#}!D8KM85$z]f!D85* (~ñ+!# A AHn" ]f! P_ / 1K*1' 0[ 0* D9)@ ñ . M τ r&(ô ñ . N τ g]^ 1*1*-,'@ (&F *n` 1' 0@n& pB 1.;!# ∅ . M τ 12&$S&K 185!DN η B^]q!D85* (Gñ@]^ 1*1*Q,'S ñ . M τ ZA'.( 7 1 1* 85*1$k#}!D8 ∅ . N τ Ab 1^#I!*1*1!)]= n.B A 7)F 3s
') ∅ . M ¢ N τ `^ 3 %85!LKM) n-Z!D852` " 1%V!D85 N%85!L%'85 $[!# n&GeJ87 1% ? kZ85õ!&(*]qk 0!D'7585/& 0 1mn 0!D%&*1FL9r%85! !#^ E-n.) A Eg!D&% $2#I!D8g*1* w ∈ ß ² σ 3r@ 1 /.> 1F 1!D E7F *1*1$G%85!LKL),.*1g%85!L%b! 9 1F 1!DC` /8 1 Z85L5- 1in E%85!)%b85F$25FOd#I85!D G]f*1*> ? !L]^\7$ D 0 0!D8585)5%V!D&(&' 0,' ]f@ $&%')(\* -,(&4 * 0/*1/'r.( 1 / 1F 1!D E7F p*1!D;D DAQ *1*1)(\n&Jó0TDW Ñ »&Ê ¸L· É ¸L· ÉFYIÏ ¼ X · ¼ W ¶ UVR ¶ ¼ Ê X !D8 è »&WWÏ Y[ZgT ¸ WsÐ ¶E· T Ñ TW º ¶E·Ñ B,.*1 $&%')f!D% $ AD]qg/'5H,b;D 1o]^ 1nkBD S,'D $%b !D&%F$ A 7 1' 0 ! Bs!D J%85!L%b! 9 1F 1!DG E=%85!)KB,.*1M`chF^ E^BD5$ks!O9-Z&) # σ .( τ BD G&)KMJB x1 σ1 , . . . , x k σk . M
σ w
σ w
¨.
! 5 !D&g*1N PAbn &D B=!5 !D&g*1N PAb.(25!oO5587 1;D&s#}!D8s> ]q87(o 1&(/. 0F 1!Dk5!L]=fn&σ→τ )q!D/8q!&(&*r/'5f.)KMd)! 5q!D*1p!# BD F$&%'M`HÄf!D'P_ /n*1$ AKM85$S]q*1*>nF$&%'L("P_ /')F 1!D"]^ 1*1*Q,'g9B Es
'L(Q` ¤ !Dn&8i]q$6s! 0!D&P 02!D&%F 1&)G]^ 1n:%85!LKB, 1*1 1 $ 34s! 0!D'9 (8 0* D9)O!#-!&(* B`Ëhn#-]q4 0!D'7 (&8on{ 0* D9O!#-#I/*1* 0* D7 *= $%b2 18787 9.> 1)PAc]^ 1n~5!DNS,'DS $%b)p%F$@.([!Dn&87N!DPAcnzn&S $%'Lr]^ z85 !D% $r 1KM85$p!(&*b85^n&rR0Ê ¸L·9·¶ R ¸ Êb%85!L%b! 9 1F 1!D.*sP/.s!L*1!D;D 1L=Äf!DbD©)Z` Ú b"=û=5 c =EÛ d:ÚÛ7 H&9 a × C5ØÙ=c 6.1
Relations over applicative structures
Q !D;D *85* ) 1!D' .BK g%85!)K 4/'0#}/* 1{n&5F/'(.$G!#qn85* )F 1!D'qs!OeJ87 1% ? -* -,(&!&(&* P` hZ4ng 0* D7 *Q!&(*QZ!D85$k!#F$&%')("* -,(.O * 0/*1/'PA'*1!D;D *Q85* ) 1!D Ep2#n 1*1$i!#^85* )F 1!D'r 1&(&0.)(@,&$\F$&%'Lg]^ 7)F E5
')r{ 0!D&(. 1 1!D[ 1%.*1$> 1&;o 0*1! 5/85g/..(8=%&%.*1 B 1!DG&(2* -,V(.S),b5587D 0 1!DC`H &-;D87*1 1P) 1!DG ! eJ87 1% ? S%&%.*1 B 1KM5585/& 0 /85)g 1KM!*1KMLJ 1&(&0. 1.;k85* ) 1!D'-,$i,b!DZ[F$&%'Lr.( %b! 7 1,.*1=]q!D85* (P` a@ ]^ 1*1*V7 1%.*1 #I$p!D/8H%85) sB 1!D,$D5/. 1&;-g
