On the Formalization of Imperative Object-based Calculiin (Co)Inductive Type Theories
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. On the Formalization of Imperative Object-based Calculi in (Co)Inductive Type Theories Alberto Ciaffaglione — Luigi Liquori — Marino Miculan. N° 4812 April 2003 THÈME 2. ISSN 0249-6399. apport de recherche.
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(44) DG@I¯£tvmw~tys0zT|su uªm;1x|m |s0wvm;Q¾£tys0zjum ¼{}Å$ï us0}j0|suz|m:tvs j0m ux£}m;É´i9jum$msN ty tvm;0 m;1tysjum utvz½¼ mÊ ml+sNtv}; x D±ÓI ·¹¸ ºª»mr jum±²x{wywvxª°"tys0zEO V \ "ML x|mrwvx} tvx|s 02043 5 8N G 8 8A> ¼{}Å A 8 > A 8 > 8NO »:9 5 8 ; . 8 - 8 ¼ x{ m G 02043 6 7 8 3 8 = m¼uwy 02043 i9jum:z|wvx|0|wã&ÖìßØ Û:tvTʱ²0s0}&tyx{sïl+u0tysuz!wyx};ª tvx|s@$ xòâ?ÛÖKç£ßÜ¿ÝÙyߪã×ØÛ&ã&É µ´wyxN¼um;  m;suxm;<£ à rmT0tvru0tywy$x±Óâ+ÛÖZçß;ܪã?|s0 ã&ÖÕ|ÝQ;P ã#J ¼|}ÅmT0¼mʱ²x{ jum?m;0} tvx|s!x±Ajum+}x{ m; @x{s0tvsuz l?mjuxÄYx£u O jum+| x£}tv m¾ª| t~uwvm;A°"t¹j!ß{êR ÛÝÖ´Ø Ûã×Ù¹ÖZã&É"m; uwym¼mNums0}m;´x±b0|tyP#O l?mjux. 9 G@ D ;. w~Ymw~ x{z|m jum;A°"ty j¼ x{ m"wvx}ªtyx{s0;¯{x{sumwvx£};ªtyx{s±²x{Am|}j+x|um;}&}x|l?Yx|sum;sN;Éäx| t~}m"j0ª¯Ntvs+x|m; xʼQÊ}wvx{ m xtyl?uwvml?m;s{ªtyx{s] m}jusut~Num;;¯, j0mmt~sux10¼m?x|±A j0mTs0x tvx|s!x±´±²x|l?|w¼u@Rt¹ tvx|sbÉ x{ m;xª ¾| m?jumÁx{@m; tvx|s0|w ml+sN t~}summu°´x l?x{ mÁ| tywvt~]0z{l?msN;¯s0|lTm;wy ¯"|s0 @x| jÄ}jum}Å£tysuzrjum"°´m;wywy½V±²x|lTmsum x|± ¼ x{ ms0?¼|}Å;¯m; @m}& tv¾|m;wy{É#sTjum"±²x|wvwvxª°"tysuz@¯ª j0m su :x|
(45) ª tyx{s
(46) 8 88
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(48) 8 YmTt~¼ tvs0}&Ér tywvtv| 0z{l?msNrm um; msN mtv
(49) s Ltyz{u m?{É L °´m;wywy½V±²x|lTmx|m$tv"0utywyr¼|¼tys0z?±² x{ljum °mwvw¹½V±²x|l?m;1ml?¼mNums0}m s0Ê|wywvx} tvsuz+ sum° }wvx{ umYx|tvs{m;?x £?±² m¼j@x{tysN m;;¯Nuxª¾£tvm jum}wvx{ um}x|sNtvs0A$°mwvwy½K±²x{ l?m;+¼{}Å J L i9jumml?°mwvwy½K±²x{ l?m;:|}ÅÄt~9uutvwyÓ tvsuz±² x{l?