Pub Date : 2022-04-26DOI: 10.48550/arXiv.2204.12311
Karol Pkak, C. Kaliszyk
The DPRM (Davis-Putnam-Robinson-Matiyasevich) theorem is the main step in the negative resolution of Hilbert's 10th problem. Almost three decades of work on the problem have resulted in several equally surprising results. These include the existence of diophantine equations with a reduced number of variables, as well as the explicit construction of polynomials that represent specific sets, in particular the set of primes. In this work, we formalize these constructions in the Mizar system. We focus on the set of prime numbers and its explicit representation using 10 variables. It is the smallest representation known today. For this, we show that the exponential function is diophantine, together with the same properties for the binomial coefficient and factorial. This formalization is the next step in the research on formal approaches to diophantine sets following the DPRM theorem.
{"title":"Formalizing a Diophantine Representation of the Set of Prime Numbers","authors":"Karol Pkak, C. Kaliszyk","doi":"10.48550/arXiv.2204.12311","DOIUrl":"https://doi.org/10.48550/arXiv.2204.12311","url":null,"abstract":"The DPRM (Davis-Putnam-Robinson-Matiyasevich) theorem is the main step in the negative resolution of Hilbert's 10th problem. Almost three decades of work on the problem have resulted in several equally surprising results. These include the existence of diophantine equations with a reduced number of variables, as well as the explicit construction of polynomials that represent specific sets, in particular the set of primes. In this work, we formalize these constructions in the Mizar system. We focus on the set of prime numbers and its explicit representation using 10 variables. It is the smallest representation known today. For this, we show that the exponential function is diophantine, together with the same properties for the binomial coefficient and factorial. This formalization is the next step in the research on formal approaches to diophantine sets following the DPRM theorem.","PeriodicalId":296683,"journal":{"name":"International Conference on Interactive Theorem Proving","volume":"2002 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128306036","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-03-06DOI: 10.48550/arXiv.2203.16344
Mar'ia In'es de Frutos-Fern'andez
The ring of adèles of a global field and its group of units, the group of idèles, are fundamental objects in modern number theory. We discuss a formalization of their definitions in the Lean 3 theorem prover. As a prerequisite, we formalize adic valuations on Dedekind domains. We present some applications, including the statement of the main theorem of global class field theory and a proof that the ideal class group of a number field is isomorphic to an explicit quotient of its idèle class group. Acknowledgements I would like to thank Kevin Buzzard for his constant support and for many helpful conversations during the completion of this project, and Ashvni Narayanan for pointing out that the finite adèle ring can be defined for any Dedekind domain. I am also grateful to Patrick Massot for making some of the topological prerequisites available in mathlib , and to Sebastian Monnet for formalizing the topology on the infinite Galois group. Finally, I thank the mathlib community for their helpful advice, and the mathlib maintainers for the insightful reviews of the parts of this project already submitted to the library.
{"title":"Formalizing the Ring of Adèles of a Global Field","authors":"Mar'ia In'es de Frutos-Fern'andez","doi":"10.48550/arXiv.2203.16344","DOIUrl":"https://doi.org/10.48550/arXiv.2203.16344","url":null,"abstract":"The ring of adèles of a global field and its group of units, the group of idèles, are fundamental objects in modern number theory. We discuss a formalization of their definitions in the Lean 3 theorem prover. As a prerequisite, we formalize adic valuations on Dedekind domains. We present some applications, including the statement of the main theorem of global class field theory and a proof that the ideal class group of a number field is isomorphic to an explicit quotient of its idèle class group. Acknowledgements I would like to thank Kevin Buzzard for his constant support and for many helpful conversations during the completion of this project, and Ashvni Narayanan for pointing out that the finite adèle ring can be defined for any Dedekind domain. I am also grateful to Patrick Massot for making some of the topological prerequisites available in mathlib , and to Sebastian Monnet for formalizing the topology on the infinite Galois group. Finally, I thank the mathlib community for their helpful advice, and the mathlib maintainers for the insightful reviews of the parts of this project already submitted to the library.","PeriodicalId":296683,"journal":{"name":"International Conference on Interactive Theorem Proving","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122209113","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-02-10DOI: 10.4230/LIPIcs.ITP.2022.10
F. Dupuis, R. Lewis, H. Macbeth
Semilinear maps are a generalization of linear maps between vector spaces where we allow the scalar action to be twisted by a ring homomorphism such as complex conjugation. In particular, this generalization unifies the concepts of linear and conjugate-linear maps. We implement this generalization in Lean's textsf{mathlib} library, along with a number of important results in functional analysis which previously were impossible to formalize properly. Specifically, we prove the Fr'echet--Riesz representation theorem and the spectral theorem for compact self-adjoint operators generically over real and complex Hilbert spaces. We also show that semilinear maps have applications beyond functional analysis by formalizing the one-dimensional case of a theorem of Dieudonn'e and Manin that classifies the isocrystals over an algebraically closed field with positive characteristic.