&L(O5585/. > F/85 h ß , ≤i Z85!D/;D!D/.dP 0 1!D4®.` ¤:à W ¶ ¼bá XÊ1T ¹¶ R ¸ ÊW7XÊ ¸ Y ¶ TDUG!)KM8feJ87 1% ? d%&%.*1 B 1KM=5585/. 0 /85) ã .e( ¦ }/'5> 1&;\n&G7N h ß , ≤i¡ 3Ni#n 1*1g$ ft¢ {R } !# 85* )F 1!D' R ⊆ A × B 1&(&0.)([,&$\F$&%'L σ &(]f!D85* ( w ∈ ß 7)F 3s#}$& 1.;Gn&O#I!L*1*1!L]^ 1.;2 ]f!i 0!D&(. > F 1!D'P`c &J
'875^ E=!D!Ds!D 0 1 $o 0!D.(. 1F 1!Dk#I!D8^$o,'D9- $%b c Ñ TDU ¡ 1%.*1 1) R a, b ¡ a, i b¡ #}!D8=*1* w ≤ w , R i ]^ 9\7$.dZ&)d]^& w ≤ w An&85* B 1!D R E- 0!D 1&)(i 1 R A!&( /*1! n-587'9 1F 1!D"#I/.' 0F 1!D'B` &g9P 0!D.(" 0!D&(. 1 1!D 5R Ñ ¼ W5X ¡ f, g ¡ ² ∀w ≥ w. ∀a, b ∈ A . R 1%.*1 1) R 5 i f ¡ a, i g ¡ b¡ , R a, b ¡ ¨D¨ σ w
σ w
σ w
c w
c w0
c w,w 0
0
c w,w 0
0
c w
0
σ→τ w
σ w0
c w0
τ w0
σ w0
σ→τ w,w 0
σ→τ w,w 0
σ w
*1*1)(6us 0!D%85&'7 1!D'wA7$.=n.Bd85* B 1KM-s!SZr#}/.' 0 1!D' )KL 1* ),.*1-#³85!D ã &( An&N85* B 1!D R 0!D 1'J*1*#}/.' 0 1!D' %% 1.;o85* BF)(\85;D/.ps !r85* BF)(85L5/*sB` &^F]q!J*1ND,'*1!)]85^%85!)KML(/'7 1.;-n&^)9F *1*1$pn 7p-85;D/.N 5^D^!D/.n*1 1&)(" 1 Ç*1! ªZ` 1 2 43Eh jå i TDUVTDYnTU ¶ R ¶ YIÏ] æ 9CX0kY f ⊆ ã × Ò X ¸ à W ¶ ¼bá XGÊ1T ¹¶ R ¸ Ê^W7XÊ ¸ Y ¶ TDU ½ 7 ºXUó0TWX â XWÏdYIÏ ¼ X σ ¸ UVÐ ¼ ¸¶ WfTsóXHTWÊÐ · w ≤ w ¶ ó R a, b¡ Y}º.XU R i a, i a@=7$r&K& 185!Dps η , η 85-W5XÊ ¸ YZX5Ð Ò lÏ fyTUoñ ¸ Y w # R η xw, η xw ¡ #}!D8=*1* x τ 12ñH` 1 2 43Eh =< å @ »UVÐ ¸Ñ XUbY ¸ Ê 9CX ÑÑO¸ æÈ3Bó f ⊆ ã × ¶3·¸ à W ¶ ¼Vá XqÊ1T ¹¶ R ¸ Ê.W5X0Ê ¸ Y ¶ TU T â XW Ñ TPÐ XÊ · ã ¸ UVÐ: $ ¸ UVÐ2XU âP¶ W5TU Ñ XUbY · η , η ¸ W5XW5X0Ê ¸ YZX7Ð Ò Ï:f£TU2ñ ¸ Y w Y}º.XUróTWgX â XWÏOYZXW Ñ ñ . M σ XfXº ¸Dâ X R ã ñ . M σ η w, S ñ . M σ η w ¡ ¤ c]^ 1n$-!#V!D/8j!DZ8c(&0
' 1F 1!D'PALn&(0
' 1 1!DN!#VeJ87 1% ? q*1!D;D *85* L> F 1!DNN$-,bq(&87 1KML(r,&$J 1 F85%85F 1.;Jn&H/'5/'*.(&0
' 1F 1!Dg 1Nn&fs!L%b! HÓX0Y `
o/'5/'*q(&0
' 1 1!D[!# *1!D;D *f85* B 1!D& f ⊆ ã × EG#n 1*1$@!#=85* )F 1!D' 5/& kn&) R ⊆A ×B » · » ¸ Ê ¡ f, g ¡ ² R ∀x ∈ A .∀y ∈ B . R x, y ¡ ⊃ R f x, gy ¡ .
