Table Of Links I-Table ye-Links ABSTRACT Ukucaciswa 1 INTRODUCTION 1 Ukulungiswa 2 DEPENDENTLY-TYPED OBJECT-ORIENTED PROGRAMMING 2 Ukulungiswa kwe-object-oriented programming 3 CASE STUDY 3 I-Case Study 4 DESIGN CONSTRAINTS AND SOLUTIONS 4 Izixhobo ze-Design kunye nezisombululo 5 FORMALIZATION 5 Ukucaciswa 6 DE/REFUNCTIONALIZATION 6 DE / Refunctionalization 7 FUTURE WORK 7 Izixhobo zangaphambili 8 RELATED WORK 8 Izixhobo ezinxulumene 9   CONCLUSION AND REFERENCES 9 Ukucaciswa kunye neengxaki Ukucaciswa kwe-object-oriented programming Iiprograms ezaziwayo ezaziwa kwiilwimi ze-object-oriented kunye ne-functional zihlanganisa ezininzi. Thina nqakraza ukuba iinkqubo ezaziwa kwiilwimi ze-object-oriented. 2.1 Method Call Syntax and Self Parameters I-object-oriented programmers ziyafumaneka kunye ne-methode call syntax o.f(e) ukufumana i-methode f ukuthatha i-argument e kwi-object o. Oku kwakhona ibonwe phantsi kwegama "i-function call syntax" njenge-notation ye-alternative ye-call f(o, e), apho i-argument yokuqala ye-methode f ibonwe yi-self-parameter. Thina ukuthatha le nqaku elula le-syntax, nqakraza indlela yokuguqula ku-dependently-typed setting, kunye neendlela yokusabela kwinkqubo yethu ye-codata types. Njengoko ukuqala, sinika i-Agda proof ukusuka kwicala 4. Kule i-proof, i-Booleans iboniswe njenge-datatype, i-negation ibonwe njenge-function non-dependent ukusuka kwi-Booleans ukuya kwi-Booleans, kwaye i-neg_-inverse ibonwe njenge-function dependent ukusuka kwi-Boolean x ukuya kwi-proof ukuba i-x ibonwe ne-neg(neg x). Kwi-system yethu, sincoma ukubonisa i-neg kunye ne-neg_inverse ngaphandle kokusebenzisa i-function non-dependent okanye i-dependent. Ukusuka kwimeko ye-non-dependent, sinokuthi ukubonisa ukuchaza kwi-Booleans: def Bool.neg: Bool { True => False, False => True } Ukuba ufuna ukuba uthetha i-function dependent neg_inverse njenge-observation kwi-Booleans ngexesha elifanayo, ke kufuneka ukongeza i-function, i-auto-parameters. Singakwazi ukuxhanjiswa ne-terms ebonakalayo kwi-variable, ebizwa ngokuba yi-self, kwaye usebenzisa le variable kwi-return type ye-observation: def (self: Bool.neg_inverse: Eq(Bool, self, self.neg) { True => Refl(Bool, True), False => Refl(Bool, False) } Ukuguqulwa kwezi zindlela nge-self-parameter ivumela i-characteristic yokuqala ye-dependent codata types: self-parameters kwi-destroyers. iimveliso ze-Bool, iimveliso ze-Bool, iimveliso ze-Bool, iimveliso ze-Bool I-auto-parameters kwi-declarations ye-codata ibonelela i-power ye-expressive yokubonisa ngokufanelekileyo i-interfaces kunye ne-class e-object-oriented style. 