{"title":"Formalized functional analysis with semilinear maps","authors":"F. Dupuis, R. Lewis, H. Macbeth","doi":"10.4230/LIPIcs.ITP.2022.10","DOIUrl":"https://doi.org/10.4230/LIPIcs.ITP.2022.10","url":null,"abstract":"Semilinear maps are a generalization of linear maps between vector spaces where we allow the scalar action to be twisted by a ring homomorphism such as complex conjugation. In particular, this generalization unifies the concepts of linear and conjugate-linear maps. We implement this generalization in Lean's textsf{mathlib} library, along with a number of important results in functional analysis which previously were impossible to formalize properly. Specifically, we prove the Fr'echet--Riesz representation theorem and the spectral theorem for compact self-adjoint operators generically over real and complex Hilbert spaces. We also show that semilinear maps have applications beyond functional analysis by formalizing the one-dimensional case of a theorem of Dieudonn'e and Manin that classifies the isocrystals over an algebraically closed field with positive characteristic.","PeriodicalId":296683,"journal":{"name":"International Conference on Interactive Theorem Proving","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130291025","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-02-11DOI: 10.4230/LIPIcs.ITP.2021.26
M. Maggesi, C. Brogi
This work presents a formalized proof of modal completeness for G"odel-L"ob provability logic (GL) in the HOL Light theorem prover. We describe the code we developed, and discuss some details of our implementation, focusing on our choices in structuring proofs which make essential use of the tools of HOL Light and which differ in part from the standard strategies found in main textbooks covering the topic in an informal setting. Moreover, we propose a reflection on our own experience in using this specific theorem prover for this formalization task, with an analysis of pros and cons of reasoning within and about the formal system for GL we implemented in our code.
{"title":"A formal proof of modal completeness for provability logic","authors":"M. Maggesi, C. Brogi","doi":"10.4230/LIPIcs.ITP.2021.26","DOIUrl":"https://doi.org/10.4230/LIPIcs.ITP.2021.26","url":null,"abstract":"This work presents a formalized proof of modal completeness for G\"odel-L\"ob provability logic (GL) in the HOL Light theorem prover. We describe the code we developed, and discuss some details of our implementation, focusing on our choices in structuring proofs which make essential use of the tools of HOL Light and which differ in part from the standard strategies found in main textbooks covering the topic in an informal setting. Moreover, we propose a reflection on our own experience in using this specific theorem prover for this formalization task, with an analysis of pros and cons of reasoning within and about the formal system for GL we implemented in our code.","PeriodicalId":296683,"journal":{"name":"International Conference on Interactive Theorem Proving","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128633006","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-02-05DOI: 10.4230/LIPIcs.ITP.2021.14
Katherine Cordwell, Yong Kiam Tan, André Platzer
We formalize the univariate fragment of Ben-Or, Kozen, and Reif’s (BKR) decision procedure for first-order real arithmetic in Isabelle/HOL. BKR’s algorithm has good potential for parallelism and was designed to be used in practice. Its key insight is a clever recursive procedure that computes the set of all consistent sign assignments for an input set of univariate polynomials while carefully managing intermediate steps to avoid exponential blowup from naively enumerating all possible sign assignments (this insight is fundamental for both the univariate case and the general case). Our proof combines ideas from BKR and a follow-up work by Renegar that are well-suited for formalization. The resulting proof outline allows us to build substantially on Isabelle/HOL’s libraries for algebra, analysis, and matrices. Our main extensions to existing libraries are also detailed.
