Emmanuel Gunther, Miguel Pagano, Pedro Sánchez Terraf
We formalize the theory of forcing in the set theory framework of Isabelle/ZF. Under the assumption of the existence of a countable transitive model of ZFC, we construct a proper generic extension and show that the latter also satisfies ZFC. In doing so, we remodularized Paulson's ZF-Constructibility library.
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We give an overview of our formalizations in the proof assistant Isabelle/HOL of certain irrationality and transcendence criteria for infinite series from three different research papers: by Erdős and Straus (1974), Hančl (2002), and Hančl and Rucki (2005). Our formalizations in Isabelle/HOL can be found on the Archive of Formal Proofs. Here we describe selected aspects of the formalization and discuss what this reveals about the use and potential of Isabelle/HOL in formalizing modern mathematical research, particularly in these parts of number theory and analysis.
A protocol-independent secrecy theorem is established and applied to several non-trivial protocols. In particular, it is applied to protocols proposed for protecting the computation results of free-roaming mobile agents doing comparison shopping. All the results presented here have been formally proved in Isabelle by building on Larry Paulson's inductive approach. This therefore provides a library of general theorems that can be applied to other protocols.
Isabelle is a generic theorem prover with a fragment of higher-order logic as a metalogic for defining object logics. Isabelle also provides proof terms. We formalize this metalogic and the language of proof terms in Isabelle/HOL, define an executable (but inefficient) proof term checker and prove its correctness w.r.t. the metalogic. We integrate the proof checker with Isabelle and run it on a range of logics and theories to check the correctness of all the proofs in those theories.
We present a formalization of basics related to infinite words in the generic proof assistant Isabelle/HOL. Furthermore, we present a formalization of purely morphic and morphic languages. Finally, we present a formalized definition of Sturmian words as lower mechanical words and prove some very elementary facts. The formalization is based on an ongoing larger project of formalization of combinatorics on words.
We present the HOLCF-Prelude, a formalization of a large part of Haskell's standard prelude in Isabelle/HOLCF. Applying this formalization to the hints suggested by HLint allows us to certify them formally.
Modern functional-logic programming languages like Toy or Curry feature non-strict non-deterministic functions that behave under call-time choice semantics. A standard formulation for this semantics is the CRWL logic, that specifies a proof calculus for computing the set of possible results for each expression. In this paper we present a formalization of that calculus in the Isabelle/HOL proof assistant. We have proved some basic properties of CRWL: closedness under c-substitutions, polarity and compositionality. We also discuss some insights that have been gained, such as the fact that left l…
We present a new software tool for teaching logic based on natural deduction. Its proof system is formalized in the proof assistant Isabelle such that its definition is very precise. Soundness of the formalization has been proved in Isabelle. The tool is open source software developed in TypeScript / JavaScript and can thus be used directly in a browser without any further installation. Although developed for undergraduate computer science students who are used to study and program concrete computer code in a programming language we consider the approach relevant for a broader audience and for…
We present an Isabelle/HOL formalization of an earlier result by Suzuki, Middeldorp, and Ida; namely that a certain class of conditional rewrite systems is level-confluent. Our formalization is basically along the lines of the original proof, from which we deviate mostly in the level of detail as well as concerning some basic definitions.
Category Theory in Isabelle/HOL as a Basis for Meta-logical Investigation
本文基于证明助手Isabelle/HOL，利用自由逻辑的浅层语义嵌入对范畴论进行元逻辑研究，形式化了初等拓扑斯与对称幺半闭范畴，并构建了易于使用的Isabelle范畴论库，为未来形式化研究奠定基础。
Formalizing Pick's Theorem in Isabelle/HOL
本研究受HOL Light中皮克定理形式化工作启发，在Isabelle/HOL中调整证明策略规避了多边形内部分割的核心证明难点，同时从基础层面扩充现有几何库，完成皮克定理的形式化并反思了策略优劣。
Proof assistants are important tools for teaching logic. We support this claim by discussing three formalizations in Isabelle/HOL used in a recent course on automated reasoning. The first is a formalization of System W (a system of classical propositional logic with only two primitive symbols), the second is the Natural Deduction Assistant (NaDeA), and the third is a one-sided sequent calculus that uses our Sequent Calculus Verifier (SeCaV). We describe each formalization in turn, concentrating on how we used them in our teaching, and commenting on features that are interesting or useful from …
A formalization of the CHSH inequality and Tsirelson's upper-bound in Isabelle/HOL
在Isabelle/HOL中形式化量子信息基本概念，证明局域隐变量假设不成立，并形式化CHSH不等式违反相关结果及Tsirelson界。
A Formalization of Martingales in Isabelle/HOL
本文在Isabelle/HOL中形式化了任意巴拿赫空间上的鞅理论，将条件期望从实数推广到巴拿赫空间，定义了随机过程及鞅类，并证明了平均定理以确立密度的唯一性。
