As the systems we have to specify and verify become larger and more complex, there is a mounting need to combine different tools and decision procedures to accomplish large proof tasks. The problem, then, is how to be sure that we can trust the correctness of a heterogeneous proof. In this work we focus on certification and synthesis of equational proofs, that are pervasive in most proof tasks ...

The Stone-Weierstrass Theorem for compact Hausdorff spaces is a basic result of functional analysis with far-reaching consequences. We introduce an equational logic ? ? associated infinitary variety and show that the consequence Beth definability property , stating every implicit definition can be made explicit. Further, we define propositional ? by means Hilbert-style calculus prove strong com...

We introduce rewriting with two sets of rules, the rst interpreted equa-tionally and the second not. A semantic view considers equational rules as deening an equational theory and reduction rules as deening a rewrite relation modulo this theory. An operational view considers both sets of rules as similar. We introduce suucient properties for these two views to be equivalent (up to diierent noti...

1.1 Introduction This paper presents an algebraic account of implementation that is applicable to the object paradigm. The key to its applicability is the notion of state: objects have local states that are observable only through their outputs. That is, objects may be viewed as abstract machines with hidden local state (as in 9]). Consequently, a correct implementation need only have the requi...

This paper introduces an axiomatisation for equational hybrid logic based on previous axiomatizations and natural deduction systems for propositional and firstorder hybrid logic. Its soundness and completeness is discussed. This work is part of a broader research project on the development a general proof calculus for hybrid logics.

Conditional equational theories can be built into logic programming using a generalized resolution rule which calls a universal uniication procedure. Such uniication procedures are often deened by sound and strongly complete sets of inference rules. In other words, to resolve upon two atoms requires to show-by means of the inference rules-that the corresponding arguments of the atoms are equal ...

A method for encoding formulas in rst-order logic with equality into some equational logics and their associated propositional logics is presented. This yields some new strategies for theorem-proving in rst-order logic with equality. We outline the techniques and give a simple illustration of their use.

The study of isomorphisms of types has been mainly carried out in an intuitionistic setting. We extend some of these works to classical logic for both call-by-name and call-by-value computations by means of polarized linear logic and game semantics. This leads to equational characterizations of these isomorphisms for all the propositional connectives.

This paper discusses the design and implementation of a set-oriented logic programming paradigm, called subset-logic programming. Subset-logic programs are built up of three kinds of program clauses: subset, equational, and relational clauses. Using these clauses, we can program solutions to a broad range of problems of interest in logic programming and deductive databases. In previous research...

This paper gives a generalisation of a version of order sorted logic and an abstract axiomatic setting for the treatment of substitution. The two ideas are shown to be related, an equational speciication of term declaration logic being a presentation of a nitary generalised compositum. The axiomatic approach to substitution is compared with some others.