Coq's Derive: Dependent Types Or Error?

This article dives into a nuanced issue within Coq's Derive functionality, specifically focusing on how it handles dependent types. We'll explore a reported bug where Derive might not be interacting with dependent types as intuitively expected, potentially leading to incorrect goal ordering or unexpected errors. Our goal is to shed light on this behavior and discuss potential solutions or clarifications.

Understanding the Problem: Goal Ordering in Derive

Let's kick off by examining the core of the issue: dependent types and their interaction with Coq's Derive command. When you use Derive, Coq attempts to generate proofs or definitions based on your input. However, it seems there might be an inconsistency in how Derive constructs the internal goal list or context when dependent types are involved. We've observed a situation where the binders (variables and their types) appear to be added to the goal list in an order that doesn't align with expectations. This unexpected ordering can lead to subsequent proof attempts failing because the context isn't set up as anticipated.

Consider this Coq snippet:

From Stdlib Require Import Derive.
Set Printing Existential Instances.
Derive (a : Type) (b : a = a) (c : b = b) (d : c = c) in (d = d) as foo.
Show.

When you run Show after this Derive command, the output reveals the state of the goal. Notice how the variables d, c, b, and a are presented:

1 focused goal (shelved: 4)
 
 d := ?Goal : c = c
 c := ?Goal0@{d:=d} : b = b
 b := ?Goal1@{d:=d; c:=c} : a = a
 a := ?Goal2@{d:=d; c:=c; b:=b} : Type
 ============================
 d = d

The issue here is that the dependencies seem to be unwound in a way that might be reversed from what one might intuitively expect when defining terms with increasing levels of dependency. The variable d is defined in terms of c, c in terms of b, and so on, but the goal display suggests a potential reversal in how these dependencies are managed internally. The hypothesis is that simply reversing the list of binders at the appropriate place within Derive's implementation might resolve this peculiar ordering and lead to more predictable behavior when working with dependent types.

Reproducing the Bug: A Concrete Example

To make this issue more concrete, let's look at a specific example that demonstrates how this goal ordering problem can manifest as a failed proof attempt. This example uses a simpler case involving a list type, which is a common scenario where dependencies can arise.

Here's the Coq code:

From Stdlib Require Import Derive.
Derive (X : Type) (l : list X) in (l = l) as foo.
Fail exact (eq_refl (@nil unit)).

In this scenario, we are using Derive with a type X and a dependent term l which is a list X. The goal is to prove l = l. The subsequent Fail exact (eq_refl (@nil unit)). command is intended to fail, but it fails due to the way Derive has set up the context. The error message provides crucial insight:

The command has indeed failed with message:
In environment
l := ?Goal : list X
X := unit : Type
The term "eq_refl" has type "nil = nil" while it is expected to have type 
"l = l" (unable to find a well-typed instantiation for "?Goal": cannot ensure that
"list unit" is a subtype of "list X").

This error message highlights a type mismatch. Coq is expecting a proof of l = l, but eq_refl can only provide a proof of nil = nil (since X is instantiated to unit). The crucial part is the mention of l := ?Goal : list X. This indicates that l is still an unknown existential variable (?Goal) within the context, and Coq cannot unify list unit with list X because X is not yet fully determined in a way that satisfies the dependency implied by l. This situation strongly suggests that the order in which Derive introduces the binders (X and l) and establishes their relationships might be contributing to this difficulty. If X were determined first, or if the dependency l : list X was handled with more care regarding the unification of X, this proof might succeed. The problem is that l is expected to be of type list X, but X is still a meta-variable, and the unification process fails. The ability to use Unshelve, admit, and Check in the subsequent lines further demonstrates the unstable or unresolvable nature of the goal introduced by Derive in this context.

Potential Solutions and Future Directions

Addressing the issues with dependent types in Coq's Derive likely involves a few key strategies. The primary suggestion, as hinted earlier, is to review and potentially refactor the internal logic of Derive that handles the ordering of binders. If the list of binders is reversed at the point where they are added to the context, it might align better with how dependencies are typically resolved in Coq. This would mean that when Derive encounters a term like l : list X, it would ensure that X is properly instantiated or constrained before or in conjunction with setting up the goal for l.

Another approach could involve enhancing Derive's type inference and unification capabilities when dealing with dependent structures. Derive could be made more robust by explicitly checking for and handling potential type ambiguities that arise from dependent types. This might involve more sophisticated constraint solving or providing clearer error messages to the user when such ambiguities are detected, guiding them on how to resolve the types manually.

Furthermore, it would be beneficial to clarify the expected behavior of Derive when dependent types are used. Documentation could be updated to explicitly state how Derive constructs its internal context for such cases, managing user expectations and providing guidance on best practices. If Derive is not intended to fully automate proofs involving complex dependent types, then it should perhaps provide a more informative error message, rather than an obscure type mismatch that requires significant debugging to unravel.

Ultimately, the goal is to make Derive a more predictable and reliable tool for a wider range of Coq applications. Improving its handling of dependent types would significantly enhance its utility, allowing users to leverage its automation capabilities with more complex inductive types and definitions.

Conclusion

In summary, the interaction between Coq's Derive command and dependent types presents an area for potential improvement. The observed issues with goal ordering and subsequent proof failures suggest that Derive might not be optimally handling the complexities of dependent structures. The proposed solutions, focusing on binder ordering, enhanced type inference, and clearer documentation, aim to make Derive more robust and user-friendly when dealing with these advanced type system features. By addressing these points, the Coq community can further strengthen this powerful automation tool.

For further exploration into Coq and its advanced features, you can refer to the official Coq documentation. Additionally, understanding dependent types is crucial, and resources like the Programming Language Foundations book offer deep insights into these concepts.