Abstract. Students of proof theory, a branch of formal logic, can benefit from computerized tools. We describe the principles behind one such tool called PROED. This tool is targeted especially at novice students, and therefore it is designed to support effortless exploratory use. We moreover argue that focusing on root-first proof construction in Sequent proof systems helps attain this effortlessness. Key words: logic, user interfaces. 1.

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ML modules are a powerful language mechanism for decomposing programs into reusable components. Unfortunately, they also have a reputation for being “complex ” and requiring fancy type theory that is mostly opaque to non-experts. While this reputation is certainly understandable, given the many non-standard methodologies that have been developed in the process of studying modules, we aim here to demonstrate that it is undeserved. To do so, we give a very simple elaboration semantics for a full-featured, higher-order ML-like module language. Our elaboration defines the meaning of module expressions by a straightforward, compositional translation into vanilla System Fω (the higher-order polymorphic λ-calculus), under plain Fω typing environments. We thereby show that ML modules are merely a particular mode of use of System Fω. We start out with a module language that supports the usual second-class modules with Standard ML-style generative functors, and includes local module definitions. To demonstrate the versatility of our approach, we further extend the language with the ability to package modules as first-class values—a very simple extension, as it turns out—and a novel treatment of OCaml-style applicative functors. Unlike previous work combining both generative and applicative functors, we do not require