How do I get started with Coq and Isabelle?

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How do i get started with coq and isabelle?

To get started with Coq and Isabelle, two popular proof assistants, you can follow the steps outlined below:

Getting Started with Coq:

1. Installation: Visit the Coq website (https://coq.inria.fr/) and download the latest stable version of Coq for your operating system. Follow the installation instructions provided.
2. Learning Resources: Coq has excellent documentation and learning resources available. You can start with the Coq tutorial provided on the Coq website. Additionally, there are several books and online courses available that can help you learn Coq, such as "Software Foundations" and "Certified Programming with Dependent Types."
3. IDEs: Coq comes with its own integrated development environment (IDE) called CoqIDE, which provides a user-friendly interface for writing and verifying proofs. Alternatively, you can use editors like Proof General or VSCode with Coq plugins for a more customized experience.
4. Practice: Practice is key to learning Coq. Start by working through small proof exercises and gradually tackle more complex problems. The Coq community provides various libraries and proof developments that you can explore to gain further insight into advanced techniques and methodologies.

Getting Started with Isabelle:

1. Installation: Visit the Isabelle website (https://isabelle.in.tum.de/) and download the latest stable version of Isabelle for your operating system. Follow the installation instructions provided.
2. Learning Resources: Isabelle also has comprehensive documentation and learning resources available. Begin with the Isabelle tutorial provided on the Isabelle website, which covers the basics of using Isabelle and building formal proofs. The website also offers a wealth of documentation and user guides for further exploration.
3. Isabelle/jEdit: Isabelle provides its own IDE called Isabelle/jEdit, which is a customized version of the jEdit text editor. It offers advanced features tailored to Isabelle development, including interactive proof editing and document preparation.
4. Practice: Similar to Coq, practice is crucial when learning Isabelle. Start by working through small examples and exercises, gradually progressing to larger proof developments. Isabelle also provides libraries and theories that you can explore to deepen your understanding.

Remember, both Coq and Isabelle are powerful tools for formal verification and proof development, but they have their own learning curves. Patience and persistence are key as you navigate these proof assistants. Engaging with their respective communities, forums, and mailing lists can also provide valuable guidance and support along your journey.

What is the difference between Coq and Isabelle?

Coq and Isabelle are both popular proof assistants used for formal verification and proof development, but they have some key differences in their underlying principles and design:

Logical Foundations:

• Coq is based on a variant of the Calculus of Inductive Constructions (CIC), which combines dependent types, higher-order logic, and inductive definitions. It allows for the formalization of mathematics and the development of certified software.
• Isabelle, on the other hand, is based on a variant of higher-order logic called Isabelle/HOL. It provides a generic framework for formal reasoning and supports various logics, including first-order logic, higher-order logic, and set theory.

Proof Styles:

• Coq emphasizes the use of interactive and tactic-based proof development. It provides a scripting language where users construct proofs by composing tactics, which are commands that guide the proof search and apply proof steps automatically.
• Isabelle focuses on declarative proofs. Users typically write proofs as structured proof scripts using Isabelle's proof language, which provides a higher-level representation of proof steps and reasoning principles.

Proof Automation:

• Coq offers a robust automation system and a wide range of built-in tactics. It has powerful automated proof procedures, such as decision procedures and tactics for handling arithmetic, that can significantly reduce the manual effort required for proving certain properties.
• Isabelle also provides automation support, but it places more emphasis on interactive proof development. It offers proof methods and tools, such as simplifiers and proof search strategies, but the user is generally more involved in guiding and constructing the proofs.

Community and Ecosystem:

• Coq has a large and active community with a rich ecosystem of libraries and developments. It is widely used in both academia and industry, particularly in areas like formal verification of software and hardware.
• Isabelle also has a strong community and is widely used in academia and industry. It has a large collection of theories and libraries built on top of its logical foundation, making it suitable for various application domains.

Choosing between Coq and Isabelle often depends on the specific requirements of the project, the preferred proof style, and the available expertise within a given community. Both tools are powerful and capable, and learning one can provide a solid foundation for understanding formal verification principles that can be applied to the other.

Which is more capable of expressing the proof of Fermat's Last Theorem, Coq or Isabelle?

Both Coq and Isabelle are capable of expressing and formalizing the proof of Fermat's Last Theorem (FLT), as it involves advanced mathematical concepts and intricate reasoning. However, it's worth noting that FLT was proven by Andrew Wiles using a different proof approach outside the scope of these proof assistants. The proof employed advanced techniques from elliptic curves and modular forms, which required additional mathematical developments.

That being said, both Coq and Isabelle are powerful tools for formal verification and theorem proving, and they have been used to formalize and verify complex mathematical theorems. The choice between Coq and Isabelle for expressing FLT (or any other mathematical proof) depends on various factors, including the specific proof approach, the familiarity and expertise of the user, and the existing libraries and theories available within each system.