!2850>n 1Z85%85Nn Ep 0!D&(. 1 1!DCAC]qOg/'5r
'875r7$i]^.BreJ89 1% ? O85* )F 1!D'g85M` ¤ o]fG.BK 29CAq@ $%'MAq!D8zuseJ89 1% ? 4PAxw Eo@#n 1*1$ A ¢ {A } !#gs 1&(&0.)(~,&$z]q!D85* (PA]^ 1Z|587'7 1 1!D~#}/.' 0 1!D' i #I!D8OBD w ≤ w ` 8PA P_ / 1K*1 n*1$A J#I/.' 0 !D8#I85!D h ß , ≤i s!N95P` ¡ hn#Q]q=&)KM^ ]f!p5/& OeJ87 1% ? 95 ¢ .( B ¢ {B } A'n&4 à W ¶ ¼Vá XrW5XÊ ¸ Y ¶ TU R ⊆ A × B EdO5/M,V9=!# A {A } ¢ A n` M` AVO#n 1*1$ R ¢ {R } !#85* )F 1!D' R ⊆ A × B A × B {A × B } 5/& "n.B 1%.*1 1) R i a, i b¡ #}!D8=*1* w ≤ w . R a, b ¡
M/' 0!D.(. 1F 1!D å Ñ TUDæ- E=,/ 1*^ 1 !Snp(0
' 1 1!D"!#ÓX0Y `HhZ#c]f-!L]Ë7$ n&)O\eJ87 1% ? G*1!D;D *85* )F 1!D~!LKM8 ã &m( ¾5&!D/* (z,b24#n 1*1$z!#JeJ87 1% ? 85* B 1!D' R ⊆ A × B 9B Es#I$ 1.; å » · » ¸ Ê æ2),b!LKMMAn&~ 1!D*1$85 1' ! 9&P ? å R7T Ñ ¼ W7XFæ ½ Eq Eq!D, 1&)(o,&$]f!D8 ? 1&;p!D/.fn&JeJ87 1% ? } n` M`VAVÓX0Y ¡ 1 Z85%85s)F 1!DG!#cn&g5 .(.87(o 0!D%85&'7 1!D2 0!D.(. 1F 1!D å » · » ¸ Ê æ ½ σ→τ w
0
a
σ w
σ w0
τ w
b
a
σ w,w 0
a
b
b
σ w
a
b
hW ,≤i
σ
σ
σ
σ→τ
σ
σ
σ
τ
w
0
A w,w 0
w
w
w
w
w
w
w0
A w,w 0
w
w
w
0
B w,w 0
hW ,≤i
σ
σ
σ
hW ,≤i
¨D¿
σ w,w 0 b
¡½
6.2
Partial equivalence relations and quotients
¤ à W ¶ XÊT ¹¶ R ¸ Ê ¸ W0Y ¶³¸ ÊXGF0» ¶}â)¸ Ê XUVRXOW7XÊ ¸ Y ¶ TU&f ã ã E-GeJ87 1% ? N*1!D;D * ´ 85* B 1!D¼bmá 7/& n.)r¼ BD R E5$&N587 S.([587⊆'9 1F× 1KMM`i &o.pk%'85 * P_ / 1K*1' 0=85* )F 1!Dk 0!DpL#³85!D§n&=#nD 0fn&) # R Ef7$N587 =.(587'7 1s> 1KMMAMn& R EqoP_/ 1K*1' 0=85* B 1!DS!Don&dH!# a ]^ 1n R a, a¡ `ca@ ]^ 1*1* B,,85K& BFOn&S 0/.-,'875!DNO%87DuseJ87 1% ? O*1!D;D *%'85 *P_/ 1KL*1' 085* L> F 1!D'ws! á Ê ¼ XWr` &J#I!L*1*1!L]^ 1.;*1N5&!L]=n&) á Ê ¼ XWrÅx^$k,'g 0!D'5585/. 0Z)( ,$k 9&! ! 9 1.;85* B 1!D'^B^,'DJF$&%'LP` 1 2 43Eh DC å Ô ¸ WY ¶³¸ Ê nF» ¶}â)¸ Ê XUVR9X ? æ 9CX)Y f ã ã Ò X ¸ à W ¶ XpÊ1T ¹¶ R ¸ ÊjW7XÊ ¸ É Y ¶ TU ½ È3órX ¸ R0º R ¶E·· Ï ÑÑ X0YIW ¶ R ¸ UVÐOYIW ¸ U ·¶ Y ¶}⊆â X .ó0TD×WpX ¸ Rº Ò5¸L· XrYI¼bÏ ¼ á X c ¸ UV?Ð HTWÊÐ ß HY}ºXU@X â XWÏ R ¶E·J· Ï ÑÑ X0YIW ¶ R ¸ UVÐSYIW ¸ U ·¶ Y ¶}â X ½ o w∈ /'d!#C%'85 *VP_/ 1K*1' 0d85* )F 1!D'f Eq 1o#}!D85 1.;_ /!D 1 q5585/& 0 /85)B` a 1n ¼ ¸ WY ¶³¸ ÊqP_ / 1K*1' 085* B 1!D'PAQ850'0. 1K& 1F$G#n 1* PA &(@5!2@*1N -&)( !D.)KMp\P_/ 1KL*1' 0 0* D9P`- &850#I!D85MAV 1J 1;D&d,bN!D85D 0/87)Zs!k *1* nL94u %'85 *C_/!DF 1sB`xwihZ# ã ¢ h ß , ≤, {A }, {ã ¼ ¼ }, {i }i E-oeJ89 1% ? N%&%.*1 B 1KMN5585/& 0 /85p.( f ⊆ ã × ã EJ á Ê ¼ X0WAZi]f(&0
