Book of Abstracts - IRIT

Universe Polymorphism and Inference in Coq
Matthieu Sozeau
INRIA Paris & PPS, Paris 7 University
matthieu.sozeau@inria.fr
Abstract
Type theories such as the Calculus of Inductive Constructions maintain a universe
hierarchy to prevent logical paradoxes. To ensure consistency while not troubling the user
with this necessary information, systems using typical ambiguity were designed, which
handle universes implicitly. Universes are seen as floating and their usage determines a
graph of constraints, which must statisfy a property of acyclicity to ensure consistency.
While this is a useful mechanism, there are a number of situations where the user would
like to have developments made polymorphic on the universes used and instantiate his
constructions at different levels. Typically, this is necessary to nest different instances of a
given structure containing universes. We present an elaboration from terms using typical
ambiguity into explicit terms which also accomodates universe polymorphism, i.e. the
ability to write a term once and use it at different universe levels. Elaboration relies on
an enhanced type inference algorithm to provide the freedom of typical ambiguity while
also supporting polymorphism, in a fashion similar to usual Hindley-Milner polymorphic
type inference. This elaboration is implemented as a drop-in replacement for the existing
universe system of Coq and has been benchmarked favorably against the previous version.
We demonstrate how it provides a solution to a number of formalization issues present in
the original system.
The Calculus of Inductive Constructions implemented in the Coq proof assistant relies on a
system of universes to ensure logical consistency. It avoids paradoxes coming from the infamous
Type : Type rule of system U- [1] by stratifying the universes used in a development, giving them
names (“levels”) and building up a graph of their inclusion relationships. It ensures that these
relationships are coherent at any given point of a development. This can be checked by showing
that there is always an assignment of natural numbers to universe levels that satisfy the universe
constraints, which implies that the stratified universe levels map to the well-founded suite of
universes Type 0 < Type 1 < .. < Type n .
To avoid forcing the user to work directly with levels, a system of typical ambiguity is used
in Coq, where one can leave out the names of universes involved in a definition and let the
system generate fresh universe variables and associated constraints. However in its current form
this system has a serious limitation as it does not allow one to make definitions that can be
instantiated at different levels: levels are always global. Let’s look at a simple example. The
polymorphic identity function id is written:
Definition id (A : Type) (a : A) := a
It takes a type A in an unspecified universe, an object of that type and returns it. The
actual CIC term constructed by this definition involves a fresh universe variable (say l) and its
type-checking judgement has conclusion:
⊢ (λ(A : Type l )(a : A), a) : Π(A : Type l ), A → A
The rule for type-checking product types in CIC allows us to derive for the type of the
identity function a judgment:
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