## Holes with Binding Power (2002)

Venue: | In Types for Proofs and Programs, Second International Workshop, TYPES 2002, Berg en Dal, The Netherlands, April 24-28, 2002, Selected Papers, H. Geuvers and F. Wiedijk, Eds. Lecture Notes in Computer Science (LNCS 2646 |

Citations: | 7 - 0 self |

### BibTeX

@INPROCEEDINGS{Jojgov02holeswith,

author = {G. I. Jojgov and Tu Eindhoven},

title = {Holes with Binding Power},

booktitle = {In Types for Proofs and Programs, Second International Workshop, TYPES 2002, Berg en Dal, The Netherlands, April 24-28, 2002, Selected Papers, H. Geuvers and F. Wiedijk, Eds. Lecture Notes in Computer Science (LNCS 2646},

year = {2002},

pages = {162--181},

publisher = {Springer}

}

### OpenURL

### Abstract

Incomplete logical proofs are the logical counterpart of the incomplete #-terms that one usually works with in an interactive theorem prover based on type theory. In this paper we extend the formalization of such incomplete proofs given in [5] by introducing unknowns that are allowed to provide temporary bindings for variables that are supposed to be bound, but whose binders are not constructed yet - a situation that typically occurs when one does forward reasoning.

### Citations

532 | Lambda Calculi with Types
- Barendregt
- 1992
(Show Context)
Citation Context ...omplete proofs from [5] because we will build further on them. The reader is referred to the paper for more details. The basic system that we start with is higher-order logic (HOL) modulo βconversion=-= [1, 6]. -=-In HOL we have domains built up from a set of basic domains B together with a special domain Prop (whose elements are called formulas or propositions) using the function space (→) operator. The term... |

428 | Isabelle: A Generic Theorem Prover - Paulson - 1994 |

124 | Unification under a mixed prefix
- Miller
- 1992
(Show Context)
Citation Context ...ed perhaps first in the setting of unification problems where while searching for unifiers, one incrementally constructs terms by replacing the holes with terms that may have holes themselves. Miller =-=[10] -=-proposes that the holes can be represented by variables of appropriate type. Since one needs to compute (β-reduce) terms with holes, the substitutions passing through the holes need to be recorded in... |

94 |
A survey of the project Automath
- Bruijn
- 1980
(Show Context)
Citation Context ... when the hole is filled. We build on this idea by separating the metalevel abstractions needed for the holes from the object-level function spaces. After the pioneering work of De Bruijn on AUTOMATH =-=[4]-=- there has been a lot of research on building systems based on type theory. Because of complexity and decidability issues most of the systems needed to be interactive and that naturally lead to the st... |

84 | Logics and Type systems
- Geuvers
- 1993
(Show Context)
Citation Context ...(6a) it occurs free while in (6b) it has been bound. This does not create the soundness problems of Example 1, but we clearly see another side of the problem. The unknown part in (6a) is replaced (5) =-=(6)sby the follow-=-ing tree in (6b): P (x) (∀y. P (y))→B ? B ⇒ P (x) ∀x. P (x) (∀y. P (y))→B B (a) (b) As we can see (7b) is not a well-formed deduction of B because the variable x that is bound by the ∀-i... |

71 | Dependently Typed Functional Programs and Their Proofs
- McBride
- 1999
(Show Context)
Citation Context ...[13] who noticed that it suffices to have explicit substitutions attached (7)sto meta-variables only. In contrast to the systems that introduce explicit substitutions to the calculus, OLEG of McBride =-=[9] ta-=-kes Miller’s functional approach and introduces it to type theory. In OLEG the scoping is managed through a special binder for meta-variables on the term level. This idea is extended in the λ[ ]-cu... |

43 | The Implementation of ALF - a Proof Editor based on Martin-Lof's Monomorphic Type Theory with Explicit Substitution", Chalmers University ofTechnology and University ofGoteborg
- Magnusson
- 1995
(Show Context)
Citation Context ...ased on type theory. Because of complexity and decidability issues most of the systems needed to be interactive and that naturally lead to the study of incomplete terms in type theory. The system ALF =-=[8]-=- of Magnusson et al was the first to have explicit representation of unknown terms. Reduction-generated substitutions were recorded by a form of explicit substitutions. This idea was further developed... |

20 | Pure type systems with definitions
- Severi, Poll
- 1994
(Show Context)
Citation Context ...riables bound by yet unconstructed binders by giving binding power to the meta-variables. The work on extensions of pure type systems with (parametric) definitions and constants by Severi, Bloo, etc. =-=[12, 2] -=-has inspired the notation that we use as well as the derivation rules. Our system can be seen as extension of those systems by hereditary parametrization. Parameters are the basis also of the λfree l... |

12 | Open proofs and open terms: A basis for interactive logic
- Geuvers, Jojgov
- 2002
(Show Context)
Citation Context ...l counterpart of the incomplete λ-terms that one usually works with in an interactive theorem prover based on type theory. In this paper we extend the formalization of such incomplete proofs given in=-= [5]-=- by introducing unknowns that are allowed to provide temporary bindings for variables that are supposed to be bound, but whose binders are not constructed yet - a situation that typically occurs when ... |

11 |
Construction and Deduction in Type Theories
- Strecker
- 1998
(Show Context)
Citation Context ...icit representation of unknown terms. Reduction-generated substitutions were recorded by a form of explicit substitutions. This idea was further developed in the system TypeLab by Luther and Strecker =-=[13]-=- who noticed that it suffices to have explicit substitutions attached (7)sto meta-variables only. In contrast to the systems that introduce explicit substitutions to the calculus, OLEG of McBride [9] ... |

7 |
Contexts in Lambda Calculus
- Bognar
- 2002
(Show Context)
Citation Context ...’s functional approach and introduces it to type theory. In OLEG the scoping is managed through a special binder for meta-variables on the term level. This idea is extended in the λ[ ]-cube of Bogn=-=ar [3]-=- where binders for meta-variables are introduced both at term and at the type levels. In all the systems however, meta-variables rely on object-level functions in order to act as binders. In some syst... |

6 |
A lambda-free logical framework
- PAL
(Show Context)
Citation Context ... system with hereditarily parameterized variables. This means that the parameters of a parameterized variable can be parameterized themselves. In that respect the system resembles the PAL + framework =-=[7].-=- It can also be seen as an extension of the λ[ ] systems in [3] with hereditarily parameterized variables. The typing system. Let M be a set of meta-variable names and V a set of variables. The set o... |

1 |
Fairouz Kamareddine, Twan Laan, and Rob Nederpelt. Parameters in Pure Type Systems
- Bloo
- 2002
(Show Context)
Citation Context ...riables bound by yet unconstructed binders by giving binding power to the meta-variables. The work on extensions of pure type systems with (parametric) definitions and constants by Severi, Bloo, etc. =-=[12, 2] -=-has inspired the notation that we use as well as the derivation rules. Our system can be seen as extension of those systems by hereditary parametrization. Parameters are the basis also of the λfree l... |