2.2 Verified Interfaces and Classes Ukubuyekeza izakhiwo zeendatha kunye neengcingo kwiilwimi ze-type-dependent, sinokukwazi ukhethe phakathi kwezindlela ezimbini ezininzi: ukubuyekezwa kwe-intrinsic kunye ne-extrinsic. Ngokusebenzisa ukubuyekeza kwe-intrinsic, sinokufuneka isakhiwo se-data okanye i-algorithm kunye neengcingo yayo yokubuyekeza. Ukubuyekeza izakhiwo ze-data kwi-intrinsic, sincoma izakhiwo ngokusebenzisa iindices ze-type, njenge-number of elements contained in a length-indexed list. Uyakwazi ukusebenzisa indlela efanayo kwi-object-oriented style usebenzisa iintlobo ze-codata ze-indexed [Thibodeau et al. 2016]. Ukusebenzisa iindices ze-type, sinokukwazi ukuqinisekisa ukuba izibuyekezo zithunyelwe kuphela kwi-object ezininzi ezininzi ezininzi. Oku kunokufumaneka kwi-example elandelayo, apho ukubuyekezwa kwe-observation ingathunyelwa kuphela kwi-buffers engapheliyo. Ukucaciswa kwimveliso ye-Bool, i-Bool ye-Bool, i-Bool ye-Bool, i-Bool ye-Bool, i-Bool ye-Bool, i-Bool ye-Bool, i-Bool ye-Bool, i-Bool ye-Bool, i-Bool ye-Bool, i-Bool ye-Bool, i-Bool ye-Bool, i-Bool ye-Bool ye-Bool, i-Bool ye-Bool ye-Bool, i-Bool ye-Bool ye-Bool, i-Bool ye-Bool ye-Bool, i-Bool ye-Bool ye-Bool, i-Bool ye-Bool ye-Bool ye-Bool, i-Bool ye-Bool ye-Bool ye Ukubonisa ukuba, njengoko ngokuvamile kwi-dependent (co)pattern matching, iingxaki ze-pattern engabonakaliyo ziyafumaneka ukuba ziyafumaneka njenge-absurd. Ngoko ke, xa sinikezela i-buffer interface ye-buffer embalwa, asikwazi ukuvelisa indlela yokufunda njengoko akuyona. Nangona kunjalo, kule nqaku, sinikezela ngaphezu kwe-indeksed codata types, ezininzi i-control style. Kwakhona nathi ukunceda indlela extrinsic, apho nathi ukwahlula izicwangciso zethu ze-specifications zabo. Jacobs [1995] inikeza nathi inkcazo yokuqala yokufumana le izicwangciso. Uyavumela inkqubo ye-coalgebraic izicwangciso ezisetyenziselwa ukulawula izicwangciso ze-object-oriented. Njengoko umzekelo, umzekelo se-5a ibonisa isicwangciso se-coalgebraic ye-buffer ye-one-element ebonisa umzekelo we-read (PR): Emva kokugcina umzekelo, akakwazi ukuguqulwa ngokusebenzisa umgangatho we-store. Kwimeko, kufuneka uqhagamshelane ne-methode emnyama ukuze uqhagamshelane ngqo i-buffer. Ukucinga kwi-buffer emnyama ibonelela i-error. Le nqakraza ye-buffer ibonwe njenge-set ye-assertions ezihambelana ne-buffer-state s. Kwi-system yethu, sinokufumana le nkqubo ngokusebenzisa i-auto-parameters kwi-destroyers, okuvumela ukuba sincoma i-specifications njenge-observations kwi-codata-types. I-codata-type e-Figure 5 ibonelela i-interface ebonakalayo ye-buffer ebonakalayo kwinkqubo yethu. Kwakhona, sinokuxhomekeke le nqakraza kwi-interfac 2.3 Dependent Functions Ngaphandle kwezinye iintlobo ze-dependent, i-π-type yeengxaki ze-dependent ayinxalenye kwi-theory yethu ye-core, kodwa inokufumaneka kwi-library. I-π-type yenzelwe njenge-type ye-codata eyenziwe kwi-type ye-family ye-p, apho sisebenzisa i-type ye-function ebonakalayo: -- Átha Non-dependent Functions codata Fun(a b: Uhlobo) { Fun(a, b).ap(a