Coq, with its underlying Calculus of Inductive Constructions (CIC) and strong automation capabilities, is often preferred for proofs involving advanced type theory, dependent types, and program extraction. It has been extensively used in formalizing mathematical results and developing certified software.

Isabelle, with its foundation in higher-order logic (Isabelle/HOL) and flexible proof scripting, is well-suited for various mathematical reasoning tasks. It has been used in formalizing a wide range of mathematical theories and proofs, including complex number theory and set theory.

In summary, both Coq and Isabelle are capable of expressing and formalizing the proof of FLT, but the choice between them would depend on factors such as the specific proof approach and the user's familiarity and preferences.

Can Fermat's Last Theorem be expressed within the terms of CIC?

Yes, Fermat's Last Theorem (FLT) can be expressed within the terms of the Calculus of Inductive Constructions (CIC), which is the logical foundation of the Coq proof assistant. CIC is a type theory that combines dependent types, higher-order logic, and inductive definitions, making it expressive enough to formalize complex mathematical theorems like FLT.

In CIC, you can define the necessary concepts, such as natural numbers, exponentiation, and equations, and then express FLT as a proposition or theorem. The statement of FLT asserts that there are no positive integer solutions (a, b, c) to the equation an + bn = cn, where n is an integer greater than 2.

Using the expressive power of dependent types in CIC, you can define the relevant functions, predicates, and proof techniques to reason about the properties of integers, exponentiation, and equality. You would need to develop the necessary mathematical machinery and strategies for proving the theorem within Coq's proof development environment.

It's important to note that formalizing FLT within Coq or any other proof assistant requires significant mathematical expertise and effort. FLT itself was proven by Andrew Wiles using advanced techniques from elliptic curves and modular forms, which would also need to be incorporated into the formalization.

So the proof of FLT can be expressed with ZFC and the Peano axioms?

Yes, Fermat's Last Theorem (FLT) can indeed be expressed using the foundational system of Zermelo-Fraenkel set theory with the axiom of choice (ZFC) and the Peano axioms.

ZFC provides a set-theoretic foundation for mathematics and is widely used as the standard foundational system in modern mathematics. It includes axioms that describe sets, functions, relations, and other fundamental mathematical concepts. Using ZFC, you can define the necessary mathematical objects and operations, such as natural numbers, exponentiation, and equations, and express FLT as a proposition or theorem.

The Peano axioms, which are a set of axioms for the natural numbers, can be used to define the basic structure of the natural numbers and arithmetic operations. These axioms provide a framework for reasoning about properties of natural numbers, including exponentiation, which is essential for formulating FLT.

In the context of ZFC and the Peano axioms, FLT can be expressed as a statement about the non-existence of positive integer solutions to the equation an + bn = cn for n greater than 2.

Formalizing and proving FLT within the framework of ZFC and the Peano axioms is a highly challenging task and requires advanced mathematical techniques. The proof of FLT by Andrew Wiles involved sophisticated mathematical concepts from elliptic curves and modular forms, which would need to be incorporated into the formalization within this foundational system.

But elliptic curves and modular forms can be expressed with ZFC and PA, amirite?

Yes, you are correct. The concepts of elliptic curves and modular forms can be expressed within the foundational system of Zermelo-Fraenkel set theory with the Peano axioms (ZFC + PA).

ZFC provides a framework for set theory, allowing the definition and manipulation of sets, functions, and relations. It includes axioms that establish the existence of sets and their properties. Within ZFC, you can define the necessary mathematical objects and operations to represent elliptic curves and modular forms.

The Peano axioms (PA) provide a set of axioms that describe the basic properties of the natural numbers, including addition and multiplication. Using these axioms, you can define the natural numbers and perform arithmetic operations required for the study of elliptic curves and modular forms.

Elliptic curves are algebraic curves defined by certain equations and have rich mathematical structures. They can be represented and studied using sets, functions, and relations within ZFC. The properties and operations on elliptic curves can be expressed using the language of set theory and arithmetic operations defined by the Peano axioms.

Modular forms are complex analytic functions with certain transformation properties. These functions can also be defined and studied within the framework of ZFC + PA. The properties of modular forms can be represented using sets, functions, and relations, and their transformation properties can be described using the language of set theory.

By using ZFC + PA, one can formalize and reason about concepts related to elliptic curves and modular forms, enabling the expression of the mathematical techniques used in the proof of Fermat's Last Theorem and other related theorems.

Which is better, Lean, Coq or Isabelle?

