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We present a typing system with non-idempotent intersection types, typing a term syntax covering three different calculi: the pure {\lambda}-calculus, the calculus with explicit substitutions {\lambda}S, and the calculus with explicit substitutions, contractions and weakenings {\lambda}lxr. In each of the three calculi, a term is typable if and only if it is strongly normalising, as it is the case in (many) systems with idempotent intersections. Non-idempotency brings extra information into typing trees, such as simple bounds on the longest reduction sequence reducing a term to its normal form. Strong normalisation follows, without requiring reducibility techniques. Using this, we revisit models of the {\lambda}-calculus based on filters of intersection types, and extend them to {\lambda}S and {\lambda}lxr. Non-idempotency simplifies a methodology, based on such filter models, that produces modular proofs of strong normalisation for well-known typing systems (e.g. System F). We also present a filter model by means of orthogonality techniques, i.e. as an instance of an abstract notion of orthogonality model formalised in this paper and inspired by classical realisability. Compared to other instances based on terms (one of which rephrases a now standard proof of strong normalisation for the {\lambda}-calculus), the instance based on filters is shown to be better at proving strong normalisation results for {\lambda}S and {\lambda}lxr. Finally, the bounds on the longest reduction sequence, read off our typing trees, are refined into an exact measure, read off a specific typing tree (called principal); in each of the three calculi, a specific reduction sequence of such length is identified. In the case of the {\lambda}-calculus, this complexity result is, for longest reduction sequences, the counterpart of de Carvalho's result for linear head-reduction sequences., Comment: 46 pages more...

Basic proof-search tactics in logic and type theory can be seen as the root-first applications of rules in an appropriate sequent calculus, preferably without the redundancies generated by permutation of rules. This paper addresses the issues of defining such sequent calculi for Pure Type Systems (PTS, which were originally presented in natural deduction style) and then organizing their rules for effective proof-search. We introduce the idea of Pure Type Sequent Calculus with meta-variables (PTSCalpha), by enriching the syntax of a permutation-free sequent calculus for propositional logic due to Herbelin, which is strongly related to natural deduction and already well adapted to proof-search. The operational semantics is adapted from Herbelin's and is defined by a system of local rewrite rules as in cut-elimination, using explicit substitutions. We prove confluence for this system. Restricting our attention to PTSC, a type system for the ground terms of this system, we obtain the Subject Reduction property and show that each PTSC is logically equivalent to its corresponding PTS, and the former is strongly normalising iff the latter is. We show how to make the logical rules of PTSC into a syntax-directed system PS for proof-search, by incorporating the conversion rules as in syntax-directed presentations of the PTS rules for type-checking. Finally, we consider how to use the explicitly scoped meta-variables of PTSCalpha to represent partial proof-terms, and use them to analyse interactive proof construction. This sets up a framework PE in which we are able to study proof-search strategies, type inhabitant enumeration and (higher-order) unification. more...

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