Formal Mathematics on Display: A Wiki for Flyspeck

The formalization of mathematics in practice relies heavily on proof assistants and automatic theorem provers, therefore we studied what are the state of the art proof assitants and their limitations to understand what are the main challenges in making formalized mathematics common practice among mathematicians. We found out that curretly the two major dificulties in formalizing mathematics with proof assistants are due to steep learning curves in how to use these tools and due to a wide gap between the notation employed in these proof assistants and the currently used mathematical notation. We also developed a C++ library to develop proof assistants with great notational flexibility.

"This paper is about our hobby." That is the first sentence of [MP93], the first report on our formal development of lambda calculus and type theory, written in autumn 1992. We have continued to pursue this hobby on and off ever since, and have developed a substantial body of formal knowledge, including Church-Rosser and standardization<br />theorems for beta reduction, and the basic theory of<br />Pure Type Systems (PTS) leading to the strengthening theorem and type checking algorithms for PTS. Some of this work is reported in [MP93, vBJMP94, Pol94b, Pol95]. In the present paper we survey this work, including some new proofs, and point out what we feel has been learned about the general issues of formalizing mathematics. On the technical side, we describe an abstract, and simplified, proof of standardization for beta reduction, not previously published, that does<br />not mention redex positions or residuals. On the general issues, we emphasize the search for formal definitions that are convenient for formal proof and convincingly represent the intended informal concepts. The LEGO Proof Development System [LP92] was used to check the work in an implementation of the Extended Calculus of Constructions<br />(ECC) with inductive types [Luo94]. LEGO is a refinement style<br />proof checker, publicly available by ftp and WWW, with a User's Manual [LP92] and a large collection of examples. Section 1.3 contains information on accessing the formal development described in this paper. Other interesting examples formalized in LEGO include program specification and data refinement [Luo91], strong normalization of System F [Alt93], synthetic domain theory [Reu95, Reu96], and operational semantics for imperative programs [Sch97].

The situation with regard to two-phonon collective states in even-even deformed nuclei is obscure. According to the Bohr-Mottelson model deformed nuclei must have two-phonon collective states; this was recently pointed out once more in [i]. According to the interacting boson model the low-lying states must include states with large two-boson components [2]. In the framework of the quasiparticle-phonon nuclear model [3] the study of the twophonon states showed [4] that allowance for the Pauli principle, i.e., allowance for the exact commutation relations between the phonons, leads to a shift of the two-phonon poles to higher energies. In [5], the corresponding secular equations were solved. Allowance for the Pauli principle leads to a shift of the centroids of the energy of the collective two-phonon states by 1-2 MeV. At excitation energies of 3-4 MeV the strength of the collective two-phonon states must be distributed over many nuclear levels. It was concluded on this basis in [5] that in even-even deformed nuclei there are no collective twophonon states. In the description of two-phonon collective states in deformed nuclei there is a contradiction between the quasiparticle-phonon model and the Bohr-Mottelson and interacting boson models. As is shown in [6], there is a contradiction between the quasiparticle-phonon model and the interacting boson model in the description of a number of other nonrotational states in deformed nuclei. We note that according to [7] allowance for the Pauli principle in even-even spherical nuclei does not lead to a large shift in the energies of the two-phonon states, in agreement with the experimental data on the twophonon collective states in spherical nuclei. An analysis of experimental data led to the conclusion in [8] that there are no twophonon collective states in deformed nuclei. New experimental data on 1~SEr in [9,10] showed that the levels previously treated as two-phonon levels have a dominant one-phonon component. The absence of two-phonon quadrupole states with energy less than 2 MeV in 168Er is explained in [i,Ii] by the large anharmonicity of the $ and ~ vibrations due to the shape of the nucleus in the form of a triaxial ellipsoid. The absence of two-phonon octupole states in a number of Ra, Th, and U isotopes [12] is explained in many studies by the existence of a stable octupole deformation. We note that the new experimental data on 168Er [9,13] agree well with calculations in the quasiparticle-phonon model and with the calculations in [14] of the structure of the single-phonon states and do not agree with the calculations in the interacting boson model. In view of the importance of clarifying the situation with regard to two-phonon collective states of even-even deformed nuclei, it is necessary to study the behavior of the two-phonon states by a different mathematical formalism. Since allowance for the Pauli principle in the two-phonon components of the wave functions plays a decisive role, it is expedient to make the calculations with phonons constructed from the operators of "true" bosons. It is necessary to establish whether there is a large shift of the twophonon poles in the problem when formulated in this way. The present paper is devoted to

