Avron, A., Honsell, F. A., Mason, I. A., and Pollack, R. (1992). Using typed lambda calculus to implement formal systems on a machine. Journal of Automated Reasoning, 9(3):309--354. A preliminary version appeared as University of Edinburgh Report ECS-LFCS-87-31.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....le teorie e le librerie) e desiderabile una alternativa piu uniforme e meno dispendiosa. E percio indispensabile, se tali applicazioni sui computer vogliono progredire, capire quali aspetti di questo compito sono comuni a tutte le logiche al fine di realizzarli una volta per tutte [AHM89] pagina 1] Questa ricerca di generalita e parzialmente gia in Isabelle (a differenza di LCF e NuPrl, che sono vincolati all uso di logiche specifiche) Isabelle non e vincolato ad una particolare logica oggetto, ma fornisce gli strumenti per definire una logica adeguata alle esigenze ....

.... Percio Isabelle adotta il tipo di rappresentazione di LCF al metalivello [ Pau89] pagina 391] Un passo sostanziale verso la definizione di un framework indipendente dalla logica usata ed in cui l utente, come primo passo, puo definire la propria logica si ha con LF 33 [RHP71, AHM89] un ambiente che permette di definire in modo uniforme diversi sistemi di logica attraverso una metateoria il cui linguaggio e definito a priori e non dipende dalla logica in cui si vuol effettuare deduzione automatica: 32 Si noti come questa osservazione sul fatto che la metalogica e ....

A. Avron, F. Honsell, and I. Mason. Using typed lambda calculus to implement formal systems on a machine. LFCS Report Series ECS-LFCS-89-72, Laboratory for the Foundations of Computer Science, Computer Science Department, University of Edinburgh, 1989.

....logics (z) Psi : L Gamma F called representation maps, that have particularly good properties such as conservativity 1 . A number of logics, particularly higher order logics based on typed lambda calculi, have been proposed as logical frameworks, including the Edinburgh logical framework LF [35, 2, 27], generic theorem provers such as Isabelle [56] Prolog [54, 25] and Elf [57] and the work of Basin and Constable [4] on metalogical frameworks. Other approaches, such as Feferman s logical framework FS 0 [24] that has been used in the work of Matthews, Smaill, and Basin [46] earlier work ....

A. Avron, F. Honsell, I. A. Mason, and R. Pollack. Using typed lambda calculus to implement formal systems on a machine. Journal of Automated Reasoning, 9(3):309--354, December 1992.

....(e.g. a term) X corresponds to X ; each x i corresponds to a place holder for the encoding of J i ; and M is a meta logic term corresponding to the encoding of . The LF logical framework consists of the calculus together with the judgements as types mechanism for representing logics [1, 9, 28]. One consequence of this method of encoding is that encoded systems inherit the structural properties of the meta logic. Now, the structural strength of LF is determined by the structural strength of the calculus which, as it stands in propositionsas types correspondence with the f ; ....

A Avron, F Honsell, IA Mason, and R Pollack. Using typed lambda calculus to implement formal systems on a machine. Journal of Automated Reasoning, 9:309-354, 1992. 45

.... complementary to those of # calculus, which has been the object of another large case study in the Theory of Context [17] Finally, some variant of the # calculus has been always taken as the traditional benchmark example of application of the many approaches to HOAS in the literature; see [2, 7, 11 14, 20] among the others. It turns out that Theory of Contexts is quite successful in handling the metatheory of # cbn . The encoding of the syntax, the semantics and the type system is straightforward, and still we delegate the # conversion to the metalevel. Only the encoding of substitution is not ....

A. Avron, F. Honsell, I. A. Mason, and R. Pollack. Using Typed Lambda Calculus to implement formal systems on a machine. Journal of Automated Reasoning, 9:309--354, 1992.

.... propositional logic are connected by the well known proposition as types paradigm [6] Stronger type theories, such as the Edinburgh Logical Framework, the Calculus of Inductive Constructions and Martin Lof s type theory, were especially designed, or can be fruitfully used, as a logical framework [13, 2, 5, 23]. In these frameworks, we can represent faithfully and uniformly all the relevant concepts of the inference process in a logical system: syntactic categories, terms, assertions, axiom schemata, rule schemata, tactics, etc. via the judgements as types, proofs as # terms paradigm [13] The key ....

