Hilbert Meets Isabelle: Formalisation of the DPRM Theorem in Isabelle

Stock, Benedikt; Pal, Abhik; Oprea, Maria Antonia; Liu, Yuanyuan; Hassler, Malte S.; Dubischar, Simon; Devkota, Prabhat; Deng, Yiping; David, Marco; Ciurezu, Bogdan; Bayer, Jonas; Aryal, Deepak

doi:10.29007/3q4s

Cited by 5 publications

(6 citation statements)

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“…Acknowledgements We want to thank the entire workgroup [19], without whose involvement we wouldn't be writing this paper; as well as Yuri Matiyasevich for initiating and guiding the project. Moreover, we would like to express our sincere gratitude to the entire welcoming and supportive Isabelle community.…”

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confidence: 99%

Beginners’ Quest to Formalize Mathematics: A Feasibility Study in Isabelle

Bayer

David

Pal

et al. 2019

Lecture Notes in Computer Science

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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.

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confidence: 99%

Beginners’ Quest to Formalize Mathematics: A Feasibility Study in Isabelle

Bayer

David

Pal

et al. 2019

Lecture Notes in Computer Science

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“…Pak formalises results regarding Pell's equation [30] and proves that Diophantine sets are closed under union and intersection [31], both as parts of the Mizar Mathematical Library. Stock et al [35,1] report on an unfinished formalisation of the DPRM theorem in Isabelle based on [27]. They cover some parts of the proof, but acknowledge for important missing results like Lucas's or "Kummer's theorem" and a "formalisation of a register machine. "…”

Section: Related Workmentioning

confidence: 99%

Hilbert's Tenth Problem in Coq (Extended Version)

Larchey-Wendling,

Forster

2020

Preprint

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We formalise the undecidability of solvability of Diophantine equations, i.e. polynomial equations over natural numbers, in Coq's constructive type theory. To do so, we give the first full mechanisation of the Davis-Putnam-Robinson-Matiyasevich theorem, stating that every recursively enumerable problem -in our case by a Minsky machine -is Diophantine. We obtain an elegant and comprehensible proof by using a synthetic approach to computability and by introducing Conway's FRACTRAN language as intermediate layer.Additionally, we prove the reverse direction and show that every Diophantine relation is recognisable by µ-recursive functions and give a certified compiler from µ-recursive functions to Minsky machines.

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“…As a consequence, in the Univalent Foundations all category-theoretic constructions and proofs are invariant under isomorphism of objects of a univalent category and under equivalence of univalent categories. In a subdirectory entitled categories (with a lower-case c, it should not be confused with the file of the same name), one proves that the category of sets as well as many categories of structured sets (monoids, groups, rings, modules, discrete fields, all standard algebraic structures formalized in the Algebra package) are univalent 11 . Moreover, any category is equivalent to a univalent category called its Rezk completion as established in the eponymous file rezk_completion.…”

Section: The Categorytheory Packagementioning

confidence: 99%

“…Note that the Isabelle proof assistant, which offers more automation, was used recently by a group of undergraduate students in Germany, under the supervision of three advisers, to formalize partly the DPRM theorem motivated by Hilbert's Tenth Problem. See their joint paper [11] presented during the FLOC 2018 conference in Oxford. 15 and great mathematicians linked by an abstract approach of space with surprisingly at the same time a feeling of its organic life.…”

Section: Architecture and Mathematicsmentioning

confidence: 99%

Univalent Foundations and the UniMath Library

Bordg

2019

Synthese Library

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We give a concise presentation of the Univalent Foundations of mathematics outlining the main ideas, followed by a discussion of the UniMath library of formalized mathematics implementing the ideas of the Univalent Foundations (section 1), and the challenges one faces in attempting to design a large-scale library of formalized mathematics (section 2). This leads us to a general discussion about the links between architecture and mathematics where a meeting of minds is revealed between architects and mathematicians (section 3).The main characters in Martin-Löf type theory (MLTT for short) are types and elements of these types. If T is a type, then the expression t : T denotes that t is an element of T . In particular, if T is a type and t, t ′ are elements of T there is a new type called the identity type of t and t ′ denoted t = T t ′ . Sometimes for convenience we will omit the type information and we will simply write t = t ′ . When one considers only a single element t, i.e. t ′ is definitionaly equal to t, the identity type t = T t has always at least one element denoted idpath t, i.e. the expression idpath t : t = T t is well-formed. This term idpath is called a constructor and the identity types belong to a particular class of types called inductive types. Indeed, besides their constructors (an inductive type can have either a single constructor or many constructors), a family of types defined inductively (like the identity types are when introduced formally) obey an induction principle. In the case of identity types, this induction principle states that given a type T , an element t : T , a family F of types indexed by an element t 0 : T and an element p 0 : t = T t 0 , if there is an element f : F t (idpath t) (the family F instantiated with the terms t and idpath t), then for any elements t ′ : T , p : t = T t ′ there is an element of the type F t ′ p, and moreover this element is f itself when t ′ and p are definitionaly equal to t and idpath t, respectively. Of course, one can iterate the process of building identity types, namely given p and q two elements of the identity type t = T t ′ , one can form the identity type p = t= T t ′ q and so on. As it happens, these identity types lead to a very rich mathematical structure and there is a surprising connection between homotopy theory and MLTT (the latter being also coined Martin-Löf dependent type theory in reference to these dependent types, i.e. dependent on previous types for their definition which may be inductive, like in the case of identity types, or not). Roughly, one can think of T as a space, two elements t and t ′ of T as points of this space, two elements p and q of the type t = T t ′ as paths from t to t ′ in the space T , and the elements of p = t= T t ′ q as homotopies between the paths p and q and so on (the elements of the successive iterated identity types being higher homotopies). Under this correspondence idpath t is the identity path between a point t and itself in the given space. Each type bearing the structure of a weak ∞-groupoi...

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Hilbert Meets Isabelle: Formalisation of the DPRM Theorem in Isabelle

Cited by 5 publications

References 1 publication

Beginners’ Quest to Formalize Mathematics: A Feasibility Study in Isabelle

Beginners’ Quest to Formalize Mathematics: A Feasibility Study in Isabelle

Hilbert's Tenth Problem in Coq (Extended Version)

Univalent Foundations and the UniMath Library

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