In many situations, the analysis of viscoelastic materials, like polymers, takes benefit from the introduction of fractional operators in the mathematical formalization. In addition, fractional differential models have been applied in a wide variety of fields, from biology to thermodynamics, from diffusion of information to dehydration/rehydration of food. Thus, a great interest is paid both in the analytical and in the numerical solution of fractional differential problems. The present paper considers a class of time-fractional diffusion problems with Dirichlet boundary conditions. Using Duhamel’s principle, the analytical solution is found. As usual in this context, the solution is given in series form and depends on the Mittag-Leffler function. We suggest a computational procedure to evaluate the solution with high accuracy, in a computing environment. Some test examples are presented both in the subdiffusion and in the superdiffusion case, to illustrate the behavior of the solution for different values of the fractional index. Test cases have been carried out in Matlab.KeywordsAnomalous diffusionFractional differential equationsAnalytical solutionDirichlet boundary conditions

Nowadays, formal methods rely on tools of different kinds: proof assistants with which the user interacts to discover a proof step by step; and fully automated tools which make use of (intricate) decision procedures. But while some proof assistants can check the soundness of a proof, they lack automation. Regarding automated tools, one still has to be satisfied with their answers Yes/No/Do not know, the validity of which can be subject to question, in particular because of the increasing size and complexity of these tools.In the context of rewriting techniques, we aim at bridging the gap between proof assistants that yield formal guarantees of reliability and highly automated tools one has to trust. We present an approach making use of both shallow and deep embeddings. We illustrate this approach with a prototype based on the CiME rewriting toolbox, which can discover involved termination proofs that can be certified by the Coq proof assistant, using the Coccinelle library for rewriting.KeywordsModule TypeAutomate ToolProof AssistantGraph CriterionStrongly Connect ComponentThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The recent introduction of the big data paradigm and advancements in machine learning and deep mining techniques have made proof guidance and automation in interactive theorem provers (ITPs) an important research topic. In this paper, we provide a learning approach based on sequential pattern mining (SPM) for proof guidance in the PVS proof assistant. Proofs in a PVS theory are first abstracted to a computer-processable corpus. SPM techniques are then used on the corpus to discover frequent proof steps and proof patterns, relationships of proof steps / patterns with each other, dependency of new conjectures on already proved facts and to predict the next proof step(s). Obtained results suggest that the integration of SPM in proof assistants can be used to guide the proof process and in the development of proof tactics/strategies.

This is a short overview of an experimental library of Mathematics formalized in the Coq proof assistant using the univalent interpretation of the underlying type theory of Coq. I started to work on this library in February 2010 in order to gain experience with formalization of Mathematics in a constructive type theory based on the intuition gained from the univalent models (see Kapulkinet al.2012).

Mutation analysis, which introduces artificial defects into software systems, is the basis of mutation testing, a technique widely applied to evaluate and enhance the quality of test suites. However, despite the deep analogy between tests and formal proofs, mutation analysis has seldom been considered in the context of deductive verification. We propose mutation proving, a technique for analyzing verification projects that use proof assistants. We implemented our technique for the Coq proof assistant in a tool dubbed mCoq. mCoq applies a set of mutation operators to Coq definitions of functions and datatypes, inspired by operators previously proposed for functional programming languages. mCoq then checks proofs of lemmas affected by operator application. To make our technique feasible in practice, we implemented several optimizations in mCoq such as parallel proof checking. We applied mCoq to several medium and large scale Coq projects, and recorded whether proofs passed or failed when applying different mutation operators. We then qualitatively analyzed the mutants, finding many instances of incomplete specifications. For our evaluation, we made several improvements to serialization of Coq files and even discovered a notable bug in Coq itself, all acknowledged by developers. We believe mCoq can be useful both to proof engineers for improving the quality of their verification projects and to researchers for evaluating proof engineering techniques.

