## A framework for defining logical frameworks (2006)

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Venue: | University of Udine |

Citations: | 4 - 1 self |

### BibTeX

@INPROCEEDINGS{Honsell06aframework,

author = {Furio Honsell},

title = {A framework for defining logical frameworks},

booktitle = {University of Udine},

year = {2006}

}

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### Abstract

Replace this file with prentcsmacro.sty for your meeting, or with entcsmacro.sty for your meeting. Both can be

### Citations

718 |
An Introduction to Modal Logic
- Hughes, Cresswell
- 1968
(Show Context)
Citation Context ...tations of modal and program logics are very problematic in standard LF. Many such systems feature rules which can be applied only to premises which depend solely on assumptions of a particular shape =-=[CH84]-=-, or whose derivation has been carried out using only certain sequences of rules. Finally, Linear or Relevance Logics appear to be encodable only using a very heavy machinery. In the past, extensions ... |

697 | A Framework for Defining Logics
- Harper, Honsell, et al.
- 1992
(Show Context)
Citation Context ...supporting the proposition-as-types paradigm, as a general metalanguage and framework for logical systems. A few researchers at LFCS joined in, and by midsummer 1986 the Framework for Defining Logics =-=[HHP93]-=- as it was presented to the LICS conference in 1987, was pretty much finalized. It was immediately clear that the higher order nature of the Dependent Typed Lambda Calculus, later to be known as LF, w... |

474 |
The Calculus of Constructions
- Coquand, Huet
- 1988
(Show Context)
Citation Context ...serving confluence and strong normalization over well typed patterns? • Our results should scale up to all the systems in [BCKL03], i.e. to systems corresponding to the full Calculus of Constructions =-=[CH88]-=-. • Is there an interesting Curry-Howard isomorphism for PLF and more generally for systems blending rewriting facilities and higher order calculi? • Instantiate GLF in order to give sharp encodings o... |

313 | System description: Twelf - a metalogical framework for deductive systems - Pfenning, Schürmann - 1999 |

215 | A linear logic framework - Cervesato, Pfenning - 1996 |

84 | An Extended Calculus of Constructions - Luo - 1990 |

82 | Using typed lambda calculus to implement formal systems on a machine
- Avron, Honsell, et al.
- 1987
(Show Context)
Citation Context ...ity of the new Frameworks. Further possible developments will be mentioned in Section 6. We assume the reader familiar with the pragmatics of Logical Frameworks. An elementary introduction appears in =-=[AHMP92]-=-. Most of the papers cited in the Introduction provide further interesting material. 5.1 Case Analysis in PLF Case analysis can be handled very easily and neatly in PLF by taking advantage of the patt... |

78 | 2003): Theorem proving modulo
- Dowek, Hardin, et al.
(Show Context)
Citation Context ...s subtle interplays between deduction and computation of definitional equality. In other words, we want to take advantage of the, so called, Poincaré principle, or the more recent Deduction Modulo of =-=[DHK03]-=-. In these cases, however, each new definitional equality has to be justified outside. GLF is a general framework along these lines. In this paper, we provide general results concerning classes of cal... |

73 | Confluence for Abstract and Higher-Order Rewriting - Oostrom - 1994 |

66 | Pure pattern calculus
- Jay, Kesner
- 2006
(Show Context)
Citation Context ...ies as in PLF, and more generally with GLF. • Among the various calculi with patterns, versions à la Curry of PLF should be explored and compared e.g. with the pattern calculus recently introduced in =-=[JK06]-=-. Acknowledgments. The last author would like to thank warmly Horatiu Cirstea and Claude Kirchner for many useful discussions, and for infecting him with the intellectual virus of blending term rewrit... |

56 | de Bruijn. A survey of the project Automath - G - 1980 |

48 | Proof Assistants using Dependent Type Systems - Barendregt, Geuvers - 2001 |

