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## Using Typed Lambda Calculus to Implement Formal Systems on a Machine (1992)

Venue: | Journal of Automated Reasoning |

Citations: | 93 - 16 self |

### Citations

995 | A Formulation of the Simple Theory of Types - Church - 1940 |

794 | A framework for defining logics
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Citation Context ...s to those of this paper and the LF. In particular the idea of having an operator T : Prop ! Type appears already in De Bruijn's earlier work, as does the idea of having several judgements. The paper =-=[24]-=- describes the basic features of the LF. In this paper we are going to provide a broader illustration of its applicability and discuss to what extent it is successful. The analysis (of the formal pres... |

560 |
The calculus of constructions.
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Citation Context ...context j \Gamma[x:A] declaration j \Gamma[x=M] global definition Table 1: Basic Syntax of LEGO EFS [23], Isabelle [37, 38], and especially from INRIA implementations of the Calculus of Constructions =-=[14, 15, 13]-=-. LEGO is coded in CAML, a version of ML from INRIA [53]. 7.1 LF in LEGO LEGO implements a variant of LF, LEGO-LF, with some pragmatic features not included in pure LF. LEGO-LF has no separate signatu... |

470 | Isabelle: A Generic Theorem Prover
- Paulson
- 1994
(Show Context)
Citation Context ...pplication j [x=A]B local definition; i.e. `let' j M:A type cast \Gamma ::= hi empty context j \Gamma[x:A] declaration j \Gamma[x=M] global definition Table 1: Basic Syntax of LEGO EFS [23], Isabelle =-=[37, 38]-=-, and especially from INRIA implementations of the Calculus of Constructions [14, 15, 13]. LEGO is coded in CAML, a version of ML from INRIA [53]. 7.1 LF in LEGO LEGO implements a variant of LF, LEGO-... |

229 |
Call-by-name, call-by-value and the -calculus
- Plotkin
- 1975
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Citation Context ...M; x)) (x : o:App(N; x))(z))(y(N))) 5.2 Call-By-Value Lambda Calculus 5.2.1 The System The call-by-value -calculus, usually referred to as thesv -calculus, was first introduced and studied in Plotkin =-=[43]-=-. In the usual formulation its syntax is identical to that of the traditional -calculus. The crucial difference manifests itself in the proof system. In particular in the formulation of the fi-reducti... |

218 |
The Lambda Calculus Its Syntax and Semantics, Revised
- BARENDREGT
- 1984
(Show Context)
Citation Context ...g the logic. 5.1 The Classical Lambda Calculus 5.1.1 The System We begin by discussing the case of the classicalscalculus, we assume the reader familiar with the standard definitions and notations in =-=[7]-=-. To this end we define a basic LF type, o, encoding the set of -terms, which we will denote by , together with a judgement, M = N , intended to encode the assertion that the term M is ff \Gamma fi eq... |

207 | An intuitionistic theory of types: predicative part, - Martin-Lof - 1973 |

143 |
On the meaning of the logical constants and the justification of the logical laws.
- Martin-Lof
- 1996
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Citation Context ...xpression constructors with arguments of functional type. The LF paradigm for specifying and handling rules and proofs is centered on the notion of judgement. This notion was introduced by Martin-Lof =-=[31]-=- and corresponds to the notion of assertion of a formal system. However the LF does not commit itself to the intuitionistic viewpoint and extends the meaning of this notion. That part of the signature... |

137 |
Syntactic control of interference, in:
- Reynolds
- 1978
(Show Context)
Citation Context ... solution to this problem is to incorporate syntactic notions explicitly into the theory. We do this by adding three new judgements, ] l ; ] i and ] o ; concerning non-interference along the lines of =-=[48]-=-, ] x is of type l ! (x ! TYPE): The intuitive meaning of the judgements can be explained, using infix notation, as follows: the judgement x] l y is interpreted as meaning that x and y denote distinct... |

124 | Programming in Martin-Löf’s Type Theory.
- Nordström, Petersson, et al.
- 1990
(Show Context)
Citation Context ...n devoted in building computer systems for checking proofs (Automath project [9, 26] and the Theory of Constructions [12]) or for developing interactively correct proofs (Edinburgh and Gothenburg LCF =-=[22, 35, 39]-=-) and Nuprl [11] in specific logical systems. However, implementing a proof environment for a specific logical system is both complex and time-consuming, this---together with the proliferation of logi... |