We present an elegant, generic and extensive formalization of Gröbner bases in Isabelle/HOL. The formalization covers all of the essentials of the theory (polynomial reduction, S-polynomials, Buchberger's algorithm, Buchberger's criteria for avoiding useless pairs), but also includes more advanced features like reduced Gröbner bases. Particular highlights are the first-time formalization of Faugère's matrix-based $F_4$ algorithm and the fact that the entire theory is formulated for modules and submodules rather than rings and ideals. All formalized algorithms can be translated into executable …
L-Mosaics and Bounded Join-Semilattices in Isabelle/HOL
本文在Isabelle/HOL中完整形式化了L-mosaics与有界并半格等价性的对象部分，采用AI辅助方法验证了主要理论结果及变换互逆性，展示了多值代数运算深度与AI增强定理证明的潜力。
Isabelle/STARK: A Formalization of zk-STARK in Isabelle/HOL
该研究在 Isabelle/HOL 中完成了 STARK 风格透明证明协议的形式化，构建了含证明者/验证者可执行模型等内容的开发成果，为相关领域提供了形式化方法层面的支撑。
Formalization of Differential Privacy in Isabelle/HOL
本文提出一个通用的 Isabelle/HOL 差分隐私形式化库，首次支持连续概率分布，并验证了标准定义、Laplace 机制和 report noisy max 机制。
The proof of the relative consistency of the axiom of choice has been mechanized using Isabelle/ZF. The proof builds upon a previous mechanization of the reflection theorem. The heavy reliance on metatheory in the original proof makes the formalization unusually long, and not entirely satisfactory: two parts of the proof do not fit together. It seems impossible to solve these problems without formalizing the metatheory. However, the present development follows a standard textbook, Kunen's Set Theory, and could support the formalization of further material from that book. It also serves as an e…
Comparing provers on a formalization of the same problem is always a valuable exercise. In this paper, we present the formal proof of correctness of a non-trivial algorithm from graph theory that was carried out in three proof assistants: Why3, Coq, and Isabelle.
We present a formalization in Isabelle/HOL of quantum projective measurements, a class of measurements involving orthogonal projectors that is frequently used in quantum computing. We also formalize the CHSH inequality, a result that holds on arbitrary probability spaces, which can used to disprove the existence of a local hidden-variable theory for quantum mechanics.
We describe a formalization of forcing using Boolean-valued models in the Lean 3 theorem prover, including the fundamental theorem of forcing and a deep embedding of first-order logic with a Boolean-valued soundness theorem. As an application of our framework, we specialize our construction to the Boolean algebra of regular opens of the Cantor space $2^{ω_2 \times ω}$ and formally verify the failure of the continuum hypothesis in the resulting model.
The Isabelle Archive of Formal Proofs (AFP) has grown to over 500 articles in late 2019. Meanwhile, finding formalizations in it has not exactly become easier. At the time of writing, the site-specific AFP google search and the Isabelle find_theories resp. find_consts commands (that only work on imported theories) are still the only tools readily available to find formalizations in Isabelle. We present FindFacts, a novel domain-specific search tool for formal Isabelle theory content. Instead of utilizing term unification, we solve the problem with a classical drill-down search engine. We put s…
Complex Bounded Operators in Isabelle/HOL
本文在Isabelle/HOL中形式化了复向量空间上的有界算子，包括复范数空间、巴拿赫空间、希尔伯特空间，定义了有界算子类型cblinfun，并发展了酉算子、投影算子、有界线性函数延拓定理、伴随算子、Loewner序、闭子空间等理论，同时为有限维情况提供了与矩阵的代码生成支持。
We present the Natural Deduction Assistant (NaDeA) and discuss its advantages and disadvantages as a tool for teaching logic. NaDeA is available online and is based on a formalization of natural deduction in the Isabelle proof assistant. We first provide concise formulations of the main formalization results. We then elaborate on the prerequisites for NaDeA, in particular we describe a formalization in Isabelle of "Hilbert's Axioms" that we use as a starting point in our bachelor course on mathematical logic. We discuss a recent evaluation of NaDeA and also give an overview of the exercises in…
We describe a natural deduction formalization of intuitionistic and classical propositional logic in the Isabelle/Pure framework. In contrast to earlier work, where we explored the pedagogical benefits of using a deep embedding approach to logical modelling, here we employ a logical framework modelling. This gives rise to simple and natural teaching examples and we report on the role it played in teaching our Automated Reasoning course in 2020 and 2021.
In interactive theorem proving, formalization quality is a key factor for maintainability and re-usability of developments and can also impact proof-checking performance. Commonly, anti-patterns that cause quality issues are known to experienced users. However, in many theorem prover systems, there are no automatic tools to check for their presence and make less experienced users aware of them. We attempt to fill this gap in the Isabelle environment by developing a linter as a publicly available add-on component. The linter offers basic configurability, extensibility, Isabelle/jEdit integratio…
Munkres' General Topology Autoformalized in Isabelle/HOL
本研究利用LLM辅助在Isabelle/HOL中自动形式化Munkres泛拓扑教材，生成85000余行代码，涵盖39章节，证明包括Tychonoff定理等核心结果，方法结合声明性证明与sledgehammer工具，展现高效形式化潜力。
We present a new framework for formalizing mathematics in untyped set theory using auto2. Using this framework, we formalize in Isabelle/FOL the entire chain of development from the axioms of set theory to the definition of the fundamental group for an arbitrary topological space. The auto2 prover is used as the sole automation tool, and enables succinct proof scripts throughout the project.