' n:F».TDY ¶ XUbY ¸ ¼M¼ Ê ¶ R ¸ Y ¶}â X · YIW»RYI»&W5X ã /f+¢ h ß , ≤, {A /R }, {ã ¼M¼ /f }, {i /f }i D#}!*1*1!L]=P` ¢ { a | R a, a¡ } Ab]^85S a E^ZoLX F» ¶}â)¸ Ê XUVR9XN 0* D • A /R a ¢ { a ∈ A | R a, a ¡ } ã ¼M¼ /f ¡ a b ¢± ã ¼M¼ a b • /f ¡ a ¢± i a • i hZN E\7 1%.*1o0.87 0 Eos!iKM87 #I$@n.BNn&k_/!DF 1N7585/& 0 /85S EN]q*1*>Z(&0
'L(QA &(GeJ87 1% ? g%%.*1 )F 1KM5585/. 0F/85M` 1 2 43Eh J È3` ó f ã ã ¶3·\¸ à W ¶ X ¸ WY ¶³¸ ÊgLX F0» ¶3â)¸ Ê1XUR9X@W7XÊ ¸ Y ¶ TU T â X0W à W ¶ ¼bá X ¸ ¼M¼ Ê ¶ R ¸ Y ¶}â X · ⊆YIW»RYI»×W7X ã Y}º.XU ã /¼bf á ·0¼ ¸ Y ¶3·3ö X · Y}º.X à W ¶ ¼bá XX5ëMYZXU ·¶ TU ¸ Ê ¶ YIÏ R7TUVÐ ¶ Y ¶ TU å X5ëDYµæ ½ σ w
σ w
σ w
c w
σ w
σ w
σ w
σ w
σ w
σ w
σ,τ w
σ w,w 0
σ w σ w
R
0
R
σ w
R
R
σ w
σ,τ w
R
0
σ,τ w
R
σ w,w 0
σ,τ w
R
R
¨Ã
σ w,w 0
σ w,w 0
¤ z 7587 1;Ds#I!D85]q87(\ 1.(/& 0 1!D@!D[Z85Op${,'o/'9L(@s!2%85!LKMn&#}!*1*1!L]^ 1.; us_ /!D 1 ^!&(* wZ!D852` 1 2 43Eh =5 È3pó f ⊆ ã × ã ¶3·q¸ ? *1%'8~T â XW à W ¶ ¼bá XqÊ ¸ÑSÒ Ð ¸-Ñ TBÐ XÊ ã Y}ºXU ã /f ¶E·f¸ à W ¶ ¼bá XHÊ ¸ÑSÒ Ð ¸dÑ TPÐ XÊ · »RºgY}º ¸ Y)ó0TWfX â XWÏJXU âB¶ W5TU Ñ XUbY η ¶ Y}º R ηxw, ηxw ¡ ó0TW ¸ Ê3Ê x τ ¶ Ukpñ BfX-º ¸â X ã /f ¡ ñ . M σ η w ¢± ã ñ . M σ ηw , cº.XW7XY}º.XXU âB¶ W5TU Ñ XUbY η óTW ã /f ¶3· Ð X ö UX5Ð Ò Ï"Y ¸ á ¶ U ¹ η xw ¢´ ηxw óTW ¸ Ê3Ê x ¸ UVÐ w ½ hZ[5&!D85PAQn&NB 1.;G!#"F85 1@ZO_/!DF 1-!(&* ã fô 3-n&OP_ / > KL*1' 0S 0* Dg!#^n&ONB 1.;G!# MM 1 ã `" Egn&!D85 Epz/ D(.%sB 1!D@!# n@us &87D 0F87 1PB 1!Din!D85Owk!#d 4 19D®LZA]^ 9i%%')87ds!k,'n&r
'875/' !#jn&- (&B` τ w
R
R
R
6.3
R
R
Kripke logical relations over classical applicative structures
a@=!)]l 0!D'9 (8eJ89 1% ? ^*1!D;D *85* B 1!D'c!LKM8 0* D97 *'%&%.*1 B 1KMq5585/& 0 /85)B`
\7 1%.*1)7S]$l !~(.!Z Ek Ek !z85;M87( 0* D7 *5585/& 0 /85)oD"5%'P 0 * ã D9G!#-eJ87 1% ? 47585/& 0 /85)P`l !@,'47%'P 0