106 | Simple consequence relations
- AVRON
- 1991
(Show Context)
Citation Context ...lways possible to infer OEs/ from the two conjuncts in these logics. The consequence relations which are directly encoded by the LF's judgements are ordinary single-conclusioned consequence relations =-=[1]-=-. A proof of a sequent OE 1 ; . . . ; OE n ` / is encoded by a term of type J(OE 1 ) ! J(OE 2 ) ! \Delta \Delta \Delta J(OE n ) ! J(/), where J is a judgement that is induced by ` : Note, however, tha... |

106 |
Edinburgh LCF: A Mechanized Logic of Computation
- Gordon, Milner, et al.
- 1979
(Show Context)
Citation Context ...n devoted in building computer systems for checking proofs (Automath project [9, 26] and the Theory of Constructions [12]) or for developing interactively correct proofs (Edinburgh and Gothenburg LCF =-=[22, 35, 39]-=-) and Nuprl [11] in specific logical systems. However, implementing a proof environment for a specific logical system is both complex and time-consuming, this---together with the proliferation of logi... |

92 | Natural Deduction: a Proof-Theoretical Study - Prawitz - 1965 |

89 | Partial polymorphic type inference and higher-order unification.
- Pfenning
- 1988
(Show Context)
Citation Context ... initialized to the empty context: # Init LF ; LF: Initial State! The syntactic categories of the call-by-value lambda calculus are declared by: 5 This feature is formally explained in [44]. See also =-=[40]-=- # [ o,v:Type ] ; declare o v This means `extend the current context with bindings o:Type and v:Type if it is valid to do so in LF'. (Notice that [o,v:Type] is an abbreviation for [o:Type][v:Type].) I... |

80 | Elf: a language for logic definition and verified metaprogramming - Pfenning |

78 |
Specifying theorem provers in a higherorder logic programming language
- Felty, Miller
- 1988
(Show Context)
Citation Context ...election of papers presented in the first Workshop are collected in the book [25]. Among the recent work most closely related to the topics presented here we mention Frank Pfenning [42] and Amy Felty =-=[17, 18, 19]-=-. This paper is organized as follows. In section 2 we introduce the LF specification language (or type theory). In section 3 we discuss the LF paradigm for specifying a logical system. The subsequent ... |

64 |
Modal logics for communicating systems.
- Stirling
- 1987
(Show Context)
Citation Context ...s use. They may be regarded as the simplest representatives of a larger class of logics that are of interest in computer science. This class includes temporal logics [27] and generalized modal logics =-=[51]-=-. Encoding these logics poses serious difficulties. These are overcome, essentially, by taking full advantage of the ability within the LF to employ, simultaneously, several different consequence rela... |

61 |
A logic covering undefinedness in program proofs
- Barringer, Cheng, et al.
- 1984
(Show Context)
Citation Context ... of a logic in the LF always depends on the existence of a reasonable natural deduction representation for its consequence relation. In the case of Kleene's logic such a formal system is described in =-=[8]-=-. Its internalization in the LF is an easy task. The resulting signature is: 3.1.2 The Signature \Sigma Kl ffl Syntactic Categories o : Type ffl Operations : : o ! os: o ! o ! os: o ! o ! o ffl Judgem... |

52 | Specifying and Implementing Theorem Provers in a Higher-Order Logic Programming Language
- Felty
- 1989
(Show Context)
Citation Context ...election of papers presented in the first Workshop are collected in the book [25]. Among the recent work most closely related to the topics presented here we mention Frank Pfenning [42] and Amy Felty =-=[17, 18, 19]-=-. This paper is organized as follows. In section 2 we introduce the LF specification language (or type theory). In section 3 we discuss the LF paradigm for specifying a logical system. The subsequent ... |

41 |
the Extended Calculus of Constructions
- ECC
- 1989
(Show Context)
Citation Context ... 52] is a system for mechanically checked formal mathematics. It supports a number of type theories, from a variant of LF to an extension of the Generalized Calculus of Constructions with strong sums =-=[28]-=-. These theories are all extensions of LF (both language and derivable judgements), and have some formal properties in common, including strong normalization of all well-typed terms, a decidable conve... |