....then represented by a type constructor of T. Moreover, substitution schemata for binding operators need not to be implemented by hand , because they are inherited from the metalanguage. This is the case, for instance, of # in First Order Logic; for further examples and discussion, we refer to [2, 8, 13]. However, in representing the proof system N # K , several di#cult issues arise. These issues escape the standard formalization paradigm, so we have to accommodate some special technique. In the following subsections, we will describe in detail these problems and the solutions we adopted. ....

A. Avron, F. Honsell, I. A. Mason, and R. Pollack. Using Typed Lambda Calculus to implement formal systems on a machine. Journal of Automated Reasoning, 9:309--354, 1992.

....#x.P as #(#x. P ) where # has a higher order functionality: # : Term # Prop) # Prop [5] It was further used by Martin Lof [19] and thoroughly expanded in the Edinburgh Logical Framework [11] This technique has been proved to work extremely well for pure functional languages (see [3] for a treatment of this in the context of # calculus and [9, 14] of more general functional languages) For example, the construct lambda x.M could be compiled to lam (#x.M ) where lam : Expr # Expr) # Expr . However, higher order syntax cannot be used directly in the case of languages with ....

....in imperative languages. For instance, the application of the # abstraction containing a command 2 lambda (#x. x : x 1]x) to 0 would be reduced to the evaluation of [0 : 0 1]0, which is meaningless. Furthermore, there is no direct representation of loops like while b do x : x 1. See [3] for more di#culties in handling Hoare s logic. Another problem is the absence of induction principles for encodings that employ higher order syntax. The di#culty of avoiding the capturing of local variables can be overcome by making explicit the textual substitution of variables in local ....

A. Avron, F. Honsell, I. A. Mason, and R. Pollack. Using Typed Lambda Calculus to implement formal systems on a machine. Journal of Automated Reasoning, 9:309--354, 1992.

.... Psi : L Gamma F called representation maps, that have particularly good properties such as conservativity 1 . A number of logics, particularly higher order logics originating in constructive type theory, have been proposed as logical frameworks, including the Edinburgh logical framework LF [30, 2, 23], meta theorem provers such as Isabelle [47] Prolog [46, 21] and Elf [48] and the work of Basin and Constable [3] on metalogical frameworks. Other approaches, such as Feferman s logical framework FS 0 [20] that has been used in the work of Matthews, Smaill, and Basin [38] earlier work by ....

A. Avron, F. Honsell, I. A. Mason, and R. Pollack. Using typed lambda calculus to implement formal systems on a machine. Journal of Automated Reasoning, 9(3):309--354, December 1992.

.... it allows higher order representation techniques so that variables, binding and locality conditions in the logic or language being represented (the object language) are encoded as variables and bindings in LF (the meta language) This allows for very concise encodings even of complicated logics [2]. The other reason is that given a proper encoding of the syntactic constructors and of the derivation rules as constant declarations in a signature , a necessary and sucient condition for a derivation to be valid is that the term that encodes the derivation has the LF type that encodes the ....

.... is that given a proper encoding of the syntactic constructors and of the derivation rules as constant declarations in a signature , a necessary and sucient condition for a derivation to be valid is that the term that encodes the derivation has the LF type that encodes the judgment being derived [2, 8]. As an example, the signature that describes the syntax and typing rules of System F is shown in Figure 6. There are three LF type constants being de ned: ty for encoding System F types, exp for encoding System F expressions, and of, a type family indexed on expressions and types such that of E ....

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A. Avron, F. A. Honsell, I. A. Mason, and R. Pollack. Using typed lambda calculus to implement formal systems on a machine. Journal of Automated Reasoning, 9(3):309-354, 1992. A preliminary version appeared as University of Edinburgh Report ECS-LFCS-87-31.

....of the calculus III: logic programming and its semantics (henceforth abbreviated to III) It is concerned with the basic model theory of the LF logical framework. Here we are concerned with logical frameworks in the original sense introduced by Avron, Harper, Honsell, Mason and Plotkin in [3, 4, 1, 11] and developed, from the point of view of encoding or representation, in [ This paper builds on the work of I, in which we present the model theory of and its internal logic. MORE The third paper in the sequence, Functorial Kripke Beth Joyal models of the calculus III: logic ....