°´m;wywy½V±²x|lTm¼ x{ mJ L °´m;wywy½V±²x|lTmò¼{}Åò};s @mÁmsutv}j0m; u0 jutvsuz x|s tyÄ x| jumÁ{ x}t~ª tvx|s Ym°mms ¿±²m; j ¾Q| t~0wymTs0] jum?m; uw¹r±²x|l?m;ÁNÁ0tvÓx±9t~tys0}$wv|@m;wvs@Át~tys@}&$@x{tysNmÓm±²m; tvsuzÄx lTmsutvsuz|±²uwbx|mwyx} tvx|s0;É ä uwbø£m;l+sN t~}"mw~ªm;´ m; l+x+ m¼uwy9tvsÊx|m;P J£t¹t~"mó@sum;1tv7s Ltvz|um nÉsTsNu jumwvw O L ôªÕªØ&ÞZÕNê&ÙvÛ m;u0}m x? j0m$ m¼uwy"t¹umsux| m;9tvs1 jum }u m;s{¼{}Å J L s¿ßN ê ÛÝÖAm;@}m;´x m¼uwy}x|s0 t~tysuzx±# ±² m¼j1}x{wywvm;} tvx|sÄx|±3wyx};ª tvx|s@¯£¼@}jÄ j@ª jumx|m tv9 m Nms0um;:±²x|{ x}t~ª tvsuz jum$l?m jux}wvx{ um;´ xT j0m; m$wyx} tvx|s0P J L GãÛÙyÛÝÖVÞZߪàÊx{@m; tvx|s+ó0¼m;@}m;´tyjux{¼9x|m}& x? m; uw¹¯N jum;s1{}& tv¾ªªm;A jum$0u x{utv m lTm jux}wvx{ u#m J L s×QõÜ{ÕªÖÛÓx|Ymª tvx|sTm;0}m;tyAx|m}&As@T jum;sT0Yu m j0mu0 x{u t~ªm¼ x{ m"wvx} tvx|s?°"t¹j s0m° l?m jux}wvx{ u#m J L ÝÙyßà0ÞÐàNèx|Ymªtyx{sm;@}m;9ty9x{m;}s0 jum;s1|wywvx} m;9±²m; j1}x|wvwym}& tvx|s:x±wyx} tvx|s09|s0 |¼x}t~ªm;A jum;l)x? jum$ m £t~¼ tvsuz?lTm jux}wvx{ um;´±²x|l jum$x|m}&  mm;}wyx{sutvsuzN8à J L ÙyÛÖ}x|s0¼ 0}&m;@}m;x jum"m; uwyx± m0}tvsuzty.Yx tvs+|}Å m Nms0um;T°"t¹jT jumYx|0s0 ¾Q| t~0wym|¼x}tv m;Ä x?jum$ m¼0w¹"x±t¹"wvx£};w ml1É Ltys@wvwy{¯{sux| t~}m"j0ª9s|wyz{x|t¹julG±²x{´m;u0}&tyx{sÄ}s+Ymm| tywvT m £ |} m;T±²x|l j0mÓuwvm;P O3ty°x|uw~ 0||wywvmw,¼s@utvl?uwvml?msN tvx|s09x|±x{m;};É C C),b&c8dfeSg.
(50) T ÞZÕÕ;èÙ ÞZßàYÛ Þ
(51) ×0ߪØ&Þ ÊÞZÝ×ÙyÕªà 2. 9 G - ;
(52) I -'5. II 9 D :' , ; 9 -<5 G - ; 5J G II
(53) I
(54) 8 V 9 ;O" J V$# T
(55). 9 D . ; 6 7 8 3 »:9 5 8 ; . 8 = 8
(56) > J 6 7 8 3 8 = 8A> 9 -
(57) 8 »#9 5 8 ; . 8 - ; 8A>
(58)
(59) U5 V! 9 VU ; 3 »:9 5 U ; . U - 9 ; VK# T DI ,:J 6 7 8 3 8AI= 8A> U 9% -<5 I 6 7 8 3 8A= 8A> ; . U J G
(60)
(61) DI DI
(62) I
(63) DI I 9 D D 7; ,E 7 U J G
(64)
(65) DI I U V 9 ; V$# ,:J 6 7 8 3 8A= 8A>
(66)
(67) I I 9 D
(68) ; ,E 7 U F »#9 5 ; . J 6 7 8 3 8
(69) = 8
(70) > 9 E U F »#9 5 ; .C- 9 ;
(71)
(72) I 8 V 9 ; 8 V , J 6 7 8 3 8
(73) = 8
(74) > T 9 DI ;" J V$#
(75)
(76) DI D. I I 9 D 7
