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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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 ¡ !#1'85085"!L]= &$@ !D ,.¯P0 #³A85!D¾ n&oF851&*q!D,.¯sP0PA!D/8(.E70/'71!Dz!# a →A ? 1i!(&* JB,V!)Kp%&%&*11L!"0!D'085Z85F)7 @0*1!)(i)Z;D!D871L-Dd]q*1*n`-hF %'85F 0/* 8PAbnp587D(.1F1!D.* 1.#I85'0p7$&5F¥Ed!Dd0!D%.*1Fr#I!D8NF -1> %&*1 )1!Do!LKM8f0!D'085F-85Z)9 20*1!)(kBF;D!D871)^]^&lu%F$.wr!D,.¯sP05f85 *1*1!)]q)(Q`
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U,V
2
Here is one example, based on the results of [Jac75]. We write M ←→ N if there is a term P with P U V =β,η M and P V U =β,η M . Then in the traditional inference system U,V
U,V
for typed lambda calculus, we have U = V ` M = N iff M ←→ . . . ←→ N . This fails when the new inference rules for empty types are added.
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2.1
Possible worlds
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0
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2.2
Applicative structures and predicate logic
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σ,τ w
σ,τ w
σ w,w 0
σ w,w 0
0
σ→τ w
σ w
σ w
τ w
σ w0
σ w
σ w,w
σ w
σ w 0 ,w 00
σ w,w 0
σ w
0
σ w,w 00
σ w
σ→τ w
τ w,w 0
σ w
σ,τ w0
Aσ→τ × Aσw0 w0
0
σ w0
σ w
σ,τ w0
σ,τ w
00
6
σ→τ w,w 0
σ w,w 0
- Aτ 0 w
6
iσ→τ × iσ
iτ
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σ w
σ,τ w
σ,τ w
0
τ w
σ w,w 0
σ w
τ w
σ,τ w
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σ→τ w
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'1F1!D4u1'9 (&Pw=^BF;D!D85$ è ,$k85;M87(.1.;Sn&g]q!D87(us7wOD=NB1.;zu!D,.¯sP0^#³85!D è wo&(uF#}/.'01!D'w DpB1.;+u!D85%E5Í#³85!D è w` /'Nl%&%.*1 B1KMo7585/&0/85S1 è 3N 0!L*1*1P01!D!#^!D,.¯sP05 {C } 1.(0.)(\,&$\F$&%'Lr.([40!*1*1P01!D!#^!D85%E5O ã } 1.(0.)(|,$m%'187 σ, τ !#-$%b)O5/.ln&) ã ¼M¼ .Dn&4(.!DN1 { ¼M¼ &(l0!&(.!DN1l;D1KMl),b!LKMM`m ![(&871KM"!D/8O(&0
'11!Dm!#JeJ871% ? 2%&%.*1 B1KM 5585/&0/85g#³85!DnE;D87* (&BAb]qr85;M87(iO%V! h ß , ≤i Ddk)Z;D!D85$G1 nd/'5/'*b]$2 I9JP0F1!DOÃ A&.(O0!D'7 (&8HZBF;D!D85$"ÓX0Y !##I/.'0!D87 #I85!D h ß , ≤i s!GsB`rhZ#H]f¡ ]q!D8 ? !D/.-]^&) ¸ ¼M¼ Ê ¶ R ¸ Y ¶3â X · YIW».RPYI»W7XNB'J1@ BF;D!D85$4!#fZp#}!D85ÍÓXY AnËus57wS]^1*1*c,'N#I/.'0!D87J.(+uF#}/.'01!D'7w &)F/87*587's#I!D85N)F1!D'B`!o]qr.(2/%{]^1n{0D0n*1$Gn&p(&0
'11!D4!#H%&%.*1 > B1KMp7585/&0/85p5%b*1*1)(i!D/.d0.%.*1 01n*1$4),b!LKMM`a@N]^1*1*7$2k*11Zn*1N!D85B,V!D/& #}/.'0s!D87J&(4&)F/87*j587's#I!D85N)1!D'=1P0F1!D{Ã'çQ5!DNp85* )ZL(i(&s1* $ ,'-#I!D/&.(o1z&0!ªAbP0F1!D"Ã .([ .D®.AÆc%.*1ré.`©N!#cÇc85=h5hZZ` hZNEN!#IZ~0!DKM1gs!{!D1Nn&k%&%.*1 B1!D[% ã ¼M¼ , ]^8711&; f x #}!D8 ã ¼M¼ f, x¡ ]^"n3(.!ML^!D=9Õ0!D.#I/'71.;'` hZSE85* )F1KM*1$zBD5$ms![/'92eJ891% ? 2%%.*1 )F1K"7585/&0/85)s!@1Z85%85o %85)(. )Zr*1!D;D r]^1n{_/'
b)F1!DG!LKM8*1*F$&%')B` ¤ ,8710#H(.370/'991!DG!#*1!D;D B^n3^%b!D1=]^1*1*C ? J1^BD718^s!!D1KL)Z-n&#}/85Z8=0!D.(.1F1!D'&)(&L("! (0
'&-eJ871% ? -* -,(&O!(&* P`ja@g]^1*1*Q/'Z-!DsB1!D wk−φ η #}!D8p#}!D85r/* φ !* (.1.;i)N]f!D85* ( w 85* )F1KMS!\K87 ),&*1"D991;Dpk }&K185!D.> p ¡ η A.("*1 ã k−φ NB wk−φ ηC#I!D8KM85$S]q!D85* ( w .("K&185!Dp η ` 0
σ→τ w 0
0
σ→τ w,w 0
0
0
σ
σ
σ
σ
σ,τ
σ→τ
σ
τ
σ
σ,τ
σ,τ
hW ,≤i
hW ,≤i
σ,τ w
σ
Æ _/')1!D'^,']fG0.%85)71!D']^1n&!D/. λ 85rBD71*1$k1F85%85Z)(QA.("]f-]^1*1* 9g&!L]+!S1Z85%85dP_/')1!D'^,']fG$%b)(G* -,(&SZ85O12n&g0.=P> F1!DC`dhn#]qrs ? gP_/'B1!D'=DBs!D J#I!D85g/* DPAVn&{0!D¯s/&'01!DCAb(.3I¯/.'01!D &("0.E5F *C_/'F
b)F1!DG85g55871;D&s#I!D85]q87(Q`j'!D8^0%.*1MA [² wk−φ ηj&( wk−ψ η . wk−φ ∧ ψ η ê=;M)F1!D[3g1F85%85ZL(,&${s ? 1&; A ]^85O,&$\(0
'11!D Ep #}!D85g/* gn&)H(.!ML!D&!* (O)H$ w¬φ` ¤ ≡H1oφ eJ⊃871⊥% ? =9 c!#Q%85!)%V!7⊥11!D&* !D8S
'875s>n!D87(&8o*1!D;D DA1%&*1 )1!D&(_/'F
b)1!D|N ? 2/'9{!#gn&4%'85 * !D87(891&;S!#]q!D85* (P`c'!D8^0%&*1MA ² ∀w ≥ w. w k−φ η1%&*11L w k−ψ η . wk−φ ⊃ ψ η '!D8n3Hs!r ? ='9MAM]fdg/'5H,bd),&*1!r85;M87(S&$NK87 ),.*1dD991;DpB Dj^K87 B,.*1qD971;DNC) A^FP9 *(&s1*M]q]^1*1*D((.85L9C,b*1!)]g` w a@q]^1*1*.1*1*1/'5587)Zn&1F85%85w)F1≥!Dw!#V_/'
b)F1!DN,$-0%.*1,'*1!)]g`j2!D85 1.#}!D85B1!D"B,V!D/&n3^1F85%85)F1!D"!#j%85)(. )ZJ*1!D;D 1{ÓQX0Y A&]^ 9kE 185*1$k5s.(&87(QA$o,'J#I!D/.&(k1&0!ªA.D®Z` 0
0
0
0
hW ,≤i
2.3
Extensionality and combinators
¤ 0* D99 *-%%.*1 )F1KMi5585/.0F/85\$|#n1*J!l,b[z!&(&*J#}!D8"]f!~85BD5!D'BA &(nL9S85BD5!D'r%&%.*1$\!ieJ871% ? S%&%.*1 B1KM5585/.0F/85LrDp]f*1*n`G &
'875 %b!71,1*11$zEOn&)]qGN$z!DO&)KMk!D/;Dl*1p5P`|'!D80%.*1MA σ→σ 1;Dj,'q%F$A) ? 1.;1j1%b!971,.*1qs!;D1KMfpB1.;s!n&f (&F1$#}/.'01!D ` &g9P0!D.("%85!D,.*1En&)%&%.*1 B1!D"N$"!D=,'r0&F'71!D.*nAn` M` λx σ.x A']qgN$k.BKM-]q!O(.E51'0=*1p5=!#c#}/.'01!D&*Q$&%'-]^ 94&)KMJn&r7N #}/.'01!D&*,'.BK&1!D8P`"Äf!D'P_/n*1$ACnSpB1.;2!#^2* -,(.2Z85 λx σ.M N$k!D^,'g(&Z851&)(G/. _/*1$`
o/'5/'*H5s)ZNg!#^0.Z'71!D.*11${Ern.B ¢ ]^&&KM8 ¢ #}!D8*1* x !#n&G%%85!L%87 )Z"F$&%'M`[hFleJ871% ? "%%.*1f )F1Kgo5585/.0F/85LPfAcx]qkgx85 0!D'087L(@!D-!D*1$4]^1Z@n,b.BK&1!D8J!#f*1N5 f, g ∈ A D#}/.'01!D' #I85!D A s! A A,/."* 5!lDk#}/.'01!D'O#³85!D A s! A #I!D8k*1* w ≥ w `
850#I!D85MA.]fJg/'75%'P0#I$On.B#I!D8=*1* f, g ∈ A A ¢ ]^&K8 f g f ¡ a ¢± i g ¡ a. ∀w ≥ w.∀a ∈ A . i
Epz,'"7 ([2*11Fn*1S!D85"71%.*1$\,&$\%&%'B*11.;{!{n&S1F85%85)1!D!# %85)(. )ZO*1!D;D (&)90871,')(B,V!)KMM`S%bP0
b*1*1$AC]q]^1*1*7$in&)gGeJ871% ? O%.> σ w
τ w
σ w0 σ→τ w
0
σ w0
σ→τ w,w 0
τ w0
σ→τ w,w 0
σ→τ w
0
