In this article we present an ongoing effort to formalise quantum algorithms and results in quantum information theory using the proof assistant Isabelle/HOL. Formal methods being critical for the safety and security of algorithms and protocols, we foresee their widespread use for quantum computing in the future. We have developed a large library for quantum computing in Isabelle based on a matrix representation for quantum circuits, successfully formalising the no-cloning theorem, quantum teleportation, Deutsch's algorithm, the Deutsch-Jozsa algorithm and the quantum Prisoner's Dilemma. We discuss the design choices made and report on an outcome of our work in the field of quantum game theory.
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.
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 …
Many problems in computer algebra and numerical analysis can be reduced to counting or approximating the real roots of a polynomial within an interval. Existing verified root-counting procedures in major proof assistants are mainly based on the classical Sturm theorem, which only counts distinct roots. In this paper, we have strengthened the root-counting ability in Isabelle/HOL by first formally proving the Budan-Fourier theorem. Subsequently, based on Descartes' rule of signs and Taylor shift, we have provided a verified procedure to efficiently over-approximate the number of real roots with…
We present the first verified implementation of a decision procedure for the quantifier-free theory of partial and linear orders. We formalise the procedure in Isabelle/HOL and provide a specification that is made executable using Isabelle's code generator. The procedure is already part of the development version of Isabelle as a sub-procedure of the simplifier.
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 present a semantic framework for the deductive verification of hybrid systems with Isabelle/HOL. It supports reasoning about the temporal evolutions of hybrid programs in the style of differential dynamic logic modelled by flows or invariant sets for vector fields. We introduce the semantic foundations of this framework and summarise their Isabelle formalisation as well as the resulting verification components. A series of simple examples shows our approach at work.
We formalise and mechanise a construtive, proof theoretic proof of Craig's Interpolation Theorem in Isabelle/HOL. We give all the definitions and lemma statements both formally and informally. We also transcribe informally the formal proofs. We detail the main features of our mechanisation, such as the formalisation of binding for first order formulae. We also give some applications of Craig's Interpolation Theorem.
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.
We describe a dataset expressing and proving properties of graph trails, using Isabelle/HOL. We formalize the reasoning about strictly increasing and decreasing trails, using weights over edges, and prove lower bounds over the length of trails in weighted graphs. We do so by extending the graph theory library of Isabelle/HOL with an algorithm computing the length of a longest strictly decreasing graph trail starting from a vertex for a given weight distribution, and prove that any decreasing trail is also an increasing one. This preprint has been accepted for publication at CICM 2020.
We present PGT, a Proof Goal Transformer for Isabelle/HOL. Given a proof goal and its background context, PGT attempts to generate conjectures from the original goal by transforming the original proof goal. These conjectures should be weak enough to be provable by automation but sufficiently strong to prove the original goal. By incorporating PGT into the pre-existing PSL framework, we exploit Isabelle's strong automation to identify and prove such conjectures.
We describe SeCaV, a sequent calculus verifier for first-order logic in Isabelle/HOL, and the SeCaV Unshortener, an online tool that expands succinct derivations into the full SeCaV syntax. We leverage the power of Isabelle/HOL as a proof checker for our SeCaV derivations. The interactive features of Isabelle/HOL make our system transparent. For instance, the user can simply click on a side condition to see its exact definition. Our formalized soundness and completeness proofs pertain exactly to the calculus as exposed to the user and not just to some model of our tool. Users can also write th…
Proof assistants, such as Isabelle/HOL, offer tools to facilitate inductive theorem proving. Isabelle experts know how to use these tools effectively; however, there is a little tool support for transferring this expert knowledge to a wider user audience. To address this problem, we present our domain-specific language, LiFtEr. LiFtEr allows experienced Isabelle users to encode their induction heuristics in a style independent of any problem domain. LiFtEr's interpreter mechanically checks if a given application of induction tool matches the heuristics, thus automating the knowledge transfer l…
We present a trustworthy connection between the Leon verification system and the Isabelle proof assistant. Leon is a system for verifying functional Scala programs. It uses a variety of automated theorem provers (ATPs) to check verification conditions (VCs) stemming from the input program. Isabelle, on the other hand, is an interactive theorem prover used to verify mathematical specifications using its own input language Isabelle/Isar. Users specify (inductive) definitions and write proofs about them manually, albeit with the help of semi-automated tactics. The integration of these two systems…
We present simple new Hoare logics and refinement calculi for hybrid systems in the style of differential dynamic logic. (Refinement) Kleene algebra with tests is used for reasoning about the program structure and generating verification conditions at this level. Lenses capture hybrid program stores in a generic algebraic way. The approach has been formalised with the Isabelle/HOL proof assistant. A number of examples explains the workflow with the resulting verification components.
