Various mechanisms are available for managing universe levels in proof assistants based on type theory. The Agda proof assistant implements a strong form of universe polymorphism in which universe levels are internalised as a type, making levels first-class objects and permitting higher-rank quantification via ordinary Π-types. We prove normalisation and decidability of equality and type-checking for a type theory with first-class universe levels inspired by Agda. We also show that level primitives can safely be erased in extracted programs. Our development is formalised in Agda itself and builds upon previous work which uses logical relations on extrinsically typed syntax.

What are mutable references; what do they mean? The answers to these questions have spawned lots of important theoretical work and form the foundation of many impactful tools. However, existing semantics collapse a key distinction: which allocations does a reference depend on? In this paper, we deconstruct the space of mutable higher-order references. We formalize a novel distinction–splitting the design space of references not only into higher-order vs (full-)ground references, but also dependency of an allocation on past vs future allocations. This distinction is fundamental to a thorny issue that arises in constructing semantic models of mutable references–the type-world circularity. The issue disappears for what we call predicative references, those that only quantify over past, not future, allocations, and for non-higher-order impredicative references. We design a syntax and semantics for each point in our newly described space. The syntax relies on a type universe hierarchy, à la dependent type theory, to kind the types of allocated terms, and stratify allocations. Each type universe corresponds to a semantic Kripke world, giving a lightweight syntactic mechanism to design and restrict heap shapes. The semantics bear a resemblance to work on regions, and suggest some connection between universe systems and regions, which we describe in some detail.

In dependent type theory, being able to refer to a type universe as a term itself increases its expressive power, but requires mechanisms in place to prevent Girard’s paradox from introducing logical inconsistency in the presence of type-in-type. The simplest mechanism is a hierarchy of universes indexed by a sequence of levels, typically the naturals. To improve reusability of definitions, they can be made level polymorphic, abstracting over level variables and adding a notion of level expressions. For even more expressive power, level expressions can be made first-class as terms themselves, and level polymorphism is subsumed by dependent functions quantifying over levels. Furthermore, bounded level polymorphism provides more expressivity by being able to explicitly state constraints on level variables. While semantics for first-class levels with constraints are known, syntax and typing rules have not been explicitly written down. Yet pinning down a well-behaved syntax is not trivial; there exist prior type theories with bounded level polymorphism that fail to satisfy subject reduction. In this work, we design an explicit syntax for a type theory with bounded first-class levels, parametrized over arbitrary well-founded sets of levels. We prove the metatheoretic properties of subject reduction, type safety, consistency, and canonicity, entirely mechanized from syntax to semantics in Lean.

A hierarchy of type universes is a rudimentary ingredient in the type theories of many proof assistants to prevent the logical inconsistency resulting from combining dependent functions and the type-in-type axiom. In this work, we argue that a universe hierarchy is not the only option for universes in type theory. Taking inspiration from Leivant’s Stratified System F, we introduce Stratified Type Theory (), where rather than stratifying universes by levels, we stratify typing judgements and restrict the domain of dependent functions to strictly lower levels. Even with type-in-type, this restriction suffices to enforce consistency. In , we consider a number of extensions beyond just stratified dependent functions. First, the subsystem employs McBride’s crude-but-effective stratification (also known as displacement) as a simple form of level polymorphism where global definitions with concrete levels can be displaced uniformly to any higher level. Second, to recover some expressivity lost due to the restriction on dependent function domains, the full includes a separate nondependent function type with a floating domain whose level matches that of the overall function type. Finally, we have implemented a prototype type checker for extended with datatypes and inference for level and displacement annotations, along with a small core library. We have proven to be consistent and to be type safe, but consistency of the full remains an open problem, largely due to the interaction between floating functions and cumulativity of judgements. Nevertheless, we believe to be consistent, and as evidence have verified the ill-typedness of some well-known type-theoretic paradoxes using our implementation.

In this paper, we explore a connection between type universes and memory allocation. Type universe hierarchies are used in dependent type theories to ensure consistency, by forbidding a type from quantifying over all types. Instead, the types of types (universes) form a hierarchy, and a type can only quantify over types in other universes (with some exceptions), restricting cyclic reasoning in proofs. We present a perspective where universes also describe where values are allocated in the heap, and the choice of universe algebra imposes a structure on the heap overall. The resulting type system provides a simple declarative system for reasoning about and restricting memory allocation, without reasoning about reads or writes. We present a theoretical framework for equipping a type system with higher-order references restricted by a universe hierarchy, and conjecture that many existing universe algebras give rise to interesting systems for reasoning about allocation. We present 3 instantiations of this approach to enable reasoning about allocation in the simply typed $𝜆$-calculus: (1) the standard ramified universe hierarchy, which we prove guarantees termination of the language extended with higher-order references by restricting cycles in the heap; (2) an extension with an impredicative base universe, which we conjecture enables full-ground references (with terminating computation but cyclic ground data structures); (3) an extension with universe polymorphism, which divides the heap into fine-grained regions.

We present a novel formulation of universe polymorphism in dependent type theory in terms of monads on the category of strict partial orders, and a novel algebraic structure, displacement algebras, on top of which one can implement a generalized form of McBride’s “crude but effective stratification” scheme for lightweight universe polymorphism. We give some examples of exotic but consistent universe hierarchies, and prove that every universe hierarchy in our sense can be embedded in a displacement algebra and hence implemented via our generalization of McBride’s scheme. Many of our technical results are mechanized in Agda, and we have an OCaml library for universe levels based on displacement algebras, for use in proof assistant implementations.

In order to avoid well-known paradoxes associated with self-referential definitions, higher-order dependent type theories stratify the theory using a countably infinite hierarchy of universes (also known as sorts), ${\text{Type}}_0:{\text{Type}}_1:\dots$. Such type systems are called cumulative if for any type A we have that $𝐴:{\text{Type}}_{𝑖}$ implies $𝐴:{\text{Type}}_{𝑖+1}$. The Predicative Calculus of Inductive Constructions (pCIC) which forms the basis of the Coq proof assistant, is one such system. In this paper we present the Predicative Calculus of Cumulative Inductive Constructions (pCuIC) which extends the cumulativity relation to inductive types. We discuss cumulative inductive types as present in Coq 8.7 and their application to formalization and definitional translations.