%&*1 )1KM7585/&0/85 ã EpX7ëDYZXU ·¶ TU ¸ Ê # ã k− ∀x σ. f x ¢ g x¡ ⊃ f ¢ g 7X5ëDY ¡ ]^85"n&oK87 ),.*1) f .( g .BKOF$&%' σ→τ `| }a@o]^1*1*(.E70/'pi5$&sD0F pP&35#I!D87%'P0#}$&1.;Sn&NF$&%')!##³85NKL89 B,.*1)1in&p&0&JP0F1!DC` ¡ hF E-S85!D/.1* 0/* )F1!Dis!kpZ&) 7X7ëDY ¡ 3JP_/1K*1s!on&N!D85N* ),b!D87)Z 5)ZN=),b!LKM]^1n20.%.*1 01_/'
b)F1!D"!LKM8%b!71,.*1-]f!D85* (B`
85[85\]f!l0!D!D+]$&k!#5%'P0#I$1.;mn&)G~0* D97 *-%&%.*1 B1KM 5585/&0/85&Dq!D/;D"*1p5fs!N1F85%85^KM85$O* -,V(.F852` SXU âB¶ W5TU'É Ñ X0UbY Ñ TPÐXÊ R9TUVÐ ¶ Y ¶ TDU2/'9Ln&-1&(/&01KM(0
'11!D"!#n&Jp)1&;M!#jZ85OPA ]^1*1dn&R7T ÑSÒ¶ U ¸ YnTWÏ Ñ TPÐXÊCR7TUVÐ ¶ Y ¶ TU"/'9LHP_/'B1!D.*1*1$>Z(&0
'&)(*1N5 K &( S *1*1)("0!D-,1&)s!D87- I9 Î89LÃ'A.Äf.%Z8=©)Q!D8J 2$. ¨ ¡ `1'0Jn&d]f! 85rP_/1KL*1p 3#}!D8,b!DnieJ871% ? r.(20* D99 *C%%.*1 )F1KJ7585/&0/85) ¡ A']fg]^1*1* (0
'&J!(&* ^/'91&;O0!D-,1.)!D87B` '!D8eJ871% ? ^%&%.*1 B1KMf7585/&0/85)PAZ^(&)70891%F1!D!# K &( S 371%.*1
')( ,$G1585!(/&01.;SZp!DF1!D{!#;D*1!D,'* *1NP` ¤ ¹ Ê1T Ò5¸ ÊX0Ê1X Ñ XUVY a σ !# ã 3d N%&%1.; w 7→ a #³85!DÕ]f!D85* (qs!*1N55/&kn&) a ∈ A .(QA]^&&KM8 A]q.)KM ¢ `HÄf!D'5s5$&-,b!* H1"*1!D;D *V#}!D85r/* Df(&!DZ w≤w ;D*1!D,'*M*1p5PçB]qi1Za85%85a0!D'5 a σ )C]q!D85* ( w D a ∈ A ` ¤ eJ871% ? %%.*1 )F1KM5585/.0/85 ã º ¸L· R7T ÑSÒ¶ U ¸ YnTW · #5AC#I!D8r&${$%b) ρ, σ, τ n&85S0.E5 ;D*1!D,'*H*1N5 K .( S !#^$%'L σ→τ →σ .( ρ→σ→τ ¡ → ρ→σ ¡ →ρ→τ 5/&"n.B ã k−K x y ¢ x K¡ ã k−S x y z ¢ x z y z ¡ . S¡ ]^85p]frD5/.pJK87 ),&*1L x, y, z 85p;D1KMGn&r%%85!L%87 )ZpF$&%')BAM` ;'`CA x σ &( y τ 1l K ¡ `hFi!D85O(&s1*nA K ¡ pB'n.)J#}!D8JKM85${]f!D85* ( w .(\*1* u*1!* wG*1N5 a ∈ A , b ∈ A AQ]q&)KM wk−K a b ¢ a `SÄf!D&(.11!D| S ¡ N$k,'g5%'*1*1L("!D/.=711* 85*1$` a@r(&0
'- à W ¶ XgÊ ¸ÑSÒ Ð ¸SÑ TPÐXÊjs!,'gOeJ871% ? g%%.*1 )F1KM5585/.0F/85 ã ]^ 9"E^0.Z'91!D&*C¼bá.(".D^0!D-,1&)s!D87P` ì í ÚHÜÞjîï[ð-N .7 : & .ÚØNÚj& H&9 w
0
σ w
w
σ w,w 0 w
w0
w
σ w
3.1
Syntax
τ w
σ w
w
¤ c/'5/'*&1p$&%')(p* -,V(.J* 0/*1/'PA]qf]^1*1*&,'1F85)5F)(1NP_/')1!D' ,']q Z85OO!#n&27p"$%bMA,/.!DS0!D'087&)(~]^1nP_/')F1!D'N,']qlF$&%')B` é
1'0i]qi]^E56s!m*1*1!L]Þ%F$l$%b)PA]fi]^1*1*J,'\0&%.*1 01"B,V!D/&kn&\F$&%') D971;D&)(2!SKL87 ),.*1)N I9S {4i.M ¡ `Äf!D'P_/n*1$A.F85Od.("n&B18dF$&%') 85f(&0
'L(g/'91&;dn&f5/M,V7 (. 85$J!DF1!Dp!#F$&%'fD71;DNP` ¤ YIÏ ¼ X ¸L·9·¶¹ U Ñ X0UbY ñ{3fg
'1ZdH!##I!D85g/* D x τ A]^1ZS! x !0/85891&;N]^ 0=1oñ` &^#I!D85g/* N$G,'N85BD(+un&pK87 ),.*1 x &Dd$%' τ `xwva@p]^891Zñ , x σ #}!D8n&p$&%' x τ D971;DN ñ , x σ ¢+ñ ∪ {x σ}, ]^85MA 1@]^871F1.;Gn3PAC]qOD95/.NpZ&) x (.!MLg!Dg%&%'B8-1ñ` 85O-]^1*1* ,'g]^871FZ212ZJ#I!D85ñ A]^ 9"$o,bg85BD(QAu &D=F$&%' 85* B1KM !Nñ`xw[1'0!L%'SF85OH. M$O(&τ0
'ku%'85F *nAxwr!D8Nu!D.;DM*1!D,'*b*1Nτ5PAxwrn85 N${,'O5!DN0!D.#I/'71!D@),b!D/.J]^&)J1-p)'J!"/'"KL87 ),.*1M`phF@0!D587D5 !Nn&d*1!D;D d!# %'85 *V*1N5f!#q '!D/VDPZA#I!D8f0%.*1MA*1*!# !D/8q0&%85L971!D' ]^1*1*&)KM0.E5ZF *H1%b!D85P`oa&[]f]^891Z x σ 1G$%bOD971;DNPA ]f pB4n&) x Ed(&0
'L(QA!D8Su0&355PAxw&(".D^$%b σ `H &r5$&-,b!*u .wSD05=D 1%&*1 )1!DS]^1n"85)7%'P0fs!N0&35Z'0MA&5!Nn&) x σ . M τ 7$.PAuF#I!D8*1* w A# x (!DF)qO*1NH!#Q$%b σ BH]f!D85* ( w AZ M Ef(&0
'L(OB w .(O(&!DZL G*1p^!#j$%b τ `xw
-]q*1*>n$%'L("Z85O=85g(&0
'&)(GDq#I!*1*1!)]=P` â)¸ W ¡ x τ .x τ ñ . M σ→τ, ñ . N σ → E¡ ñ . MN τ ñ , x σ . M τ → I¡ ñ . λx σ.M σ→τ ñ .M τ ¸ ÐMÐ â)¸ W ¡ ñ , x σ . M τ ¤ GBD5$S1&(/&01!D"5!L]=qn.Bq#jñ . M σ E]q*1*>n$%'L(QA&nGñ@r/'5pF1!D KM85$O#³85-K87 B,.*1-!# M ` hZdE0!DKM1s!S!D1dn&r%$k$&%'pD71;DN=]^&4]^8711.;o0*1!)( Z85OP`hF\D((.11!DCAC91'0NZ$%bp!#^"0*1!9L(\Z85tEJ/. _/*1${(&Z851&)(QA ]f5!DNF1N)d!D1-n&F$&%'DJ]f*1*n`p'!D8J0.%.*1MA1JE-0!D&KM1s!"]^871Z 1'5ZBD("!# ∅ . λx σ.x σ→σ ` λx σ.x a1nG$&%'dD71;DN5fDq%'85!# n&J5$D0 #I!D85g/* )F1!Dk!# Z85OBA1qE &)F/87*Qs!]^871FgP_/')1!D'1GnJ#I!D85 ñ .M ¢ N τ ª
]^85p]frD5/.p-Z&)ñ . M τ &(2ñ . N τ 85r,V!Dn4]q*1*>n$%'L(Q`^'!D8=$&%b!> ;D87% *Q85BD5!D'PA.1=3=5!DNF1N)&*1%.#}/*s!*1B)KMJ!²@n-$&%')^!#cn&-Z85OPA ]^871F1.;Oñ ¢ 1'5ZBD("!#ñ ¢ `ca@-]^1*1*Q]^891Zo M #I!D8qZ 85)5/*!#5./,bM51F/.N1&; N #I!D8^#³85r.!M0/8585N'0pτ!# x 1 M `^hF4(&0
'N/x 1.;{ M A ]f2r/'5O,'4850#I/*=s![85.pG,V!D/.&(lK87 ),.*1)O1 M s!)KM!D (~%N/x F/85MAD /'5/'*n` a@d.)KM^n&d/'5/'*L&1!DO#}!D8f85&1.;N,b!D/..(OK87 B,.*1LPAK*1/')F1.;-#I/.'> F1!D2%&%.*1 B1!Do,&$o7/M,b51F/.1!DCA.&(kP_/'B1.;N0.Z'91!D&*1*1$oP_/'*#I/.'0F1!D'B` α¡ ñ . λx σ.M ¢ λy σ y/x M, fò or y 6∈ F V M ¡ β¡ ñ . λx σ.M ¡ N ¢± N/x M η¡ ñ . λx σ.M x ¢ M, fò or x 6∈ F V M ¡ a@g* 5!L(G850'0&1K&1$SL&1!D W5X}ó ¡ ñ .M ¢ M σ &(6KM87*J1.#}85'0\85/*1)B`´ &\N1+1.#I85'0@85/*1)G85[7$N585$.( 587'711K1$ ñ .M ¢ N σ ·Ï Ñ ¡ ñ .N ¢ M σ ñ . M ¢ N σ, ñ . N ¢ P σ sYIW ¸ U · ¡ ñ .M ¢ P σ D^]f*1*QD=0!D.;D85/'0-]^1nG85L5%'P0=s!O%&%.*1 B1!Dk.("* -,V(.),b5587D01!D ñ . M ¢ M σ→τ, ñ . N ¢ N σ 5R7TU ¹ ¡ ñ .M N ¢ M N τ ñ , x σ . M ¢ N τ ξ¡ ñ . λx σ.M ¢ λx σ.N σ→τ . 1'0$&%'-D971;DN5q85J0&%.*1 01n*1$O1'0*1/'(&L(o1"P_/'B1!D'PA]fJ* 5!NL(kn 85/*1 ñ ¢ ¸ ÐMÐ â)¸ W ¡ ñ , x .τ M. M N¢ Nσ σ
Er*15p/'pD((zD((.1F1!D.*H$&%1&;4$&%b!Dn&))rs!4P_/'B1!D'P`"a@S]^891Z E ` ñ . M ¢ N σ #jn&-P_/'B1!DGñ . M ¢ N σ 3^%85!)K),&*1#³85!Dn&-P_/'B1!D' 1 E ` D 1
2
1
1
1
2
2
2
¤ /'0#}/*V#nD0^B,V!D/&$%1.;&(kP_/'B1!D.*V85BD7!D1&;EZ&)f#cñ E ]f*1*>n$%b)(QA.&( M β, η>n85L(/.0)^s! N A.Z2ñ . N σ E^* 5!]f*1*>n$&%'.)M(Q` σ & 0!DKM87J#n1* PA'&!L]fKM8PA91'0g]^& >n85L(/.0)s! A'n&gZ85 N$ &)KM!D85-#³85-KL87 ),.*1)P`c &850#}!D85MAMñ . β,N ησ (.!ML=!D^1N%.*1$oñ . M σM`