We present a proof procedure for univariate real polynomial problems in Isabelle/HOL. The core mathematics of our procedure is based on univariate cylindrical algebraic decomposition. We follow the approach of untrusted certificates, separating solving from verifying: efficient external tools perform expensive real algebraic computations, producing evidence that is formally checked within Isabelle's logic. This allows us to exploit highly-tuned computer algebra systems like Mathematica to guide our procedure without impacting the correctness of its results. We present experiments demonstrating…
We introduce a language, PSL, designed to capture high level proof strategies in Isabelle/HOL. Given a strategy and a proof obligation, PSL's runtime system generates and combines various tactics to explore a large search space with low memory usage. Upon success, PSL generates an efficient proof script, which bypasses a large part of the proof search. We also present PSL's monadic interpreter to show that the underlying idea of PSL is transferable to other ITPs.
Many facts possess symmetrical counterparts that often require a separate formal proof, depending on the nature of the involved symmetry. We introduce a method in Isabelle/HOL which produces such a symmetrical fact for the list datatype and the symmetry induced by the list reversal mapping. The method is implemented as an attribute and its result is based on user-declared symmetry rules. Besides general rules, we provide rules that are aimed to be applied in the domain of Combinatorics on Words.
We have formalised Szemerédi's Regularity Lemma and Roth's Theorem on Arithmetic Progressions, two major results in extremal graph theory and additive combinatorics, using the proof assistant Isabelle/HOL. For the latter formalisation, we used the former to first show the Triangle Counting Lemma and the Triangle Removal Lemma: themselves important technical results. Here, in addition to showcasing the main formalised statements and definitions, we focus on sensitive points in the proofs, describing how we overcame the difficulties that we encountered.
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.
Simulation and formal verification are important complementary techniques necessary in high assurance model-based systems development. In order to support coherent results, it is necessary to provide unifying semantics and automation for both activities. In this paper we apply Interaction Trees in Isabelle/HOL to produce a verification and simulation framework for state-rich process languages. We develop the core theory and verification techniques for Interaction Trees, use them to give a semantics to the CSP and Circus languages, and formally link our new semantics with the failures-divergenc…
We formalize a multivariate quantifier elimination (QE) algorithm in the theorem prover Isabelle/HOL. Our algorithm is complete, in that it is able to reduce any quantified formula in the first-order logic of real arithmetic to a logically equivalent quantifier-free formula. The algorithm we formalize is a hybrid mixture of Tarski's original QE algorithm and the Ben-Or, Kozen, and Reif algorithm, and it is the first complete multivariate QE algorithm formalized in Isabelle/HOL.
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.
Interactive theorem provers have developed dramatically over the past four decades, from primitive beginnings to today's powerful systems. Here, we focus on Isabelle/HOL and its distinctive strengths. They include automatic proof search, borrowing techniques from the world of first order theorem proving, but also the automatic search for counterexamples. They include a highly readable structured language of proofs and a unique interactive development environment for editing live proof documents. Everything rests on the foundation conceived by Robin Milner for Edinburgh LCF: a proof kernel, usi…
In complex analysis, the winding number measures the number of times a path (counter-clockwise) winds around a point, while the Cauchy index can approximate how the path winds. We formalise this approximation in the Isabelle theorem prover, and provide a tactic to evaluate winding numbers through Cauchy indices. By further combining this approximation with the argument principle, we are able to make use of remainder sequences to effectively count the number of complex roots of a polynomial within some domains, such as a rectangular box and a half-plane.