我们形式化了Isabelle/HOL中一阶实数算法的Ben-Or, Kozen, and Reif (BKR)决策过程的单变量片段。BKR算法具有良好的并行化潜力，并设计用于实际应用。它的关键洞察力是一个聪明的递归过程，它为一组单变量多项式的输入计算所有一致的符号赋值集合，同时小心地管理中间步骤，以避免天真地枚举所有可能的符号赋值而导致指数爆炸(这种洞察力对于单变量情况和一般情况都是基本的)。我们的证明结合了BKR的想法和Renegar的后续工作，非常适合形式化。由此产生的证明大纲允许我们在Isabelle/HOL的代数、分析和矩阵库的基础上进行大量构建。我们对现有库的主要扩展也有详细说明。
{"title":"A Verified Decision Procedure for Univariate Real Arithmetic with the BKR Algorithm","authors":"Katherine Cordwell, Yong Kiam Tan, André Platzer","doi":"10.4230/LIPIcs.ITP.2021.14","DOIUrl":"https://doi.org/10.4230/LIPIcs.ITP.2021.14","url":null,"abstract":"We formalize the univariate fragment of Ben-Or, Kozen, and Reif’s (BKR) decision procedure for first-order real arithmetic in Isabelle/HOL. BKR’s algorithm has good potential for parallelism and was designed to be used in practice. Its key insight is a clever recursive procedure that computes the set of all consistent sign assignments for an input set of univariate polynomials while carefully managing intermediate steps to avoid exponential blowup from naively enumerating all possible sign assignments (this insight is fundamental for both the univariate case and the general case). Our proof combines ideas from BKR and a follow-up work by Renegar that are well-suited for formalization. The resulting proof outline allows us to build substantially on Isabelle/HOL’s libraries for algebra, analysis, and matrices. Our main extensions to existing libraries are also detailed.","PeriodicalId":296683,"journal":{"name":"International Conference on Interactive Theorem Proving","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134428645","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-02-04DOI: 10.4230/LIPIcs.ITP.2021.18
Floris van Doorn
We describe the formalization of the existence and uniqueness of Haar measure in the Lean theorem prover. The Haar measure is an invariant regular measure on locally compact groups, and it has not been formalized in a proof assistant before. We will also discuss the measure theory library in Lean's mathematical library textsf{mathlib}, and discuss the construction of product measures and the proof of Fubini's theorem for the Bochner integral.
{"title":"Formalized Haar Measure","authors":"Floris van Doorn","doi":"10.4230/LIPIcs.ITP.2021.18","DOIUrl":"https://doi.org/10.4230/LIPIcs.ITP.2021.18","url":null,"abstract":"We describe the formalization of the existence and uniqueness of Haar measure in the Lean theorem prover. The Haar measure is an invariant regular measure on locally compact groups, and it has not been formalized in a proof assistant before. We will also discuss the measure theory library in Lean's mathematical library textsf{mathlib}, and discuss the construction of product measures and the proof of Fubini's theorem for the Bochner integral.","PeriodicalId":296683,"journal":{"name":"International Conference on Interactive Theorem Proving","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125427600","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-10-03DOI: 10.4230/LIPICS.ITP.2021.7
Christoph Benzmüller, D. Fuenmayor
As quantum computing steadily progresses from theory to practice, programmers are faced with a common problem: How can they be sure that their code does what they intend it to do? This paper presents encouraging results in the application of mechanized proof to the domain of quantum programming in the context of the SQIR development. It verifies the correctness of a range of a quantum algorithms including Simon's algorithm, Grover's algorithm, and quantum phase estimation, a key component of Shor's algorithm. In doing so, it aims to highlight both the successes and challenges of formal verification in the quantum context and motivate the theorem proving community to target quantum computing as an application domain.
{"title":"Value-Oriented Legal Argumentation in Isabelle/HOL","authors":"Christoph Benzmüller, D. Fuenmayor","doi":"10.4230/LIPICS.ITP.2021.7","DOIUrl":"https://doi.org/10.4230/LIPICS.ITP.2021.7","url":null,"abstract":"As quantum computing steadily progresses from theory to practice, programmers are faced with a common problem: How can they be sure that their code does what they intend it to do? This paper presents encouraging results in the application of mechanized proof to the domain of quantum programming in the context of the SQIR development. It verifies the correctness of a range of a quantum algorithms including Simon's algorithm, Grover's algorithm, and quantum phase estimation, a key component of Shor's algorithm. In doing so, it aims to highlight both the successes and challenges of formal verification in the quantum context and motivate the theorem proving community to target quantum computing as an application domain.","PeriodicalId":296683,"journal":{"name":"International Conference on Interactive Theorem Proving","volume":"75 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127398195","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-02-14DOI: 10.4230/LIPIcs.ITP.2021.13
L. Ciccone
Theorem provers are tools that help users to write machine readable proofs. Some of this tools are also interactive. The need of such softwares is increasing since they provide proofs that are more certified than the hand written ones. Agda is based on type theory and on the propositions-as-types correspondence and has a Haskell-like syntax. This means that a proof of a statement is turned into a function. Inference systems are a way of defining inductive and coinductive predicates and induction and coinduction principles are provided to help proving their correctness with respect to a given specification in terms of soundness and completeness. Generalized inference systems deal with predicates whose inductive and coinductive interpretations do not provide the expected set of judgments. In this case inference systems are enriched by corules that are rules that can be applied at infinite depth in a proof tree. Induction and coinduction principles cannot be used in case of generalized inference systems and the bounded coinduction one has been proposed. We first present how Agda supports inductive and coinductive types highlighting the fact that data structures and predicates are defined using the same constructs. Then we move to the main topic of this thesis, which is investigating how generalized inference systems can be implemented and how their correctness can be proved.