!r%&D91^n=(.²85'0=,']q!D/8%85!!#C5$.5Zô.(n&^%85!!#Q85/*1) n&)r%&%.*1$4]^&[F$&%'Lg85OD95/.N)(\!D-%$AQ1g3-]f!D85Z[pF1!D1.;Gn.) ]fg(.!GUVTMYV.)KMdZ-85/*1 ñ , x σ . M ¢ N τ UVTUX Ñ ¼ YIÏ ¡ ñ . M ¢ N τ x 6∈ F V M, N ¡ 71'0-nE=1.#}85'0-E5!D/.&(k!D*1$k#n&85g0.E55^;D*1!D,'*Q*1p^!#$%b σ ` hZHE1F85)51.;rs!-!D,b85KM^n.B]q=.BKqgeJ891% ? 0>n*1 ? =5585/.0F/85]^1n1Sn 5$L{!#fZ85OJ!D8JP_/'B1!D'P`-a@$2n1 ? !#qk$%'D71;DNJñ|Dn u%b!71,.*1p]q!D85* (wO1i]^ 2n&pK87 ),&*1Ld%&%'B891&;o1iñ°u0.E5PAxwo14n&9' !# b!D/VDPZAD!D8pus85(&0
'&)(Q`xw{a@=N$Nn&85BD(Oñ . M σ D7$&1&;'ACu M 0.E55 &(z&Dp$%' σ )p]q!D85* (mñ`xwÞ o&)F/87*f!D87(&871.;i!Dm$%'kD991;DpsrE 0!Ds1pPA.(\85/*1 å ¸ ÐMÐ â)¸ WFæo'5/85L-n.Bg# M Er(&0
'&)([.(@.D-$%' )]f!D85* (lñHAn& us0!DF1M/L7w\s!\,'2\Z85õ!#F$&%' BKM85$z]f!D85* ( σ ñ ≥ ñ`[a@kl* 5M!i1'0!D85%b!D87)Z"P_/'B1!D'PAc&(z85BD(~ñ .σM ¢ N τ DPA u M .( N (0
'&Jn&J9p*1p!#$%' τ B]q!D85* (oñH`xw1'0-!D85F85O ,'4(&0
'&)(l]^&|]q2.BKMG!D854KL87 ),.*1)PAq1O3S0*1B8on&)S&$mñ ñ ]^1*1*q.)KMo)p*1)D5nk*1N5N!#dñ`@a&)EN%'85&%V*1)9N!D,&K1!D/'pE≥n.) ]^1n+85)5%bP0G!l085s16* -,V(.ln&!D871)BA]q\N$&)KM4!D85@P_/')F1!D'kB ñ ñH`f ! ? gO71%.*1g0%.*1MA'5/%&%b!-]fr&)KM-O0!D'5s AV.( ¢ ∅ Aj]q *1 ≥E ,'on&S71.;D*1OP_/'B1!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 EpEN!5pBD71*1$[(&!D'5587)ZL(,&$@ 9 "85;D/&NPAqD1vP01!D®.` Ã'` ¡ <!)]qKM8PAH1EOBD7$s!kZ&)OB ]f!D85* (zñ ¢ {z τ } ≥ ñAj]fO.)KM E ` ñ . c ¢ λy σ.y `G M/'pnS%85!)%'851) !# n&d587'91F1!Do#}/.'01!D' i 85]q*1*>n!DF1KBF)(S,$O%85!L%'851)f!# n&d%85!!# 5$&7Z°#I!D8$%'L(* -,V(.p* 0/*1/'P` a@]^1*1*b/'^F$&%'D71;DN5Dru%V!991,&*1 ]f!D85* (7wN12%85!LK&1&;Nn&g0!D%.*1Z&)9=n!D852` 0
0
0
0
0
σ w,w 0
3.2
Environments and meanings of terms
¤ ~X0U âP¶ WsTU Ñ X0UbY #I!D8OeJ891% ? r%&%&*1 )1KM-5585/&0/85 ã EdO%'85F *C%&%1&; #I85!DÕK87 ),.*1)=.η("]f!D85* (s!*1N5!# ã 5/.9"n.B 7XU â ¡ hZ# ηxw ∈ A &( w ≥ w, n& ¢ i ηxw ¡ . ηxw P¨ 0
σ w
0
σ w,w 0
hZF/11KM*1$Ajz&K&185!Dp %b4K87 ),.*1 s!\6u%'85 *q*1p7w ]^ 9g$J0&35q }!D8,'f(&0
'&)η( ¡ )5!DN]q!D85* (PAB,x/.C!DP0L97891*1$J*1*]f!D85* (ηxB` 1'0=-F$&%'$r,'=%$p)!D&q]q!D85* (.(n!D&%F$p* BF8PAM]qL(! &)KM&K185!DN55/&gn.) ηxw E /..(&0
'L(r)5!DN w AL.(gZ\u,'P0!DpL7w (0
'&)(~){* )Z8 w ≥ w `[a@"]^1*1*85/87~!\n3%b!D1OL#IZ8(&0
'1.;{Z pB1.;M!#cZ85OP` hn# η Ed4&K185!DNd&( a ∈ A A']qg]^871F η a/xj#I!D8n&g&K185!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 B1KM=5585/.0F/85 ã A&(oñ{3qrF$&%'D971;D.> pPA.]fg7$ w ·0¸ Y ¶E·}ö X · ñ ¸ Y η A']^871ZF wk−ño η # #}!D8=*1* x σ ∈ ñ . ηxw ∈ A ê=!DZrn&)^# wk−ño η.( w ≥ w An& w k−ñS ηZ` '!D8j$geJ871% ? H!(&* ã &(rK&185!Dp −ñS ηZA]fq(0
'&HZfpB1.; ñ . M σ ηw !#cZ85ñ . M σ 1"K&185!Dpwk η )^]f!D85* ( w ,&$S1.(/.01!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'71!D.*11F$N;D/'87FdZ&)1On&ñ . λx σ.M σ→τ DMA 0&E7sc.(r3/. _/M` EjEc%85!LKML(rDj1n0* D99 *Z1.;'A)/'91.;d587'5* L> d F1!Do1s!N0!D-,1.)!D87J Î89LÃ'An!D87(&85L(o h ß , ≤i D^N)Z;D!D85$S1"n&/'5/'*]q$` %'P0
b*1*1$A&n&d!D,.¯sP05H!# ZEBF;D!D85$o85n*1N5q!#pßä&(Sn&853q /& _/iu*1L9s>nZ&.>n!D8s>nP_/'*>ns!MwO8585!)] ` #³85!D w s! w ² w ≤ w `d1'0p BF;D!D85$4g/'5-.)KMr (&11)J&(i,'O0*1!9L({/..(&80!D%V!711!DCAQ]qp*1 ` ,'gn&J (F1$S!D w &("(&0
'&J0!D%V!711!DG,&$ ¢ ` . ` ◦` .M
A
A
0
w,w 0
0
w,w
w 0 ,w 00
w,w 0
Ã
w,w 00
¬1KM"eJ871% ? J%%.*1 )F1KMd5585/&0/85 ã A&1f3BD5$Os!9n.BBDo$&%' σ (F851L=N#I/&'0s!D8ý #³85!D h ß , ≤i !O5P`%'P0
b*1*1$A.]qs ? ý w¡ ¢ A ý ` ¡ ¢ i &(m/'"0!D.(.11!D' å ¶ Ðæ2&( å R9T Ñ ¼ æG1ln2(&0
'11!Dz!#-eJ871% ? G%&%.*1 B1KM 5585/&0/85Os!45&!L]ôn.BgZEr%[Er#I/&'0s!D87 *n`ka1*1S1r$@9Ý71%.*1)5 !l/'i#}/.'0s!D872ý DG!D,.¯sP05o!# AZEG$ (&#}$F$&%')"1+Z@D ]^85 σ 6¢ τ 5$D0 *1*1$Ab,/& A C¢ A &%&%'\s!o,bn7N9P`p1'0 ]fo]f!D/* (@!DpP0L97891*1$i]ps!4 (&#}$\%&%&*1 )1!D\#}/.'01!D'g!DZO]q! F$&%')BAQnEg0!D/* ([*1BD(\s!G/&&P0)785$40!D.#I/'71!DC` &850#}!D85MAC]f]^1*1*/'n F$&%'-0.%85)991!D'^Dn&-!D,.¯P05!# C ` 1'0BDO$%b(&Z851&)Hg#}/.'0s!D8PAD]q=]^1*1*b/'=.B/87*587's#I!D85N)1!D' D=n&-!D85%E5O^!# C `Hb!D8^KM85$k%'18=!#j$&%') σ .( τ A'0!D.(.1F1!D å U ¸ Yµæg1 n-(&0
'11!Do!#ceJ871% ? -%%.*1 )F1K5585/.0F/85J7$.fZ&)n% w 7→ ã ¼¼ A ]^ 9+]q[5.*1*-]^871Zm71%.*1$vD ã ¼M¼ AJE2l.)F/87*-587's#I!D85B1!Dv#I85!D ý × ý s!ý `mþ91&; ã ¼M¼ AH]fGlkZ&)KM85$[;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'91!D&*H%&%.*1 B1KM5585/.0F/85Lo I.(@&'0O!(&* A1rErBD5$\s!4 n&) #']q!=;D*1!D,'**1N5 a .( b (&F851fnq7N&)/¡87*587's#I!D85N)1!DCA n a ¢ b )-KM85$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)F1!D' ν Cý →ý 1.(/&0L({,&$4;D*1!D,'*H*1psg!# ã !#HF$&%' σ→τ .(4*1J0!D%b!711!Di!#!D85%35O,'!D87(.1&85$20!D%V!711!D\!# &)F/87*Q587'5#}!D85B1!D'q1\ÓX0Y ` ¤ 85!D/.F1&f* 0/* )1!DN5&!L]= n.)# ã EcJeJ871% ? q* -,V(.d!(&*nAn& E d)Z;D!D85$-]^1nN!D,.¯P0 #}!D8BD9pF$&%'MAL&(rn&85fEjd!D0>n!D&H0!D8585)7%b!D.(C'0 ,'F]q;D*1!D,'*.*1p5!#V$%b σ→τ 1 A &(!D85%E7Oj#I85!D σ s! τ 1 C ` hZND((.1F1!DCA)1 EBD5$-s!7!L][n.B C Ej85ZL7 0*1!)(g# ã .D %85!&(/.05 .( Z851.*C!D,.¯sP0P` -85* )1!D'51%",'F]qGn&gBF;D!D87 *C1Z85%85sB1!D2!# Z85OPA&DH1\&0!ªL A.(n&NB1.;g#I/.'0F1!DS]fd&)KM^;D1KMO3f5/.871)( σ
σ w σ w,w 0
σ σ
w,w 0
A
σ
σ w
τ w
A
A
σ,τ w
σ,τ
σ→τ
σ
σ,τ
τ
σ
σ,τ w
w
w
τ
w
w
A
σ
τ
hW ,≤i
A
A
A
3
3
There is a minor source of confusion in [Sco80, page 413]. In assigning an arrow of a category to an open term M , we must decide which variables to consider free in M . In particular, we may want to consider some variables “vacuously” free. Scott’s slightly informal discussion does not address this point. However, in the formalism of the present paper, we have explicit type assignments, and so we simply treat all variables in Γ as occurring free in Γ . M : σ.
