We present a dependent-type system for a λ-calculus with explicit substitutions. In this system, meta-variables, as well as substitutions, are first-class objects. We show that the system enjoys properties like type uniqueness, subject reduction, soundness, confluence, and weak normalization.

In this abstract we will describe research in progress on the problem of extracting information from proofs. Here we will concentrate our attention on semiconstructive calculi, which is a kind of calculus that is of interest in the framework of program synthesis and formal verification. We will discuss the notion of uniformly semiconstructive calculus, introduce our information extraction mechanism and apply it to two calculi extending Intuitionistic Arithmetic.

We introduce a method to associate calculi of proof terms and rewrite rules with cut elimination procedures for logical deduction systems (i.e., Gentzen-style sequent calculi) in the case of intuitionistic logic. We illustrate this method using two different versions of the cut rule for a variant of the intuitionistic fragment of Kleene's logical deduction system G3.

Our systems are in fact calculi of explicit substitution, where the cut rule introduces an explicit substitution and the left-→ rule introduces a binding of the result of a function application. Cut propagation steps of cut elimination correspond to propagation of explicit substitutions, and propagation of weakening (to eliminate it) corresponds to propagation of index-updating operations. We prove various subject reduction, termination, and confluence properties for our calculi.

Our calculi improve on some earlier calculi for logical deduction systems in a number of ways. By using de Bruijn indices, our calculi qualify as first-order term rewriting systems (TRS's), allowing us to use correctly certain results for TRS's about termination. Unlike in some other calculi, each of our calculi has only one cut rule and we do not need unusual features of sequents.

We show that the substitution and index-updating mechanisms of our calculi work the same way as the substitution and index-updating mechanisms of Kamareddine and Ríos' λs and λt, two well-known systems of explicit substitution for the standard λ-calculus. By a change in the format of sequents, we obtain similar results for a known λ-calculus with variables and explicit substitutions, Rose's λbxgc.

Since Melliès showed that λσ (a calculus of explicit substitutions) does not preserve the strong normalization of the β-reduction, it has become a challenge to find a calculus satisfying the following properties: step-by-step simulation of the β-reduction, confluence on terms with metavariables, strong normalization of the calculus of substitutions and preservation of the strong normalization of the λ-calculus. We present here such a calculus. The main novelty of this calculus (given with de Bruijn indices) is the use of labels that represent updating functions and correspond to explicit weakening. A typed version is also presented.