CoLoR: a Coq library on well-founded rewrite relations and its application to the automated verification of termination certificates

Termination is an important property of programs; notably required for programs formulated in proof assistants. It is a very active subject of research in the Turing-complete formalism of term rewriting systems, where many methods and tools have been developed over the years to address this problem. Ensuring reliability of those tools is therefore an important issue. In this paper we present a library formalizing important results of the theory of well-founded (rewrite) relations in the proof assistant Coq. We also present its application to the automated verification of termination certificates, as produced by termination tools

Extending the Calculus of Constructions with Tarski's fix-point theorem

We propose to use Tarski's least fixpoint theorem as a basis to define recursive functions in the calculus of inductive constructions. This widens the class of functions that can be modeled in type-theory based theorem proving tool to potentially non-terminating functions. This is only possible if we extend the logical framework by adding the axioms that correspond to classical logic. We claim that the extended framework makes it possible to reason about terminating and non-terminating computations and we show that common facilities of the calculus of inductive construction, like program extraction can be extended to also handle the new functions

Termination of rewrite relations on λ\lambda-terms based on Girard's notion of reducibility

In this paper, we show how to extend the notion of reducibility introduced by Girard for proving the termination of $\beta$-reduction in the polymorphic $\lambda$-calculus, to prove the termination of various kinds of rewrite relations on $\lambda$-terms, including rewriting modulo some equational theory and rewriting with matching modulo $\beta$$\eta$, by using the notion of computability closure. This provides a powerful termination criterion for various higher-order rewriting frameworks, including Klop's Combinatory Reductions Systems with simple types and Nipkow's Higher-order Rewrite Systems

Higher-Order Termination: from Kruskal to Computability

Termination is a major question in both logic and computer science. In logic, termination is at the heart of proof theory where it is usually called strong normalization (of cut elimination). In computer science, termination has always been an important issue for showing programs correct. In the early days of logic, strong normalization was usually shown by assigning ordinals to expressions in such a way that eliminating a cut would yield an expression with a smaller ordinal. In the early days of verification, computer scientists used similar ideas, interpreting the arguments of a program call by a natural number, such as their size. Showing the size of the arguments to decrease for each recursive call gives a termination proof of the program, which is however rather weak since it can only yield quite small ordinals. In the sixties, Tait invented a new method for showing cut elimination of natural deduction, based on a predicate over the set of terms, such that the membership of an expression to the predicate implied the strong normalization property for that expression. The predicate being defined by induction on types, or even as a fixpoint, this method could yield much larger ordinals. Later generalized by Girard under the name of reducibility or computability candidates, it showed very effective in proving the strong normalization property of typed lambda-calculi..

A Comparative Study of Coq and HOL

This paper illustrates the differences between the style of theory mechanisation of Coq and of HOL. This comparative study is based on the mechanisation of fragments of the theory of computation in these systems. Examples from these implementations are given to support some of the arguments discussed in this paper. The mechanisms for specifying definitions and for theorem proving are discussed separately, building in parallel two pictures of the different approaches of mechanisation given by these systems

The computability path ordering

This paper aims at carrying out termination proofs for simply typed higher-order calculi automatically by using ordering comparisons. To this end, we introduce the computability path ordering (CPO), a recursive relation on terms obtained by lifting a precedence on function symbols. A first version, core CPO, is essentially obtained from the higher-order recursive path ordering (HORPO) by eliminating type checks from some recursive calls and by incorporating the treatment of bound variables as in the com-putability closure. The well-foundedness proof shows that core CPO captures the essence of computability arguments \'a la Tait and Girard, therefore explaining its name. We further show that no further type check can be eliminated from its recursive calls without loosing well-foundedness, but for one for which we found no counterexample yet. Two extensions of core CPO are then introduced which allow one to consider: the first, higher-order inductive types; the second, a precedence in which some function symbols are smaller than application and abstraction

Computability and analysis: the legacy of Alan Turing

We discuss the legacy of Alan Turing and his impact on computability and analysis.Comment: 49 page

Inductive and Coinductive Components of Corecursive Functions in Coq

In Constructive Type Theory, recursive and corecursive definitions are subject to syntactic restrictions which guarantee termination for recursive functions and productivity for corecursive functions. However, many terminating and productive functions do not pass the syntactic tests. Bove proposed in her thesis an elegant reformulation of the method of accessibility predicates that widens the range of terminative recursive functions formalisable in Constructive Type Theory. In this paper, we pursue the same goal for productive corecursive functions. Notably, our method of formalisation of coinductive definitions of productive functions in Coq requires not only the use of ad-hoc predicates, but also a systematic algorithm that separates the inductive and coinductive parts of functions.Comment: Dans Coalgebraic Methods in Computer Science (2008