{"title":"Flexible Coinduction in Agda","authors":"L. Ciccone","doi":"10.4230/LIPIcs.ITP.2021.13","DOIUrl":"https://doi.org/10.4230/LIPIcs.ITP.2021.13","url":null,"abstract":"Theorem provers are tools that help users to write machine readable proofs. Some of this tools are also interactive. The need of such softwares is increasing since they provide proofs that are more certified than the hand written ones. Agda is based on type theory and on the propositions-as-types correspondence and has a Haskell-like syntax. This means that a proof of a statement is turned into a function. Inference systems are a way of defining inductive and coinductive predicates and induction and coinduction principles are provided to help proving their correctness with respect to a given specification in terms of soundness and completeness. Generalized inference systems deal with predicates whose inductive and coinductive interpretations do not provide the expected set of judgments. In this case inference systems are enriched by corules that are rules that can be applied at infinite depth in a proof tree. Induction and coinduction principles cannot be used in case of generalized inference systems and the bounded coinduction one has been proposed. We first present how Agda supports inductive and coinductive types highlighting the fact that data structures and predicates are defined using the same constructs. Then we move to the main topic of this thesis, which is investigating how generalized inference systems can be implemented and how their correctness can be proved.","PeriodicalId":296683,"journal":{"name":"International Conference on Interactive Theorem Proving","volume":"94 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115821789","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-09-08DOI: 10.4230/LIPICS.ITP.2019.8
F. Bréhard, A. Mahboubi, D. Pous
We present a library to verify rigorous approximations of univariate functions on real numbers, with the Coq proof assistant. Based on interval arithmetic, this library also implements a technique of validation a posteriori based on the Banach fixed-point theorem. We illustrate this technique on the case of operations of division and square root. This library features a collection of abstract structures that organise the specfication of rigorous approximations, and modularise the related proofs. Finally, we provide an implementation of verified Chebyshev approximations, and we discuss a few examples of computations.
{"title":"A Certificate-Based Approach to Formally Verified Approximations","authors":"F. Bréhard, A. Mahboubi, D. Pous","doi":"10.4230/LIPICS.ITP.2019.8","DOIUrl":"https://doi.org/10.4230/LIPICS.ITP.2019.8","url":null,"abstract":"We present a library to verify rigorous approximations of univariate functions on real numbers, with \u0000the Coq proof assistant. Based on interval arithmetic, this library also implements a technique of \u0000validation a posteriori based on the Banach fixed-point theorem. We illustrate this technique on \u0000the case of operations of division and square root. This library features a collection of abstract \u0000structures that organise the specfication of rigorous approximations, and modularise the related \u0000proofs. Finally, we provide an implementation of verified Chebyshev approximations, and we discuss \u0000a few examples of computations.","PeriodicalId":296683,"journal":{"name":"International Conference on Interactive Theorem Proving","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123799805","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-09-08DOI: 10.4230/LIPIcs.ITP.2019.18
Armaël Guéneau, Jacques-Henri Jourdan, A. Charguéraud, F. Pottier
We study a state-of-the-art incremental cycle detection algorithm due to Bender, Fineman, Gilbert, and Tarjan. We propose a simple change that allows the algorithm to be regarded as genuinely online. Then, we exploit Separation Logic with Time Credits to simultaneously verify the correctness and the worst-case amortized asymptotic complexity of the modified algorithm.
{"title":"Formal Proof and Analysis of an Incremental Cycle Detection Algorithm","authors":"Armaël Guéneau, Jacques-Henri Jourdan, A. Charguéraud, F. Pottier","doi":"10.4230/LIPIcs.ITP.2019.18","DOIUrl":"https://doi.org/10.4230/LIPIcs.ITP.2019.18","url":null,"abstract":"We study a state-of-the-art incremental cycle detection algorithm due to Bender, Fineman, Gilbert, and Tarjan. We propose a simple change that allows the algorithm to be regarded as genuinely online. Then, we exploit Separation Logic with Time Credits to simultaneously verify the correctness and the worst-case amortized asymptotic complexity of the modified algorithm.","PeriodicalId":296683,"journal":{"name":"International Conference on Interactive Theorem Proving","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127947199","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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