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Self-Formalisation of Higher-Order Logic

Ramana Kumar, Rob Arthan, Magnus O. Myreen, Scott Owens
2016 Journal of automated reasoning  
Soundness of the logic extends to soundness of a theorem prover, because we also show that a synthesised implementation of the kernel in CakeML refines the inference system.  ...  Our semantics supports defined constants directly via a context, and we find this approach cleaner than our previous work formalising Wiedijk's Stateless HOL.  ...  We are grateful to John Harrison and Freek Wiedijk for providing inspiration for this project. We also thank the anonymous reviewers for carefully reading our submission and suggesting improvements.  ... 
doi:10.1007/s10817-015-9357-x fatcat:3dcm7pdklzcrhc37yryjzm3f7u

A Mechanized Theory of Regular Trees in Dependent Type Theory [chapter]

Régis Spadotti
2015 Lecture Notes in Computer Science  
Proof assistants are tools developed by computer scientists in order to ease formal reasoning. In this sense, they provide a framework to express statements and properties.  ...  Then, by using the proof rules of the underlying logic, theorems are proved and mechanically checked by the machine.  ...  First, we give a short introduction to the dependent type theory that serves as the theoretical foundation of all definitions and results presented in the subsequent chapters.  ... 
doi:10.1007/978-3-319-22102-1_27 fatcat:yoldarn4rbflph37rh3wvwfjy4

To be or not to be constructive, that is not the question

Sam Sanders
2018 Indagationes mathematicae  
Out of intuitionism grew intuitionistic logic and the associated Brouwer-Heyting-Kolmogorov interpretation by which 'there exists x' intuitively means 'an algorithm to compute x is given'.  ...  To this end, we identify a part of classical mathematics, namely classical Nonstandard Analysis, and show it inhabits the twilight-zone between the constructive and non-constructive.  ...  An example involving Transfer . 4.3.1. Introduction.  ... 
doi:10.1016/j.indag.2017.05.005 fatcat:ssfs6tio7vfxdpbhxz4ka5gcsy

A Refinement of de Bruijn's Formal Language of Mathematics

Fairouz Kamareddine, Rob Nederpelt
2004 Journal of Logic, Language and Information  
We provide a syntax and a derivation system for a formal language of mathematics called Weak Type Theory (WTT). We give the metatheory of WTT and a number of illustrative examples.  ...  The derivation system allows one to establish that a book written in WTT is well-formed following the syntax of WTT, and has great resemblance with ordinary mathematics books.  ...  , a set theory (ZF? FM?), etc. The choice of the proof checker: Automath [14] , Isabelle [15] , Coq [2] , PVS [18] , Mizar [17] , etc.  ... 
doi:10.1023/b:jlli.0000028393.47593.b8 fatcat:4fvkefihbngpddqzcjc5qb7q7a

Computability in constructive type theory [article]

Yannick Forster, Universität Des Saarlandes
2022
We give a formalised and machine-checked account of computability theory in the Calculus of Inductive Constructions (CIC), the constructive type theory underlying the Coq proof assistant.  ...  Since then, my journey through academia has brought me to many great places and introduced me to many great people.  ...  He verifies a universal term and proves Rice's theorem. Forster and Smolka [83] formalise the weak call-by-value λ-calculus in Coq.  ... 
doi:10.22028/d291-35758 fatcat:deb5muacebhsnnmk5p25mgc7ia

Change actions: from incremental computation to discrete derivatives [article]

Mario Alvarez-Picallo
2020 arXiv   pre-print
Second, to establish and clarify the connection between derivatives in the incremental sense and derivatives in the analytic sense, that is to say, to provide a common definition of derivative of which  ...  To this end we define and explore the notions of change actions and differential maps between change actions and show how these notions relate to incremental computation through the concrete example of  ...  On the other hand, a causal map f is ε-linear if and only if it is a group homomorphism in every component except the first. zf ([a i ] + [b i ]) = zf ([a i ]) + zf ([b i ]).  ... 
arXiv:2002.05256v2 fatcat:o6zzymbz5rcgbgamuqve2fkyou

Evidence for Set-Theoretic Truth and the Hyperuniverse Programme [chapter]

Sy-David Friedman
2017 The Hyperuniverse Project and Maximality  
For example, the formalisation in Coq of ever more abstract mathematics has turned out to require a deep revision of the concept of identity in Martin-Löf type theory.  ...  A widely used proof assistant is Coq, named after Coquand. For formalising mathematics proof theory is indispensable.  ...  This is not immediately obvious since it appears we need induction up to ψ θ (ε Ω+1 ), which we do not have access to in KP(P).  ... 
doi:10.1007/978-3-319-62935-3_4 fatcat:kqy4pyoxtzepjejot5cxup3qtu

LIPIcs, Volume 39, TYPES'14, Complete Volume [article]

Hugo Herbelin, Pierre Letouzey, Matthieu Sozeau
2015
I also thank the anonymous referees for their many suggestions that led to major improvements in this paper. This research was partially supported by French ANR project COQUAS (12 JS02 006 01).  ...  I want to thank the referees, especially the first one for numerous pertinent and helpful remarks and suggestions. Typeful Normalization by Evaluation Acknowledgements.  ...  We show that ZF ε is a conservative extension of ZF, and that the model N satisfies the axioms of ZF ε , which means that each one of these axioms is realized by a proof-like term.  ... 
doi:10.4230/lipics.types.2014 fatcat:3zwqtan44jcxfheghkucefqbh4

Organization Programme Committee Organizing Committee

Andrea Asperti, Ulrich Berger, Edwin Brady, Herman Geuvers, Radboud, Christine Paulin-Mohring, Bas Spitters, Tiina Laasma, Monika Perkmann, Tarmo Uustalu
2015 21st International Conference on Types for Proofs and Programs TYPES 2015   unpublished
The TYPES meetings are a forum to present new and ongoing work in all aspects of type theory and its applications, especially in formalized and computer assisted reasoning and computer programming.  ...  The TYPES conferences are of open and informal character. Selection of contributed talks is based on short abstracts; reporting work in progress and work presented or published elsewhere is welcome.  ...  This work was supported by IUT20-56 "Computational models for Estonian", the Swedish Institute, and the European Regional Development Fund through the Estonian Center of Excellence in Computer Science,  ... 
fatcat:ej5sa3lm7bc6tdhhjd3w3hwos4