b DAH*1 ã ¢ ,'4 0* D97 *f%%.*1 )F 1KM5585/. 0 /85MAC n` M`HA {A } 3p2 0!*1*1P 0 h{A 1!D!},#={sg¼ ¼ 1&(&0}i.)(@,&$ F$&%').( {ã ¼M¼ } E\ 0!L*1*1P 0 1!D~!#%&%.*1 B 1!D[#I/.' 0F 1!D'P`[a@"(&0
'&oZ eJ87 1% ? g5585/. 0F/85 ã ¢ h ß , ≤, {A }, {ã ¼M¼ }, {i }i ,$d ? 1.;5 A ¢ A AB%&%.*1 B 1!Dd#}/.' 0 1!D' ã ¼M¼ ¢ ã ¼M¼ .(J587'7 1 1!D #}/.' 0 1!D' i ¢ ¶ ÐXUbY ¶ YIÏ'`phFJ EJBD5$2s!G P ? n.B ã E-GeJ87 1% ? N* -,V(. !(&* ]^&K 8 ã 3d 0* D97 *D* -,(. !&(*nA.(-n.B nfp) 1&; !#b^Z85 1 ã 3dL9 *1*1$Gn&9pNDdZNB 1.;o!# M 1 ã `ghZ\ )Z;D!D87 * M Z85OPA ã E 2%%.*1 )F 1KM5585/. 0F/85J!# 0!D'5 ^%85)5&B)KM)B` a@S7q$ fÕ¢ {R } Ep à W ¶ ¼bá XSÊ1T ¹¶ R ¸ ÊW7XÊ ¸ Y ¶ TDU~T â XWkR0Ê ¸L··¶ R ¸ Ê ¸ ¼M¼ Ê ¶ R ¸ Y ¶}â X · YIW».RYI»&W7X · ã .`( ô X# fË E=eJ87 1% ? -*1!D;D *85* ) 1!Dk!LKM8 ã &+( `c E !D/&5Hs!gn&d9p=Z 1&;NDfÇ*1!D ? 1CÅxf(0
' 1 1!DO!#jÈÉFW5XÊ ¸ Y ¶ TUcAD0. 0%H#}!D8Hn 1!D8g(. ²85' 0n.BJ]q&)KMr/')(\%&%.*1 B 1KMp7585/& 0 /85) 1'5FBD(\!#f!&(* I 0 #s` Ç*1! ª ` Îq$-Qd®.`©.A)]fH r%85!(/& 0qeJ87 1% ? !&(* ,$s ? 1.; eJ87 1% ? _ /!D 1 5!#¡ 0* D97 *!&(* B` σ,τ
σ
σ
σ,τ
W
σ w,w 0
σ w
σ w
σ,τ w
σ w,w 0
σ,τ w
σ
σ,τ
W
W
W
σ w
W
¨©
W
r Sstst3 uvh Dh È3ó ã ¶3·J¸ RÊ ¸L··¶ R ¸ ÊYIÏ X5ÐÊ ¸ÑSÒ Ð ¸Ñ TBÐ XÊ ¸ UV%Ð f ¶E·J¸ ? *1%b8±T â XW ã HY}º.XU ã /f ¶3·r¸ à W ¶ ¼Vá X-Ê ¸ÑSÒ Ð ¸SÑ TP¼ Ð XÊ ½
Eq;D 1KM)f/'fr#n 185*1$O7 1%.*1J 0* D9f!#C!(&* H]^ 1Zo 1F/ 1 1!D E5 =%85!L%'85 1)P`chF #³D 0PA]qp \5&!)] n&) K 85$G* -,V(.SZ!D85$G En&Nn!D85$G!#Ho!(&*j!#n E #}!D852` ÿ wh6 tx 9CXY E Ò X ¸ U'Ï · X0YqT5óNLX F0» ¸ Y ¶ TU · R0Ê1T · X7Ðk»UVÐXW ` ½ 7 º.XW7XX5ë ¶3· Y ·p¸ R0Ê ¸·9·¶ R ¸ ÊqÊ ¸ÑSÒ Ð ¸iÑ TPÐ XÊ ã ¸ UÐ ¸ à W ¶ X ¸ W0Y ¶³¸ ÊLX F» ¶}â)¸ Ê XUVR9X2W7XÊ ¸ Y ¶ TAU f · »Rº Y}º ¸ Y ã /f ·0¸ Y ¶E·}ö X · ¼ W5X5R ¶E· XÊ ÏkY}º&T · XpLX F» ¼b¸ á Y ¶ T¼U · Y}º ¸ Y Ò XÊ1TU ¹ YnT E ½ N 6O;PIRSyT8UzBW Q A ,brn&reJ87 1% ? rZ85£!(&*Q#I!D8 E AbD 14n&g%85!!# !# !D85Õ©.` Ã'`ja@d]^ 1*1*Q5&!L]n.B A Eq E5!D!D85% ds!NN_ /!D 1 q!#Cn&d!L%' Z85!&(* B !# β, η>Z 0!DK 877 1!DC` a@p]^ 1*1*j85K& 1]:n&p 0* D7 * F85!&(&*j 0!D'5585/. 0 1!D2,87 10'$ `Qdñ ,' m 1.