©
1On&^#}!*1*1!L]^1.;Z!D852`cê=!DFn.)]^1nO%85!(/.0F$&%')BAM$ñ . M τ 3BD91*1$ 587's#}!D85N)(~1s!@9 *1*1$P_/1K*1SëC σ . M τ ]^1n|!D*1$m!D"#³85 KL87 ),.*1M`d n1%.*1$k85%&* D0rng0!*1*1P01!DG!#jK87 ),.*1)=12ñ,$"71.;D*1gK87 ),.*1 !#jn&g%&%85!)%87 )Z-%85!&(/.0F$&%'M` ¡ ÿ È3ó ã ¶E·2¸ à W ¶ ¼bá X2Ê ¸ÑSÒ Ð ¸[Ñ TPÐX0Ê ¶ Y}º ¼ W5TPлRY ·4¸ UVÐzYZXW Ѷ U ¸ Ê YIÏ ¼ X JY}ºXUlY}º.X ¶ UbYZX0W ¼ W7X0Y ¸ Y ¶ TUlTsópë σ . M τ ¶ U C ¸L· ÐX ö UX5Ð ¶ U nÓR7T ¶E· Y}º.XdU ¸ YI»&W ¸ ÊbYIW ¸ U · óTW Ѹ Y ¶ TUdóW5T Ñ ý YnT-ý ¶ UVлR9X5Ð Ò ÏY}º.X ¹ Ê1T Ò5¸ ÊQXÊ X Ñ XUbY w 7→ λx σ.M σ→τ ∅ w cº.X0W5X ∅ ¶3· Y}º.XpX Ñ ¼ YIÏkXU âP¶ W5TU Ñ XUbY ½
850#I!D85gGP_/')1!Dk&!* (1 ã ²\1^!* (1 C `hF=5&!D/* (o,bg%b!D1Z)(2!D/& n&)Zp#}/.'0s!D8^#³85!D C s!2ÓXY Ab%%1.; τ s!Gý AVE=#n1n.#I/*.(2%850> 985KML%85!(/&05PA&,/.^1^$S!D^,'J#I/*1*Q!D8^N$o!D^%85)85KMJ#I/&'01!DG5%'D0)B`
85BD7!D@EJn.BgGeJ871% ? * -,V(.G!(&*E^¯/'5-k
'875-!D87(&8g5585/.0/85N1 s!L%b!o!#r%85)7B)KMLP`: &850#}!D85MAZi1F85%85)F1!DCAý !# &)( !D^,'pý →ý ç&0&F'71!D.*11F$o!D*1$S85P_/185)n.)=ý ,'g@us5/M,.#Iσ→τ /&'0s!D87wN!# ý →ý ` a@!)]° ? i"Nn&!(i#I!D8-(0
'1.;GGeJ871% ? * -,(&2!(&*j#I85!D&$ 5N*1*q85ZL7 z0*1!9L()Z;D!D85$`G4!D85o5%'P0
b*1*1$A ]qoD5/.p]qS85S;D1K 6D5!0 )1!D!#r$%bi0!D'5s5os!z!D,.¯sP05k.(|F85ø0!D'5s5Ss!~8585!)]= !#^{5*1*85Z)9 m0*1!9L([BF;D!D85$ D `2/&[zD5!0 )1!D[(F851Lr 1Z85%85s)F1!DO!#$%'L(* -,V(.g* 0/*1/'H1 D A1On&=9'!#&0!ªLZ`a@=]^1*1* 5!L]Z&)Z85q0&355deJ871% ? q* -,V(.!&(&* 69BEs#I$1.;^n&7pfP_/')1!D' D D `hFon&-5%'P0 *QDZ&)!D/8)Z;D!D89 *Q1Z85%85s)F1!D"!# $%b)("* -,V(. * 0/*1/'Eqn1F87.*Q* &;D/';D!# I .D®) A&nE^0!D'5585/.0F1!Do;D1KM)q/' eJ871% ? -* -,V(.!&(* ô]^ k3DP_/1K*1^s! ¡ D }1Gn&J/'5/'*C)Z;D!D87 * 9' ¡ A&,/&^!D^&P0)7871*1$oE5!D!D85% D`
85g85Jn85-5F%bq#³85!D5N*1* Ä^Ä^Äs!NeJ891% ? J* -,V(.!(&*n` &
'875o5F%|587's#I!D85O!D/8S)Z;D!D87 *1Z85%85s)F1!D|1 D 1s![)Z;D!D87 * 1Z85%85s)F1!D@1mÓX0Y An&s!)%V!!#f%85L5)BKML!)KM8 D `N &NP0!D&(is ? ) $-)Z;D!D87 *M1Z85%85sB1!Dr1p=s!)%V!Q!#%85)7B)KMLcÓX0Y .(-%85!(/.0) %%.*1 )F1KM^5585/.0/85q1Sn=9p=)Z;D!D85$`j Ef%%.*1 )F1KM5585/.0F/85=9BEs
'L A &({0.Z'71!D.*11$Ab]^ {85*1*j
'875s>n!D87(&8d0.%85)71,.*1M`J Nn187( 5KZ%o¡
'&S(¡ fo*1Ns871*1$P_/1KL*1f%&%&*1 )1KM=5585/.0F/85=1{ÓX0Y A]^85 W E=%b!9^]^1n2*1)D5^*1pP`
{
'875G5F%v/')kn&{Á!DL(&z-,'L((.1&; !# 1s!~nis!)%V!k!# %85)5&B)KM)H!)KM8 D `c Ef%85!(/&0LfpBF;D!D87 *V1YZ85%85sDB1!Dk1Sn&s!L%b!f!# %85)5&B)KM)d!)K8 D ADJ5%'*1*1L(i!D/&J1l&0!ªLZ`-'!D81'5s'0MAV# d En&p1Z8s> %85)F1!DN!#bn&q,'D9fF$&%' σ 1 D ALn&]fq/' Y d¡ DZq1F85%85)F1!DN!# σ 1Gn-s!L%b!=!#%85)7B)KMLPA.(G711* 85*1$o#I!D8qZ-1Z85%85sB1!D'^!#j0!D'7sB` ® 0
A
σ
τ
A
A
hW ,≤i
τ
σ→τ
σ
σ
τ
σ→τ
τ
D op
D op
W
D
D
E0&F&(f/. _/*1$S!*1*F$&%'d0&%85L971!D'.(SF85OP`H1'0 %85)85KM) n-85F)7 20*1!9L(o5585/.0/85MAn&J1F85%85)F1!D"!#&$O$%b.(kY Z85Õ1kn !L%b!^!#%85L5)BKMLqE^n&-1N;DS }K& Y ¡ !#j15^1F85%85)F1!DG1 D `
g)Z;D!D89 *C1F85%85)F1!DG1GZ-s!L%b!^!#c%85)7B)KMLf!LKM8 ;D1K)^/' S%%.*1 )F1Kf5585/.0F/85 ã 1n&=7ps!L%b!P`%'P0
b*1*1$A ã EZ=D%&%.*1 B1KM 5585/&0/85r]^1Z{n&NF$&%' σ 1F85%85Z)(@DJS#}/.'0s!D8Jý #I85!D D s!4ÓXYA.( ã ¼M¼ DnkK*1/'B1!Dz!D85%35 eval #I85!Dø ný →ý ¡ × ý s!ý ` I F/85 ã 7BEs
'LnOL.∼1!DOO→ K ¡ AH S ¡ .(\¡ n0.Z'91!D&*11$40!D.(.1F1!D å X5ëDYµæA D0!D87(.1&;s!n&geJ871% ? 0"> !M!)$*NF q!#j#}!D85g/* D1\ÓQX0Y I9S .D®) ¡ ` a@g!L]+&)KMJ* -,V(.!&(* 1@ÓX0Y A.,/.=!D^&P0)7871*1$" à W ¶ X * -,V(.N!(&*91'0 D $O!Dq,'Ap%b!9P`c &dn187(k5F%o!# ZJ0!D'5585¼b/.á > F1!Di/'9Ln$ # D0!DL70/\0!)KM8P`J &p;D&87*c0!D'5585/&01!D4;D1K{1\Æc%&*1 ¨ `.(NÄf!D85!*1* 85$ ¿ ` ¿ !# !M!Dbª%85!&(/.0)j%V! .(r^#I/.'0s!D8qÐ 5/&Gn.B=&$o%&%.*1 B1KM-5585/.0F/85 ã 1@ÓX0Y EW=*1p871*1$kP_/1KLW*1→D ^! 2%%.*1 )F1KM5585/.0F/85% ±1\ÓXY !D,s1L(",$o0!D%b!91&;)D9k#I/&'0s!D8^.( &)F/87*f587's#I!D85N)F1!Dz1 ã ]^1nËÐ W →D `\hF%'85 0/* 8PA ã &&( Þ7)s> Es#I$n&27NoP_/'B1!D',']f~$%b)(m* -,V(.@F85OP`l KL*1-! C .( C EJE5!DD!D85% Cs! C ` 1&*!D,.¯P0k!# D `± !z%85!&(/.0 C E5!D!D85% {s! C A]f{ ? W 1Z (0
'11!D2!)# §!S,'rn&g%V!9!#
'1Fp0!D%b!7B,.*1rP_/'0)^!#!D85%E5O 1 D A1'0*1/'(.1.;\n&k%$P_/'0 ⊥ `m E9NE%'85 *1*1$[!D87(&85)(~,&$ ²Gn&dP_/'0 w EqS11 *V;Dp!# w `j 3%b!9 W E P w ≤w !#Æc%.*1 ¨ `k1 !D!DbªLZAQ00%-n.BJ]q&)KMD((L(\n&N%${P_/'0M`
#I/&'0s!D8-ÐkE(&0
'L(kD1[ !D!DbªLZA.0.Z.(&)(os! ⊥ ,&$O%%1.;NnJ%$ 9P_/'0=s!rn&F851&*!D,.¯sP0 !# D `jê!DZn.BZ85E0D0n*1$!D&=]$Ns! 0&F&(\Ðo!N!D85%E5Of#³85!D ⊥ s!$ w ∈ W A,bPP/' Eqn&11 *!D,.¯P0 1 D `o2!D85O0.%.*1 01n*1$AHÐ\3-n&#}/.'0s!D8J#³85!D W ! D 5/.9@Z&)J#}!D8g&$ 9P_/'0 f , . . . , f !#c0!D%b!7),.*1-N%Vg }!D,.¯P0^!# 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
σ→τ
σ
σ
τ
σ
τ
τ
D op
D op
op
op
D op
W
op
σ→τ
σ
τ
B
A
A
B
B
A
op
1
2
1
op
2
op
op
n
1
n
1
0
n
1
n
op
`w,w0 w → w0
!#
s!
