We present a simple and effective methodology for equational reasoning in proof checkers. The method is based on a two-level approach distinguishing between syntax and semantics of mathematical theories. The method is very general and can be carried out in any type system with inductive and oracle types. The potential of our two-level approach is illustrated by some examples developed in Lego.

The last fifteen years have seen an explosion in work on explicit substitution, most of which is done in the style of the oe-calculus. In (Kamareddine & R'ios, 1995a), we extended the -calculus with explicit substitutions by turning de Bruijn's meta-operators into objectoperators offering a style of explicit substitution that differs from that of oe. The resulting calculus, s, remains as close as possible to the -calculus from an intuitive point of view and, while preserving strong normalisation (Kamareddine & R'ios, 1995a), is extended in this paper to a confluent calculus on open terms: the se-caculus. Since the establishment of these results, another calculus, i, came into being in (Mu~noz Hurtado, 1996) which preserves strong normalisation and is itself confluent on open terms. However, we believe that se still deserves attention because, while offering a new style to work with explicit substitutions, it is able to simulate one step of classical fi-reduction, whereas i is not. To ...

. We present a technique to study relations between weak and strong fi-normalisations in various typed -calculi. We first introduce a translation which translates a -term into a I-term, and show that a -term is strongly fi-normalisable if and only if its translation is weakly fi-normalisable. We then prove that the translation preserves typability of -terms in various typed -calculi. This enables us to establish the equivalence between weak and strong fi-normalisations in these typed -calculi. This translation can deal with Curry typing as well as Church typing, strengthening some recent closely related results. This may bring some insights into answering whether weak and strong fi-normalisations in all pure type systems are equivalent. 1 Introduction In various typed -calculi, one of the most interesting and important properties on -terms is how they can be fi-reduced to fi-normal forms. A -term M is said to be weakly fi-normalisable (WN fi (M )) if it can be fi-reduced to a fi-n...

We generalize Exel’s notion of partial group action to monoids. For partial monoid actions that can be defined by means of suitably well-behaved systems of generators and relations, we employ classical rewriting theory in order to describe the universal induced global action on an extended set. This universal action can be lifted to the setting of topological spaces and continuous maps, as well as to that of metric spaces and non-expansive maps. Well-known constructions such as Shimrat’s homogeneous extension are special cases of this construction. We investigate various properties of the arising spaces in relation to the original space; in particular, we prove embedding theorems and preservation properties concerning separation axioms and dimension. These results imply that every normal (metric) space can be embedded into a normal (metrically) ultrahomogeneous space of the same dimension and cardinality.