' 1F"F$&%'"D97 1;DN Nn.BN%85!LK (&)p 1.
' 1Z*1$zN&$@K87 B,.*1L!# BD F$&%'M`j'!D8BD O $%' σ A *1 B ,' nd 0!L*1*1P 0 1!D!#C*1*P_ / 1K*1' 0= 0* D9) {M } ]^ 1n4ñ . M σ #}!D8=5!DN
' 1Zrñ ⊆ ñ A.( ¢ ñ ¢ {M } {N | ` . M N σ} #}!D85!DNoñ ⊆ ñ `\Q B ¢ h{B }, {ã ¼M¼ }i ,'kn&k%&%.*1 B 1KMO5585/. 0F/85 ]^ 1n ã ¼M¼ {M }{N } ¢ {M N }
E^ EBD9 1*1$o5&!L]^ks!,'g"!D87(. 1.85$o $%b)(k* -,V(.!(&*n`4!D85g(s 1* !# n-F85!(&*C 0!D'5585/. 0 1!Dk$k,'J#}!D/..(k 1 '87 3©)ZA&#I!D80.%.*1M` þ 9 1&; E A.]qg 2(&0
'J á Ê ¼ XW R !)K 8 B ` 1' 0-n-%V! 99 1,.*1]q!D85* (!# A 85p $%bpD99 1;Dp5PA]qr]^ 1*1*j/'9pF$&%'pD7 1;DN 5 Ddn&r]q!D85* (=!# R `'!D8 BD 9 σ .(GñA.*1 ²@ñ . M ¢ N σ ∈ E, R {M }, {N } ¡ ]^85 M En& β, η>n!D85*j#}!D85Þ!# M A&(i7 1 1* 85*1$4#}!D8 N `g E EJ]q*1* (0
'&)(9 1' 0^BD {M } .DJ/& _/ β, η>n!D85*#I!D852AD,&$rn&dÄfM/87 .>Z«^! 98 n!D852`m 1' 0k%85!LKB,.*1SP_/'*1 1 $@ E5$&p587 o&(z587'7 1 1KMo]^ 1n~85)5%'P 0 !@&$zñA R E 0*1B85*1$m\%'85F *P_/ 1KL*1' 0k85* B 1!DC`6 IhZ~;D&87*nA R ]^ 1*1* !Dg,'O850'0. 1KM!D B Aj7 1' 0O5!DN M N$i85P_/ 185OKL89 B,.*1)g!Dg 1[ 1ñ` ¡ Îf$"85/*1 å ¸ ÐMÐ â)¸ WZ[æ 7) Es
')=np!D!Ds!D 0 1F$" 0!D&(. 1 1!D å Ñ TDUDæ ½ &g%85!!# n&) R 7B Es
'L å R7T Ñ R¼ W7XFæg E7 1 1* 8s!Nn%85! !#jn.BeJ87 1% ? JZ85¦!&(* 85 0&F'7 1!D.*n` ¨® ∞
σ
∞
σ,τ
σ
∞
σ,τ
σ Γ
0
0
0
0
σ Γ
σ
0
hZS85N 1')D5$zs![5&!L]Õn&) 3O E5!D!D85% "s! `ÆKM85$ P_ / 1K*1' 0 0* DH 3^ 9.87D 0Z87 1L(k,A$O 0!*1*1P 0F 1!Dk!# !D85B/R *b#I!D85Oqn.BB85/R *1* ]f*1*>n $%b)(- 1Nño.(-%85!LKL),.*1$ P_/'*M/'7 1.; E ` M/'Q#I!D8CBD B /R P_/ 1K*1' 0 0* D9 AMn&85= Ef-/. _ /r ñ `Äf!D&KM87*1$ A*1*'!#Z > !D85N{M*.#I!D}/R 85Oj 1O$o ñ .M σ']^ 1*1*&.,bM^P_ /σ 1K∈*1A c!(/*1! R AD&(5!J]qqβ,.)KMη S5587 1;D&s#}!D85]87(G, ¯sP 0F 1!D4,'F]q A &( B /R `=hF 3 BD7$ks!k5!L]Ën.) %%.*1 )F 1!Di,'&)KM)-%&%85!L%87 BF*1$ A&(\5!"]q&)KM@ E5!D!D85% E5Þ,' ]q .( B/R ` 3%85!)KMLn&-n!D852` A !D85% J !O&$oeJ89 1% ? -_ /!D 1 ^!