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Outline

Title
Abstract
Introduction
Programming Methodology
Motivations and Overview of the Thesis
Related Work
Quasi Normalization: A Summary
Girard-Tait's Reducibility Method'
Discussions
Existential Types and Discussion
Conclusions and Further Research
References
All Topics
Computer Science

An extended calculus of constructions

Profile image of Zhaohui LuoZhaohui Luo

1990

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200 pages

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Abstract

This thesis presents and studies a unifying theory of dependent types ECC-Extended Calculus of Constructions. ECC integrates Coquand-Huet's (impredicative) calculus of constructions and Martin-Löf's (predicative) type theory with universes, and turns out to be a strong and expressive calculus for formalization of mathematics, structured proof development and program specification.

Related papers

Universes for generic programs and proofs in dependent type theory
Peter Dybjer

2003

We show how to write generic programs and proofs in Martin-Löf type theory. To this end we consider several extensions of Martin-Löf's logical framework for dependent types. Each extension has a universes of codes (signatures) for inductively defined sets with generic formation, introduction, elimination, and equality rules. These extensions are modeled on Dybjer and Setzer's finitely axiomatized theories of inductive-recursive definitions, which also have a universe of codes for sets, and generic formation, introduction, elimination, and equality rules. However, here we consider several smaller universes of interest for generic programming and universal algebra. We thus formalize one-sorted and many-sorted term algebras, as well as iterated, parameterized, generalized, and indexed inductive definitions. We also show how to extend the techniques of generic programming to these universes. Most of the definitions in the paper have been implemented using the proof assistant Alfa for dependent type theory.

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The Implicit Calculus of Constructions as a Programming Language with Dependent Types
Bruno Bernardo

2008

In this paper, we show how Miquel’s Implicit Calculus of Constructions (ICC) can be used as a programming language featuring dependent types. Since this system has an undecidable type-checking, we introduce a more verbose variant, called ICC* which fixes this issue. Datatypes and program specifications are enriched with logical assertions (such as preconditions, postconditions, invariants) and programs are decorated with proofs of those assertions. The point of using ICC* rather than the Calculus of Constructions (the core formalism of the Coq proof assistant) is that all of the static information (types and proof objects) is transparent, in the sense that it does not affect the computational behavior. This is concretized by a built-in extraction procedure that removes this static information. We also illustrate the main features of ICC* on classical examples of dependently typed programs.

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Imperative Object-Based Calculi in Co-inductive Type Theories
Luigi Liquori

Lecture Notes in Computer Science, 2003

We discuss the formalization of Abadi and Cardelli's impς, a paradigmatic object-based calculus with types and side effects, in Co-Inductive Type Theories, such as the Calculus of (Co)Inductive Constructions (CC (Co)Ind ). Instead of representing directly the original system "as it is", we reformulate its syntax and semantics bearing in mind the proof-theoretical features provided by the target metalanguage. On one hand, this methodology allows for a smoother implementation and treatment of the calculus in the metalanguage. On the other, it is possible to see the calculus from a new perspective, thus having the occasion to suggest original and cleaner presentations. We give hence a new presentation of impς, exploiting natural deduction semantics, (weak) higher-order abstract syntax, and, for a significant fragment of the calculus, coinductive typing systems. This presentation is easier to use and implement than the original one, and the proofs of key metaproperties, e.g. subject reduction, are much simpler. Although all proof developments have been carried out in the Coq system, the solutions we have devised in the encoding of and metareasoning on impς can be applied to other imperative calculi and proof environments with similar features.

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Constructor Subtyping in the Calculus of Inductive Constructions
Femke van Raamsdonk

Lecture Notes in Computer Science, 2000

The Calculus of Inductive Constructions (CIC) is a powerful type system, featuring dependent types and inductive definitions, that forms the basis of proof-assistant systems such as Coq and Lego. We extend CIC with constructor subtyping, a basic form of subtyping in which an inductive type σ is viewed as a subtype of another inductive type τ if τ has more elements than σ. It is shown that the calculus is wellbehaved and provides a suitable basis for formalizing natural semantics in proof-development systems.

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Working with Mathematical Structures in Type Theory
Claudio Sacerdoti Coen

Types for Proofs and Programs, 2007

We address the problem of representing mathematical struc- tures in a proof assistant which: 1) is based on a type theory with de- pendent types, telescopes and a computational version of Leibniz equal- ity; 2) implements coercive subtyping, accepting multiple coherent paths between type families; 3) implements a restricted form of higher order unication and type reconstruction. We show how

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An overview of type theories
Nino Guallart

Axiomathes, 2014

Pure type systems arise as a generalisation of simply typed lambda calculus. The contemporary development of mathematics has renewed the interest in type theories, as they are not just the object of mere historical research, but have an active role in the development of computational science and core mathematics. It is worth exploring some of them in depth, particularly predicative Martin-Löf's intuitionistic type theory and impredicative Coquand's calculus of constructions. The logical and philosophical differences and similarities between them will be studied, showing the relationship between these type theories and other fields of logic.