The International Mathematical Olympiad (IMO) is perhaps the most celebrated mental competition in the world and as such is among the greatest grand challenges for Artificial Intelligence (AI). The IMO Grand Challenge, recently formulated, requires to build an AI that can win a gold medal in the competition. We present some initial steps that could help to tackle this goal by creating a public repository of mechanically checked solutions of IMO Problems in the interactive theorem prover Isabelle/HOL. This repository is actively maintained by students of the Faculty of Mathematics, University o…
Automated theorem provers are now commonly used within interactive theorem provers to discharge an increasingly large number of proof obligations. To maintain the trustworthiness of a proof, the automatically found proof must be verified inside the proof assistant. We present here a reconstruction procedure in the proof assistant Isabelle/HOL for proofs generated by the satisfiability modulo theories solver veriT which is part of the smt tactic. We describe in detail the architecture of our improved reconstruction method and the challenges we faced in designing it. Our experiments show that th…
In this paper we present the verification of an imperative implementation of the ubiquitous B+-tree data structure in the interactive theorem prover Isabelle/HOL. The implementation supports membership test, insertion and range queries with efficient binary search for intra-node navigation. The imperative implementation is verified in two steps: an abstract set interface is refined to an executable but inefficient purely functional implementation which is further refined to the efficient imperative implementation.
The Isabelle/HOL proof assistant has a powerful library for continuous analysis, which provides the foundation for verification of hybrid systems. However, Isabelle lacks automated proof support for continuous artifacts, which means that verification is often manual. In contrast, Computer Algebra Systems (CAS), such as Mathematica and SageMath, contain a wealth of efficient algorithms for matrices, differential equations, and other related artifacts. Nevertheless, these algorithms are not verified, and thus their outputs cannot, of themselves, be trusted for use in a safety critical system. In…
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 …
Special Relativity is a cornerstone of modern physical theory. While a standard coordinate model is well-known and widely taught today, several alternative systems of axioms exist. This paper reports on the formalisation of one such system which is closer in spirit to Hilbert's axiomatic approach to Euclidean geometry than to the vector space approach employed by Minkowski. We present a mechanisation in Isabelle/HOL of the system of axioms as well as theorems relating to temporal order. Proofs and excerpts of Isabelle/Isar scripts are discussed, particularly where the formal work required addi…
We extend a semantic verification framework for hybrid systems with the Isabelle/HOL proof assistant by an algebraic model for hybrid program stores, a shallow expression model for hybrid programs and their correctness specifications, and domain-specific deductive and calculational support. The new store model yields clean separations and dynamic local views of variables, e.g. discrete/continuous, mutable/immutable, program/logical, and enhanced ways of manipulating them using combinators, projections and framing. This leads to more local inference rules, procedures and tactics for reasoning w…
Proof by induction plays a central role in formal verification. However, its automation remains as a formidable challenge in Computer Science. To solve inductive problems, human engineers often have to provide auxiliary lemmas manually. We automate this laborious process with template-based conjecturing, a novel approach to generate auxiliary lemmas and use them to prove final goals. Our evaluation shows that our working prototype, TBC, achieved 40 percentage point improvement of success rates for problems at intermediate difficulty level.
Freie Universität Berlin（柏林自由大学）；Cambridge University（剑桥大学）
We present an automated verification of the well-known modal logic cube in Isabelle/HOL, in which we prove the inclusion relations between the cube's logics using automated reasoning tools. Prior work addresses this problem but without restriction to the modal logic cube, and using encodings in first-order logic in combination with first-order automated theorem provers. In contrast, our solution is more elegant, transparent and effective. It employs an embedding of quantified modal logic in classical higher-order logic. Automated reasoning tools, such as Sledgehammer with LEO-II, Satallax and …
CIIRC, Czech Technical University in Prague（捷克布拉格技术大学捷克信息学、机器人学与控制论研究所）；University of Innsbruck（因斯布鲁克大学）
Inductive theorem proving is an important long-standing challenge in computer science. In this extended abstract, we first summarize the recent developments of proof by induction for Isabelle/HOL. Then, we propose united reasoning, a novel approach to further automating inductive theorem proving. Upon success, united reasoning takes the best of three schools of reasoning: deductive reasoning, inductive reasoning, and inductive reasoning, to prove difficult inductive problems automatically.
