Deduction modulo is an extension of first-order predicate logic where axioms are replaced by rewrite rules. For instance, the axiom P ⇔ (Q ⇒ R) is replaced by the rule P −→ (Q ⇒ R). These rules define an equivalence relation and, in a proof, a proposition can be replaced by an equivalent one at any time. A presentation of Simple Type Theory in Deduction modulo, calledHOL, has been given in [3]....

This paper studies, with techniques of Abstract Algebraic Logic, the effects of putting a bound on the cardinality of the set of side formulas in the Deduction Theorem, viewed as a Gentzen-style rule, and of adding additional assumptions inside the formulas present in Modus Ponens, viewed as a Hilbert-style rule. As a result, a denumerable collection of new Gentzen systems and two new sententia...

Herewith a fairly old concept is published first time and named “Lucas-Interpretation”, which has been implemented in a prototype, which has been proved useful in educational practice and which gains academic relevance with an emerging generation of educational math assistants (EMA) based on Computer Theorem Proving (CTP). Automated Theorem Proving (ATP), i.e. deduction is the appropriate techn...

‎In this paper, generalized convex contractions on orthogonal metric spaces are stablished in whath  might be called their  definitive versions. Also, we show that there are examples which show that our main theorems are  genuine generalizations of Theorem 3.1 and 3.2 of [M.A. Miandaragh, M. Postolache and S. Rezapour,  {it Approximate fixed points of generalized convex contractions}, Fixed Poi...

in this paper, we establish an existence result of the solution for an generalized strong vector variational inequality already considered in the literature and as applications we obtain a new coincidence point theorem in hilbert spaces.

Appeared in: N.V. Murray, editor, Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX’99), Lecture Notes in Artificial Intelligence 1617, Springer, pages 247–262, 1999. We present a tableau calculus for reasoning in fragments of natural language. We focus on the problem of pronoun resolution and the way in which it complicates automated theorem proving for natural language ...

We introduce an encoding of the set theory of the B method using polymorphic types and deduction modulo, which is used for the automated verification of proof obligations in the framework of the BWare project. Deduction modulo is an extension of predicate calculus with rewriting both on terms and propositions. It is well suited for proof search in theories because it turns many axioms into rewr...

In a propositional calculus based on combinatory logic it is necessary to have a restriction on the deduction theorem for implication as otherwise Curry's paradox results (see [5]). In [1] and [2] we restricted the deduction theorem for implication as follows: where Δ is any sequence of obs and HX stands for "X is a proposition". Motivation for this deduction theorem was given in [2] using the ...

Aquarius is a distributed theorem prover for first order logic with equality, developed for a network of workstations. Given in input a theorem proving problem and the number n of active nodes, Aquarius creates n deductive processes, one on each workstation, which work cooperatively toward the solution of the problem. Aquarius realizes a specific variant of a general methodology for distributed...