b: Uhlobo, x: a): b } -- Dependent Functions codata Π(a: Uhlobo, p: a -> Uhlobo) { Π(a, p.dap(a: Uhlobo, p: a -> Uhlobo, x: a): p.ap(a, Uhlobo, x) } I-Designers ye-Java uye wabelane le nqakraza xa wabelane i-lambda abstractions njengeimeko ze-interfaces ezisebenzayo [Goetz et al. 2014; Setzer 2003] kwi-Java 8. Oku kubonisa ukuba ingxelo lethu ayinxalenye, kwaye sincoma ukuba iyasetyenziselwa. Ukongezelela ukunciphisa i-complexity ye-language core, zibonisa i-situation ukuba kukho i-type ye-function ezininzi. Oku kwimeko ye-substructural apho i-functions ye-linear kunye ne-non-linear. Umzekelo, kwi-language ye-programming ye-Rust, kukho iimpawu ze-function ezininzi ezahlukileyo ze-Fn, FnOnce, kunye ne-FnMut, ezininzi kwi-modality ye-receiver3 . 2.4 Weak and Strong Dependent Pairs Kwi-section ezidlulileyo, sinikezela indlela yokufaka i-π-type. Kwi-π-type, asikwazi ukucacisa njenge-codata type4 , kodwa kwi- Σ-type sinokufumana ukuba ukuguqulwa njenge-data okanye i-codata type. Le ukhetho ukwahlula i-sweak and strong Σ-types [Howard 1980]: i-sweak Σ-types zihlanganiswa njenge-codata type nge-projection ezimbini, apho i-projection yesibini ibonise imiphumela ye-projection yokuqala kwi-return type yayo; i-sweak Σ-types, ngokuqhelekileyo, zihlanganiswa njenge-data type nge-constructor eyodwa ebonakalayo i-element yok Iintlobo ezimbini zinokufumaneka ngamnye ngokusebenzisa i-defunctionalization kunye ne-refunctionalization. Oku kwenziwa nangokuxhomekeke i-pairs engabikhoyo kwi-S-type. Ngokusho ne-P-type kwi-Section 2.3, i-S-type ibekwe yi-type T. Njengomtype yedatha, ibekwe yi-Constructor Pair eyodwa ebonakalayo i-T-type, i-element x ye-type A kunye ne-witness w njenge-arguments. Njengomtype ye-codata, sinazo izibonelelo ezimbini ze- π1 kunye ne- π2 njengoko ngexesha le-pairs engabikhoyo. Kodwa i-projection yesibini isebenzisa ngoku i-self-param Kwimeko, i-Agda inokufumana i- Σ-types kwiintlobo ezimbini. Kodwa ingxelo elinye: I-Agda ayenzelwe ekuqaleni nge-codata iintlobo emangalisayo, kwaye i-codata iintlobo zayo zikhuselwa kwi-records ezihambelana, nto leyo ivimbele ukuba iintlobo ze-codata ziyafumaneka. Umzekelo, umzekelo we-destroyers kwi-codata iintlobo iintlobo we-Agda, ngoko asikwazi ukutshintsha i-projection yokuqala kunye ne-second. Kwi-system yethu, i-destroyers ye-codata iintlobo ayinokufumaneka kwaye inokufumaneka ngokufanelekileyo, nto leyo ibonise ngokufanelekileyo indlela Kodwa ngoko kufuneka siphinde ezimbini encodings ezahlukileyo ze- Σ-type? Khetha, umzekelo, Eisenberg et al. [2021] owaziwa ukongeza iintlobo zonyango kwi-Haskell. Njengoko i-Haskell iyiphi na ingxaki kunye nokuxhomekeka ukunciphisa iintlobo, Eisenberg et al. ziye zithunyelwe kwi-design enokusetyenziswa iintlobo zonyango ezininzi. Thina ndingathanda ukuba usebenzisa i-framework yeendaba kunye neentlobo ze-codata, sinokwenza iintlobo ezifanelekileyo ngaphezulu. 