43 | Pure patterns type systems
- Barthe, Cirstea, et al.
- 2003
(Show Context)
Citation Context ...By so doing, we allow for a generalized form of pattern lambda calculi, and also go beyond. The papers which are most influential for our proposal and which we are most indebted with are [CKL01b] and =-=[BCKL03]-=-. The former is the paper which first puts to use the decomposition of the rule (O·Appl ′ ) in special cases. It presents a collection of type systems for a typed variant of the Rewriting Calculus 4 ,... |

43 | Abstract Data Type Systems - Jouannaud, Okada - 1997 |

37 |
Parallel Reductions in λ-calculus
- Takahashi
- 1995
(Show Context)
Citation Context ..., we can prove: Lemma 3.12 (Substitution) If T ↦ →β T ′ and θ ↦ →β θ ′ , then T θ ↦ →β T ′ θ ′ . The proof of confluence is a suitable application of the usual argument based on parallel reduction of =-=[Tak89]-=-. As pointed out above, confluence holds for raw terms, provided they satisfy the suitable restrictions on patterns introduced so far. Theorem 3.13 (Confluence) The relation ↦→β is confluent. ✷ 3.4 PL... |

32 | Executable higher-order algebraic specification languages - Jouannaud, Okada - 1991 |

31 | The Rho Cube
- Cirstea, Kirchner, et al.
- 2001
(Show Context)
Citation Context ...en the “λ” and “Π” operators is not new, see e.g. [dB80, KBN99], but what we do here is to capitalize on it, in the type system, as was done in the work by Cirstea, Kirchner, and Liquori The Rho Cube =-=[CKL01b]-=-. By so doing, we allow for a generalized form of pattern lambda calculi, and also go beyond. The papers which are most influential for our proposal and which we are most indebted with are [CKL01b] an... |

30 | Matching Power - Cirstea, Kirchner, et al. - 2001 |

27 | Lambda calculus with patterns
- Oostrom
- 1990
(Show Context)
Citation Context ...her substitutions are provided. E.g., for an algebraic constant f of type a → a, M ≡ (λ(f y):[y:a].y) x is stuck, but (λ(f x):[x:a].M) (f (f 3)) ↦ →β 3. As it is well known, since the seminal work of =-=[Oos90]-=-, in untyped calculi, variables in patterns can be bound only if they occur linearly (i.e. at most once) and not actively (i.e. not in functional position), otherwise confluence is lost. For this reas... |

24 | Finitary inductively presented logics - FEFERMAN - 1989 |

23 | Strong normalizability for the combined system of the typed lambda calculus and an arbitrary convergent term rewrite system - Okada |

22 | M.: A logical framework with dependently typed records - Coquand, Pollack, et al. - 2005 |

22 | Intuitionistic Type Theory, volume 1 - Martin-Löf |

20 | Adding algebraic rewriting to the untyped lambda calculus - Dougherty - 1992 |

20 |
Truth of a proposition, evidence of a judgement, validity of a proof, Synthese 73
- Martin-Löf
- 1987
(Show Context)
Citation Context ...temming from the Automath tradition [dB80, NGe94], Constructive Type Theory [Mar84,CH88] and it capitalized on the notion of judgment as discussed by MartinLöf in a series of papers in the mid ’80’s, =-=[ML85]-=-. The Logical Framework game, triggered by LF, became rapidly quite popular in the formal proof development community and many authors [Fef88, CH90] played it on their systems. Since then, Logical Fra... |

17 | π-calculus in (co)inductive type theories - Honsell, Miculan, et al. - 2001 |

14 | Encoding modal logics in logical frameworks
- Avron, Honsell, et al.
- 1998
(Show Context)
Citation Context ... Π-abstraction in rules of derivation. 28sHonsell, Lenisa, Liquori We shall not develop here the encodings of all the plethora of modal logics, in Hilbert and Natural Deduction style, which appear in =-=[AHMP98]-=-. By way of example, we shall only give the signature for classical S4 in Hilbert style, which features necessitation as a rule of proof, namely ∅ ⊢ φ ∅ ⊢ ✷φ (NEC) The predicate Closedx △ = “x is a te... |