....are called rules of derivation. 5 In fact, we can make a stronger claim: that presentations based on bare propositions are formally inadequate. However, the development of the supporting argument is beyond our present scope. 5 discussed in [ It also applies to the following systems from [1], and some minor variations thereon: Kleene s three valued logic; classical rst order logic with (a version of) Hilbert s choice operator; classical calculus; call by value calculus; and, with care, Hoare s logic. An abstract de nition of the class of systems we consider is provided in xx3.2 ....

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A. Avron, F. Honsell, I. Mason and R. Pollack. Using typed lambda calculus to implement formal systems on a machine. J. Automated Reasoning

....similar to that for models. However, since the base categorlatex y is concerned only with terms, rather than with propositions as well, we can begin with the satisfaction of propositions, rather than of consequences 15 Note that if X = then [ X) w; KJ is an object of KJ (w) [1]] w; KJ ) 16 Note that here X ranges over the whole syntax of L T sequents (cf. De nition 3.3) 43 De nition 8.6 (k satisfaction) Let hK J ; KJ i be a Kripke model of L T . The satisfaction (forcing) relation w ; k KJ T is de ned, by induction on the structure of ....

A. Avron, F. Honsell, I. Mason and R. Pollack. Using typed lambda calculus to implement formal systems on a machine. J. Automated Reasoning

....is true. The resemblance between the meta level axioms and the rules should be regarded as a happy coincidence. An axiom formalizes not the syntax of a rule but its semantic justification. The resemblance diminishes in first order logic (Section 5) The formalization of modal logic by Avron et al. [3] (in their meta logic) reflects Kripke semantics rather than the syntax of the rules. An obvious question is whether the object logic is faithfully represented. The definition below is oriented towards natural deduction: it concerns entailments rather than theorems. Definition 1 Let L be a logic ....

.... [24] In a notation resembling automath s he has produced succinct descriptions of first order logic and Constructive Type Theory [25] Most recently, people at the University of Edinburgh have elaborated Martin Lof s ideas into the Edinburgh Logical Framework [15] and formalized diverse logics [3]. automath and its descendants exploit the interpretation of propositions astypes. Each proposition A is interpreted as the type (or set) # A of proof objects, and A is true if it has a proof object, namely if there exists a : # A. Here is a summary of propositions as types [18, 27] ....

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A. Avron, F. A. Honsell, and I. A. Mason, Using typed lambda calculus to implement formal systems on a machine, Report ecs-lfcs-87-31, Computer Science Department, University of Edinburgh (1987).

....both in lf and tl, and we discuss and compare these encoding. Final remarks and directions for future work are in Section 4. Bruni et al. 1 The two Metalanguages and Encoding Protocols 1. 1 Logical Frameworks based on Type Theory Type Theories, such as the Edinburgh Logical Framework lf [15, 2] or the Calculus of (Co)Inductive Constructions [19] were especially designed, or can be fruitfully used, as a general logic speci cation language, i.e. as a Logical Framework (LF) In an LF, we can represent faithfully and uniformly all the relevant concepts of the inferential process in a ....

....of the calculus with lazy operational semantics in lf. The signature corresponding to calculus is in Figure 8. Our encoding is full HOAS, i.e. the set of variables does not have a corresponding type in lf, and the operation of substitution is delegated to the metalanguage like in [15,2]. In particular, in rules ( and ( the higher order is fully exploited, since term substitution is rendered via the application of the metalanguage. 16 Bruni et al. One can easily de ne an encoding function for types, and a family of encoding functions double indexed over types and ....

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Avron, A., F. Honsell, I. Mason and R. Pollack, Using Typed Lambda Calculus to implement formal systems on a machine, J. Aut. Reas. 9 (1992), pp. 309-354.

....both in lf and tl, and we discuss and compare these encoding. Final remarks and directions for future work are in Section 4. Bruni et al. 1 The two Metalanguages and Encoding Protocols 1. 1 Logical Frameworks based on Type Theory Type Theories, such as the Edinburgh Logical Framework lf [14, 2] or the Calculus of (Co)Inductive Constructions [18] were especially designed, or can be fruitfully used, as a general logic specification language, i.e. as a Logical Framework (LF) In an LF, we can represent faithfully and uniformly all the relevant concepts of the inferential process in a ....

