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In Praise of Impredicativity: A Contribution to the Formalisation of Meta-Programming [article]

François Bry
2019 arXiv   pre-print
Reflective Predicate Logic, a conservative extension of first-order logic, which provides a simple formalisation of meta-programming.  ...  In logic programming, there is a gap between the meta-programming practice and its theory: The formalisations of meta-programming do not explicitly address its impredicativity and are not fully adequate  ...  Formalisations interpreting meta-programs as first-order theories are based on reifying firstorder formulas by encoding them as first-order terms.  ... 
arXiv:1807.06051v3 fatcat:7b3yzhf5u5awngbvqndvarfbhm

A Framework for the Formalisation of Pi Calculus Type Systems in Isabelle/HOL [chapter]

Simon J. Gay
2001 Lecture Notes in Computer Science  
of the Isabelle theories and proof scripts to variations on the language and other type systems.  ...  The use of a uniform encoding of pi calculus syntax in a meta language, the development of a general theory of type environments, and the structured formalisation of the main proofs, facilitate the adaptation  ...  First, we formalise an operational semantics based on a reduction relation rather than a labelled transition system; this is because the language which we set out to formalise uses the reduction semantics  ... 
doi:10.1007/3-540-44755-5_16 fatcat:i6h4la3xyjemvfajisizdkknlm

Experiments with ZF set theory in HOL and Isabelle [chapter]

Sten Agerholm, Mike Gordon
1995 Lecture Notes in Computer Science  
The advantages and disadvantages of higher-order set theory versus firstorder set theory are explored experimentally.  ...  This study also provides a comparison of the proof infrastructure of HOL and Isabelle.  ...  Francisco Corella, whose PhD thesis [6] contains many insights into mechanizing set theory (including the use of simple type theory as the underlying logic) has had a significant influence on the second  ... 
doi:10.1007/3-540-60275-5_55 fatcat:3ormlf77dvdp7ld77ltd7kulhm

Experiments with ZF Set Theory in HOL and Isabelle

Sten Agerholm, Mike Gordon
1995 BRICS Report Series  
<br />The advantages and disadvantages of higher-order set theory versus first-order<br />set theory are explored experimentally.  ...  This study also provides a<br />comparison of the proof infrastructure of HOL and Isabelle.  ...  Francisco Corella, whose PhD thesis [6] contains many insights into mechanizing set theory (including the use of simple type theory as the underlying logic) has had a significant influence on the second  ... 
doi:10.7146/brics.v2i37.19940 fatcat:4ptzes5o2vbmhh3m3wmwjn6ake

A Formalised Theorem in the Partition Calculus [article]

Lawrence C. Paulson
2021 arXiv   pre-print
The work is part of a project on formalising the partition calculus.  ...  A paper on ordinal partitions by Erdős and Milner (1972) has been formalised using the proof assistant Isabelle/HOL, augmented with a library for Zermelo-Fraenkel set theory.  ...  Thanks to Mirna Džamonja (who proposed the project in the first place) and to Angeliki Koutsoukou-Argyraki for discussions and help.  ... 
arXiv:2104.11613v1 fatcat:w6svnczwgzaqdk2we7t62ba7vm

Formalising Exact Arithmetic in Type Theory [chapter]

Milad Niqui
2005 Lecture Notes in Computer Science  
In this work we focus on a formalisation of the algorithms of lazy exact arithmeticà la Potts and Edalat [1] . We choose the constructive type theory as our formal verification tool.  ...  We discuss an extension of the constructive type theory with coinductive types that enables one to formalise and reason about the infinite objects.  ...  Thus in order to formalise these algorithms we choose a type theory extended with coinductive types.  ... 
doi:10.1007/11494645_46 fatcat:alsonnqjonfebi7b3ickdcuzru

On formalisation of the goal concept in law

Vytautas Čyras
2007 Engineering applications of artificial intelligence  
The paper raises a problem of formalising several key concepts in legal theory, namely, goal, function and value. The law is viewed from a perspective of computer science in law.  ...  I find a formalisation to have different levels depending on a distinct meaning of the term "law". E.g., legal drafting is closer to engineering than to legal reasoning.  ...  Bench-Capon (2002) models a theory as a partial order.  ... 
doi:10.1016/j.engappai.2006.11.010 fatcat:m7qjcfrr7jd5texxggogvm2pom

A pluralist approach to the formalisation of mathematics

2011 Mathematical Structures in Computer Science  
We present a programme of research for pluralist formalisations, that is, formalisations that involve proving results in more than one foundation.  ...  If we know that a translation Φ exists between two logic-enriched type theories (LTTs) S and T , and we have formalised a proof of a theorem α in S , we may wish to make use of the fact that Φ(α) is a  ...  Acknowledgements We would like to thank the anonymous referees for their very helpful and detailed comments, and particularly for making us aware of related work in the area of proof reuse.  ... 
doi:10.1017/s0960129511000156 fatcat:pl5ccfooanaifmyoetq5um3nxi