(77) @CB D ; ? 7A@CB D 9 , ; J 6 7 8 3 8 = 8A> 9 - 8 9 8 ;<; 8A>
(78) I
(79) DI I DI , J G 9% G ; .NJ G
(80) I
(81) DI DI
(82) -'5 I I I
(83) DI I 9 D DCG/; 7
(84) DCG 5H3O,SJ B .J G
(85) I I
(86) DI I L tyz{u m n%.O äªuw3ø£m;l?|sN t~} ÂR\ £> ôx±Ó·¹¸ ºª» 5UV! T 9 ; 9 B G :' ; 9 B G ; , -<50 8 " J VS# 0 0 9
(87) D . ; 9% . , B T ; A 8 > 67 8 0 8 = 0 8 " J VS$# #&% 9% . . ; 9% . "D ! 7; 0 6 7 8 0 8A= 8A> ( ') 0 6 7 8 0 8A= 8A> L tyz{umr¡E.O tvwyt~i´N0tysuz+"0wym )**. +-,. -0/ <.,(CM4 /N>. i9jum£Ym ¼¼ m;l)±²x|· ¸ ºQ»t~Tó0¼¼½ìx|m£@m¼¼ ml °"t¹j]¼0£utys0z0É´i9j0m$x|suwvÄ£Ym }x{s0 @}& x{ t~Ä6ju= mÁx{sumʱ²x|1x|um;}&NYm;;¯ ÞÐÉÐÛÉ O - 0 0 3 6 7 8 0 8
(88) = 8A> É7äxtv}m] j@ªjum¿x|suwvòz{ x{us0 £Ym t~ O t¹]}|s Ym®0¼m {1 ¼ tvsuzò@x{tysN±²x|uutvw~tys0z x{m;}¼½V£@mÉ jum1z|x|0s0 £Ym;;¯r{ }|sE}|s @m¿|umEª1°"tywvwVÉ7i9jum]±²x{ l+|w£utvsuzï £¼ m;l tvz|tv¾|ms £ j0m ±²x|wvwvxª°"- tysuz!- ±²x|0Ó- @z|l?- EmEsNE K O - s@ - s0 0 - |s0 , 0 E i9jum1£utys0z ms£¾£tv x{sul?msN }x{s0¼t~¼rx±´:wvtv¼$x±´{ ul? tvx|s@±²x{¾Q| t~0wym¯ m;{}j]x±´ jumT±²x{ l Éi9jum£utys0z 0z{l?msN"mw~ªm; xT j0x{ mÓjumm90z{l?msN"| m$}x{wywvm;} mtv
(89) s L#tvz|0 m$¡uÉsÊTs£ ju5mwv0 Vw O UAÛÙÐÙ é ߪØ&â+ÛÜ]ÖVëõuÛ:Ûà0ôÞÐØ ßªà@â?Ûà0Ö O i9j0m L i9jumTó0¼"0z{l?msN ! m } tv@mjuxª°GxÊuutvwv®! ml?Tms£¾£tyx|sul?m;s{ tvA°´m;wywy½V±²x|lTm J£$sum° £Ymutvs0tvsuz 50 m £ m;s0u$°mwvw¹½V±²x|l?m;?ms£¾£tv x{s½ lTm;sN90 xª¾£t~m; j0 5 tv±²m; jÊs0 t~"°mwvwy½K±²x{ l?m;£@m{É UAÛÙÐÙ é ߪØ&â+ÛÜÖVëõuÛ tvs j0m ms£¾£tyx|s0lTm;sN ÄÉ9s L i9jum ¼m}x{s00z|l?m;s{8A > ¼ m;" j0 t~7 x|m}&£@m 6 7 8 0 8
(90) = ?t~$°mwvw¹½V±²x|7 8 l?m;!tys®jum+ms£¾£tv x{sul?msN u xª¾£t~m;Á j@ª m;{}j 8 t~$°´m;wyw ±²x|lTmtvs |s0: j0" jum$w~Ymw~ mtv¼ tvs0};É L i9jumÄjuty0uz|l?msNtvsN x0}m jums0x tvx|sïx±$ã&×@êã&×âÓõ@ÖVÞZߪà@¯°"j0tv}j t~tys0u0}m £¿ ã&×@ê&ÖVëõÛ m;wv tvx|s 0 @m°´m;ms x|m}&TNYm;;É s x|m}&+@m;wyx{suz|tvsuz! x<!z|tv¾|msòx|m}&?£@mÊ|wv x . . /. 012123 545687 697 /:8;29 3. <. =. -. -. >?. @. $. . A. . B. C. . . $. . A. . . D. . <. ?. E. HCWFE.
