Rewrite Tactic Fails When Using A Simplification Rule With Binders

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Introduction

In the realm of automated theorem proving and proof assistants, the rewrite tactic is a powerful tool for simplifying complex expressions. However, when attempting to use this tactic with a simplification rule that includes binders, such as a universal quantifier, the rewrite tactic may fail to find a matching subterm. This issue can be particularly frustrating when working with complex mathematical proofs. In this article, we will delve into the problem of using the rewrite tactic with binders and explore possible solutions.

The Problem

When using the rewrite tactic with a simplification rule that includes binders, the tactic may fail to find a matching subterm. This is because the binder, such as a universal quantifier, introduces a new scope for the variable, making it difficult for the rewrite tactic to identify a matching subterm.

For example, consider the simplification rule forallSimp, which equates ∀ x, p with p. When attempting to apply this rule using the rewrite tactic, the tactic may fail to find a matching subterm, resulting in the error message "No subterm ... matches ...".

A Concrete Example

To illustrate this issue, let us consider a concrete example. Suppose we have the following expression:

∀ x, P x ∧ Q x

We would like to apply the forallSimp rule to simplify this expression. However, when attempting to use the rewrite tactic, we encounter the following error message:

No subterm ∀ x, P x ∧ Q x matches ∀ x, p

This error message indicates that the rewrite tactic is unable to find a matching subterm for the forallSimp rule.

Possible Solutions

There are several possible solutions to this issue. One approach is to use a different simplification rule that does not involve binders. For example, we could use the simp rule, which is a more general simplification rule that does not involve binders.

Another approach is to use a tactic that is specifically designed to handle binders, such as the forallSimp tactic. This tactic is designed to simplify expressions involving universal quantifiers and can be used in conjunction with the rewrite tactic to simplify complex expressions.

Rewrite Tactic with Binders

In some cases, it may be possible to use the rewrite tactic with binders by modifying the simplification rule to avoid the use of binders. For example, we could rewrite the forallSimp rule to use a different notation, such as ∀ x ∈ A, p instead of ∀ x, p.

However, this approach can be cumbersome and may not always be possible. In such cases, it may be necessary to use a different tactic or simplification rule that is specifically designed to handle binders.

Conclusion

In conclusion, the rewrite tactic can fail to find a matching subterm when used with a simplification rule that includes binders. This issue can be particularly frustrating when working with complex mathematical proofs. However, by using a different simplification rule or tactic, it may be possible to overcome this issue and simplify complex expressions.

Future Work

Future work on this issue may involve developing new tactics or simplification rules that are specifically designed to handle binders. Additionally, it may be possible to modify the rewrite tactic to better handle binders and simplify complex expressions.

References

Appendix

The following is a list of additional resources that may be helpful in understanding this issue:

Introduction

In our previous article, we discussed the issue of using the rewrite tactic with a simplification rule that includes binders, such as a universal quantifier. This issue can be particularly frustrating when working with complex mathematical proofs. In this article, we will answer some frequently asked questions (FAQs) related to this issue.

Q: What is the rewrite tactic?

A: The rewrite tactic is a powerful tool for simplifying complex expressions in automated theorem proving and proof assistants. It allows users to replace subterms with simpler expressions, making it easier to prove theorems.

Q: What is a binder?

A: A binder is a construct that introduces a new scope for a variable. Examples of binders include universal quantifiers (∀), existential quantifiers (∃), and lambda abstractions (λ).

Q: Why does the rewrite tactic fail when using a simplification rule with binders?

A: The rewrite tactic fails when using a simplification rule with binders because the binder introduces a new scope for the variable, making it difficult for the rewrite tactic to identify a matching subterm.

Q: How can I overcome this issue?

A: There are several possible solutions to this issue. One approach is to use a different simplification rule that does not involve binders. Another approach is to use a tactic that is specifically designed to handle binders, such as the forallSimp tactic.

Q: Can I modify the simplification rule to avoid the use of binders?

A: Yes, it may be possible to modify the simplification rule to avoid the use of binders. However, this approach can be cumbersome and may not always be possible.

Q: What are some common pitfalls to avoid when using the rewrite tactic with binders?

A: Some common pitfalls to avoid when using the rewrite tactic with binders include:

  • Using a simplification rule that involves binders without properly handling the binder.
  • Failing to identify the correct subterm to replace.
  • Not properly handling the scope of the binder.

Q: How can I debug the rewrite tactic when it fails to find a matching subterm?

A: To debug the rewrite tactic when it fails to find a matching subterm, you can try the following:

  • Check the simplification rule to ensure that it is properly handling the binder.
  • Verify that the subterm being replaced is correctly identified.
  • Use a tactic that is specifically designed to handle binders, such as the forallSimp tactic.

Q: Are there any tools or resources available to help me overcome this issue?

A: Yes, there are several tools and resources available to help you overcome this issue. Some examples include:

  • The forallSimp tactic, which is specifically designed to handle universal quantifiers.
  • The simp rule, which is a more general simplification rule that does not involve binders.
  • The rewrite tactic, which can be used in conjunction with the forallSimp tactic to simplify complex expressions.

Conclusion

In conclusion, the rewrite tactic can fail to find a matching subterm when used with a simplification rule that includes binders. However, by understanding the issue and using the right tools and resources, you can overcome this challenge and simplify complex expressions.

Additional Resources