23 |
A generalized framework and
- unknown authors
- 1982
(Show Context)
Citation Context ...nder the auspices of the ESPRIT Basic Research Action 3245 "Logical Frameworks: Design, Implementation and Experiment". A selection of papers presented in the first Workshop are collected in=-= the book [25]-=-. Among the recent work most closely related to the topics presented here we mention Frank Pfenning [42] and Amy Felty [17, 18, 19]. This paper is organized as follows. In section 2 we introduce the L... |

23 |
Mathematical Logic
- Schoenfield
- 1967
(Show Context)
Citation Context ... logic we 2 The splitting of the of the : rules into introduction and elimination schemas, for its combination with other connectives, is a known idea also in the context of classical logic, see e.g. =-=[50]-=-, p. 66. present here illustrates, with three examples, how one handles binding operators in the LF. Apart from the quantifiers 9 and 8 we include ffl; a version of Hilbert's choice operator. If x is ... |

22 |
Type' is not a type: preliminary report
- Meyer, Reinhold
- 1986
(Show Context)
Citation Context ...tion language is a weak constructive type theory, more specifically a \Pi-typed -calculus, closely related to AUT-PI and AUT-QE [9], to Martin Lof's early type theories and to Meyer and Reinhold'ss�=-=� [34]-=-. The expressive power of this language suffices to specify the language of a logic, its axioms, rules and its proofs. A typed -calculus can be used as a specification language for formal systems beca... |

22 |
The Theory of LEGO
- Pollack
- 1994
(Show Context)
Citation Context ...a proof is unique. In other words the search space for these subsystems is linear, and so extremely suitable for automation, perhaps behind the naive users back. 7 A Machine Implementation of LF LEGO =-=[29, 44, 52]-=- is a system for mechanically checked formal mathematics. It supports a number of type theories, from a variant of LF to an extension of the Generalized Calculus of Constructions with strong sums [28]... |

21 | What is an inference rule?
- Fagin, Halpern, et al.
- 1992
(Show Context)
Citation Context ...ning a deduction in the given system. In particular the way one of its two rules can be used in such deductions. In the present case (of modal logic) both relations have been investigated previously, =-=[1, 16]-=-. They have a clear semantical interpretation (see below). In the LF, however, the less standard one arises in a natural way, and the LF has the power to handle them simultaneously. Moreover, even if ... |

21 |
Temporal Logic of Programs. Volume 8
- Kroger
- 1987
(Show Context)
Citation Context ...ain axioms and rules controlling ts use. They may be regarded as the simplest representatives of a larger class of logics that are of interest in computer science. This class includes temporal logics =-=[27]-=- and generalized modal logics [51]. Encoding these logics poses serious difficulties. These are overcome, essentially, by taking full advantage of the ability within the LF to employ, simultaneously, ... |

21 | Hoare’s logic in the LF
- Mason
- 1987
(Show Context)
Citation Context ... (note that it cannot occur bound). The corrected version of the assignment axiom may be written as follows. Ass : \Pi x:l;t:i \Phi:i!o x] o 8\Phi ! (` h f\Phi(t)gx := tf\Phi(x!)g) This solution, see =-=[32]-=-, takes the notion of a free variable as primitive, another solution is to encode substituting a term for all free occurrences of a banged location in terms and formulas. This would involve introducin... |

17 |
A Programming System for Type Theory
- Petersson
- 1982
(Show Context)
Citation Context ...n devoted in building computer systems for checking proofs (Automath project [9, 26] and the Theory of Constructions [12]) or for developing interactively correct proofs (Edinburgh and Gothenburg LCF =-=[22, 35, 39]-=-) and Nuprl [11] in specific logical systems. However, implementing a proof environment for a specific logical system is both complex and time-consuming, this---together with the proliferation of logi... |

15 |
Une th'eorie des constructions. Th`ese de troisi`eme cycle, Universit'e Paris VII
- Coquand
(Show Context)
Citation Context ...omputers as an aid for correctly manipulating formal systems. Much research has been devoted in building computer systems for checking proofs (Automath project [9, 26] and the Theory of Constructions =-=[12]-=-) or for developing interactively correct proofs (Edinburgh and Gothenburg LCF [22, 35, 39]) and Nuprl [11] in specific logical systems. However, implementing a proof environment for a specific logica... |