13 | A higher-order specification of the π-calculus - Despeyroux |

13 | Pi-calculus in (co)inductive type theory - Honsell, Miculan, et al. |

13 |
Typage et déduction dans le calcul de réécriture. Thèse de doctorat, Université Henri Poincaré - Nancy I, Oct. 2005. 17 Houtmann, Wack A Con uence proofs Let us recall the rst theorem we wish to prove here: Theorem 2.9 (Con uence of left-linear ρd-calculu
- Wack
(Show Context)
Citation Context ...ituted during reductions, as the variable y in the following term (λy:a.λ(f x y):[x:a].y) z ↦→β λ(f x z):[x:a].z But this causes problems when combined with untypable fix-points, since, as noticed in =-=[Wac05]-=-, the non-linear term N in item (ii) above can be mimicked in this setting, even under the linearity pattern condition. Namely, let M △ = λx:a.λx:∅.g. Then M behaves as N of item (ii), since M N1 N2 ↦... |

13 | A Natural Deduction Approach to Dynamic Logics
- Honsell, Miculan
- 1995
(Show Context)
Citation Context ...roved to be particularly successful in dealing with metavariables, variable scoping and binding, Higher Order Abstract Syntax and, with a little effort, also with names [Des00, HMS01], program logics =-=[HM96]-=- and modal logics [AHMP98, AHMP92]. Enough for introductory personal views on history, and let’s start we the paper proper. Although LF, very rightly so, allows to encode rules as functions from proof... |

12 |
Interactive theorem proving with Cambridge LCF – a user’s manual
- Paulson
- 1985
(Show Context)
Citation Context ...roof assistant which could provide a large number of proof editing, proof checking, and proof searching facilities for an arbitrary logical system as were available, at the time, in tools such as LCF =-=[Pau85]-=- or NuPrl [Con86], only for specific formal systems. The challenge was that of not having to duplicate the implementation effort each time an interactive environment for a new logic was needed. The id... |

12 | Modularity of Strong Normalisation and Confluence in the Algebraic λ-Cube - Barbanera, Fernández, et al. - 1997 |

11 | Nuprl as a general logic - Constable, Howe - 1990 |

11 | On π-conversion in the λ-cube and the combination with abbreviations. APAL - Kamareddine, Bloo, et al. - 1999 |

5 |
Call by name, call by value and the λ calculus
- Plotkin
- 1975
(Show Context)
Citation Context ... have a type of a given form. This format can also recover many existing calculi in the literature such as LF, the Rewriting Calculus [CKL01a, CKL01b], and the Plotkin’s call-by-value lambda calculus =-=[Plo75]-=-. In all cases, an application of the “type equality” rule can be used to recover, conservatively, the effect of successful β-reductions: (λP:∆.M) N −→ M � P(N) provided P(N) holds and � P(N) is a sub... |

3 | A Correspondence between Martin-Löf Type Theory, the Ramified Theory of Types and Pure Type Systems - Kamareddine, Laan |

2 |
A Language for Verification and Manipulation of Web Documents
- Liquori, Honsell, et al.
- 2007
(Show Context)
Citation Context ...e, we will provide only one example encoding to illustrate how patterns can increase the usability of Dependent Type Theory as a metalanguage for encoding logical systems. Another encoding appears in =-=[LHR05]-=-. Plotkin’s call-by-value lambda calculus (λv-calculus) [Plo75] differs from the traditional lambda calculus in the formulation of theβv-reduction rule, namely (λx.M) N →βv M[N/x] provided that N is a... |

2 | Canonical Typing and Π-conversion in the λ-cube - Kamareddine, Nederpelt - 1996 |

1 | On π-conversionontheλ-cube and the Combination with Abbreviations - Kamareddine, Bloo, et al. - 1999 |