Czech Technical University in Prague（布拉格捷克理工大学）；University of Innsbruck（因斯布鲁克大学）
Proof assistants offer tactics to facilitate inductive proofs. However, it still requires human ingenuity to decide what arguments to pass to those induction tactics. To automate this process, we present smart_induct for Isabelle/HOL. Given an inductive problem in any problem domain, smart_induct lists promising arguments for the induct tactic without relying on a search. Our evaluation demonstrated smart_induct produces valuable recommendations across problem domains.
We formalize in the proof assistant Isabelle essential basic notions and results in financial mathematics. We provide generic formal definitions of concepts such as markets, portfolios, derivative products, arbitrages or fair prices, and we show that, under the usual no-arbitrage condition, the existence of a replicating portfolio for a derivative implies that the latter admits a unique fair price. Then, we provide a formalization of the Cox-Rubinstein model and we show that the market is complete in this model, i.e., that every derivative product admits a replicating portfolio. This entails t…
Formal verification of cyber-physical and robotic systems requires that we can accurately model physical quantities that exist in the real-world. The use of explicit units in such quantities can allow a higher degree of rigour, since we can ensure compatibility of quantities in calculations. At the same time, improper use of units can be a barrier to safety and therefore it is highly desirable to have automated sanity checking in physical calculations. In this paper, we contribute a mechanisation of the International System of Quantities (ISQ) and the associated SI unit system in Isabelle/HOL.…
Czech Technical University in Prague（布拉格捷克理工大学）；University of Innsbruck（因斯布鲁克大学）；University of Sydney（悉尼大学）
Deciding which sub-tool to use for a given proof state requires expertise specific to each ITP. To mitigate this problem, we present PaMpeR, a Proof Method Recommendation system for Isabelle/HOL. Given a proof state, PaMpeR recommends proof methods to discharge the proof goal and provides qualitative explanations as to why it suggests these methods. PaMpeR generates these recommendations based on existing hand-written proof corpora, thus transferring experienced users' expertise to new users. Our evaluation shows that PaMpeR correctly predicts experienced users' proof methods invocation especi…
An Isabelle/HOL formalisation of Gödel's two incompleteness theorems is presented. The work follows Świerczkowski's detailed proof of the theorems using hereditarily finite (HF) set theory. Avoiding the usual arithmetical encodings of syntax eliminates the necessity to formalise elementary number theory within an embedded logical calculus. The Isabelle formalisation uses two separate treatments of variable binding: the nominal package is shown to scale to a development of this complexity, while de Bruijn indices turn out to be ideal for coding syntax. Critical details of the Isabelle proof are…
University of Innsbruck（因斯布鲁克大学）；Czech Technical University（捷克技术大学）
Mechanized theorem proving is becoming the basis of reliable systems programming and rigorous mathematics. Despite decades of progress in proof automation, writing mechanized proofs still requires engineers' expertise and remains labor intensive. Recently, researchers have extracted heuristics of interactive proof development from existing large proof corpora using supervised learning. However, such existing proof corpora present only one way of proving conjectures, while there are often multiple equivalently effective ways to prove one conjecture. In this abstract, we identify challenges in d…
We present a short proof of the Church-Rosser property for the lambda-calculus enjoying two distinguishing features: Firstly, it employs the Z-property, resulting in a short and elegant proof; and secondly, it is formalized in the nominal higher-order logic available for the proof assistant Isabelle/HOL.
Czech Technical University in Prague（布拉格捷克理工大学）；University of Innsbruck（因斯布鲁克大学）
Recently, a growing number of researchers have applied machine learning to assist users of interactive theorem provers. However, the expressive nature of underlying logics and esoteric structures of proof documents impede machine learning practitioners, who often do not have much expertise in formal logic, let alone Isabelle/HOL, from achieving a large scale success in this field. In this data description, we present a simple dataset that contains data on over 400k proof method applications along with over 100 extracted features for each in a format that can be processed easily without any kno…