14 |
Encoding a dependent-type -calculus in a logic programming Polymorphic Lemmas and De in Prolog and Twelf 39 language
- Felty, Miller
- 1990
(Show Context)
Citation Context ...election of papers presented in the first Workshop are collected in the book [25]. Among the recent work most closely related to the topics presented here we mention Frank Pfenning [42] and Amy Felty =-=[17, 18, 19]-=-. This paper is organized as follows. In section 2 we introduce the LF specification language (or type theory). In section 3 we discuss the LF paradigm for specifying a logical system. The subsequent ... |

13 |
How to use lego: A preliminary user's manual
- Luo, Pollack, et al.
- 1989
(Show Context)
Citation Context ...a proof is unique. In other words the search space for these subsystems is linear, and so extremely suitable for automation, perhaps behind the naive users back. 7 A Machine Implementation of LF LEGO =-=[29, 44, 52]-=- is a system for mechanically checked formal mathematics. It supports a number of type theories, from a variant of LF to an extension of the Generalized Calculus of Constructions with strong sums [28]... |

12 |
The semantics and proof theory of linear logic’, Theoret
- Avron
- 1988
(Show Context)
Citation Context ...xtend \Phi to a compositional surjection over proof terms. The above signature can be taken as a basis for the LF encoding of the external consequence relation of the minimal fragment of linear logic =-=[2]-=- [21]. 6 Program Logics Program logics such as Hoare's logic and dynamic logic exhibit an unusual overloading of variables. In both these logics variables play two roles, behaving in some instances as... |

12 | An Overview of the Edinburgh Logical Framework
- Avron, Honsell, et al.
- 1989
(Show Context)
Citation Context ...t at the LFCS in Edinburgh. An early version of this paper (written by the first three authors) circulated in 1987 as an LFCS Technical Report [6]. A gentle abridged version of that report appears in =-=[5]-=-. At that time the only implementation of the LF was a version written by Timothy Griffin in the Cornell Synthesizer Generator [23]. Since then Robert Pollack implemented the LF in his LEGO system, de... |

11 |
Constructions: a higher-order proof system for mechanizing mathematics
- Coquand, Huet
- 1985
(Show Context)
Citation Context ...context j \Gamma[x:A] declaration j \Gamma[x=M] global definition Table 1: Basic Syntax of LEGO EFS [23], Isabelle [37, 38], and especially from INRIA implementations of the Calculus of Constructions =-=[14, 15, 13]-=-. LEGO is coded in CAML, a version of ML from INRIA [53]. 7.1 LF in LEGO LEGO implements a variant of LF, LEGO-LF, with some pragmatic features not included in pure LF. LEGO-LF has no separate signatu... |

11 |
Natural deduction proof as higher-order resolution
- Paulson
- 1986
(Show Context)
Citation Context ...pplication j [x=A]B local definition; i.e. `let' j M:A type cast \Gamma ::= hi empty context j \Gamma[x:A] declaration j \Gamma[x=M] global definition Table 1: Basic Syntax of LEGO EFS [23], Isabelle =-=[37, 38]-=-, and especially from INRIA implementations of the Calculus of Constructions [14, 15, 13]. LEGO is coded in CAML, a version of ML from INRIA [53]. 7.1 LF in LEGO LEGO implements a variant of LF, LEGO-... |

10 |
An Environment for Formal Systems
- Griffin
- 1987
(Show Context)
Citation Context ...nical Report [6]. A gentle abridged version of that report appears in [5]. At that time the only implementation of the LF was a version written by Timothy Griffin in the Cornell Synthesizer Generator =-=[23]-=-. Since then Robert Pollack implemented the LF in his LEGO system, described by him in the seventh section of this paper. Since this paper was submitted (in 1987) for publication a great deal of resea... |

10 |
Checking Landau’s grundlagen in the AUTOMATH system
- JUTTING
- 1977
(Show Context)
Citation Context ...e has been a growing interest in using computers as an aid for correctly manipulating formal systems. Much research has been devoted in building computer systems for checking proofs (Automath project =-=[9, 26]-=- and the Theory of Constructions [12]) or for developing interactively correct proofs (Edinburgh and Gothenburg LCF [22, 35, 39]) and Nuprl [11] in specific logical systems. However, implementing a pr... |

