System F, also known as the polymorphic λ-calculus, is a typed λcalculus independently discovered by the logician Jean-Yves Girard and the computer scientist John Reynolds. We consider F ωµ , which adds higher-order kinds and iso-recursive types. We present the first complete, intrinsically typed, executable, formalisation of System F ωµ that we are aware of. The work is motivated by verifying the core language of a smart contract system based on System F ωµ. The paper is a literate Agda script [14].

We provide a Lawvere-style definition for partial theories, extending the classical notion of equational theory by allowing partially defined operations. As in the classical case, our definition is syntactic: we use an appropriate class of string diagrams as terms. This allows for equational reasoning about the class of models defined by a partial theory. We demonstrate the expressivity of such equational theories by considering a number of examples, including partial combinatory algebras and cartesian closed categories. Moreover, despite the increase in expressivity of the syntax we retain a well-behaved notion of semantics: we show that our categories of models are precisely locally finitely presentable categories, and that free models exist.

We identify the algebraic structure of the material histories generated by concurrent processes. Specifically, we extend existing categorical theories of resource convertibility to capture concurrent interaction. Our formalism admits an intuitive graphical presentation via string diagrams for proarrow equipments.

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Lambda calculi are often used as intermediate representations for compilers. However, they require extensions to handle higherlevel features of programming languages. In this paper we show how to construct an IR based on System F µ ω which supports recursive functions and datatypes, and describe how to compile it to System F µ ω . Our IR was developed for commercial use at the IOHK company, where it is used as part of a compilation pipeline for smart contracts running on a blockchain.