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!711!DG!# f , . . . , f A.!Dn&85]^E9M` 1'0NZN!(.
'L(4#I/.'0s!D8Ð →D 51*1*j9BEs
'Ln&0!D&(.11!D'!#^Äf!D85!L*> * 85$ ¿ ` ¿ 1y !D!DbªLZAZ2!&(.W
')* ( ¾85N1'S*1p871*1$~P_/1KL*1Os! ã ` 2!D85!LKM8PAn&g;D*1!D,'* *1N!#Hý 1@ÓXY 85r12.B/87*C0!D8585)7%b!D.('0 ]^1nn&O;D*1!D,'**1N5-!#dý 1lÓX0Y `khZ@#nD0PA nEr0!D'7585/&01!D@&Dgn #}!*1*1!L]^1&;%85!L%'851)P ÓXY , ý ¡ ∼¢ ý ⊥¡ ,&$SZJÁ!D&)(.QA ý ⊥¡ ¢Ëý ¡ ,'PP/'rý ¢+ý ◦ Ð".([ÐQ ⊥¡ ¢ A ÓX0Y , ý ¡ ∼¢ ý ¡ ;M1G,$onJÁ!D&)(.Q`
850#I!D85rn&r;D*1!D,'**1p5=!#$"$%bý 1 ã 85r1{!D0>ns!>n!D&r0!D8585)s> %b!D.('0]^1n"n&d;D*1!D,'**1N5q!# n&0!D8585L5%b!D.(.1.;p$&%'-ý 1+ `H1'0 nJ]f!)Z;D!D871L C &( C &)KMn-7NJ9q!#!D,.¯sP05PA]qJN$k0!D'0*1/'(& n&) C &( C 85{E5!D!D85% D`Ë%'P0
b*1*1$Afn&4!D85%E7O#³85!D σ s! τ 14B1Z8BF;D!D85$"85p12!D&0>n!Dg0!D8585L5%b!D.(&'0r]^1n2Zp;D*1!D,'*C*1p5=!# ý Aq&(m]q ? !L]Õn.Bn&20!D8585)7%b!D.(.1&;[5!#;D*1!D,'*^*1N5O85G1 !D0>ns!>n!D&J0!D8585L5%b!D.(&'0MA.,$on&g85;D/&N=),b!LKMM`
J]f!85ZL7 20*1!)(")Z;D!D871L C &( D 85JP_/1KL*1qK& pn-0!D8s> 85)5%V!D&(&'0n&)qN%bqk!D,.¯P0 σ !# C s!pZ1F85%85)F1!D@ σ !# σ 1 D ` hZ=3^BD5$ks!S9-Z&)dý ¢ Y 5 σ ¡ A'91'0gnEE=0*1B85*1$"585/J#I!D8=,'D9-$&%') &({%85)85KM)(\)d1;D&8JF$&%'Ld,bPP/'ý ¢ôý →ý ¢ Y 5 σ→τ ¡ `gÎq$ (0
'11!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&S85F)7 m0*1!)([)Z;D!D871L &( 85OP_/1KL*1P`"1'0 E E5!D!D85% -s! C AbnE0!D%&*1F)dn&gC%85!!#n.D)=n&r!D871;D1.*)Z;D!D85$ DC &( ng)Z;D!D85$ C (&Z851&)(",&$oeJ871% ? -* -,(.!(&* °85rP_/1K*1P` î .Ü£ØNÛ7Új&Ú ÚVy &' 0 s& H&7 , - = Ú/ þ71.;OZg85* )1!D'51%b,']f{eJ871% ? -* -,V(.S!(&* =.(285ZL7 40*1!)( BF;D!D871)d(L70871,'L("1"n-* D7P01!DCAn&g5!D/..(.L9q&("0!D%.*1Z&)n0> !D85O^#I!D8=eJ871% ? g* -,(&S!(&* ^$",'r(&871KML("#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!D871)" I94 j.D®.A Çc85rhZ ¡ ` U 9CX ÑÑO¸ ?æ 9 X0Y ã Ò X ¸ à W ¶ ¼bá XÊ ¸ÑSÒ Ð ¸GÑ TPÐXÊ ¸ 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$os!%85!)KM-5!D/..(.L9q,&$S1.(/.0F1!Dk!DGP_/'B1!D&*%85!!#IP` 1 2 43456 DC å ÓTD»UVÐDUX ·9· 4 æ 9CX0Y Ò X ¸~· X0YOTsEó fXÊ}Ê ÉYIÏ X5ÐGX F0» ¸ Y ¶ TU ·½ È3ó ñ . M ¢ N σ Y}º.XU[X â X0WÏ Ñ TBÐEXÊ ·0¸ Y ¶E· óÏ ¶ U ¹ E ¸ Ê · T ·¸ ¼ Y ¶E·}ö X · ñ . M ¢ N Eσ ½ ` '!D8=eJ871% ? * -,(.!(&* PA&]q-%85!)KMJ(&)(/&01KMJ0!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&UVÐXW ` ½ 7 ºXW5X ¶E·p¸ à W ¶ ¼bá XrÊ ¸ÑSÒ Ð ¸kÑ TPÐX0Ê 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,$N0!D'5585/.0F1.;-JZ85 !(&* ã ¢ h ß , ≤, {A }, {ã ¼M¼ }, {i }i 1Gn&J#I!L*1*1!L]^1.;]q$` ß ErnS%b!g!#q
'1ZS$&%'OD991;Dpsrñ|!D87(&85)(,$i1'0*1/'71!DC`khZ • ]^&)#}!*1*1!L]=PA]f-]^1*1*Q]^871Frñ\#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%bP0^! 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 hZpEgBD7$is!4&P ? n.BgZk(0
'11!D[ ? S'9MA&([n&)g]qO.BKM;D*1!D,'* *1ps K .( S B=*1*C%&%85!L%87 BF-F$&%'LP`c'!D80.%.*1MA ¢± λx σ.λy τ.x K
G%85!!#!#J0&F'71!D.*11F$zE\*11ZZ*12!D85G1F85)51.;'`/%%V!9GZ&)4 ñ . f.(| ñ H.BKNZO7N#I/&'01!D.*j,b.BK&1!D8PAQn` M`AC#}!D8g*1* M σ→τ ñ ≥ ñ[&(Gñ . P. σNA']fσ→τ J.BKM ñ . M P τ ¢± ñ . N P τ
CA.1G%'85 0/* 8PA#}!D8=ñ ≡ ñ , x σ ]^1n x !D^12ñA.]q-.BKM ñ , x σ . M x τ ¢° ñ , x σ . N x τ &(z5!4,&$@85/*1i ξ ¡ .(L.1!Dü η ¡ Ac]qS.BKM4 ñ . M σ→τ ¢¾ ñ . N σ→τ Z`
M/' A 3=eJ871% ? J* -,V(.!&(&*n` hZH85N1'H!p5!L]ln.B 7)Es
')%85P0E9*1$Nn&=P_/'B1!D','*1!D.;D1.;gs! ` a@r,';D1{,$"85* )F1.;On&g1AZ85%85sB1!D4!#OF85¥s!S15P_/1K*1'0r0* D9B`qEhZ# ñ[E^&$S$%bgD971;DNPA&]q-N$k(&0
'-G&K185!DN η ,$ ñ ¢ ñ . x σ !D# n&x 85σ]^∈E ñ ⊆ ñ ηx undef ined ¤ 75871;Ds#I!D85]q87(\1.(/&01!D@!D[Z85ON5&!)]=gn&)-#I!D8r&$\ñ ⊇ ñ ⊇ ñAj]f &)KM ñ . M σ η ñ ¢± ñ . M σ hZ|%'85 0/* 8PA]^&&KM8 A 7)Es
')S|P_/')1!D~ñ . M ¢ N σ Aq]qG&)KM ñ k−ñf η ,&$o0!D'5585/.01!Dk!# η A&(G5! ñ . M σ¢° ñ . N σ 1'0nE-85BD7!D1&;G%&%.*11)Js!GKM85$4ñACKM85$4P_/'B1!D\7BEs
'L(i,$ A g/'5 ,'g%85!LKL),.*1#³85!D E ` a1*1p1=E^%V!991,&*1r!O7!L]+n.B A 7BEs
'L^KM85$kP_/')1!Dk1 E (.185P0n*1$A ,$711* 885BD7!D1&;'AD085s1O0!D%.*1 B1!D'c$p,'=)KM!D (&)(N,&$r85L5587 01&;-!D/8 BFZ1!D2s!S0*1!)(2Z85OP` &85gE=!o*1!9=!#c;D&87*11$o14(.!D1.;S7!'A'71'0g1 E^BD5$os!%85!)KMJn&g0*1!)(kP_/'B1!D ¢ ∅ . λx σ . . . . λx σ .M λx σ . . . . λx σ .N ¨ª 0
• iσΓ,Γ0
0
0
0
0
0
0
0
0
0
00
0
1
1
00
k
k
00
1
1
k
k
0
#I85!DÕZ-P_/'B1!D
¢ N ,'F]q\!)%b@Z85OPAC.(iK 0KM879`pb!D8J&${0*1!9L({P_/')F1!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$z]f!D85* (~ñ+!# A AHn"]f! P_/1K*1'0[0* D9)@ ñ . M τ r&(ô ñ . N τ g]^1*1*-,'@ (&F *n`1'0@n& pB1.;!# ∅ . M τ 12&$S&K185!DN η B^]q!D85* (Gñ@]^1*1*Q,'S ñ . M τ ZA'.( 711* 85*1$k#}!D8 ∅ . N τ Ab1^#I!*1*1!)]=n.B A 7)F3s
') ∅ . M ¢ N τ `^ 3%85!LKM) n-Z!D852` "1%V!D85N%85!L%'85$[!#n&GeJ871% ? kZ85õ!&(*]qk0!D'7585/&01mn 0!D%&*1FL9r%85!!#^E-n.) A Eg!D&%$2#I!D8g*1* w ∈ ß ² σ 3r@1/.> 1F1!DE7F *1*1$G%85!LKL),.*1g%85!L%b!91F1!DC` /81Z85L5-1inE%85!)%b85F$25FOd#I85!D G]f*1*> ? !L]^\7$D0 0!D8585)5%V!D&(&'0,']f@$&%')(\* -,(&4* 0/*1/'r.( 1/1F1!DE7F p*1!D;D DAQ*1*1)(\n&Jó0TDW Ñ »&Ê ¸L· É ¸L· ÉFYIÏ ¼ X · ¼ W ¶ UVR ¶ ¼ Ê X !D8 è »&WWÏ Y[ZgT ¸ WsÐ ¶E· T Ñ TW º ¶E·Ñ B,.*1$&%')f!D%$AD]qg/'5H,b;D1o]^1nkBDS,'D$%b!D&%F$A71'0! Bs!D J%85!L%b!91F1!DGE=%85!)KB,.*1M`chF^E^BD5$ks!O9-Z&)# σ .( τ BDG&)KMJB x1 σ1 , . . . , x k σk . M
σ w
σ w
¨.
