Abstract Recursive types and bounded quantification are prominent features in many modern programming languages, such as Java, C#, Scala, or TypeScript. Unfortunately, the interaction between recursive types, bounded quantification, and subtyping has shown to be problematic in the past. Consequently, defining a simple foundational calculus that combines those features and has desirable properties, such as decidability , transitivity of subtyping, conservativity , and a sound and complete algorithmic formulation, has been a long-time challenge. This paper shows how to extend $F_{\le}$ with iso-recursive types in a new calculus called $F_{\le}^{\mu}$ . $F_{\le}$ is a well-known polymorphic calculus with bounded quantification. In $F_{\le}^{\mu}$ , we add iso-recursive types and correspondingly extend the subtyping relation with iso-recursive subtyping using the recently proposed nominal unfolding rules. In addition, we use so-called structural folding/unfolding rules for typing iso-recursive expressions, inspired by the structural unfolding rule proposed by Abadi et al . (1996). The structural rules add expressive power to the more conventional folding/unfolding rules in the literature, and they enable additional applications. We present several results, including: type soundness; transitivity; the conservativity of $F_{\le}^{\mu}$ over $F_{\le}$ ; and a sound and complete algorithmic formulation of $F_{\le}^{\mu}$ . We study two variants of $F_{\le}^{\mu}$ . The first one uses an extension of the $\textrm{kernel}~F_{\le}$ (a well-known decidable variant of $F_{\le}$ ). This extension accepts equivalent rather than equal bounds and is shown to preserve decidable subtyping. The second variant employs the $\textrm{full}~F_{\le}$ rule for bounded quantification and has undecidable subtyping. Moreover, we also study an extension of the kernel version of $F_{\le}^{\mu}$ , called $F_{\le\ge}^{\mu\wedge}$ , with a form of intersection types and lower bounded quantification . All the properties from the kernel version of $F_{\le}^{\mu}$ are preserved in $F_{\le\ge}^{\mu\wedge}$ . All the results in this paper have been formalized in the Coq theorem prover.

There are two well-known formulations of recursive types: iso-recursive and equi-recursive types. Abadi and Fiore [LICS 1996] have shown that iso- and equi-recursive types have the same expressive power. However, their encoding of equi-recursive types in terms of iso-recursive types requires explicit coercions. These coercions come with significant additional computational overhead , and complicate reasoning about the equivalence of the two formulations of recursive types. This paper proposes a generalization of iso-recursive types called full iso-recursive types. Full iso-recursive types allow encoding all programs with equi-recursive types without computational overhead. Instead of explicit term coercions, all type transformations are captured by computationally irrelevant casts, which can be erased at runtime without affecting the semantics of the program. Consequently, reasoning about the equivalence between the two approaches can be greatly simplified. We present a calculus called λ F i μ , which extends the simply typed lambda calculus (STLC) with full iso-recursive types. The λ F i μ calculus is proved to be type sound, and shown to have the same expressive power as a calculus with equi-recursive types. We also extend our results to subtyping, and show that equi-recursive subtyping can be expressed in terms of iso-recursive subtyping with cast operators.