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The intersection type assignment system has been designed directly as deductive system for assigning formulae of the implicative and conjunctive fragment of the intuitionistic logic to terms of lambda-calculus. But its relation with the logic is not standard. Between all the logics that have been proposed as its foundation, we consider ISL, which gives a logical interpretation of the intersection by splitting the intuitionistic conjunction into two connectives, with a local and global behaviour respectively, being the intersection the local one. We think ISL is a logic interesting by itself, and in order to support this claim we give a sequent calculus formulation of it, and we prove that it enjoys the cut elimination property.
A genoid is a category of two objects such that one is the product of itself with the other. A genoid may be viewed as an abstract substitution algebra. It is a remarkable fact that such a simple concept can be applied to present a unified algebraic approach to lambda calculus and first order logic.
Proof assistants and programming languages based on type theories usually come in two flavours: one is based on the standard natural deduction presentation of type theory and involves eliminators, while the other provides a syntax in equational style
Nakanos later modality, inspired by G{o}del-L{o}b provability logic, has been applied in type systems and program logics to capture guarded recursion. Birkedal et al modelled this modality via the internal logic of the topos of trees. We show that th
Type theories with higher-order subtyping or singleton types are examples of systems where computation rules for variables are affected by type information in the context. A complication for these systems is that bounds declared in the context do not
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 defi