Plato on the foundations of Modern Theorem Provers

Is it possible to achieve such a proof that is independent of both acts and dispositions of the human mind? Plato is one of the great contributors to the foundations of mathematics. He discussed, 2400 years ago, the importance of clear and precise definitions as fundamental entities in mathematics, independent of the human mind. In the seventh book of his masterpiece, The Republic, Plato states “arithmetic has a very great and elevating effect, compelling the soul to reason about abstract number, and rebelling against the introduction of visible or tangible objects into the argument” (525c). In the light of this thought, I will discuss the status of mathematical entities in the twentieth first century, an era when it is already possible to demonstrate theorems and construct formal axiomatic derivations of remarkable complexity with artificial intelligent agents --- the modern theorem provers

Learning-Assisted Automated Reasoning with Flyspeck

The considerable mathematical knowledge encoded by the Flyspeck project is combined with external automated theorem provers (ATPs) and machine-learning premise selection methods trained on the proofs, producing an AI system capable of answering a wide range of mathematical queries automatically. The performance of this architecture is evaluated in a bootstrapping scenario emulating the development of Flyspeck from axioms to the last theorem, each time using only the previous theorems and proofs. It is shown that 39% of the 14185 theorems could be proved in a push-button mode (without any high-level advice and user interaction) in 30 seconds of real time on a fourteen-CPU workstation. The necessary work involves: (i) an implementation of sound translations of the HOL Light logic to ATP formalisms: untyped first-order, polymorphic typed first-order, and typed higher-order, (ii) export of the dependency information from HOL Light and ATP proofs for the machine learners, and (iii) choice of suitable representations and methods for learning from previous proofs, and their integration as advisors with HOL Light. This work is described and discussed here, and an initial analysis of the body of proofs that were found fully automatically is provided

Formalising Mathematics – in Praxis; A Mathematician’s First Experiences with Isabelle/HOL and the Why and How of Getting Started

AbstractThis is an account of a mathematician’s first experiences with the proof assistant (interactive theorem prover) Isabelle/HOL, including a discussion on the rationale behind formalising mathematics and the choice of Isabelle/HOL in particular, some instructions for new users, some technical and conceptual observations focussing on some of the first difficulties encountered, and some thoughts on the use and potential of proof assistants for mathematics.</jats:p

Case Studies in Proof Checking

The aim of computer proof checking is not to find proofs, but to verify them. This is different from automated deduction, which is the use of computers to find proofs that humans have not devised first. Currently, checking a proof by computer is done by taking a known mathematical proof and entering it into the special language recognized by a proof verifier program, and then running the verifier to hopefully obtain no errors. Of course, if the proof checker approves the proof, there are considerations of whether or not the proof checker is correct, and this has been complicated by the fact that so many systems have sprung into being. The two main challenges in using a proof checker today are the time needed to learn the syntax and general usage of the system and the time needed to formalize a proof in the system even when the user is already proficient with it. As mathematicians are not yet using proof checkers regularly, we wanted to evaluate the validity of this reluctance by analyzing these main obstacles. Judging by Dr. Wiedijk’s Formalizing 100 Theorems list, which gives an overview of the headway various proof systems have made in mathematics, Coq and Mizar are two of the most successful systems in use today (Wiedijk, 2007). I simultaneously formalized two fairly involved theorems in these two systems while I was at approximately the same level of familiarity with each. I kept track of my experiences with learning the systems and analyzed their comparative strengths and weaknesses. The analysis and summary of experiences should also give a general idea of the current state of computer-aided proof checking

Em direção à formalização das propriedades de normalização do sistema λex

Trabalho de Conclusão de Curso (graduação)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Ciência da Computação, 2016.O cálculo /\ é um sistema formal, capaz de expressar o processo computacional. Pela sua simplicidade e expressividade, este cálculo é usado como modelo teórico para o paradigma de programação funcional. Em consequência disto, uma grande quantidade de extensões do cálculo foi proposta, com o objetivo de obter um sistema formal intermediário entre o cálculo /\ e suas implementações. O objeto de estudo deste trabalho é uma destas variantes, chamada /\ex, um cálculo com substituições explicitas proposto por Delia Kesner. Este cálculo é um dos primeiros a possuir a preservação da normalização forte enquanto permite composição completa de substituições explícitas. Continuamos o trabalho de formalização deste cálculo, no assistente de prova Coq, iniciado em 2014, e que tem por objetivo fornecer uma prova mecância e construtiva da propriedade de normalização forte para o cálculo /\ex. Mais especificamente, iniciamos a prova da propriedade IE, chave para a prova da preservação da normalização forte do cálculo /\ex. Isto foi feito seguindo a estratégia de prova no artigo da Kesner: estendemos a formalização para marcar alguns termos que não inserem problemas de normalização e definimos regras de redução para lidar com tais termos. Por fim, provamos a equivalência dessas novas regras com a regra original do sistema./\-calculus is a formal system, capable of expressing the computational process. Because of its simplicity and expressiveness, this calculus is used as a theorical model for the paradigm of functional programming. Consequently, a great variety of extensions were proposed, with the goal of obtaining an intermediate formal system between the /\-calculus and its implementations. The object of study of this work is one of these variants, called /\ex, a calculus with explicit subsititutions, proposed by Delia Kesner. This calculus is one of the first to preserve strong normalization of terms while permitting full composition of explicit substitutions. We continued the work in the formalization of this calculus, in the Coq proof assistant, initiated in 2014, with the goal of providing a mechanical and constructive proof of the strong normalization property for the /\ ex calculus. More specifically, we began the proof of the IE property, key to the demonstration of the preservation of strong normalization of the _ex-calculus. This was done following the strategy on Kesner’s paper: we extended the formalization to mark some terms that do not insert normalization issues and define reduction rules to deal with such terms. Finally, we prove the equivalence of these new rules with the original reduction rule of the system