Determining which proof assistant is "better" among Lean, Coq, and Isabelle depends on various factors, including your specific requirements, background, and personal preferences. Each proof assistant has its own strengths and characteristics. Here's a brief overview of each:

Lean:

• Lean is a modern proof assistant developed by Microsoft Research.
• It has a strong emphasis on programming and theorem proving in the context of computer science and formal verification.
• Lean has a powerful and expressive type system, and it supports dependent types and higher-order logic.
• It is designed to be highly efficient and has good support for automation and metaprogramming.
• Lean has a growing community and is gaining popularity for formalizing mathematics, computer science, and formal verification.

Coq:

• Coq is one of the oldest and most well-established proof assistants.
• It has a rich ecosystem and a large and active community.
• Coq is known for its interactive proof development and tactic-based approach, which allows for fine-grained control over the proof process.
• It supports dependent types, higher-order logic, and inductive definitions.
• Coq has excellent automation capabilities, including built-in proof procedures and tactics.
• It is widely used in academia and industry for formal verification and certified programming.

Isabelle:

• Isabelle is a versatile proof assistant that has been under development for several decades.
• It has a strong foundation in higher-order logic (Isabelle/HOL) and supports various logics and reasoning styles.
• Isabelle emphasizes declarative proof development, allowing users to write structured proof scripts.
• It provides powerful automation tools, including simplifiers, proof methods, and proof search strategies.
• Isabelle has an extensive library of theories and is well-suited for formalizing mathematics and reasoning about complex systems.
• It has a dedicated user community and is widely used in academia and industry.

Ultimately, the "best" proof assistant depends on your specific needs and preferences. It can be helpful to explore the features, documentation, and available libraries of each proof assistant, try out some examples, and consider the community support and resources available. It's also worth noting that learning one proof assistant can often facilitate learning others, as they share many fundamental concepts and principles.

Which of these 3 has the biggest market penetration among the target audience?

As an AI language model, I don't have access to real-time data on market penetration or user statistics for Lean, Coq, and Isabelle. The popularity and market penetration of proof assistants can vary over time and depend on various factors, such as the specific target audience and the application domains where these tools are used.

However, historically, Coq has had significant market penetration and a large user community. It has been widely adopted in both academia and industry for formal verification, certified programming, and theorem proving. Coq has been around for several decades and has an extensive ecosystem, including libraries, tools, and documentation.

Isabelle is also a well-established proof assistant with a dedicated user community. It has gained popularity and market penetration, particularly in academia, where it is used for formalizing mathematics, reasoning about complex systems, and interactive theorem proving.

Lean, being a relatively newer proof assistant, has been gaining attention and adoption in recent years. It has been developed with a focus on programming, formal verification, and computer science applications. Lean has gained popularity in various research communities and is actively used in academia and industry.

It's important to note that market penetration can vary across different regions, application domains, and research communities. The best way to assess the current market penetration of these proof assistants and their popularity within a specific target audience is to consult recent surveys, research papers, and online forums dedicated to formal methods and proof assistants.

What are the sources of funding for Lean, Coq and Isabelle?

The funding sources for Lean, Coq, and Isabelle come from various organizations and institutions. Here is some information regarding the funding sources for each proof assistant:

Lean:

• Lean is developed and maintained by a team at Microsoft Research.
• The development of Lean has been supported by Microsoft Research and other funding sources within Microsoft.
• Additionally, Lean has received research grants and support from academic institutions and funding agencies.

Coq:

• Coq is an open-source project and is developed by a community of researchers and developers.
• The Coq development team at INRIA (French National Institute for Research in Computer Science and Automation) has received funding from INRIA itself and other research organizations.
• Coq has also received funding from various research projects, European Union grants, and collaborations with academic institutions and industry partners.

Isabelle:

• Isabelle is also an open-source project, developed and maintained by a community of researchers and developers.
• The development of Isabelle has been supported by funding from universities and research institutions.
• Isabelle has received research grants from funding agencies, including the European Research Council (ERC) and other national and international research projects.

It's important to note that the funding sources for these proof assistants can vary over time and depend on specific projects, collaborations, and research grants. The support for these tools often comes from a combination of public funding, research institutions, industry partnerships, and community contributions.

For the most up-to-date and specific information on the funding sources of Lean, Coq, and Isabelle, it is recommended to refer to the respective project websites, research papers, and announcements from the development teams and institutions involved.

Are the developers of these packages considered colloquially and among their colleagues to be logicians, mathematicians, computer scientists or phrenologists?

The developers of Lean, Coq, and Isabelle are generally considered to be a diverse mix of logicians, mathematicians, and computer scientists. These proof assistants have been developed by teams of researchers, developers, and contributors who often have interdisciplinary backgrounds and expertise.

Given the nature of these tools, which combine logical foundations, formal methods, and computer science, the developers typically have a strong foundation in mathematical logic and formal reasoning. They are knowledgeable in areas such as type theory, proof theory, formal semantics, and automated theorem proving.

