In this paper we show that the intuitionistic monotone modal logic iM has the uniform Lyndon interpolation property (ULIP). The logic iM is a non-normal modal logic on an intuitionistic basis, and the property ULIP is a strengthening of interpolation in which the interpolant depends only on the premise or the conclusion of an implication, respecting the polarities of the propositional variables. Our method to prove ULIP yields explicit uniform interpolants and makes use of a terminating sequent calculus for iM that we have developed for this purpose. As far as we know, the results that iM has ULIP and a terminating sequent calculus are the first of their kind for an intuitionistic non-normal modal logic. However, rather than proving these particular results, our aim is to show the flexibility of the constructive proof-theoretic method that we use for proving ULIP. It has been developed over the last few years and has been applied to substructural, intermediate, classical (non-)normal modal and intuitionistic normal modal logics. In light of these results, intuitionistic non-normal modal logics seem a natural next class to try to apply the method to, and we take the first step in that direction in this paper.

We introduce a Gentzen-style framework, called layered sequent calculi, for modal logic K5 and its extensions KD5, K45, KD45, KB5, and S5 with the goal to investigate the uniform Lyndon interpolation property (ULIP), which implies both the uniform interpolation property and the Lyndon interpolation property. We obtain complexityoptimal decision procedures for all logics and present a constructive proof of the ULIP for K5, which to the best of our knowledge, is the first such syntactic proof. To prove that the interpolant is correct, we use modeltheoretic methods, especially bisimulation modulo literals.⋆ Supported by a UKRI Future Leaders Fellowship, 'Structure vs Invariants in Proofs', project reference MR/S035540/1. ⋆⋆ Acknowledges the support of the Netherlands Organization for Scientific Research under grant 639.073.807 and the Czech Science Foundation Grant No. 22-06414L. ⋆ ⋆ ⋆ Supported by the Austrian Science Fund (FWF) ByzDEL project (P33600).

In this paper, we investigate the proof complexity of a wide range of substructural systems. For any proof system P at least as strong as Full Lambek calculus, FL, and polynomially simulated by the extended Frege system for some infinite branching super-intuitionistic logic, we present an exponential lower bound on the proof lengths. More precisely, we will provide a sequence of P-provable formulas tA n u 8n"1 such that the length of the shortest P-proof for A n is exponential in the length of A n . The lower bound also extends to the number of proof-lines (proof-lengths) in any Frege system (extended Frege system) for a logic between FL and any infinite branching superintuitionistic logic. We will also prove a similar result for the proof systems and logics extending Visser's basic propositional calculus BPC and its logic BPC, respectively. Finally, in the classical substructural setting, we will establish an exponential lower bound on the number of proof-lines in any proof system polynomially simulated by the cut-free version of CFL ew .

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