We consider a classical (propositional) version, CBI, of O’Hearn and Pym’s logic of bunched implications (BI) from a modeland prooftheoretic perspective. We present a class of classical models of BI which extend the usual BI-models, based on partial commutative monoids, with an algebraic notion of “resource negation”. This class of models gives rise to natural definitions of multiplicative fals...

The main method of proving the Craig Interpolation Property (CIP) constructively uses cut-free sequent proof systems. Until now, however, no such method has been known for proving the CIP using more general sequent-like proof formalisms, such as hypersequents, nested sequents, and labelled sequents. In this paper, we start closing this gap by presenting an algorithm for proving the CIP for moda...

Whilst results from Structural Proof Theory can be couched in many formalisms, it is the sequent calculus which is the most amenable of the formalisms to metamathematical treatment. As such, we present some tools and techniques so that formalisations of results in sequent calculi can proceed more smoothly. The proof assistant Isabelle is used, along with the package for reasoning about first-or...

General conditions on sequent calculi are formulated under which the intermediate and modal logics corresponding to the calculi have uniform interpolation. It follows that the intuitionistic versions of K and KD, as well as several other intuitionistic modal logics, have uniform interpolation. It also follows that no intermediate or intuitionistic modal logic without uniform interpolation has a...

A system of rules consists of (possibly labelled) sequent rules connected to each other by some variables and subject to the condition of appearing in a certain order in the derivation. The formalism of systems of rules is quite powerful and allows, e.g., the definition of analytic labelled sequent calculi for intermediate and modal logics characterised by frame conditions beyond the geometric ...

In this paper we investigate, for intuitionistic implicational logic, the relationship between normalization in natural deduction and cut-elimination in a standard sequent calculus. First we identify a subset of proofs in the sequent calculus that correspond to proofs in natural deduction. Then we define a reduction relation on those proofs that exactly corresponds to normalization in natural d...

Deduction modulo is a way to combine computation and deduction in proofs, by applying the inference rules of a deductive system (e.g. natural deduction or sequent calculus) modulo some congruence that we assume here to be presented by a set of rewrite rules. Using deduction modulo is equivalent to proving in a theory corresponding to the rewrite rules, and leads to proofs that are often shorter...