Mathematicians play a significant role in the development of these proof assistants as they provide mathematical expertise and contribute to formalizing mathematical theories, proving theorems, and exploring advanced mathematical concepts within these systems.

Computer scientists also play a crucial role in the development of these proof assistants, as they contribute to the design and implementation of the software, efficiency optimizations, automation techniques, and integration with programming languages and tools.

The specific backgrounds of the developers can vary, and many individuals involved in the development of these proof assistants have a combination of expertise in logic, mathematics, and computer science.

In summary, the developers of Lean, Coq, and Isabelle are typically considered to be a mix of logicians, mathematicians, and computer scientists, reflecting the interdisciplinary nature of these tools and their applications.

But if you went to a typical mathematics department, would anybody care about the formalization of proofs that people already know and accept? Isn't that work redundant? Why pay for it?

The perception and value of formalizing proofs within a typical mathematics department can vary among individuals and academic institutions. While there may be mathematicians who view the formalization of known and accepted proofs as redundant, there are several reasons why formalization work is still valued and pursued:

1. Rigor and Certainty: Formalizing proofs provides an opportunity to establish a higher level of rigor and certainty in mathematics. By translating mathematical arguments into a formal system, potential ambiguities, hidden assumptions, or errors can be exposed and resolved. This contributes to the overall reliability and trustworthiness of mathematical knowledge.
2. Verification and Validation: Formalized proofs enable verification and validation by providing a machine-checkable representation of mathematical arguments. This allows for rigorous checking of the correctness of proofs, eliminating human errors and enhancing confidence in the validity of results. Formal verification can be particularly relevant in critical domains where correctness is of utmost importance, such as safety-critical systems or cryptography.
3. Deepening Understanding: The process of formalizing known proofs often involves a deep examination of the underlying mathematical concepts, assumptions, and reasoning. It can lead to a deeper understanding of the proofs themselves, uncovering new insights, connections, or generalizations that may not have been apparent before. Formalization work can also facilitate the exploration of alternative proof strategies or the discovery of new results.
4. Pedagogy and Education: Formalized proofs can serve as valuable educational tools. They can be used to teach mathematics, logic, and formal reasoning, offering students an opportunity to engage with foundational concepts and develop strong analytical skills. Formalization work can also contribute to the development of interactive learning environments and educational resources.
5. Bridging the Gap: Formalization work aims to bridge the gap between mathematics and computer science, fostering collaboration and cross-pollination between these disciplines. It allows for the integration of formal methods and automated reasoning techniques into mathematical practice, leading to new avenues for research and applications.

It's worth noting that formalization work is often pursued by researchers, institutions, and funding agencies with a particular interest in foundational research, formal methods, and the development of reliable software and hardware systems. While not all mathematicians may engage in formalization directly, there is recognition of the value and impact that formalization can have on the field of mathematics and related domains.

Ultimately, the decision to invest resources in formalization work depends on the specific goals, priorities, and perspectives of individual mathematicians, academic institutions, and funding agencies.

But would a real mathematician (somebody who does complex analysis and stuff) consider a logician to be a real mathematician?

The question of whether a mathematician considers a logician to be a "real mathematician" can vary among individuals and depend on personal perspectives and biases. Mathematics is a broad and diverse field, encompassing various subfields, including pure mathematics, applied mathematics, mathematical logic, and more. Different mathematicians may have different views on what constitutes "real mathematics" and the boundaries of the field.

However, it's important to note that mathematical logic, which is a branch of mathematics concerned with formal systems, proof theory, and mathematical reasoning, is widely recognized and respected within the mathematical community. Mathematical logic plays a fundamental role in foundational research, formal methods, and the study of mathematical structures and proofs.

Many mathematicians value and appreciate the contributions of logicians, recognizing the importance of formal logic and mathematical reasoning as essential tools for advancing mathematical knowledge. Logicians often contribute to deepening our understanding of mathematical concepts, developing new proof techniques, and exploring the foundations of mathematics itself.

In academia, mathematicians and logicians often collaborate and engage in interdisciplinary research, recognizing the mutual benefits and synergies between their respective fields. There are mathematicians who specialize in areas such as set theory, model theory, proof theory, or computability theory, which are firmly rooted in mathematical logic.

While individual opinions can vary, it is generally acknowledged that mathematicians and logicians are both valuable contributors to the broader mathematical enterprise. Their expertise, perspectives, and collaborations contribute to the growth and advancement of mathematics as a whole.

Thank you, you have refined my thinking!

You're welcome! I'm glad I could assist you and contribute to refining your thinking. If you have any more questions or need further clarification, feel free to ask. Happy exploring and learning!