Lean: Types With Numeric Quantifiers Produce Unknown Identifiers

Mar 10, 2025 by ADMIN 65 views

Introduction

Lean is a powerful proof assistant and programming language that is widely used in mathematics, computer science, and other fields. However, like any complex system, it is not immune to errors and bugs. In this article, we will explore a specific issue that has been reported in Lean, where types with numeric quantifiers produce unknown identifiers. We will delve into the details of this issue, its causes, and potential solutions.

The Problem

The issue at hand is related to the use of numeric quantifiers in Sail types. Sail is a type theory that is used in Lean to define the syntax and semantics of the language. Numeric quantifiers are used to specify the number of elements in a type. However, when these quantifiers are used in certain contexts, they can produce unknown identifiers. This is a problem because unknown identifiers are not valid in Lean, and they can cause errors and crashes.

Example Use Cases

Let's take a look at two example use cases that illustrate this issue.

`config_vec_list`

The first example is the config_vec_list test, which is failing due to unknown identifiers in numerically quantified Sail types. The code is as follows:

let xs : {'n, 'n >= 2. vector('n, string)} = config foo.bar;

This code is trying to create a vector of strings with a length that is specified by the n quantifier. However, when Lean tries to compile this code, it produces an error:

✖ [7/12] Building Out
trace: .> LEAN_PATH=././.lake/build/lib DYLD_LIBRARY_PATH= /Users/hamlinb/.elan/toolchains/leanprover--lean4-nightly---nightly-2025-02-05/bin/lean --tstack=400000 ././././Out.lean -R ./././. -o ././.lake/build/lib/Out.olean -i ././.lake/build/lib/Out.ilean -c ././.lake/build/ir/Out.c --json
error: ././././Out.lean:99:26: unknown identifier 'k_n'
error: Lean exited with code 1

As we can see, the error message indicates that there is an unknown identifier k_n in the code.

`type_if_bits`

The second example is the type_if_bits test, which is also failing due to unknown identifiers in numerically quantified Sail types. The code is as follows:

let xs : {'n, 'n >= 2. vector('n, bool)} = config foo.bar;

This code is trying to create a vector of booleans with a length that is specified by the n quantifier. However, when Lean tries to compile this code, it produces an error:

✖ [7/12] Building Out
trace: .> LEAN_PATH=././.lake/build/lib DYLD_LIBRARY_PATH= /Users/hamlinb/.elan/toolchains/leanprover--lean4-nightly---nightly-2025-02-05/bin/lean --tstack=400000 ././././Out.lean -R ./././. -o ././.lake/build/lib/Out.olean -i ././.lake/build/lib/Out.ilean -c ././.lake/build/ir/Out.c --json
error: ././././Out.lean:99:26: unknown identifier 'k_n'
error: Lean exited with code 1

As we can see, the error message is the same as in the previous example.

Causes of the Issue

So, what is causing this issue? The problem lies in the way that Lean handles numeric quantifiers in Sail types. When Lean encounters a numeric quantifier, it tries to infer the type of the quantifier. However, in some cases, the type of the quantifier is not clear, and Lean produces an unknown identifier.

Potential Solutions

There are several potential solutions to this issue. One possible solution is to add more type annotations to the code to make it clear what type the quantifier is supposed to be. For example:

let xs : {'n, 'n >= 2. vector('n, string)} = config foo.bar;

becomes:

let xs : {'n, 'n >= 2. vector('n, string)} (k_n : Nat) = config foo.bar;

This code adds a type annotation to the k_n quantifier, making it clear that it is supposed to be a Nat.

Another possible solution is to use a different type theory that does not have this issue. For example, Lean has a type theory called hott that does not have this issue.

Conclusion

In conclusion, the issue of types with numeric quantifiers producing unknown identifiers is a complex problem that requires a deep understanding of Lean and its type theory. While there are several potential solutions to this issue, they require careful consideration and experimentation to determine the best approach. We hope that this article has provided a useful overview of this issue and its potential solutions.

Future Work

There are several areas of future work that could be explored to address this issue. One possible area of research is to develop a more robust type inference system for Lean that can handle numeric quantifiers more effectively. Another possible area of research is to develop a new type theory that does not have this issue.

References

Acknowledgments

Introduction

In our previous article, we explored the issue of types with numeric quantifiers producing unknown identifiers in Lean. We discussed the causes of this issue and potential solutions. In this article, we will answer some frequently asked questions about this issue.

Q: What is the cause of this issue?

A: The cause of this issue is the way that Lean handles numeric quantifiers in Sail types. When Lean encounters a numeric quantifier, it tries to infer the type of the quantifier. However, in some cases, the type of the quantifier is not clear, and Lean produces an unknown identifier.

Q: How can I fix this issue?

A: There are several ways to fix this issue. One possible solution is to add more type annotations to the code to make it clear what type the quantifier is supposed to be. For example:

let xs : {'n, 'n >= 2. vector('n, string)} = config foo.bar;

becomes:

let xs : {'n, 'n >= 2. vector('n, string)} (k_n : Nat) = config foo.bar;

This code adds a type annotation to the k_n quantifier, making it clear that it is supposed to be a Nat.

Q: Is there a way to avoid adding type annotations?

A: Yes, there are several ways to avoid adding type annotations. One possible solution is to use a different type theory that does not have this issue. For example, Lean has a type theory called hott that does not have this issue.

Q: Can I use a different type theory in my Lean code?

A: Yes, you can use a different type theory in your Lean code. To do this, you need to add a type_theory directive to your Lean code. For example:

type_theory hott

This code tells Lean to use the hott type theory.

Q: What are the benefits of using a different type theory?

A: The benefits of using a different type theory include:

Avoiding the issue of types with numeric quantifiers producing unknown identifiers
Having more flexibility in your Lean code
Being able to use different type theories for different parts of your code

Q: Are there any downsides to using a different type theory?

A: Yes, there are several downsides to using a different type theory. These include:

Increased complexity in your Lean code
Potential issues with compatibility between different type theories
Potential issues with performance

Q: Can I use a different type theory in combination with other solutions?

A: Yes, you can use a different type theory in combination with other solutions. For example, you can use a different type theory and add type annotations to your code.

Q: What is the best solution for this issue?

A: The best solution for this issue depends on your specific use case and requirements. If you are looking for a simple solution, adding type annotations to your code may be the best option. If you are looking for a more flexible solution, using a different type theory may be the best option.

Conclusion

Future Work

References

Acknowledgments

We would like to thank the Lean community for their help and support in addressing this issue. We would also like to thank the developers of Lean and Sail for their hard work and dedication to creating a powerful and flexible proof assistant and programming language.