#c$" 0* D99 *%&%.*1 B 1KM5585/& 0 /85M` ÿ wh6 D{ 7 º.XW7X ¶E·g¸ à W ¶ ¼bá XgÊ ¸ÑSÒ Ð ¸OÑ TPÐ X0|Ê }cº ¶ Rº ¶E· UVTDYqXÊ1X Ñ X0UbY ¸ W ¶ ÊµÏ LX F0» ¶3â)¸ Ê1XUVYQYnT ã /fLóTW ¸ U'ÏRÊ ¸L··¶ R ¸ Ê ¸ ¼M¼ Ê ¶ R ¸ Y ¶3â X · YIW».RPYI»W7X ã ¸ UÐ à W ¶ ¼Vá X ¼ ¸ W0Y ¶³¸ Ê LX F0» ¶3â)¸ Ê1XUR9XNW7XÊ ¸ Y ¶ T` U f ½ N 6O'PRSTUVXW a@;D 1KMNo#I!D85g/* φ 1\n&%85)(. )Z*1!D;D r#}!D8-%&%&*1 ) 1KM 5585/& 0 /85)q]^ 1nG,'DF$&%') p .( q ]^ 9k EK*1 (S 1G*1*C_/!DF 15 ã /f:,/.q E !D^KL*1 (o 1GKM85$SeJ87 1% ? * -,(.!(&~* `hF / 1F 1K *1$ A φ 3nJ#I!D85g/* Nn.B 7$&B # .( ¬¬inhabited p→q ¡ A.n inhabited p→q ¡ A • empty p ¡ ]^85 inhabited τ ¡ ≡ ∃x τ.x ¢ x¡ &( empty τ ¡ ≡ ¬inhabited τ ¡ hFc EjBD5$ !O &P ? Z&)^n E^&!* ( 14&$o_/!DF 1q!# 0* D99 *Q%%.*1 )F 1KM5585/. 0F/85M`
!\5&!)]§n&) Er!DN 1 / 1 1!D E5 *1*1${K*1 (QAc]fk 0!D'7 (&8rZS#}!*1*1!L]^ 1&; eJ87 1% ? -* -,V(.!&φ(&* ß E^n&-%b!9]^ 1n2 ]q!N*1N 5^ª < • §) E^n&J#}/*1* $%bg5585/& 0 /85]^ 1n2n&-,'D9J $%b) p ¢ ∅ .( q ¢ ω • :)HªJ En&= 1Z87 1!D8f!# :B A n` M`AMn&=%&%&*1 ) 1KMq7/M,.>Z5585/. 0F/85]^&! 9 • *1ps 85- 1 Z85%85s)F 1!D'!# 0*1! )( λ>nF85O Z-587'9 1F 1!D"#I/.' 0F 1!D i E=n- 1' 0*1/'9 1!D •
A 0!D s 1'f0.D 0n*1$S!D*1pJ }Z%F$#I/.' 0F 1!D ¡ A&.(On&850#}!D85 Ì7)F E5
') ¬¬inhabited p→q ¡ çJ,/. A Ek%F$ },bP P/'{n&85@85{! 0*1! 9L("Z85O !#n.B^ $&%' ¡ A.5(! °(.!ML=!D=7) Es#I$ inhabited p→q ¡ ` σ
σ
σ Γ
σ Γ
0
σ Γ
0
σ
0,1
p→q 1
p→q 0
¨
σ
σ Γ
σ Γ
σ Γ
6.4
A counter-model to implication (∗)
¤ i %&%&*1 ) 1!DË!#OeJ87 1% ? z_/!D 1 5PAJ]q[]^ 1*1*p5&!L]!L]s! 0!D'7585/& 04 0!D/&Z8s>n!&(* !\n&G 1%.*1 B 1!DË ∗¡ ;D 1KM~ 1mZ2P 0!D&(z%'87;D87%l!#dZ %'%'8P`\a@"]^ 1*1*q 0!D'5585/. 0p\< ? 1m!(&* ã ]^ 1n~,'D9oF$&%') .( &( ;D 1KM-NeJ89 1% ? J*1!D;D *%'85 *P_ / 1K*1' 0J85* B 1!D/f ⊆ ã × ã 5/& aSZ&) 1bkZ _ /!D 1 ^!&(* ã /fNA]q-]^ 1*1*Q.BKM ¢ λx a.f π ,/.!D f π ¢ f π . λx a.f π a@o*1 ã ,'"{ 0* D7 *HF85Ì!