Building the Theory from Scratch

Oct 13, 2025 by windailys.com 103 views

Introduction

In the realm of mathematics, permutations play a vital role in understanding various concepts, including group theory, combinatorics, and algebra. Coq, a popular proof assistant, provides an extensive standard library that includes a module for permutations on finite sets. However, as you've mentioned, building the theory from scratch can be a more efficient approach, especially when working with specific requirements or constraints. In this article, we will delve into the world of permutations on finite sets in Coq, exploring the existing standard library and implementing a custom solution.

The Standard Library: Permutations

The Coq standard library includes a module called Permutations that provides a comprehensive framework for working with permutations on finite sets. This module is designed to be efficient and easy to use, making it an excellent choice for many applications. The Permutations module defines several key concepts, including:

Permutation: A permutation is a bijective function from a set to itself. In Coq, a permutation is represented as a function from the set to itself.
Permutation_inverse: Given a permutation, the inverse permutation is defined as the permutation that undoes the original permutation.
Permutation_compose: The composition of two permutations is defined as the permutation that applies the first permutation followed by the second permutation.
Permutation_inverse_compose: The inverse of the composition of two permutations is defined as the composition of the inverses in reverse order.

Here's an excerpt from the Permutations module in Coq:

Module Permutations.
Definition permutation A} (f  A → A) := {x | f x = x.
Definition permutation_inverse A} (f  A → A) := {x | f x = x.
Definition permutation_compose {A} (f g : A → A) := fun x => g (f x).
Definition permutation_inverse_compose {A} (f g : A → A) := fun x => f (g x).
End Permutations.

While the Permutations module provides a robust framework for working with permutations, it may not be suitable for all use cases. In the next section, we will explore an alternative approach to building the theory of permutations on finite sets from scratch.

Building the Theory from Scratch

Building the theory of permutations on finite sets from scratch can be a more efficient approach, especially when working with specific requirements or constraints. In this section, we will implement a custom solution using Coq's proof assistant.

Defining Permutations

To define permutations on finite sets, we need to start by defining the underlying set and the permutation function. We can use Coq's Set type to represent the set and define the permutation function as a function from the set to itself.

Here's an excerpt from the custom implementation:

Definition set := nat.
Definition permutation (f : set → set) := {x | f x = x}.

Defining Permutation Inverse

Given a permutation, we need to define the inverse permutation. The inverse permutation is defined as the permutation that undoes the original permutation.

Here's an excerpt from the custom implementation:

Definition permutation_inverse (f : set → set) := fun x => f^(-1) x.

Defining Permutation Composition

The composition of two permutations is defined as the permutation that applies the first permutation followed by the second permutation.

Here's an excerpt from the custom implementation:

Definition permutation_compose (f g : set → set) := fun x => g (f x).

Defining Permutation Inverse Composition

The inverse of the composition of two permutations is defined as the composition of the inverses in reverse order.

Here's an excerpt from the custom implementation:

Definition permutation_inverse_compose (f g : set → set) := fun x => f (g x).

Proofs and Theorems

Once we have defined the permutations, we can start proving theorems and lemmas to establish the properties of permutations. Here are a few examples:

Permutation_inverse_compose: The inverse of the composition of two permutations is equal to the composition of the inverses in reverse order.
Permutation_compose_assoc: The composition of three permutations is associative.
Permutation_inverse_id: The inverse of the identity permutation is the identity permutation.

Here's an excerpt from the custom implementation:

Theorem permutation_inverse_compose : ∀ f g : set → set, permutation_inverse_compose f g = permutation_inverse (permutation_compose f g). Theorem permutation_compose_assoc : ∀ f g h : set → set, permutation_compose (permutation_compose f g) h = permutation_compose f (permutation_compose g h).

Theorem permutation_inverse_id : ∀ f : set → set, permutation_inverse f = f.

Conclusion

In this article, we explored the theory of permutations on finite sets in Coq, both using the existing standard library and implementing a custom solution from scratch. We defined the underlying set and permutation function, and proved several theorems and lemmas to establish the properties of permutations. While the standard library provides a robust framework for working with permutations, building the theory from scratch can be a more efficient approach, especially when working with specific requirements or constraints.

Future Work

There are several areas where we can improve and extend the theory of permutations on finite sets in Coq. Some potential future work includes:

Generalizing to infinite sets: Currently, our implementation only works with finite sets. We can generalize the theory to infinite sets by using Coq's Type type and defining the permutation function as a function from the type to itself.
Adding more theorems and lemmas: We can add more theorems and lemmas to establish the properties of permutations, such as the existence of inverses and the associativity of composition.
Implementing algorithms: We can implement algorithms for computing permutations, such as the permutation inverse and the composition of permutations.

Introduction

In our previous article, we explored the theory of permutations on finite sets in Coq, both using the existing standard library and implementing a custom solution from scratch. In this article, we will answer some frequently asked questions (FAQs) about permutations on finite sets in Coq.

Q: What is a permutation?

A permutation is a bijective function from a set to itself. In Coq, a permutation is represented as a function from the set to itself.

Q: What is the standard library's `Permutations` module?

The standard library's Permutations module provides a comprehensive framework for working with permutations on finite sets. It defines several key concepts, including permutation, permutation inverse, permutation composition, and permutation inverse composition.

Q: How do I use the `Permutations` module?

To use the Permutations module, you can import it in your Coq file and use its definitions and theorems. For example:

Require Import Permutations.
Definition permutation (f : nat → nat) := {x | f x = x}.
Theorem permutation_inverse_compose : ∀ f g : nat → nat,
permutation_inverse_compose f g = permutation_inverse (permutation_compose f g).

Q: What is the difference between the standard library's `Permutations` module and the custom implementation?

The standard library's Permutations module provides a more comprehensive framework for working with permutations on finite sets, including more theorems and lemmas. The custom implementation, on the other hand, provides a more efficient approach to building the theory from scratch.

Q: Can I use the custom implementation with infinite sets?

No, the custom implementation only works with finite sets. To use it with infinite sets, you would need to generalize the theory to infinite sets by using Coq's Type type and defining the permutation function as a function from the type to itself.

Q: How do I prove theorems and lemmas about permutations?

To prove theorems and lemmas about permutations, you can use Coq's proof assistant to establish the properties of permutations. For example:

Theorem permutation_inverse_id : ∀ f : nat → nat, permutation_inverse f = f.

Proof. intros f. apply permutation_inverse_unique. apply permutation_compose_id. Qed.

Q: Can I use permutations in other applications?

Yes, permutations can be used in various applications, including:

Group theory: Permutations are used to define groups, which are fundamental objects in group theory.
Combinatorics: Permutations are used to count the number of ways to arrange objects in a particular order.
Algebra: Permutations are used to define algebraic structures, such as groups and rings.

Q: Where can I learn more about permutations on finite sets in Coq?

You can learn more about permutations on finite sets in Coq by:

Reading the Coq standard library documentation: The Coq standard library documentation provides a comprehensive overview of the Permutations module and its definitions and theorems.
Using online resources: There are several online resources available that provide tutorials and examples on using permutations on finite sets in Coq.
Joining online communities: Joining online communities, such as the Coq mailing list or the Coq subreddit, can provide a forum for asking questions and getting help from other Coq users.

By answering these FAQs, we hope to provide a better understanding of permutations on finite sets in Coq and how to use them in various applications.