Pure Type Systems without Explicit Contexts

We present an approach to type theory in which the typing judgments do not have explicit contexts. Instead of judgments of shape “Gamma |- A : B”, our systems just have judgments of shape “A : B”. A key feature is that we distinguish free and bound variables even in pseudo-terms. Specifically we give the rules of the “Pure Type System” class of type theories in this style. We prove that the typing judgments of these systems correspond in a natural way with those of Pure Type Systems as traditionally formulated. I.e., our systems have exactly the same well-typed terms as traditional presentations of type theory. Our system can be seen as a type theory in which all type judgments share an identical, infinite, typing context that has infinitely many variables for each possible type. For this reason we call our system “Gamma_infinity”. This name means to suggest that our type judgment “A : B” should be read as “Gamma_infinity |- A : B”, with a fixed infinite type context called “Gamma_infinity”.

💡 Research Summary

The paper introduces a novel formulation of Pure Type Systems (PTS) that eliminates the need for explicit contexts in typing judgments. Traditional PTS presentations rely on judgments of the form “Γ ⊢ A : B”, where Γ records the assumptions about free variables and their types. Managing Γ explicitly adds meta‑theoretical complexity and requires additional bookkeeping in implementations such as proof assistants or type checkers.

The authors propose to replace every contextual judgment with a context‑free judgment “A : B”. To retain the information normally carried by Γ, they distinguish free variables from bound variables directly in the syntax of pseudo‑terms. Each variable carries an annotation indicating whether it is free (available from an implicit, infinite context) or bound (introduced by a λ‑ or Π‑binder). The underlying idea is to imagine a fixed, infinite typing context Γ∞ that contains an unbounded supply of variables for every possible type. In practice, Γ∞ is never written; it is implicitly assumed whenever a free variable appears.

The paper proceeds in three main stages. First, it reviews the standard definition of a PTS: a set of sorts S, a set of axioms, and a set of product rules R. It recalls the usual typing rules (variable, abstraction, application, product, conversion) that all reference Γ explicitly. Second, it defines the Γ∞‑based system. The syntax is extended with annotations for free versus bound occurrences, but the term language itself remains unchanged apart from this tagging. The typing rules are then reformulated so that they never mention Γ. For example, the variable rule simply states that a free variable x of sort s can be typed as “x : s” because x is assumed to be present in Γ∞; a bound variable is typed only within the scope of its binder. All other rules (Π‑formation, λ‑abstraction, application, conversion) are adapted in the same way, using the free/bound distinction to control scope.

The core technical contribution is a pair of translation functions establishing a bijective correspondence between derivations in the traditional PTS and derivations in the Γ∞ system. One direction maps a Γ∞‑derivation to a conventional derivation by inserting the appropriate Γ that contains exactly the free variables used in the judgment. The opposite direction extracts from a conventional derivation the set of free variables, verifies that they belong to Γ∞, and then discards the explicit Γ, yielding a context‑free derivation. The authors prove that these translations preserve typing, conversion, and reduction properties, thereby showing that the two systems have exactly the same set of well‑typed terms.

Beyond the theoretical equivalence, the paper discusses practical implications. By removing explicit contexts, a type checker no longer needs to maintain a mutable environment of assumptions; it only needs to track the free/bound status of each variable. This simplifies the implementation of proof assistants, reduces the overhead of context merging and weakening, and can improve performance in large developments where context manipulation is a bottleneck. The infinite context assumption also eliminates the need for ad‑hoc name generation strategies: any fresh variable can be drawn from Γ∞, provided it is given a unique identifier.

The authors acknowledge limitations. Although Γ∞ is a convenient abstraction, any concrete implementation must still allocate a finite pool of identifiers, so a systematic naming scheme (e.g., de Bruijn indices or globally unique IDs) is required to avoid collisions. Moreover, the notion of an “infinite context” may be less intuitive for newcomers to type theory, suggesting a need for pedagogical material to bridge the conceptual gap.

In conclusion, the paper demonstrates that the explicit handling of contexts is not essential for the expressive power of Pure Type Systems. By re‑encoding context information into the syntax of terms and assuming an implicit, infinite environment, the authors obtain a system that is both theoretically equivalent to traditional PTS and potentially more ergonomic for implementation. This work opens a new line of research into context‑free formulations of more advanced type theories, such as dependent type theories with universes, and suggests that many existing proof‑assistant architectures could be simplified by adopting a Γ∞‑style approach.