In this paper, we introduce a new type system, the Implicit Calculus of Constructions, which is a Curry-style variant of the Calculus of Constructions that we extend by adding an intersection type binder| called the implicit dependent product. Unlike the usual approach of Type Assignment Systems, the implicit product can be used at every place in the universe hierarchy. We study syntactical properties of this calculus such as the -subject reduction property, and we show that the implicit product induces a rich subtyping relation over the type system in a natural way. We also illustrate the specicities of this calculus by revisitting the impredicative encodings of the Calculus of Constructions, and we show that their translation into the implicit calculus helps to reect the computational meaning of the underlying terms in a more accurate way.

A typechecker for a typed -calculus having implicit arguments is presented. The typechecker works in such a way that the uniqueness of implicit arguments is always preserved during reduction. Consequently, when it compares two terms by reduction, it can reduce them without inferring implicit arguments. Before describing the typechecker, we analyze various situations where the uniqueness of implicit arguments is not preserved by naively defined reduction. ANY OTHER IDENTIFYING INFORMATION OF THIS REPORT The preliminary version of this paper appeared in Logic, Language and Computation, Lecture Notes in Computer Science, Vol.792, 1994, pp.10-30. DISTRIBUTION STATEMENT First issue 40 copies. This technical report is available via anonymous FTP from ftp.is.s.u-tokyo.ac.jp (directory /pub/tech-reports). SUPPLEMENTARY NOTES REPORT DATE January 1, 1995 TOTAL NO. OF PAGES 31 WRITTEN LANGUAGE English NO. OF REFERENCES 12 DEPARTMENT OF INFORMATION SCIENCE Faculty of Science, Universit...

The open calculus of constructions integrates key features of Martin-Löf's type theory, the calculus of constructions, Membership Equational Logic, and Rewriting Logic into a single uniform language. The two key ingredients are dependent function types and conditional rewriting modulo equational theories. We explore the open calculus of constructions as a uniform framework for programming, specification and interactive verification in an equational higher-order style. By having equational logic and rewriting logic as executable sublogics we preserve the advantages of a first-order semantic and logical framework and especially target applications involving symbolic computation and symbolic execution of nondeterministic and concurrent systems.

. This paper concerns techniques for providing a convenient syntax for object languages implemented via a type-theoretic Logical Framework, and reports on work in progress. We first motivate the need for a type-theoretic logical framework. Firstly, we take the logical framework seriously as a meta-language for implementing object languages (including object type theories). Another reason is the goal of building domain-specific reasoning tools which are implemented using type theory technology but do not require great expertise in type theory to use productively. We then present several examples of bi-directional translations between an encoding in the framework language and a more convenient syntax. The paper ends by discussing several techniques for implementing the translations and properties that we may require for the translation. Coercive subtyping is shown to help in the translation. 1

Proof assistants based on type theories, such as Coq and Lego, allow users to omit subterms on input that can be inferred automatically. While those mechanisms are well known, ad-hoc algorithms are used to suppress subterms on output. As a result, terms might be printed identically although they di#er in hidden parts. Such ambiguous representations may confuse users. Additionally, terms might be rejected by the type checker because the printer has erased too much type information. This paper addresses these problems by proposing e#ective erasure methods that guarantee successful term reconstruction, similar to the ones developed for the compression of proof-terms in Proof-Carrying Code environments. Experiences with the implementation in Typelab proved them both e#cient and practical. 1 Implicit Syntax Type theories are powerful formal systems that capture both the notion of computation and deduction. Particularly the expressive theories, such as the Calculus of Constructions (CC) [CH88] which is investigated in this paper, are used for the development of mathematical and algorithmic theories since proofs and specifications are representable in a very direct way using one uniform language. There is a price to pay for this expressiveness: abstractions have to be decorated with annotations, and type applications have to be written explicitly, because type abstraction and type application are just special cases of #-abstraction and application. For example, to form a list one has to provide the element type as an additional argument to instantiate the polymorphic constructors cons and nil as in (cons IN1(nil IN))). Also, one has to annotate the abstraction of n in #n:IN .n+1 with its type IN although this type is determined by the abstraction body. These excessive ...

Although type reconstruction for dependently typed languages is common in practical systems, it is still ill-understood. Detailed descriptions of the issues around it are hard to find and formal descriptions together with correctness proofs are non-existing. In this paper, we discuss a one-pass type reconstruction for objects in the logical framework LF, describe formally the type reconstruction process using the framework of contextual modal types, and prove correctness of type reconstruction. Since type reconstruction will find the most general types and may leave free variables, we in addition describe abstraction which will return a closed object where all free variables are bound at the outside. We also implemented our algorithms as part of the Beluga language, and the performance of our type reconstruction algorithm is comparable to type reconstruction in existing systems such as the logical framework Twelf. 1