Abstract. Deduction modulo is a way to remove computational arguments from proofs by reasoning modulo a congruence on propositions. Such a technique, issued from automated theorem proving, is of much wider interest because it permits to separate computations and deductions in a clean way. The first contribution of this paper is to define a sequent calculus modulo that gives a proof theoretic account of the combination of computations and deductions. The congruence on propositions is handled via rewrite rules and equational axioms. Rewrite rules apply to terms and also directly to atomic propositions. The second contribution is to give a complete proof search method, called Extended Narrowing and Resolution (ENAR), for theorem proving modulo such congruences. The completeness of this method is proved with respect to provability in sequent calculus modulo. An important application is that higher-order logic can be presented as a theory modulo. Applying the Extended Narrowing and Resolution method to this presentation of higher-order logic subsumes full higher-order resolution.

The lambda-Pi-calculus allows to express proofs of minimal predicate logic. It can be extended, in a very simple way, by adding computation rules. This leads to the lambda-Pi-calculus modulo. We show in this paper that this simple extension is surprisingly expressive and, in particular, that all functional Pure Type Systems, such as the system F, or the Calculus of Constructions, can be embedded in it. And, moreover, that this embedding is conservative under termination hypothesis.

. We show how higher-order rewriting may be encoded into

Abstract. This paper presents on-going researches on theoretical and practical issues of combining rewriting based automated theorem proving and user-guided proof development, with the strong constraint of safe cooperation of both. In practice, we instantiate the theoretical study on the Coq proof assistant and the ELAN rewriting based system, focusing first on equational and then on inductive proofs. Different concepts, especially rewriting calculus and deduction modulo, contribute to define and to relate proof search, proof representation and proof check.

In 1973, Parikh proved a speed-up theorem conjectured by Gödel 37 years before: there exist arithmetical formulæ that are provable in first order arithmetic, but whose shorter proof in second order arithmetic is arbitrarily smaller than any proof in first order. On the other hand, resolution for higher order logic can be simulated step by step in a first order narrowing and resolution method based on deduction modulo, whose paradigm is to separate deduction and computation to make proofs clearer and shorter. We prove that i+1-th order arithmetic can be linearly simulated into i-th order arithmetic modulo some confluent and terminating rewrite system. We also show that there exists a speed-up between i-th order arithmetic modulo this system and i-th order arithmetic without modulo. All this allows us to prove that the speed-up conjectured by Gödel does not come from the deductive part of the proofs, but can be expressed as simple computation, therefore justifying the use of deduction modulo as an efficient first order setting simulating higher order.

Abstract. We extend the notion of Heyting algebra to a notion of truth values algebra and prove that a theory is consistent if and only if it has a B-valued model for some non trivial truth values algebra B. A theory that has a B-valued model for all truth values algebras B is said to be super-consistent. We prove that super-consistency is a model-theoretic sufficient condition for strong normalization. 1

Abstract not found

Abstract. We present a restriction of Resolution modulo where the rewrite rules are such that a clause always rewrites to a clause. This way, the reduct of a clause needs not be further transformed into clause form. Restricting Resolution modulo this way requires to extend it in another way and distinguish the rules that apply to negative and to positive atomic propositions. As an example, we show how this method applies to a first-order presentation of Simple type theory. Finally, we show that this method can be seen as a restriction of Equational resolution that mixes clause selection restrictions and literal selection restrictions, but unlike many restrictions of Resolution, it is not an instance of Ordered resolution. 1

Deduction modulo is a way to combine computation and deduction in proofs, by applying the inference rules of a deductive system (e.g. natural deduction or sequent calculus) modulo some congruence that we assume here to be presented by a set of rewrite rules. Using deduction modulo is equivalent to proving in a theory corresponding to the rewrite rules, and leads to proofs that are often shorter and more readable. However, cuts may be not admissible anymore. We define a new system, the unfolding sequent calculus, and prove its equivalence with the sequent calculus modulo, especially w.r.t. cut-free proofs. It permits to show that it is even undecidable to know if cuts can be eliminated in the sequent calculus modulo a given rewrite system. Then, to recover the cut admissibility, we propose a procedure to complete the rewrite system such that the sequent calculus modulo the resulting system admits cuts. This is done by generalizing the Knuth-Bendix completion in a non-trivial way, using the framework of abstract canonical systems. These

Peter Andrews has proposed, in 1971, the problem of finding an analog of the Skolem theorem for Simple Type Theory. A first idea lead to a naive rule that worked only for Simple Type Theory with the axiom of choice and the general case has only been solved, more than ten years later, by Dale Miller [9, 10]. More recently, we have proposed with Thérèse Hardin and Claude Kirchner [7] a new way to prove analogs of the Miller theorem for different, but equivalent, formulations of Simple Type Theory. In this paper, that does not contain new technical results, I try to show that the history of the skolemization problem and of its various solutions is an illustration of a tension between two points of view on Simple Type Theory: the logical and the theoretical points of view.