(91) à]ÖKç£Û3ßØ&â?ÕÙ ÞÕªÖVÞZߪà<ßâõuÛØÕªÖVÞÐôQÛ ê ÛÝÖKé êÕQãÛÜ
(92) .ÕÙyÝ×Ù¹Þ.ÞÐà , 0. . . 0. K , 0 8
(93) > -<5 8 0 6 7 8 0 8
(94) = . 8 0 8 6 7 8 3 »:9 5 8 ; . 8 = 8
(95) > 0 6 7 8 , 0 6 7 8 0 8 = 8A> ? 7
(96) @CB D 9 , ; 0 6 7 8 0 8
(97) = 8
(98) > ,K0 6 7 8 0 . . . " J VK# 8
(99) > 0 8= . I -'50 - I I 9 ' , 7 (' , ; I -'50 - I I 5 8
(100) > 0 ,S0 6 7 8 0 8 = V$# 9 ' , 77 DC7; ,E 7 U 0 U ,S0 -<50 . 0 9 ' , 7 DCG/; 7
(101) DCG 5H3O,SJ B . 0 0 U V$# ' 7
(102) ; 9 , . 9 ' , 77 . ;. . ¢. . 9 ' , 77 . ;. 9 ' , 7 7A@CB D ;. . . . 8
(103) = 8A> -<50 6 7 8 0 8A= 8A> . ,E 7 U F »:9 5 ; . 0 6 7 8 0 8A= 8
(104) > L tyz{u mXOi9j0m iA£Ym µ´j0m;}Å|m;9±²x|Ó·¹¸ ºª». @m;wyx{suz{ xÊs£Ê u@m;¼£Ym?x±.j0ª£@m+|s0Á};s! u0 ul?m?x|m}&Ótvs]jum+¼0@m;¼£Ym|¯bYm;};0 m jum¼mj0Q¾{mrTl?x|mÓwvtvlTty m1u x| x}x{wKÉi9jumró0¼°x? 0wym| mrjum0| tv}ruwvm;9x± mKQ0m tv¾£t¹Ä|s0 s0 t¹ty¾£ty|É3i9j0m jutvx|sumA|wywvxª°b"wvx|s0z|m,x{m;}b£@m. xYm´ u£Ym.x±u jux{6 7¼8 mbx{su8 = m|8
(105) >É3äxtv}m 6 7 j08 x|8 = 8
(106) > m}& £Ym; | m8 ÞÐà@ôQÕ8Ø&ÞZÕªà0Ö tys®jumtv }x{l?@x{sumsN £Ym;P O j0m¼0£utys0z 0 0 m;Nutv m ±²x{wvw É.i9j0ª"t~¯ux{m;}"£@m"mrs0mty jum;}xª¾ªtv|sNsum;t¹jum }x|sN0 Q¾ªt~sNtvs+jumtv´} x|l?@x{sumsN.£J YVSm;P # J|jut~´}x|s0ty tvx|s+t~.s0m;}m tvs+x|m; x z{0sNmm9 j0m ¼x{us0sum x|±#jum£@m t~}tvuwytvsum{É +¥ ú «§Ä§ û i9jum l+|tys<ÖÛØ&â ÖVëõ@ÞÐàNH è &×0Ü;èâ+Ûà0Ö ,H0 ªm;9j0ª , j0{9£@m tys O´ j0m £@m1 uwvm;"| mrum; msN mtv
(107) s Ltyz{u m@É m 9 ' 7 ; }x|susum}&jumN0tysuz x jum?¼0£utys0Ez O|s]x|m}&$}|s]mluwv mTsux| jum; L i9jum uwvN x|m}&9 j0j0, |±²m;°´. m;"l?m j0x£0¯utVÉ m|É?l?x|mÓwvtyl?ty mu x| x}x{w J L i9jumuwvm 9 ' , 7 (' , ; t~´0¼m? x m £ |}|s|¼0lTu tvx|s?±²x|l sÄm;s£¾Ntvx|sul?msN¯{°"jum; m 5 x}}u ¼x{lTm;°"jummrtys J 8A> L "}}x{utysuz] x 9 ' , 77 . ; ¯Asïx|m}&TNYm 6 7 8 0 8 = };s Ym1{ tvz|sum¿ x®Á}x{wywvm;} tvx|s x|± B lTm juxu°"jux{ m$Yx£utymj0Q¾{mNYm; - E E E - 6 7 8 Éäx8= m$8