How difficult are interactive theorem provers to use? We respond by reviewing the formalization of Hilbert's tenth problem in Isabelle/HOL carried out by an undergraduate research group at Jacobs University Bremen. We argue that, as demonstrated by our example, proof assistants are feasible for beginners to formalize mathematics. With the aim to make the field more accessible, we also survey hurdles that arise when learning an interactive theorem prover. Broadly, we advocate for an increased adoption of interactive theorem provers in mathematical research and curricula.
Gödel's ontological proof has been analysed for the first-time with an unprecedent degree of detail and formality with the help of higher-order theorem provers. The following has been done (and in this order): A detailed natural deduction proof. A formalization of the axioms, definitions and theorems in the TPTP THF syntax. Automatic verification of the consistency of the axioms and definitions with Nitpick. Automatic demonstration of the theorems with the provers LEO-II and Satallax. A step-by-step formalization using the Coq proof assistant. A formalization using the Isabelle proof assistant…
We describe a proof of the Central Limit Theorem that has been formally verified in the Isabelle proof assistant. Our formalization builds upon and extends Isabelle's libraries for analysis and measure-theoretic probability. The proof of the theorem uses characteristic functions, which are a kind of Fourier transform, to demonstrate that, under suitable hypotheses, sums of random variables converge weakly to the standard normal distribution. We also discuss the libraries and infrastructure that supported the formalization, and reflect on some of the lessons we have learned from the effort.
Teaching Higher-Order Logic Using Isabelle
本文基于 Isabelle/Pure 构建了一个小巧可读的高阶逻辑形式化，提供示例证明、公理规则说明及课堂教学经验，作为学习证明助手的轻量入门。
We lay the ground for an Isabelle/ZF formalization of Cohen's technique of forcing. We formalize the definition of forcing notions as preorders with top, dense subsets, and generic filters. We formalize the definition of forcing notions as preorders with top, dense subsets, and generic filters. We formalize a version of the principle of Dependent Choices and using it we prove the Rasiowa-Sikorski lemma on the existence of generic filters. Given a transitive set $M$, we define its generic extension $M[G]$, the canonical names for elements of $M$, and finally show that if $M$ satisfies the axiom…
We present an Isabelle/HOL formalization of a characterization of confluence for quasi-reductive strongly deterministic conditional term rewrite systems, due to Avenhaus and Loría-Sáenz.
Formalizing MLTL Formula Progression in Isabelle/HOL
本文在Isabelle/HOL中形式化了MLTL的语法、语义、关键属性库及公式推进算法，并验证其正确性与可执行性。
The formal verification of the ctm approach to forcing
本研究借助Isabelle证明助手与Isabelle/ZF库，形式化验证了ZFC可数传递模型的力迫兼纳扩张构造，并明确了替换公理模式中21个关键实例的作用，为连续统假设相关独立性结果提供了机器验证基础。
This paper presents a formalization of decreasing diagrams in the theorem prover Isabelle. It discusses mechanical proofs showing that any locally decreasing abstract rewrite system is confluent. The valley and the conversion version of decreasing diagrams are considered.
Formalization of Fragments of the Theory of Hereditarily Finite Sets
该研究在Isabelle/HOL中形式化遗传有穷集理论片段，采用locale层级结构对应不同公理片段，用一阶可定义谓词的归纳定义处理公理模式，分析等价的有穷性与正则性公理，并通过模型形式化公理独立性结论。
When faced with the question of how to represent properties in a formal proof system any user has to make design decisions. We have proved three of the theorems from Maskin's 2004 survey article on Auction Theory using the Isabelle/HOL system, and we have produced verified code for combinatorial Vickrey auctions. A fundamental question in this was how to represent some basic concepts: since set theory is available inside Isabelle/HOL, when introducing new definitions there is often the issue of balancing the amount of set-theoretical objects and of objects expressed using entities which are mo…
A theory of recursive definitions has been mechanized in Isabelle's Zermelo-Fraenkel (ZF) set theory. The objective is to support the formalization of particular recursive definitions for use in verification, semantics proofs and other computational reasoning. Inductively defined sets are expressed as least fixedpoints, applying the Knaster-Tarski Theorem over a suitable set. Recursive functions are defined by well-founded recursion and its derivatives, such as transfinite recursion. Recursive data structures are expressed by applying the Knaster-Tarski Theorem to a set, such as V[omega], that…
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 co…
We report on our formalization of matrix-interpretation in Isabelle/HOL. Matrices are required to certify termination proofs and we wish to utilize them for complexity proofs, too. For the latter aim, only basic methods have already been integrated, and we discuss some upcoming problems which arise when formalizing more complicated results on matrix-interpretations, which are based on Cayley-Hamilton's theorem or joint-spectral radius theory.
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