(&*f!# β, η>Z 0!DK 877 1!DCAD(&)7 087 1,b)(z 1zn& %85! !# !#= &!D85Í®.` &`i 1' 0 f %&%'B87g 1mn&SP_/'B 1!D'g),b!LKMMA]fo 1' 0*1/'( Z85OJ]^ 1n\ 0!D'5s f P a→a→a¡ →b 1@ 0!D'5585/. 0 1&; ã `O I &r 1 F85%85 ) 1!D !# f 1 A 3^ 15P_ / 1K*1' 0- 0* D9PA.!&( /*1! ` ` ¡ hZH85N 1'H !N(&0
'^n& 85* ) 1!De fvBH,'D9= $&%')PA7 1' 0=Z EH]^ 1*1*V(F85 1 fl) 1;D&8 $&%')P`c 1' 0fZ ¯ /'5
b )F 1!Dp!# (&%'.( !DNF$&%' ,'B 1.;B 1n&8 ;D*1!D,'*1*1$N% $r!D8H;D*1!D,'*1*1$p!D&%F$A]f^]^ 1*1∗*'¡ ? a %F$B!Da&^]q!D85* (O&( !D% $4)!Dn&8P`-a@*1\ß ¢ {ª , } ]^ 1n@ª ≤ .({ ? R ¢ ∅ &( n&- (& 1F$O85* ) 1!D A `ê !)]gA.]q-] ^s!O7)F E5#}$P_/'B 1!D R E¡ ¢ λx a.f π λx a.f π BJ,b!DZ@]f!D85* (P`p EJ EJBD7${)]q!D85* (iªAC7 1' 0 EJ% $ `-a@O \s ? !G,bOn& (F 1 $285* )F 1!D@!D A ` !27B Es#I$ Ea¡ )-]q!D85* ( ]^85 a E-R!D %F$ A.]qgg/'7=P_ /')Z f π .( f π ` ¤ 4BD7$o]$os!S(.!Sn E= 3¯s/'5=s!Os ? ¢ A × A 5!n.B A /R .D!D*1$k!DJ*1N P`hF^ E^BD5$os!KM87 #I$Sn.B R λx a.f π , λx a.f π ¡ R B^,V!Dn2]q!D85* ( w ¢vª , A'.("5!n&)-Z85O=85-P_/'*Q 1Gn&g_ /!D 1 !(&*n` Z) ! #BÅx $&%'Gn&!D85$+ I 0 #5` ÎqB ¨ ¡ `6hZ ESBD5$~s![94n&)SeJ87 1% ? * -,V(.m!(&* o85i!D85i;D87*dn&6 0* D99 * * -,(&z!(&* PA 7 1' 0{&$ 0* D97 * * -,V(.S!&(&*CN$G,br,'p85;M87(L(4DdSeJ87 1% ? r* -,V(.O!&(*!)K 8 o4ß 0!D'7 E7F 1.;O!#Ho7 1.;D*1r%V! 99 1,&*1N]q!D85* (Q`=eJ87 1% ? p!(&* ]^ 1Z4%85!(/. 0s .(pdZ85 1.*.F$&%' 85* 5!JJ5%'P 0 * ? 1.(r!# 85Z)9 0*1! 9L(p )Z;D!D85$ σ ×τ &(QA& 0!DK 879*1$ AD&$O 85ZL7 " 0*1! )(S )Z;D!D85$$,'d-,')((&)(k 1kpeJ87 1% ? !(&*n` ¤ *n&!D/;D2]qJ(&0
'&)("eJ87 1% ? J!&(* ]^ 1Z!D/.^/'7 1.;g/. 9G )Z;D!D85$on!D85$A !DJ]$Os!NK& 1]6!D/8^(K *1!)%N q E^D[u ]f!D8 ? L(S0%.*1PwN 1kZJ/'!#n 1 Z89&** .;D/';D!#fks!)%V! P`%'P 0
b *1*1$ A!D/8-eJ87 1% ? * -,V(."!(&* 85)7/*µ #I85!D 1 F85%85F 1.;On&r5s.(&87(G 0* D7 *C(&0
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