!5!D&g*1NPAbn &DB=!5!D&g*1NPAb.(25!oO55871;D&s#}!D8s> ]q87(o1&(/.0F1!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_/')F1!D"]^1*1*Q,'g9BEs
'L(Q` ¤ !Dn&8i]q$6s!0!D&P02!D&%F1&)G]^1n:%85!LKB,1*11$34s!0!D'9 (8 0* D9)O!#-!&(* B`Ëhn#-]q40!D'7 (&8on{0* D9O!#-#I/*1*0* D7 *=$%b2187879.> 1)PAc]^1n~5!DNS,'DS$%b)p%F$@.([!Dn&87N!DPAcnzn&S$%'Lr]^ z85 !D%$r1KM85$p!(&*b85^n&rR0Ê ¸L·9·¶ R ¸ Êb%85!L%b!91F1!D.*sP/.s!L*1!D;D1L=Äf!DbD©)Z` Ú b"=û=5 c =EÛ d:ÚÛ7 H&9 a × C5ØÙ=c 6.1
Relations over applicative structures
Q !D;D *85* )1!D'.BKg%85!)K4/'0#}/*1{n&5F/'(.$G!#qn85* )F1!D'qs!OeJ871% ? -* -,(&!&(&* P` hZ4ng0* D7 *Q!&(*QZ!D85$k!#F$&%')("* -,(.O* 0/*1/'PA'*1!D;D *Q85* )1!D Ep2#n1*1$i!#^85* )F1!D'r1&(&0.)(@,&$\F$&%'Lg]^ 7)FE5
')r{0!D&(.11!D[1%.*1$> 1&;o0*1!5/85g/..(8=%&%.*1 B1!DG&(2* -,V(.S),b5587D01!DC`H &-;D87*11P)1!DG! eJ871% ? S%&%.*1 B1KM5585/&0/85)g1KM!*1KMLJ1&(&0.1.;k85* )1!D'-,$i,b!DZ[F$&%'Lr.( %b!71,.*1=]q!D85* (P` a@]^1*1*V71%.*1#I$p!D/8H%85)sB1!D,$D5/.1&;-g
&L(O5585/.> F/85 h ß , ≤i Z85!D/;D!D/.dP01!D4®.` ¤:à W ¶ ¼bá XÊ1T ¹¶ R ¸ ÊW7XÊ ¸ Y ¶ TDUG!)KM8feJ871% ? d%&%.*1 B1KM=5585/.0/85) ã .e( ¦ }/'5> 1&;\n&G7N h ß , ≤i¡ 3Ni#n1*1g$ ft¢ {R } !#85* )F1!D' R ⊆ A × B 1&(&0.)([,&$\F$&%'L σ &(]f!D85* ( w ∈ ß 7)F3s#}$&1.;Gn&O#I!L*1*1!L]^1.;2]f!i0!D&(.> F1!D'P`c &J
'875^E=!D!Ds!D 01$o0!D.(.1F1!Dk#I!D8^$o,'D9-$%b c Ñ TDU ¡ 1%.*11) R a, b ¡ a, i b¡ #}!D8=*1* w ≤ w , R i ]^ 9\7$.dZ&)d]^& w ≤ w An&85* B1!D R E-0!D1&)(i1 R A!&(/*1! n-587'91F1!D"#I/.'0F1!D'B` &g9P0!D.("0!D&(.11!D 5R Ñ ¼ W5X ¡ f, g ¡ ² ∀w ≥ w. ∀a, b ∈ A . R 1%.*11) R 5 i f ¡ a, i g ¡ b¡ , R a, b ¡ ¨D¨ σ w
σ w
σ w
c w
c w0
c w,w 0
0
c w,w 0
0
c w
0
σ→τ w
σ w0
c w0
τ w0
σ w0
σ→τ w,w 0
σ→τ w,w 0
σ w
*1*1)(6us0!D%85&'71!D'wA7$.=n.Bd85* B1KM-s!SZr#}/.'01!D')KL1* ),.*1-#³85!D ã &(An&N85* B1!D R 0!D1'J*1*#}/.'01!D'%%1.;o85* BF)(\85;D/.ps !r85* BF)(85L5/*sB` &^F]q!J*1ND,'*1!)]85^%85!)KML(/'71.;-n&^)9F *1*1$pn 7p-85;D/.N5^D^!D/.n*11&)("1 Ç*1!ªZ` 1 2 43Eh jå i TDUVTDYnTU ¶ R ¶ YIÏ] æ 9CX0kY f ⊆ ã × Ò X ¸ à W ¶ ¼bá XGÊ1T ¹¶ R ¸ Ê^W7XÊ ¸ Y ¶ TDU ½ 7 ºXUó0TWX â XWÏdYIÏ ¼ X σ ¸ UVÐ ¼ ¸¶ WfTsóXHTWÊÐ · w ≤ w ¶ ó R a, b¡ Y}º.XU R i a, i a@=7$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 TPÐXÊ · 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$-!#V!D/8j!DZ8c(&0
'1F1!D'PALn&(0
'11!DN!#VeJ871% ? q*1!D;D *85* L> F1!DNN$-,bq(&871KML(r,&$J1F85%85F1.;Jn&H/'5/'*.(&0
'1F1!Dg1Nn&fs!L%b!HÓX0Y `
o/'5/'*q(&0
'11!D[!#*1!D;D *f85* B1!D& f ⊆ ã × EG#n1*1$@!#=85* )F1!D' 5/&kn&) R ⊆A ×B » · » ¸ Ê ¡ f, g ¡ ² R ∀x ∈ A .∀y ∈ B . R x, y ¡ ⊃ R f x, gy ¡ .
!2850>n1Z85%85NnEp0!D&(.11!DCAC]qOg/'5r
'875r7$i]^.BreJ891% ? O85* )F1!D'g85M` ¤ o]fG.BK29CAq@$%'MAq!D8zuseJ891% ? 4PAxwEo@#n1*1$ A ¢ {A } !#gs 1&(&0.)(~,&$z]q!D85* (PA]^1Z|587'711!D~#}/.'01!D' i #I!D8OBD w ≤ w ` 8PA P_/1K*1n*1$AJ#I/.'0!D8#I85!D h ß , ≤i s!N95P` ¡ hn#Q]q=&)KM^]f!p5/&OeJ871% ? 95 ¢ .( B ¢ {B } A'n&4 à W ¶ ¼Vá XrW5XÊ ¸ Y ¶ TU R ⊆ A × B EdO5/M,V9=!# A {A } ¢ An` M` AVO#n1*1$ R ¢ {R } !#85* )F1!D' R ⊆ A × B A × B {A × B } 5/&"n.B 1%.*11) R i a, i b¡ #}!D8=*1* w ≤ w . R a, b ¡
M/'0!D.(.1F1!D å Ñ TUDæ-E=,/1*^1!Snp(0
'11!D"!#ÓX0Y `HhZ#c]f-!L]Ë7$ n&)O\eJ871% ? G*1!D;D *85* )F1!D~!LKM8 ã &m( ¾5&!D/* (z,b24#n1*1$z!#JeJ871% ? 85* B1!D' R ⊆ A × B 9BEs#I$1.; å » · » ¸ Ê æ2),b!LKMMAn&~1!D*1$851'! 9&P ? å R7T Ñ ¼ W7XFæ ½ EqEq!D,1&)(o,&$]f!D8 ? 1&;p!D/.fn&JeJ871% ? }n` M`VAVÓX0Y ¡ 1Z85%85s)F1!DG!#cn&g5.(.87(o0!D%85&'71!D20!D.(.1F1!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-GeJ871% ? N*1!D;D * ´ 85* B1!D¼bmá 7/&n.)r¼ BD R E5$&N587 S.([587⊆'91F×1KMM`i &o.pk%'85 * P_/1K*1'0=85* )F1!Dk0!DpL#³85!D§n&=#nD0fn&)# R Ef7$N587 =.(587'71s> 1KMMAMn& R EqoP_/1K*1'0=85* B1!DS!Don&dH!# a ]^1n R a, a¡ `ca@]^1*1* B,,85K& BFOn&S0/.-,'875!DNO%87DuseJ871% ? O*1!D;D *%'85 *P_/1KL*1'085* L> F1!D'ws! á Ê ¼ XWr` &J#I!L*1*1!L]^1.;*1N5&!L]=n&) á Ê ¼ XWrÅx^$k,'g0!D'5585/.0Z)( ,$k9&!!91.;85* B1!D'^B^,'DJF$&%'LP` 1 2 43Eh DC å Ô ¸ WY ¶³¸ Ê nF» ¶}â)¸ Ê XUVR9X? æ 9CX)Y f ã ã Ò X ¸ à W ¶ XpÊ1T ¹¶ R ¸ ÊjW7XÊ ¸ É Y ¶ TU ½ È3órX ¸ R0º R ¶E·· Ï ÑÑ X0YIW ¶ R ¸ UVÐOYIW ¸ U ·¶ Y ¶}⊆â X .ó0TD×WpX ¸ Rº Ò5¸L· XrYI¼bÏ ¼ á X c ¸ UV?Ð HTWÊÐ ß HY}ºXU@X â XWÏ R ¶E·J· Ï ÑÑ X0YIW ¶ R ¸ UVÐSYIW ¸ U ·¶ Y ¶}â X ½ o w∈ /'d!#C%'85 *VP_/1K*1'0d85* )F1!D'fEq1o#}!D851.;_/!D1q5585/&0/85)B` a1n ¼ ¸ WY ¶³¸ ÊqP_/1K*1'085* B1!D'PAQ850'0.1K&1F$G#n1* PA &(@5!2@*1N-&)( !D.)KMp\P_/1KL*1'00* D9P`- &850#I!D85MAV1J1;D&d,bN!D85D0/87)Zs!k*1* nL94u%'85 *C_/!DF1sB`xwihZ# ã ¢ h ß , ≤, {A }, {ã ¼¼ }, {i }i E-oeJ891% ? N%&%.*1 B1KMN5585/&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 XUVR9XN0* D • A /R a ¢ { a ∈ A | R a, a ¡ } ã ¼M¼ /f ¡ a b ¢± ã ¼M¼ a b • /f ¡ a ¢± i a • i hZNE\71%.*1o0.870Eos!iKM87#I$@n.BNn&k_/!DF1N7585/&0/85SEN]q*1*>Z(&0