An LCF style tactic can be used to verify that the conclusion is a logical consequence of the premises, however, in mathematical practice, we often construct a conclusion in a proof step. This paper proposes a calculational style of tactic, and illustrates the characteristic and the realization issues of the tactics on the top of a theorem prover.

Persistent homology is one of the most active branches of computational algebraic topology with applications in several contexts such as optical character recognition or analysis of point cloud data. In this article, we report on the formal development of certified programs to compute persistent Betti numbers , an instrumental tool of persistent homology, using the C oq proof assistant together with the SSR eflect extension. To this aim it has been necessary to formalize the underlying mathematical theory of these algorithms. This is another example showing that interactive theorem provers have reached a point where they are mature enough to tackle the formalization of nontrivial mathematical theories.

Practical Reflection for Sequent Logics

Algorithmic methods can successfully automate the proof, and even the discovery, of a large class of identities involving sums of hypergeometric terms. In particular, the Wilf-Zeilberger (WZ) algorithm is a uniform framework for a substantial class of hypergeometric summation problems. This algorithm can produce a rational function certificate that can, on the face of it, be used to verify the result by routine algebraic manipulations, independently of the working of the algorithm that discovered it. It is therefore very natural to consider using this certificate to produce, by automated means, a rigorous deductive proof in an interactive theorem prover. However, naive presentations of the WZ method tend to gloss over trivial-looking but rather knotty questions about zero denominators, which makes their rigorous formalization tricky and their ultimate logical justification somewhat obscure. We describe how we have handled these difficulties to produce rigorous WZ proofs inside the HOL Light theorem prover.

Toward prompt gamma spectrometry for monitoring boron distributions during extra corporal treatment of liver metastases by boron neutron capture therapy: A Monte Carlo simulation study

Proof searching and proof automation are the two most desired properties in interactive theorem provers (ITPs) as they generally require manual user guidance, which can be quite cumbersome. In this paper, we provide an evolutionary proof searching approach for the HOL4 proof assistant, where a genetic algorithm (GA) with different crossover and mutation operators is used to search and optimize the proofs in different HOL theories. Random proof sequences are first generated from a population of frequently occurring HOL4 proof steps that are discovered with sequential pattern mining. Generated proof sequences are then evolved with GA operators (three crossover and two mutation) till their fitness match the fitness of the target proof sequences. Various crossover and mutation operators are used to compare their effect on the performance of GAs in proof searching. Obtained results suggest that integrating GAs with HOL4 allows us to efficiently support proof finding and optimization.

The Isabelle/HOL proof assistant provides quite advanced and reliable support for discharging proof goals with external SMT solvers by reconstructing the resulting proof tree within the Isabelle/Pure inference kernel. In particular, currently Isabelle fully and efficiently supports proof reconstruction for the quantifier-free fragments of the theories of equality and uninterpreted functions (QF_UF) and linear integer arithmetic (QF_LIA). Yet there are a number of practically relevant theories that can be decided using relatively efficient algorithms, but are not supported either by the Isabelle proof reconstruction procedure or by the SMT solvers themselves. Meanwhile, if the decision procedure for an extended logical theory can be expressed as a complete quantifier instantiation strategy that reduces the given decision problem into the supported logical fragment (QF_UFLIA), the theory can be fully supported within the existing Isabelle/HOL infrastructure with relatively little additional effort. The goal of our work was to develop a fully formal completeness proof for an efficient quantifier instantiation strategy that translates a satisfiability problem in the logic of bounded linear integer arithmetic (we call it QF_BLIA) into the equisatisfiable problem in the logic of linear integer arithmetic and uninterpreted functions (QF_UFLIA). In this paper we first give the initial informal presentation of the proof, then discuss the most relevant issues of our resulting corresponding formal development of the completeness proof within Isabelle/HOL. The resulting formal proofs tightly follow the provided informal presentation, but at the same time they also contain quite significant amount of auxiliary developments that can potentially be generalized for other similar proof formalizations and automated using theory generation facilities of Isabelle/ML.