Inductive types in the Calculus of Algebraic Constructions

In a previous work, we proved that an important part of the Calculus of Inductive Constructions (CIC), the basis of the Coq proof assistant, can be seen as a Calculus of Algebraic Constructions (CAC), an extension of the Calculus of Constructions with functions and predicates defined by higher-order rewrite rules. In this paper, we prove that almost all CIC can be seen as a CAC, and that it can be further extended with non-strictly positive types and inductive-recursive types together with non-free constructors and pattern-matching on defined symbols.Comment: Journal version of TLCA'0

Building Decision Procedures in the Calculus of Inductive Constructions

It is commonly agreed that the success of future proof assistants will rely on their ability to incorporate computations within deduction in order to mimic the mathematician when replacing the proof of a proposition P by the proof of an equivalent proposition P' obtained from P thanks to possibly complex calculations. In this paper, we investigate a new version of the calculus of inductive constructions which incorporates arbitrary decision procedures into deduction via the conversion rule of the calculus. The novelty of the problem in the context of the calculus of inductive constructions lies in the fact that the computation mechanism varies along proof-checking: goals are sent to the decision procedure together with the set of user hypotheses available from the current context. Our main result shows that this extension of the calculus of constructions does not compromise its main properties: confluence, subject reduction, strong normalization and consistency are all preserved

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..

The computability path ordering: the end of a quest

In this paper, we first briefly survey automated termination proof methods for higher-order calculi. We then concentrate on the higher-order recursive path ordering, for which we provide an improved definition, the Computability Path Ordering. This new definition appears indeed to capture the essence of computability arguments \`a la Tait and Girard, therefore explaining the name of the improved ordering.Comment: Dans CSL'08 (2008

From formal proofs to mathematical proofs: a safe, incremental way for building in first-order decision procedures

We investigate here a new version of the Calculus of Inductive Constructions (CIC) on which the proof assistant Coq is based: the Calculus of Congruent Inductive Constructions, which truly extends CIC by building in arbitrary first-order decision procedures: deduction is still in charge of the CIC kernel, while computation is outsourced to dedicated first-order decision procedures that can be taken from the shelves provided they deliver a proof certificate. The soundness of the whole system becomes an incremental property following from the soundness of the certificate checkers and that of the kernel. A detailed example shows that the resulting style of proofs becomes closer to that of the working mathematician

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

Coq without Type Casts: A Complete Proof of Coq Modulo Theory

International audienceIncorporating extensional equality into a dependent intensional type system such as the Calculus of Constructions provides with stronger type-checking capabilities and makes the proof development closer to intuition. Since strong forms of extensionality lead to undecidable type-checking, a good trade-off is to extend intensional equality with a decidable first-order theory T, as done in CoqMT, which uses matching modulo T for the weak and strong elimination rules, we call these rules T-elimination. So far, type-checking in CoqMT is known to be decidable in presence of a cumulative hierarchy of universes and weak T-elimination. Further, it has been shown by Wang with a formal proof in Coq that consistency is preserved in presence of weak and strong elimination rules, which actually implies consistency in presence of weak and strong T-elimination rules since T is already present in the conversion rule of the calculus. We justify here CoqMT's type-checking algorithm by showing strong normalization as well as the Church-Rosser property of β-reductions augmented with CoqMT's weak and strong T-elimination rules. This therefore concludes successfully the meta-theoretical study of CoqMT

Réalisabilité et paramétricité dans les systèmes de types purs

Cette thèse porte sur l adaptation de la réalisabilité et la paramétricité au cas des types dépendants dans le cadre des Systèmes de Types Purs. Nous décrivons une méthode systématique pour construire une logique à partir d un langage de programmation, tous deux décrits comme des systèmes de types purs. Cette logique fournit des formules pour exprimer des propriétés des programmes et elle offre un cadre formel adéquat pour développer une théorie de la réalisabilité au sein de laquelle les réalisateurs des formules sont exactement les programmes du langage de départ. Notre cadre permet alors de considérer les théorèmes de représentation pour le système T de Gödel et le système F de Girard comme deux instances d'un théorème plus général.Puis, nous expliquons comment les relations logiques de la théorie de la paramétricité peuvent s'exprimer en terme de réalisabilité, ce qui montre que la logique engendrée fournit un cadre adéquat pour développer une théorie de la paramétricité du langage de départ. Pour finir, nous montrons comment cette théorie de la paramétricité peut-être adaptée au système sous-jacent à l'assistant de preuve Coq et nous donnons un exemple d'application original de la paramétricité à la formalisation des mathématiques.This thesis focuses on the adaptation of realizability and parametricity to dependent types in the framework of Pure Type Systems. We describe a systematic method to build a logic from a programming language, both described as pure type systems. This logic provides formulas to express properties of programs and offers a formal framework that allows us to develop a theory of realizability in which realizers of formulas are exactly programs of the starting programming language. In our framework, the standard representation theorems of Gödel's system T and Girard's system F may be seen as two instances of a more general theorem. Then, we explain how the so-called logical relations of parametricity theory may be expressed in terms of realizability, which shows that the generated logic provides an adequate framework for developping a general theory of parametricity. Finally, we show how this parametricity theory can be adapted to the underlying type system of the proof assistant Coq and we give an original example of application of parametricity theory to the formalization of mathematics.LYON-ENS Sciences (693872304) / SudocSudocFranceF

Inductive Types in the Calculus of Algebraic Constructions

International audienceIn a previous work, we proved that almost all of the Calculus of Inductive Constructions (CIC), which is the basis of the proof assistant Coq, can be seen as a Calculus of Algebraic Constructions (CAC), an extension of the Calculus of Constructions with functions and predicates defined by higher-order rewrite rules. In this paper, we not only prove that CIC as a whole can be seen as a CAC, but also that it can be extended with non-free constructors, pattern-matching on defined symbols, non-strictly positive types and inductive-recursive types