'L(QA &(GeJ871% ? g%%.*1 )F1KM5585/.0F/85M` 1 2 43Eh J È3` ó f ã ã ¶3·\¸ à W ¶ X ¸ WY ¶³¸ ÊgLX F0» ¶3â)¸ Ê1XUR9X@W7XÊ ¸ Y ¶ TU T â X0W à W ¶ ¼bá X ¸ ¼M¼ Ê ¶ R ¸ Y ¶}â X · ⊆YIW»RYI»×W7X ã Y}º.XU ã /¼bf á ·0¼ ¸ Y ¶3·3ö X · Y}º.X à W ¶ ¼bá XX5ëMYZXU ·¶ TU ¸ Ê ¶ YIÏ R7TUVÐ ¶ Y ¶ TU å X5ëDYµæ ½ σ w
σ w
σ w
c w
σ w
σ w
σ w
σ w
σ w
σ w
σ,τ w
σ w,w 0
σ w σ w
R
0
R
σ w
R
R
σ w
σ,τ w
R
0
σ,τ w
R
σ w,w 0
σ,τ w
R
R
¨Ã
σ w,w 0
σ w,w 0
¤ z 75871;Ds#I!D85]q87(\1.(/&01!D@!D[Z85Op${,'o/'9L(@s!2%85!LKMn&#}!*1*1!L]^1.; us_/!D1^!&(* wZ!D852` 1 2 43Eh =5 È3pó f ⊆ ã × ã ¶3·q¸ ? *1%'8~T â XW à W ¶ ¼bá XqÊ ¸ÑSÒ Ð ¸-Ñ TBÐXÊ ã Y}ºXU ã /f ¶E·f¸ à W ¶ ¼bá XHÊ ¸ÑSÒ Ð ¸dÑ TPÐXÊ · »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· ÐX ö UX5Ð Ò Ï"Y ¸ á ¶ U ¹ η xw ¢´ ηxw óTW ¸ Ê3Ê x ¸ UVÐ w ½ hZ[5&!D85PAQn&NB1.;G!#"F85 1@ZO_/!DF1-!(&* ã fô3-n&OP_/> KL*1'0S0* Dg!#^n&ONB1.;G!# MM 1 ã `" Egn&!D85Epz/ D(.%sB1!D@!# n@us&87D0F871PB1!Din!D85Owk!#d 419D®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@=!)]l0!D'9 (8eJ891% ? ^*1!D;D *85* B1!D'c!LKM80* D97 *'%&%.*1 B1KMq5585/&0/85)B`
\71%.*1)7S]$l!~(.!ZEkEk!z85;M87(0* D7 *5585/&0/85)oD"5%'P0 * ã D9G!#-eJ871% ? 47585/&0/85)P`l !@,'47%'P0
bDAH*1 ã ¢ ,'4 0* D97 *f%%.*1 )F1KM5585/.0/85MACn` M`HA {A } 3p20!*1*1P0h{A 1!D!},#={sg¼¼ 1&(&0}i.)(@,&$ F$&%').( {ã ¼M¼ } E\0!L*1*1P01!D~!#%&%.*1 B1!D[#I/.'0F1!D'P`[a@"(&0
'&oZ eJ871% ? g5585/.0F/85 ã ¢ h ß , ≤, {A }, {ã ¼M¼ }, {i }i ,$d ? 1.;5 A ¢ A AB%&%.*1 B1!Dd#}/.'01!D' ã ¼M¼ ¢ ã ¼M¼ .(J587'711!D #}/.'01!D' i ¢ ¶ ÐXUbY ¶ YIÏ'`phFJEJBD5$2s!GP ? n.B ã E-GeJ871% ? N* -,V(. !(&*]^&K8 ã 3d0* D97 *D* -,(.!&(*nA.(-n.B nfp)1&;!#b^Z85 1 ã 3dL9 *1*1$Gn&9pNDdZNB1.;o!# M 1 ã `ghZ\)Z;D!D87 * M Z85OPA ã E2%%.*1 )F1KM5585/.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 .`( ôX# fËE=eJ871% ? -*1!D;D *85* )1!Dk!LKM8 ã &+( `c E !D/&5Hs!gn&d9p=Z1&;NDfÇ*1!D ? 1CÅxf(0
'11!DO!#jÈÉFW5XÊ ¸ Y ¶ TUcAD0.0%H#}!D8Hn 1!D8g(.²85'0n.BJ]q&)KMr/')(\%&%.*1 B1KMp7585/&0/85)1'5FBD(\!#f!&(* I0#s` Ç*1!ª ` Îq$-Qd®.`©.A)]fHr%85!(/&0qeJ871% ? !&(* ,$s ? 1.;eJ871% ? _/!D15!#¡ 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 Sstst3uvh Dh È3ó ã ¶3·J¸ RÊ ¸L··¶ R ¸ ÊYIÏ X5ÐÊ ¸ÑSÒ Ð ¸Ñ TBÐXÊ ¸ 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 ÐXÊ ½
Eq;D1KM)f/'fr#n185*1$O71%.*1J0* D9f!#C!(&* H]^1Zo1F/11!DE5 =%85!L%'851)P`chF #³D0PA]qp\5&!)] n&)K85$G* -,V(.SZ!D85$GEn&Nn!D85$G!#Ho!(&*j!#nE #}!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 TPÐXÊ 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&reJ871% ? rZ85£!(&*Q#I!D8 E AbD14n&g%85!!# !# !D85Õ©.` Ã'`ja@d]^1*1*Q5&!L]n.B A EqE5!D!D85% ds!NN_/!D1q!#Cn&d!L%' Z85!&(* B !# β, η>Z0!DK8771!DC` a@p]^1*1*j85K&1]:n&p0* D7 * F85!&(&*j0!D'5585/.01!D2,8710'$`Qdñ ,' m1.
'1F"F$&%'"D971;DNNn.BN%85!LK (&)p1.
'1Z*1$zN&$@K87 B,.*1L!#BD F$&%'M`j'!D8BDO$%' σ A*1 B ,'nd0!L*1*1P01!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 B1KMO5585/.0F/85 ]^1n ã ¼M¼ {M }{N } ¢ {M N }
E^EBD91*1$o5&!L]^ks!,'g"!D87(.1.85$o$%b)(k* -,V(.!(&*n`4!D85g(s1* !# n-F85!(&*C0!D'5585/.01!Dk$k,'J#}!D/..(k1 '873©)ZAI!D80.%.*1M` þ91&; E A.]qg2(&0
'J á Ê ¼ XW R !)K8 B `1'0-n-%V!991,.*1]q!D85* (!# A 85p$%bpD991;Dp5PA]qr]^1*1*j/'9pF$&%'pD71;DN5Ddn&r]q!D85* (=!# R `'!D8 BD9 σ .(GñA.*1 ²@ñ . M ¢ N σ ∈ E, R {M }, {N } ¡ ]^85 M En& β, η>n!D85*j#}!D85Þ!# M A&(i711* 85*1$4#}!D8 N `g EEJ]q*1* (0
'&)(91'0^BD {M } .DJ/& _/ β, η>n!D85*#I!D852AD,&$rn&dÄfM/87.>Z«^!98 n!D852`m1'0k%85!LKB,.*1SP_/'*11$@E5$&p587 o&(z587'711KMo]^1n~85)5%'P0 !@&$zñA R E0*1B85*1$m\%'85F *P_/1KL*1'0k85* B1!DC`6 IhZ~;D&87*nA R ]^1*1* !Dg,'O850'0.1KM!D B Aj71'0O5!DN M N$i85P_/185OKL89 B,.*1)g!Dg1[1ñ` ¡ Îf$"85/*1 å ¸ ÐMÐ â)¸ WZ[æ 7)Es
')=np!D!Ds!D 01F$"0!D&(.11!D å Ñ TDUDæ ½ &g%85!!# n&) R 7BEs
'L å R7T Ñ R¼ W7XFægE711* 8s!Nn%85!!#jn.BeJ871% ? JZ85¦!&(* 85 0&F'71!D.*n` ¨® ∞
σ
∞
σ,τ
σ
∞
σ,τ
σ Γ
0
0
0
0
σ Γ
σ
0
hZS85N1')D5$zs![5&!L]Õn&) 3OE5!D!D85% "s! `ÆKM85$ P_/1K*1'00* DH3^9.87D0Z871L(k,A$O0!*1*1P0F1!Dk!# !D85B/R *b#I!D85Oqn.BB85/R *1* ]f*1*>n$%b)(-1Nño.(-%85!LKL),.*1$P_/'*M/'71.; E ` M/'Q#I!D8CBD B /R P_/1K*1'0 0* D9 AMn&85=Ef-/. _/r ñ `Äf!D&KM87*1$A*1*'!#Z > !D85N{M*.#I!D}/R 85Oj1O$o ñ .M σ']^1*1*&.,bM^P_/σ1K∈*1Ac!(/*1! R AD&(5!J]qqβ,.)KMη S55871;D&s#}!D85]87(G, ¯sP0F1!D4,'F]q A &( B /R `=hF3BD7$ks!k5!L]Ën.) %%.*1 )F1!Di,'&)KM)-%&%85!L%87 BF*1$A&(\5!"]q&)KM@E5!D!D85%E5Þ,']q .( B/R ` 3%85!)KMLn&-n!D852` A !D85% J!O&$oeJ891% ? -_/!D1^!#c$"0* D99 *%&%.*1 B1KM5585/&0/85M` ÿ wh6 D{ 7 º.XW7X ¶E·g¸ à W ¶ ¼bá XgÊ ¸ÑSÒ Ð ¸OÑ TPÐX0|Ê }cº ¶ Rº ¶E· UVTDYqXÊ1X Ñ X0UbY ¸ W ¶ ÊµÏ LX F0» ¶3â)¸ Ê1XUVYQYnT ã /fLóTW ¸ U'ÏRÊ ¸L··¶ R ¸ Ê ¸ ¼M¼ Ê ¶ R ¸ Y ¶3â X · YIW».RPYI»W7X ã ¸ UÐ à W ¶ ¼Vá X ¼ ¸ W0Y ¶³¸ Ê LX F0» ¶3â)¸ Ê1XUR9XNW7XÊ ¸ Y ¶ T` U f ½ N 6O'PRSTUVXW a@;D1KMNo#I!D85g/* φ 1\n&%85)(. )Z*1!D;D r#}!D8-%&%&*1 )1KM 5585/&0/85)q]^1nG,'DF$&%') p .( q ]^ 9kEK*1 (S1G*1*C_/!DF15 ã /f:,/.qE !D^KL*1 (o1GKM85$SeJ871% ? * -,(.!(&~* `hF/1F1K*1$A φ 3nJ#I!D85g/* Nn.B 7$&B # .( ¬¬inhabited p→q ¡ A.n inhabited p→q ¡ A • empty p ¡ ]^85 inhabited τ ¡ ≡ ∃x τ.x ¢ x¡ &( empty τ ¡ ≡ ¬inhabited τ ¡ hFcEjBD5$ !O&P ? Z&)^nE^&!* (14&$o_/!DF1q!#0* D99 *Q%%.*1 )F1KM5585/.0F/85M`