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A Type-Theoretic Framework for Formal Reasoning with Different Logical Foundations
Zhaohui Luo

Lecture Notes in Computer Science, 2007

A type-theoretic framework for formal reasoning with different logical foundations is introduced and studied. With logic-enriched type theories formulated in a logical framework, it allows various logical systems such as classical logic as well as intuitionistic logic to be used effectively alongside inductive data types and type universes. This provides an adequate basis for wider applications of type theory based theorem proving technology. Two notions of set are introduced in the framework and used in two case studies of classical reasoning: a predicative one in the formalisation of Weyl's predicative mathematics and an impredicative one in the verification of security protocols.

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The Calculus of Algebraic Constructions
Mitsuhiro Okada

Eprint Arxiv Cs 0610063, 2006

This paper is concerned with the foundations of the Calculus of Algebraic Constructions (CAC), an extension of the Calculus of Constructions by inductive data types. CAC generalizes inductive types equipped with higher-order primitive recursion, by providing definitions of functions by patternmatching which capture recursor definitions for arbitrary non-dependent and non-polymorphic inductive types satisfying a strictly positivity condition. CAC also generalizes the first-order framework of abstract data types by providing dependent types and higher-order rewrite rules.

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A simple model construction for the Calculus of Constructions
Milena Stefanova

Lecture Notes in Computer Science, 1996

We present a model construction for the Calculus of Constructions CC where all dependencies are c arried out in a set-theoretical setting. The Soundness Theorem is proved and as a consequence o f i t Strong Normalization for CC is obtained. Some other applications of our model constructions a r e: showing that CC + Classical logic is consistent by constructing a model for it and showing that the Axiom of Choice is not derivable in CC by constructing a m o del in which the type that represents the Axiom of Choice is empty.

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A Refinement of de Bruijn's Formal Language of Mathematics
Fairouz Fairouz

Journal of Logic, Language and Information, 2004

We provide a syntax and a derivation system for a formal language of mathematics called Weak Type Theory (WTT). We give the metatheory of WTT and a number of illustrative examples. WTT is a refinement of de Bruijn's Mathematical Vernacular (MV) and hence: WTT is faithful to the mathematician's language yet is formal and avoids ambiguities. WTT is close to the usual way in which mathematicians express themselves in writing. WTT has a syntax based on linguistic categories instead of set/type theoretic constructs. More so than MV however, WTT has a precise abstract syntax whose derivation rules resemble those of modern type theory enabling us to establish important desirable properties of W TT such as strong normalisation, decidability of type checking and subject reduction. The derivation system allows one to establish that a book written in WTT is well-formed following the syntax of WTT, and has great resemblance with ordinary mathematics books. WTT (like MV) is weak as regards correctness: the rules of W TT only concern linguistic correctness, its types are purely linguistic so that the formal translation into W TT is satisfactory as a readable, well-organized text. In WTT, logico-mathematical aspects of truth are disregarded. This separates concerns and means that WTT can be easily understood by either a mathematician, a logician or a computer scientist. acts as an intermediary between the language of mathematicians and that of logicians.

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ECC, an extended calculus of constructions
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[1989] Proceedings. Fourth Annual Symposium on Logic in Computer Science

We present a higher-order calculus ECC which can be seen as an extension of the calculus of constructions CH88] by adding strong sum types and a fully cumulative type hierarchy. ECC turns out to be rather expressive so that mathematical theories can be abstractly described and abstract mathematics may be adequately formalized. It is shown that ECC is strongly normalizing and has other nice prooftheoretic properties. An !?Set (realizability) model is described to show how the essential properties of the calculus can be captured set-theoretically.

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Information and Computation, 1992

Luo's Extended Calculus of donstructions (ECC) is a higher order functional calculus based on Coquand's and Huet's Calculus of Constructions, but has in addition strong sums and a predicative cumulative type hierarchy. In this paper I introduce inductive types on the predicative type levels of ECC. I also show how the o-Set model for ECC can be extended to a model for this augmented calculus.

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A higher-order calculus and theory abstraction
Zhaohui Luo

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We present a higher-order calculus ECC which naturally combines Coquand-Huet's calculus of constructions and Martin-L of's type theory with universes. ECC is very expressive, both for structured abstract reasoning and for program speci cation and construction. In particular, the strong sum types together with the type universes provide a useful module mechanism for abstract description of mathematical theories and adequate formalization of abstract mathematics. This allows comprehensive structuring of interactive development of speci cations, programs and proofs. After a summary of the meta-theoretic properties of the calculus, an !?Set (realizability) model of ECC is described to show how its essential properties can be captured set-theoretically. The model construction entails the logical consistency of the calculus and gives some hints on how to adequately formalize abstract mathematics. Theory abstraction in ECC is discussed as a pragmatic application.

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The calculus of constructions is a higher-order formalism for constructive proofs in natural deduction style. Every proof is a A-expression, typed with propositions of the underlying logic. By removing types we get a pure l-expression, expressing its associated algorithm. Computing this A-expression corresponds roughly to cut-elimination.

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