First-Order Logic Formalisation of Impossibility Theorems in Preference Aggregation

Umberto Grandi, Ulle Endriss
2012 Journal of Philosophical Logic  
In the long run, we hope that our approach to formalisation can serve as the basis for a fully automated proof of classical and new theorems in social choice theory.  ...  We define a first-order language for social welfare functions and we give a complete axiomatisation for this class, without having the number of individuals or alternatives specified in the language.  ...  Acknowledgements Earlier versions of this paper have been presented at the Third Workshop on Decision, Games and Logic in 2009 in Lausanne and at the Second Workshop of Logic, Rationality and Interaction  ... 
doi:10.1007/s10992-012-9240-8 fatcat:rbwlzyh4bnhinnjwpqndbczhri

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

Angeliki Koutsoukou-Argyraki
2020 Jahresbericht der Deutschen Mathematiker-Vereinigung (Teubner)  
This 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  ...  To view a copy of this licence, visit  ... 
doi:10.1365/s13291-020-00221-1 fatcat:pt4sfc3hlvfwlant3b4v2varwu

Just What Is Full-Blooded Platonism?†

2003 Philosophia Mathematica  
They are either too strong, with untoward consequences we all reject, or too weak, not providing a distinctive brand of platonism strong enough to do the work Balaguer requires of it. * Thanks to  ...  Mark Balaguer has, in his Platonism and Anti-Platonism in Mathematics [Balaguer, 1998 ], given us an intriguing new brand of platonism, which he calls, plenitudinous platonism, or more colourfully, full-blooded  ...  After all, Balaguer takes his attempts at formalisations of FBP to be true. 2 On page 6, he formalises a part of FBP as follows: (∀Y) ♦(∃x)(Mx ∧ Yx) ⊃ (∃x)(Mx ∧ Yx) (1) This is a second-order modal sentence  ... 
doi:10.1093/philmat/11.1.82 fatcat:fnjbjzymkrbq5dsxcbfgqwt47i

Towards a Readable Formalisation of Category Theory

Greg O'Keefe
2004 Electronical Notes in Theoretical Computer Science  
We formally develop category theory up to Yoneda's lemma, using Isabelle/HOL/Isar, and survey previous formalisations.  ...  By using recently added Isabelle features, we have produced a formal text that more closely approximates informal mathematics.  ...  We also note the survey on formalised category theory of Takahisi Mohri [17] . Development The full text of the formal development, produced automatically by Isabelle, is 27 pages long.  ... 
doi:10.1016/j.entcs.2003.12.014 fatcat:c5pslpe3sjaubby3li36t34avi

Productivity of Edalat-Potts Exact Arithmetic in Constructive Type Theory

Milad Niqui
2007 Theory of Computing Systems  
In this work we focus on a formalisation of the algorithms of lazy exact arithmeticà la Edalat-Potts in type theory.  ...  Furthermore, we show how to employ this semantic information to formalise a restricted case of normalisation algorithm in our type theory. 1 The reason for considering such a restricted form is that it  ...  Coq Formalisation of Semantical Method  ... 
doi:10.1007/s00224-006-1342-7 fatcat:z7aoy5nanjgybnoan7hstgppkq

Heterogeneous Proofs: Spider Diagrams Meet Higher-Order Provers [chapter]

Matej Urbas, Mateja Jamnik
2011 Lecture Notes in Computer Science  
We provide a mechanisation of the theory of abstract spider diagrams and establish a formal link between diagrammatic concepts and the existing theories in Isabelle/HOL.  ...  Sentential Reasoner Our first step was to provide a formalisation of the theory of spider diagrams within Isabelle/HOL (files available from  ...  We formalised the theory of spider diagrams in Isabelle/HOL (see Section 3.1).  ... 
doi:10.1007/978-3-642-22863-6_29 fatcat:u5hbevxi7fagjcnk5sfcts4opu

Formalisation of Selected Results from Group Theory

Joseph Johannes Thommes, Manuel Eberl, Tobias Nipkow
2021 Zenodo  
Additionally, some results from the field of character groups are included, as well as work on formalising the direct product, the internal direct product and more group-theoretic lemmas – on both the  ...  The results include the long-known and well understood fundamental theorem of finitely generated abelian groups characterising the structure of finitely generated abelian groups as a uniquely determined  ...  to formalise some theory on these subjects first.  ... 
doi:10.5281/zenodo.4785363 fatcat:ggo4zojijrgk5kfuufsxxsfsgu
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