!\5&!)]§n&) Er!DN1/11!DE5 *1*1${K*1 (QAc]fk0!D'7 (&8rZS#}!*1*1!L]^1&; eJ871% ? -* -,V(.!&φ(&* ß E^n&-%b!9]^1n2]q!N*1N5^ª < • §) E^n&J#}/*1*$%bg5585/&0/85]^1n2n&-,'D9J$%b) p ¢ ∅ .( q ¢ ω • :)HªJEn&=1Z871!D8f!# :B An` M`AMn&=%&%&*1 )1KMq7/M,.>Z5585/.0F/85]^&!9 • *1ps85-1Z85%85s)F1!D'!#0*1!)( λ>nF85O Z-587'91F1!D"#I/.'0F1!D i E=n-1'0*1/'91!D •
A 0!Ds1'f0.D0n*1$S!D*1pJ }Z%F$#I/.'0F1!D ¡ A&.(On&850#}!D85 Ì7)FE5
') ¬¬inhabited p→q ¡ çJ,/. A Ek%F$ },bPP/'{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!#OeJ871% ? z_/!D15PAJ]q[]^1*1*p5&!L]!L]s!0!D'7585/&04 0!D/&Z8s>n!&(*!\n&G1%.*1 B1!DË ∗¡ ;D1KM~1mZ2P0!D&(z%'87;D87%l!#dZ %'%'8P`\a@"]^1*1*q0!D'5585/.0p\< ? 1m!(&* ã ]^1n~,'D9oF$&%') .( &( ;D1KM-NeJ891% ? J*1!D;D *%'85 *P_/1K*1'0J85* B1!D/f ⊆ ã × ã 5/&aSZ&)1bkZ _/!D1^!&(* ã /fNA]q-]^1*1*Q.BKM ¢ λx a.f π ,/.!D f π ¢ f π . λx a.f π a@o*1 ã ,'"{0* D7 *HF85Ì!(&*f!# β, η>Z0!DK8771!DCAD(&)70871,b)(z1zn& %85!!#!#= &!D85Í®.` &`i1'0 f %&%'B87g1mn&SP_/'B1!D'g),b!LKMMA]fo1'0*1/'( Z85OJ]^1n\0!D'5s f P a→a→a¡ →b 1@0!D'5585/.01&; ã `O I &r1F85%85)1!D !# f 1 A 3^15P_/1K*1'0-0* D9PA.!&(/*1! ` ` ¡ hZH85N1'H!N(&0
'^n&85* )1!De fvBH,'D9=$&%')PA71'0=ZEH]^1*1*V(F851 fl)1;D&8$&%')P`c1'0fZ ¯/'5
b)F1!Dp!# (&%'.( !DNF$&%' ,'B1.;B1n&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)FE5#}$P_/'B1!D R E¡ ¢ λx a.f π λx a.f π BJ,b!DZ@]f!D85* (P`p EJEJBD7${)]q!D85* (iªAC71'0 EJ%$`-a@O\s ? !G,bOn& (F1$285* )F1!D@!D A ` !27BEs#I$ Ea¡ )-]q!D85* ( ]^85 a E-R!D %F$A.]qgg/'7=P_/')Z f π .( f π ` ¤ 4BD7$o]$os!S(.!SnE=3¯s/'5=s!Os ? ¢ A × A 5!n.B A /R .D!D*1$k!DJ*1NP`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_/'*Q1Gn&g_/!D1!(&*n` Z) ! #BÅx$&%'Gn&!D85$+ I0#5` ÎqB ¨ ¡ `6hZESBD5$~s![94n&)SeJ871% ? * -,V(.m!(&* o85i!D85i;D87*dn&60* D99 ** -,(&z!(&* PA71'0{&$ 0* D97 * * -,V(.S!&(&*CN$G,br,'p85;M87(L(4DdSeJ871% ? r* -,V(.O!&(*!)K8 o4ß 0!D'7E7F1.;O!#Ho71.;D*1r%V!991,&*1N]q!D85* (Q`=eJ871% ? p!(&* ]^1Z4%85!(/.0s .(pdZ851.*.F$&%' 85* 5!JJ5%'P0 * ? 1.(r!#85Z)9 0*1!9L(p)Z;D!D85$ σ ×τ &(QA&0!DK879*1$AD&$O85ZL7 "0*1!)(S)Z;D!D85$$,'d-,')((&)(k1kpeJ871% ? !(&*n` ¤ *n&!D/;D2]qJ(&0
'&)("eJ871% ? J!&(* ]^1Z!D/.^/'71.;g/.9G)Z;D!D85$on!D85$A !DJ]$Os!NK&1]6!D/8^(K*1!)%NqE^D[u]f!D8 ? L(S0%.*1PwN1kZJ/'!#n 1Z89&** .;D/';D!#fks!)%V!P`%'P0
b*1*1$A!D/8-eJ871% ? * -,V(."!(&* 85)7/*µ #I85!D1F85%85F1.;On&r5s.(&87(G0* D7 *C(&0
'11!DG!#c$&%')(G* -,(.O!(&*C1 n*1!D;D N!#fks!)%V!J!#H%85L5)BKML!LKM8Jk%b!P`rhFiD((.11!DCAQDJ%b!D1ZL(\!D/& !/'O,&$mÆ(.r/&.(«=!D,1'5!D.(|Ç1!«^!5!*11nAf!D/8o0!D%&*1FL9Sn!D85 N$z,'G(891KM)(m/'71.;@0!D&P0F1!D'N,']f~85ZL7 |0*1!)(~)Z;D!D871LO.( %85)5&BL#Hs!L%b!)P` /8-5/'(.$G!#feJ871% ? N*1!D;D *j85* B1!D'$4* 5!k,'K&1]fL( nE^]$o/'91.;NZg5s.(&87(k!D1!D"!#j85* )F1!D"1Gn&-1F87.* *1!D;D D` /8c,8710#b1&KM)51;M)F1!Dg!#eJ871% ? H*1!D;D *&85* )F1!D'j5/;D;D)7sjn.)$g0* D5> 7 *c!(&*>nn&!D85F pZP _/)N${,'D(.%Z)(\s!"eJ871% ? * -,V(."!&(* BA &(m(&!D'5587BF)Nn.BNÇ*1!D ? 1CÅxNhZ>n85* )F1!D'p%85!)K& (&S{/'90#I/*f0* Dp!#ou1F/.> 1F1!DE7F wJ* -,(&r!(&* P`a@=.BKM=7!L]^n.BKM85$N$%'L(* -,V(.rn!D85$NE nn&!D85$r!#5!DNqeJ871% ? _/!D1c!#J0* D99 ** -,(.J!(&* PA,/.cn&)cnE (.!M)!Dd8585$G!LKM8=s!k_/'F
'L(G#I!D85g/* DP`d%'P0
b*1*1$A']qg#I!D/&.(2O#}!D85g/* ]^ z3KL*1 (z1leJ871% ? G_/!D1!&(* BA,/.p!D1l*1*^eJ871% ? "* -,V(. φ !(&* P` hZ:;D87*nAJ!D/8G1+#I!0/'2.D",' !D+n!D85 *gD5%'P05G!#OeJ871% ? * -,V(.k!(&* P`<)K1.;o#}!D/..(4eJ871% ? p!(&* 85* B1KM*1$".B/87*.(2BD7$G! ]f!D8 ? ]^1nCA&1qEq]f!D85Z"D ? 1.;r]^&n&8qeJ891% ? d* -,(.p!&(&* q85J%&%85!)%87 )Z !G0!D%/.Z8g901'0N%&%.*1 B1!D'P`-'!D80%&*1MAQ(.!GeJ871% ? 0>n*1 ? p!&(&* 89E9 &)F/87*1*1$41@nNF d!#f%85!D;D871.;G* .;D/';D)8
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0 !DZ0.5PAj!D8pZ)18pNB1.;MP`{¬1K[n&k(.²85'0)p,']fmZ5!D85ZL(OP_/'L> 1!D&**1!D;D D`qÓ.zÈ H)Ô 9 ã TDY ¶ R9X · A &é &A éD ¨ ` !'`¬!D;D/r.%( !'`)G);D/8P`«^8 ? !Dr858 ? Q!DpN&$>Z5!D85ZL( P_/'B1!D.**1!D;D DB` =»Ê}Ê X0Y ¶ +U n ã 7 è ÓVA ¿ ª®D® . ¿ A éDD®.` `< ? 1C`§Äf!D%.*1Z&)k1vZin&!D85$|!#p$%b)P ` TD»WU ¸ Ê-Tsó ÓbÏ ÑSÒ TÊ ¶ R 9QT ¹D¶ RA © ¨ ¡ A !D/. éD©ª`c%';DLd >Fé ` ay`<!)]87(Q`{ &O#I!D85g/* Ds>ZD5>nF$&%')p!D1!D!#0!D'5585/.0F1!DC`ihF 7 KT Z ½ ½ è »&WWÏ ^Bn ··0¸ Ï · TU è T ÑSÒ¶ U ¸ YnTDWÏ 9QT ¹¶ _R I9 ¸ÑSÒ Ð ¸ É è ¸ Ê1R0»Êµ» · ¸ UÐ @ TDW ÑO¸ Ê ¶E·Ñ A%';DL^Ãé ÃéªA ¤ D(& JÇ85L9PA éDª` !'`x«r`V<1&(.*1$2.+ ( !'`xÇc`Q* (.1C`ÈUbYIW5TPлRY ¶ TU[TW è T ÑSÒ¶ U ¸ YnTW ·p¸ UVÐ 9 ¸DÑOÒ Ð ¸ è ¸ ÊR0»&ʵ» · `Q!D.(.!D"{BZB * !01$A éDD®.` ¿ª
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