2.5 Codata Encodings of Natural Numbers Ukususela ekubeni kwe-lambda calculus, abacwaningi wamkelekile kwi-encodings ezisebenzayo zeendaba ezifana ne-booleans, iinombolo yemvelo kunye neenombolo. Iimeko ezizodwa zeencodings ezisebenzayo ziquka i-Church, Scott kunye ne-Parigot encodings zeendaba (cf. Geuvers [2014]; Koopman et al. [2014]). Izixhobo ziquka kwinkcukacha lethu kuphela isibonelo esizodwa se-codata-type, ngoko sincoma kwinkcukacha ezininzi ze-codata kunokuba yi-codings ezisebenzayo. I-Codata ye-codata ye-datypes ingatholakala ngokufanisa i-data type nge-observation efanelekileyo. I-Church encoding ingatholakala ngokufanisa i-program nge-Number ye-Peano kunye ne-iter function yaziwa kwakhona ngu-Ostermann kunye neJabs [2018]; thina ubhalise lo mzekelo kwi-Figure 6. Uyakwazi ukucinga ukuba i-codata ye-type kwi-Figure 6b ebonakalayo i-Church encoding yeenombolo yamanzi ayikho yi-recursive. Oku kuqhagamshelane ne-theorem ebonakalayo ukuba i-coding ye-Church ingasetyenziswa kwi-pure system F. Ukuba sisetyenzise indlela efanayo ukufumana i-Scott okanye i-Parigot encoding yeenombolo yamanzi, ke sinokufumana ukuba i-codata ye-type ebonakalayo. Oku kuqhagamshelane ne-theorem ebonakalayo ukuba le encoding ayikho ku-Pure System F kwaye kufuneka i-types ebonakalayo. Ngoko ke sinokufumana ngexesha elinye. I-Geuvers [2001] ibonise ukuba le encodings ezidlulileyo ayikwazi ukubonisa induction okanye ukunciphisa ngokufanelekileyo. Enye indlela yokufumana i-codings functional ebonakalayo eyenza i-induction yinto ukongeza uhlobo lwe-self-types kwi-system; le uhlobo le-coding yenzelwe ngu-Fu kunye ne-Stump [2014]. Nangona kunzima ukuba ibonise ukuxhomekeka ngokufanelekileyo, sincoma ukuba i-idea ebalulekileyo ye-coding ye-Fu kunye ne-Stump ingathunyelwa kwinkqubo yethu kwi-Figure 7 kunye ne-Figure 8. Kwi-Figure 7, sisebenzisa i-codata ye-helper ye-StepFun eyenza i-step ye-induction ye-predicate ye-P yeenombolo yamanzi. I-induction yaye ibonisa njengoko i-observation ind kwi-nombolo yamanzi eyenza i-case ye-base kunye ne-induction step ye-induction njenge-arguments. I-argument ye-n apho sincoma i-observation ibonakala kwakhona kwifomu ye-return. Ukuguqulwa kwinkqubo leyo kuholele kwi-Figure 8. Thina ndingathanda ukuba oku kubonisa ukuba ama-auto-parameters etholakalayo kwinkqubo ziyafumaneka ngokwemvelo xa sinxulumene kwi-non-dependent kwi-dependent setting. 
 
 
 Umbhali: 
 
 
 
 
 UDavid Binder Inkampani Ingo U-Tim Overkrüb U-Klaus Ostermann Authors: 
 
 
 
 
 UDavid Binder Inkampani Ingo U-Tim Overkrüb U-Klaus Ostermann 
 
 Oku kunokwenzeka kwi-archiv phantsi kolawulo lwe-CC 4.0. Oku kunokwenzeka kwi-archiv phantsi kolawulo lwe-CC 4.0. Mzantsi

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