Thierry Coquand

The calculus of constructions

ion

Constructions: A Higher Order Proof System for Mechanizing Mathematics

We present an extensive set of mathematical propositions and proofs in order to demonstrate the power of expression of the theory of constructions.

Infinite objects in type theory

We show that infinite objects can be constructively understood without the consideration of partial elements, or greatest fixed-points, through the explicit consideration of proof objects. We present then a proof system based on these explanations. According to this analysis, the proof expressions should have the same structure as the program expressions of a pure functional lazy language: variable, constructor, application, abstraction, case expressions, and local let expressions.

Inheritance as implicit coercion

Abstract We present a method for providing semantic interpretations for languages with a type system featuring inheritance polymorphism. Our approach is illustrated on an extension of the language Fun of Cardelli and Wegner, which we interpret via a translation into an extended polymorphic lambda calculus. Our goal is to interpret inheritances in Fun via coercion functions which are definable in the target of the translation. Existing techniques in the theory of semantic domains can be then used to interpret the extended polymorphic lambda calculus, thus providing many models for the original language. This technique makes it possible to model a rich type discipline which includes parametric polymorphism and recursive types as well as inheritance. A central difficulty in providing interpretations for explicit type disciplines featuring inheritance in the sense discussed in this paper arises from the fact that programs can type-check in more than one way. Since interpretations follow the type-checking derivations, coherence theorems are required: that is, one must prove that the meaning of a program does not depend on the way it was type-checked. Proofs of such theorems for our proposed interpretation are the basic technical results of this paper. Interestingly, proving coherence in the presence of recursive types, variants, and abstract types forced us to reexamine fundamental equational properties that arise in proof theory (in the form of commutative reductions) and domain theory (in the form of strict vs. non-strict functions).

Inductively generated formal topologies

Formal topology aims at developing general topology in intuitionistic and predicative mathematics. Many classical results of general topology have been already brought into the realm of constructive mathematics by using formal topology and also new light on basic topological notions was gained with this approach which allows distinction which are not expressible in classical topology. Here we give a systematic exposition of one of the main tools in formal topology: inductive generation. In fact, many formal topologies can be presented in a predicative way by an inductive generation and thus their properties can be proved inductively. We show however that some natural complete Heyting algebra cannot be inductively defined.

Intuitionistic model constructions and normalization proofs

The traditional notions of strong and weak normalization refer to properties of a binary reduction relation. In this paper we explore an alternative approach to normalization, in which we bypass the reduction relation and instead focus on the normalization function, that is, the function that maps a term to its normal form. We work in an intuitionistic metalanguage, and characterize a normalization function as an algorithm that picks a canonical representative from the equivalence class of convertible terms. This means that we also get a decision algorithm for convertibility. Such a normalization function can be constructed by building an appropriate model and a function quote, which inverts the interpretation function. The normalization function is then obtained by composing the quote function with the interpretation function. We also discuss how to get a simple proof of the property that constructors are one-to-one, which is usually obtained as a corollary of Church–Rosser and normalization in the traditional sense. We illustrate this approach by showing how a glueing model (closely related to the glueing construction used in category theory) gives rise to a normalization algorithm for a combinatory formulation of Godel System T. We then show how the method extends in a straightforward way when we add cartesian products and disjoint unions (full intuitionistic propositional logic under a Curry–Howard interpretation) and transfinite inductive types such as the Brouwer ordinals.

An algorithm for type-checking dependent types

We present a simple type-checker for a language with dependent types and let expressions, with a simple proof of correctness.

Cubical Type Theory: A Constructive Interpretation of the Univalence Axiom

This paper presents a type theory in which it is possible to directly manipulate n-dimensional cubes (points, lines, squares, cubes, etc.) based on an interpretation of dependent type theory in a cubical set model. This enables new ways to reason about identity types, for instance, function extensionality is directly provable in the system. Further, Voevodskys univalence axiom is provable in this system. We also explain an extension with some higher inductive types like the circle and propositional truncation. Finally we provide semantics for this cubical type theory in a constructive meta-theory.

A Logical Framework with Dependently Typed Records

Our long term goal is a system to formally represent complex structured mathematical objects, and proofs and computation on such objects; e.g. a foundational computer algebra system. Our approach is informed by the long development of module systems for functional programming based on dependent record types as signatures [22]. For our logical purposes, however, we want a dependently typed base language. In this paper we propose an extension of Martin-Lof’s logical framework [25, 21] with dependently typed records, and present the semantic foundation and the typechecking algorithm of our system. Some of the work is formally checked in Coq [8].1 We have also implemented and experimented with several related systems. Our proposal combines a semantic foundation, provably sound typechecking, good expressiveness (e.g. subtyping, sharing) and first-class higher-order modules. The development of functional programming modules has addressed many aspects of the problem, such as use of manifest or transparent fields to control the information available in a signature, signature strengthening, type abstraction, sharing and subtyping [19, 15, 20]. The problem of modularity is not, however, closed, with much current research into first-class higher-order modules, recursive modules and mixins. There has also been work on dependently typed records over dependent base languages. A first practical implementation is described in [3], however without semantic foundation. An original extension of Martin-Lof’s logical framework is given in [6], however it lacks manifest fields to express sharing, and lacks metamathematical analysis. A general approach to adding modules on top of a Pure Type System

Extensional models for polymorphism

We present a general method for constructing extensional models for the polymorphic lambda calculus—the polymorphic extensional collapse. The method yields models that satisfy additional, computationally motivated constraints like having only two polymorphic booleans and having only the numerals as polymorphic integers. Moreover the method yields models that prove that the polymorphic lambda calculus can be conservatively added to arbitrary algebraic data type specifications, even with complete transfer of the computational power to the added data types.