....of the # calculus with lazy operational semantics in lf. The signature # # corresponding to # calculus is in Figure 8. Our encoding is full HOAS, i.e. the set of variables does not have a corresponding type in lf, and the operation of substitution is delegated to the metalanguage like in [14,2]. In particular, in rules (#) and ( the higher order is fully exploited, since term substitution is rendered via the application of the metalanguage. 16 Bruni et al. One can easily define an encoding function # for types, and a family of encoding functions # # double indexed over types and ....

[Article contains additional citation context not shown here]

Avron, A., F. Honsell, I. Mason and R. Pollack, Using Typed Lambda Calculus to implement formal systems on a machine, J. Aut. Reas. 9 (1992), pp. 309-354.

....that its notion of higher order rules and its method of treating the elimination rules were incorporated into the Edinburgh LF (A general logical framework for implementing logical formalisms on a computer, which was developed in the computer science department of the university of Edinburgh. See [4], 5] We shall show that the notions of S.H. that are the most difficult to handle (discharge functions and subrules) become redundant in the Gentzen type version. The complex normalization proof of S.H. in [2] can be replaced therefore by a standard cut elimination proof. Moreover, the unusual ....

Avron A., Honsell F. and Mason I., Using typed lambda calculus to implement formal systems on a machine, Technical Report, Laboratory for the Foundations of Computer Science, Edinburgh University, 1987, ECSLFCS -87-31. 4 Similar characterization of the definability of logical connectives is given in [3]. 10

....can easily be done in Feferman s FS 0 . Moreover: in this framework the availability of the simplest operation suffices for having all types of inductive definitions (and corresponding inductive principles) 1 Introduction One of the most serious drawbacks of the LF style Logical Frameworks ([7, 1, 8]) is their inability to prove metatheorems about encoded systems in a direct way (if at all) Meta level reasoning is, however, very important in mathematical discourse. Many theorems concerning a mathematical theory are not theorems of the theory, but theorems about the theory. Such theorems are ....

Avron A., Honsell F. A,, Mason I.A., and Pollack R., Using Typed Lambda Calculus to Implement Formal Systems on a Machine, Journal of Automated Deduction, vol. 9 (1992) pp. 309-354.

.... proof development environment for a particular system which can play the role of a logic specification language, i.e. of a Logical Framework [18, 12] Since the 80 s, higher order predicative, or impredicative, intuitionistic type theories have been successfully experimented as Logical Frameworks [12, 1, 6, 22]. In these theories one can represent (formalize) faithfully and uniformly all the relevant notions and aspects of the inference process in an arbitrary system: syntactic categories, terms, assertions, axiom schemata, rule schemata, tactics. The basic idea is the judgements as types, ....

....search much like he would do informally. At every step, the type checking algorithm ensures the soundness of the proof. When the proof term is completed, it can be saved (by the command Qed) for future applications. 2. 3 Logical Frameworks Type Theories, such as the Edinburgh Logical Framework [12, 1] or the Calculus of (Co)Inductive Constructions [2] were especially designed, or can be fruitfully used, as a general logic specification language, i.e. as a Logical Framework (LF) In an LF, we can represent faithfully and uniformly all the relevant concepts of 8 the inferential process in a ....

A. Avron, F. Honsell, I. A. Mason, and R. Pollack. Using Typed Lambda Calculus to implement formal systems on a machine. Journal of Automated Reasoning, 9:309--354, 1992.

....third version of S4) We introduce and study various encodings, in dependent typed # calculus, of Hilbert and ND style systems for both the consequence relations of validity and truth of K, KT, K4, KT4 (S4) KT45 (S5) KJ1. In particular, we extend and generalize the methodology developed in [3], by using judgements on proofs or exploiting the underlying # calculus structure of the metalanguage. For each encoding we state the appropriate faithfulness and adequacy theorem. The reason for considering first Hilbert style systems is that, in this more elementary setting vis a vis the ....