7 |
de Bruijn. A survey of the project AUTOMATH
- Nicolas
- 1980
(Show Context)
Citation Context ...e has been a growing interest in using computers as an aid for correctly manipulating formal systems. Much research has been devoted in building computer systems for checking proofs (Automath project =-=[9, 26]-=- and the Theory of Constructions [12]) or for developing interactively correct proofs (Edinburgh and Gothenburg LCF [22, 35, 39]) and Nuprl [11] in specific logical systems. However, implementing a pr... |

4 |
A proof of the Church-Rosser property for the Edinburgh LF with j-conversion
- Salvesen
- 1990
(Show Context)
Citation Context ...sponding notion of well typed beta-eta contraction. This, in turn, is defined in the obvious way between objects of the same type in a given context and types of the same kind in a given context. See =-=[49]-=- or [24] for more details. N ! fij P for some term P . The following theorem from [24] summarizes the basic theoretical facts about LF (here ff ranges over the basic assertions of the type theory): Th... |

3 |
The Calculus of Constructions: Documentation and user's guide
- Coquand, Dowek, et al.
- 1989
(Show Context)
Citation Context ...context j \Gamma[x:A] declaration j \Gamma[x=M] global definition Table 1: Basic Syntax of LEGO EFS [23], Isabelle [37, 38], and especially from INRIA implementations of the Calculus of Constructions =-=[14, 15, 13]-=-. LEGO is coded in CAML, a version of ML from INRIA [53]. 7.1 LF in LEGO LEGO implements a variant of LF, LEGO-LF, with some pragmatic features not included in pure LF. LEGO-LF has no separate signatu... |

2 |
Furio Honsell and Simonetta Ronchi della Rocca. Models for Theories of Functions Strictly Depending on all their Arguments
- Dezani
- 1986
(Show Context)
Citation Context ...have the variables of the LF stand also for schematic variables ranging oversv -terms. The solution to this problem, that is presented here, was inspired by the denotational semantics of the calculus =-=[20]-=-. Let D be a domain such that D ? j ! i / [D !? D]; where i ffi j 62 [D !? D] while j ffi i = Id [D!?D] ; The set D !? D consists of all strict continuous functions from D to D. Such a D can easily be... |

2 |
Playing with LEGO: Some Examples of Developing Mathematics in the Calculus of Constructions
- Taylor
- 1989
(Show Context)
Citation Context ...a proof is unique. In other words the search space for these subsystems is linear, and so extremely suitable for automation, perhaps behind the naive users back. 7 A Machine Implementation of LF LEGO =-=[29, 44, 52]-=- is a system for mechanically checked formal mathematics. It supports a number of type theories, from a variant of LF to an extension of the Generalized Calculus of Constructions with strong sums [28]... |

1 |
Solutions to -Term Equations Under a Mixed Prefix
- Miller
- 1989
(Show Context)
Citation Context ...bound by implicit existential quantifiers, but in LEGO, as in Isabelle [38], there can be quantifier alternation, and the unification algorithm must account for this if resolution is to be sound. See =-=[33, 38, 44, 47]-=- for more discussion of this point. Coming back to the example, we use the built-in resolution tactic, Refine, to resolve goal ?1 by the transitivity rule E2. # Refine E2 ?3 : o ?5 : eq (shriek M) ?3 ... |

1 | On Natural Deduction Style Semantics, Environments and Stores - Ore - 1989 |

1 |
Effective Search for the Logical Framework
- Pym, Wallen
- 1989
(Show Context)
Citation Context ...bound by implicit existential quantifiers, but in LEGO, as in Isabelle [38], there can be quantifier alternation, and the unification algorithm must account for this if resolution is to be sound. See =-=[33, 38, 44, 47]-=- for more discussion of this point. Coming back to the example, we use the built-in resolution tactic, Refine, to resolve goal ?1 by the transitivity rule E2. # Refine E2 ?3 : o ?5 : eq (shriek M) ?3 ... |

1 |
The CAML Reference Manual, Version 2.6. Projet Formel, INRIA-ENS
- Weis
- 1989
(Show Context)
Citation Context ...n Table 1: Basic Syntax of LEGO EFS [23], Isabelle [37, 38], and especially from INRIA implementations of the Calculus of Constructions [14, 15, 13]. LEGO is coded in CAML, a version of ML from INRIA =-=[53]-=-. 7.1 LF in LEGO LEGO implements a variant of LF, LEGO-LF, with some pragmatic features not included in pure LF. LEGO-LF has no separate signature, but instead merges the notions of signature and cont... |