(108) > jumrö¼}ty}uw~ty|÷tysN x0}m;£Äjum mwy± 0l?mmf O"tys¿x|m;x:z{ty¾{m:¾Q|wy0mT£@m 0 jum? m t~ms0}mx|±´¾ª|wyumTx±A jum+l?m NYmtv|¼ul?m J 8
(109) > L i9jum$ 0wym 9 ' , 7 D 7 ; m;wyw~9j0ª¯0°"jum;sTl?mjux 7 U x|±|s:x{m;}"£@m 6 7 8 0 8 = t~"tys£¾{x|Å|m,¯ty u x@}m;´jum }x{ m; @x{s0tvsuzT m¼uwy9£@m U J ; um; m¾|m jum"£Ymx|±Yjumx{m;}. j0At~.u uªm; Ojum"£@mÓx± j0m L mjux?uY0ª m 9 ' , 77 x|m}&}|susux"Ym wvwyxª°m;Ä x+}j0|suz|m{¯@m}@¼mxjum"l?m juxu{ ul?mt¹f J L i9jumwv{9°xTuwvm; 9 ' , 7 7A@CB D ; |s0 9 ' , 7 DG/; | mx|£¾£tvx|0;É ú ¦ ²§®© ÿ)¦;§ þ ¨ ú § û i9jum1£utvsuz!x|±Óm; uwyTtv+umwvtv};ª m{¯AYm;};0 m:m; uwy?m1@x{tysN m;x j0mÄx|m|¯3|s0®x|m;rl+QÁ}x{s{tvs!wvx£x|0;É+i9j£0$t~sux Yx{¼tvuwvmT x]m ml?tys0m? jum?£YmTx|±91m; uwy m u|lTtvsutvsuz?t¹ u0¼ 0}&u m m}u¼tv¾|m;wy{É i9jumã&ÖßªØ Û:ÖVëõÛãÄwvwyxª° x£Ym+ jum m¼uwy tys@mYms0umsN wv!x|±"0 t~}uw~ ¼ x{ mK O jut~ t~$Yx{¼tvuwvm Ym;}|0 m £@m½ì x|0s0]}x{l?u tvx|s0Óxs0x$x|m m¼0w¹x|±At¹À mmsNÓNYm;Ótvs] j0mTl?m wvx}ªtyx{sb6 7 8É ¼ 8x|= 8AmÄ> £Ym1{U x}t~ªm;T6 7 8 â+ÛÖKç£ß8 Ü¿= 8A> ÖKëõuÛ1x!m|}jï¼ x{ m:wyx} tvx|sbÉïU Êm j0x£ £@mj@Q¾|mÄjum:±²x|l 3 . ¯´°"j0mm 3 ÊtvTjum£@mx±¼m;w¹±$s@ t~T jumÊm; uwy?NYm|É ì+t~+0 m±²uw9x . . . . . . . . . . . . . . . .
(110) . . . . C C),b&c8dfeSg. ('). . . . .
(111) . T ÞZÕÕ;èÙ ÞZßàYÛ Þ
(112) ×0ߪØ&Þ ÊÞZÝ×ÙyÕªà. 8 3 " J V$# V$# 9 G@ DM
(113) D ; 9 DCG2
(114) MD#; 8
(115) > 8N 8 3 6 7 8 0 8 = 8
(116) > U 3 3 3 8 0 8 9 8 ; 3 6 7 8 0 #9 8 ;
(117) = 8A> " J VS# 8 -'5 8 0 9 8 ; . 8 0 :9 8 ;" J V$# 9 D T . ; 9 G@ DM
(118) J B ; 3 6 7 8 3 8 = 8A> 0 6 7 8 0 #9 8 ; = 8
(119) > 3
(120) 8 »#9 5 8 ; . 8 - 8 8A> 3 3 0 3 G 0 5UV T 9 ; 9 G :'
(121) ; 9 G , ?
(122) ; , ? , 3 0 3 -<5 G 0 -'50 L tyz{u m_%O."uwvm;±²x|ø£ x|m$iA£utvsuz@É tvs{ x@}m$w~¼xTjum$uxm;} tvx|s0´±²x{"lTm jux:£@mÉ.wywbNYm$ @}& 0 m"mÓjum±²x|wvwvxª°"tysuzXO 8
(123) > 9 ; Q 6 7 8 0 8 = 8
(124) > ty± 9 ; 3 6 7 8 0 8 = 8
(125) > U x|m£Ym 02043
(126) 8 8 8
(127) > :9 ; Q U ty± 9 ; 3 6 7 8 0 8 = 8
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