....here a detailed presentation of them, because we feel that their working is selfevident; see [24, Chapter 3] for further details. We need them in order to capture systems for validity for logics weaker than S4. Moreover, they allow to achieve a sharpening of the adequacy theorems appearing in [3]. In Section 5.2.5 we briefly outline how to introduce multiple CR systems for truth, extending those for validity. All the systems for truth appearing elsewhere in the paper are classical. 5 # I ## # # ## # ## ## E # # #(# # #) # # ## # # ## # E # # ## # # # # # I # # # ....

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A. Avron, F. Honsell, I. A. Mason, and R. Pollack. Using Typed Lambda Calculus to implement formal systems on a machine. J. Automated Reasoning, 9:309--354, 1992.

....useful, and probably necessary, in using logical systems for reasoning about programs. In fact, the amount of (often trivial and repetitive) routine details involved in using program logics renders error prone the activity of a human prover. Type Theories, such as the Edinburgh Logical Framework [9, 3] or the Calculus of Inductive Constructions [5, 27] were especially designed, or can be fruitfully used, as a general logic specification language, i.e. as a Logical Framework (LF) Thus they can streamline the process of generating interactive proof development environments tailored to the ....

....noninterference judgments a la Reynolds [24] as side conditions of the rules. These are judgments which generalize side conditions such as x ## FV(A) See [20] for further details. 2 Encoding ND style Systems for DL In this section we apply and generalize the methodology developed in [9, 3] and define an encoding of SND(DL) and of S a ND (DL) within the Calculus of Inductive Constructions, as it is implemented by the Coq V5.10 proof assistant [14] X, Te, B, C, P : Set b : B # B # b , # b : B # B # B : P # P #, # : P # P # P isId : X # Te 0, 1 : Te ....

[Article contains additional citation context not shown here]

A. Avron, F. Honsell, I. A. Mason, and R. Pollack. Using Typed Lambda Calculus to implement formal systems on a machine. Journal of Automated Reasoning, 9:309--354, 1992.

.... proof development environment for a particular system which can play the role of a logic specification language, i.e. of a Logical Framework [18, 12] Since the 80 s, higher order predicative, or impredicative, intuitionistic type theories have been successfully experimented as Logical Frameworks [12, 1, 6, 22]. In these theories one can represent (formalize) faithfully and uniformly all the relevant notions and aspects of the inference process in an arbitrary system: syntactic categories, terms, assertions, axiom schemata, rule schemata, tactics. The basic idea is the judgements as types, ....

....search much like he would do informally. At every step, the type checking algorithm ensures the soundness of the proof. When the proof term is completed, it can be saved (by the command Qed) for future applications. 2. 3 Logical Frameworks Type Theories, such as the Edinburgh Logical Framework [12, 1] or the Calculus of (Co)Inductive Constructions [2] were especially designed, or can be fruitfully used, as a general logic specification language, i.e. as a Logical Framework (LF) In an LF, we can represent faithfully and uniformly all the relevant concepts of 8 the inferential process in a ....

A. Avron, F. Honsell, I. A. Mason, and R. Pollack. Using Typed Lambda Calculus to implement formal systems on a machine. Journal of Automated Reasoning, 9:309--354, 1992.

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Avron, A., Honsell, F. A., Mason, I. A., and Pollack, R. (1992). Using typed lambda calculus to implement formal systems on a machine. Journal of Automated Reasoning, 9(3):309--354. A preliminary version appeared as University of Edinburgh Report ECS-LFCS-87-31.

No context found.

Arnon Avron, Furio Honsell, Ian A. Mason, and Robert Pollack. Using typed lambda calculus to implement formal systems on a machine. Journal of Automated Reasoning, 9:309--354, 1992.

No context found.

Arnon Avron, Furio Honsell, Ian A. Mason, and Robert Pollack. Using typed lambda calculus to implement formal systems on a machine. Journal of Automated Reasoning, 9(3):309--354, December 1992.

No context found.

Arnon Avron, Furio Honsell, Ian A. Mason, and Robert Pollack. Using typed lambda calculus to implement formal systems on a machine. Journal of Automated Reasoning, 9:309--354, 1992.

No context found.

A. Avron, F. Honsell, I. A. Mason, and R. Pollack. Using Typed Lambda Calculus to implement formal systems on a machine. Journal of Automated Reasoning, 9:309--354, 1992.

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A. Avron, F. Honsell, I. Mason and R. Pollack. Using typed lambda calculus to implement formal systems on a machine. J. Automated Reasoning

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