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Validating Mathematical Structures [chapter]

Kazuhiko Sakaguchi
2020 Lecture Notes in Computer Science  
When combined with mechanisms for implicit coercions and unification hints, packed classes enable automated structure inference and subtyping in hierarchies, e.g., that a ring can be used in place of a  ...  We implement our algorithms as tools for the Coq proof assistant, and show that they significantly improve the development process of Mathematical Components, a library for formalized mathematics.  ...  We are deeply grateful to Karl Palmskog for his careful proofreading and comments.  ... 
doi:10.1007/978-3-030-51054-1_8 fatcat:wytjsiyfj5bvvf4mcbcp5v5kya

Validating Mathematical Structures [article]

Kazuhiko Sakaguchi
2020 arXiv   pre-print
When combined with mechanisms for implicit coercions and unification hints, packed classes enable automated structure inference and subtyping in hierarchies, e.g., that a ring can be used in place of a  ...  We implement our algorithms as tools for the Coq proof assistant, and show that they significantly improve the development process of Mathematical Components, a library for formalized mathematics.  ...  We are deeply grateful to Karl Palmskog for his careful proofreading and comments.  ... 
arXiv:2002.00620v2 fatcat:jk5rbny4vfhk5mdgajzvofccy4

Formal Verification of Monad Transformers [article]

Brian Huffman
2012 arXiv   pre-print
It also provides automation for instantiating those classes, and for defining further subclasses.  ...  We present techniques for reasoning about constructor classes that (like the monad class) fix polymorphic operations and assert polymorphic axioms.  ...  Thanks also to Jasmin Blanchette for reading an early draft and providing helpful comments.  ... 
arXiv:1207.3208v1 fatcat:4i25rjaosncsdftbfndlyv75ci

The TPTP Typed First-Order Form with Arithmetic [chapter]

Geoff Sutcliffe, Stephan Schulz, Koen Claessen, Peter Baumgartner
2012 Lecture Notes in Computer Science  
The TPTP World is a well established infrastructure supporting research, development, and deployment of Automated Theorem Proving systems.  ...  Acknowledgments: Alexandre Riazanov did the analysis of TPTP CNF problems for implicit type information. Michael Schick and Peter Watson produced many of the TFF and arithmetic problems.  ...  The TFF arithmetic language aims to provide a comprehensive basis for automated reasoning with arithmetic.  ... 
doi:10.1007/978-3-642-28717-6_32 fatcat:ktx5dc5odbd43pgp7jgwt5666m

The Lean Theorem Prover (System Description) [chapter]

Leonardo de Moura, Soonho Kong, Jeremy Avigad, Floris van Doorn, Jakob von Raumer
2015 Lecture Notes in Computer Science  
It aims to bridge the gap between interactive and automated theorem proving, by situating automated tools and methods in a framework that supports user interaction and the construction of fully specified  ...  Lean is an ongoing, long-term effort, and much of the potential for automation will be realized only gradually over time.  ...  Most automation available in Lean is integrated into the system as tactics. For example, Lean contains a rewrite tactic that provides a basic mechanism for performing rewriting.  ... 
doi:10.1007/978-3-319-21401-6_26 fatcat:kr7ri6irxre5fmzd4wgz5rlfim

Hierarchy Builder: Algebraic hierarchies Made Easy in Coq with Elpi (System Description)

Cyril Cohen, Kazuhiko Sakaguchi, Enrico Tassi, Zena M. Ariola
2020 International Conference on Formal Structures for Computation and Deduction  
We implement the HB language in the hierarchy-builder addon for the Coq system using the Elpi extension language.  ...  The key concepts are the ones of factory, builder and abbreviation that let the hierarchy developer describe an actual interface for their library.  ...  Finally, we thank the anonymous reviewers of this paper for their useful comments.  ... 
doi:10.4230/lipics.fscd.2020.34 dblp:conf/fscd/CohenST20 fatcat:6xond3vrmzh67huwtyokmhfvfe

Fixed point semantics and partial recursion in Coq

Yves Bertot, Vladimir Komendantsky
2008 Proceedings of the 10th international ACM SIGPLAN symposium on Principles and practice of declarative programming - PPDP '08  
This is only possible if we extend the logical framework by adding some axioms of classical logic.  ...  Acknowledgments The authors are grateful to the anonymous referees for their detailed comments.  ...  Moreover, note that a coercion from D c → D to Type requires only one step, whereas a coercion from D C → D to Type would have required two.  ... 
doi:10.1145/1389449.1389461 dblp:conf/ppdp/BertotK08 fatcat:sue2j7pbhzdqlc3d2ac3kffc64

Interfacing Coq + SSReflect with GAP

Vladimir Komendantsky, Alexander Konovalov, Steve Linton
2012 Electronical Notes in Theoretical Computer Science  
We allow for a manual definition of Coq terms that provide helper information for the automated tactic that performs data exchange between the two systems.  ...  A coercion path is given by the composition of k elementary coercions, for k ≥ 0, that is, f 1 • · · · • f k .  ... 
doi:10.1016/j.entcs.2012.06.003 fatcat:ncjcjjli4jbotje66h7ahi6rtm

Proof-Search in Natural Deduction Calculus for Classical Propositional Logic [chapter]

Mauro Ferrari, Camillo Fiorentini
2015 Lecture Notes in Computer Science  
Γ A ⇑ Proof-search strategy for Nc We have three possible ways to expand a leaf Γ K ⇑. (1.1) If K is prime, we can apply coercion. Γ K ↓ ↓⇑ Γ K ⇑ . . .  ...  Proof-search strategy for Nc (2) ↓-expansion To prove Γ K ↓: Select H ∈ Γ (head formula) Starting from the axiom sequent Γ H ↓ apply ∧, →-elimination rules with the goal to extract K from H.  ... 
doi:10.1007/978-3-319-24312-2_17 fatcat:7d7vk72rufcczeg2y4ialdk2ii

Page 7 of Mathematical Reviews Vol. , Issue 2002A [page]

2002 Mathematical Reviews  
Linton and artin, Automated theorem ; symbolic definite Delphine Bouc r diffe nec line US 11Nneal erential Automated coercion for Symbolic computation in Java inte- Cit l 595 f“auion and Marc Gaeta displ  ...  sition (213-219 (electronic Dominique Villard and Michael B Monagan, ADrien: an implementation of automatic different tion in Maple (221-228 Nicolas J Doy e tronic Laurent Bernardin Corless, electronic axiom  ... 

Packaging Mathematical Structures [chapter]

François Garillot, Georges Gonthier, Assia Mahboubi, Laurence Rideau
2009 Lecture Notes in Computer Science  
Interfaces for the structures enjoy the convenience of a classical setting, without requiring any axiom.  ...  This alternative to telescopes in particular supports multiple inheritance, maximal sharing of notations and theories, and automated structure inference.  ...  Module Coercions are mostly useful for establishing generic theorems for abstract structures.  ... 
doi:10.1007/978-3-642-03359-9_23 fatcat:vkan4oitzberffz5sd6b3ukkju

Gradual certified programming in coq

Éric Tanter, Nicolas Tabareau
2015 SIGPLAN notices  
Casts can be integrated with the implicit coercion mechanism of Coq to support implicit cast insertionà la gradual typing.  ...  A simple mix of type classes and axioms makes it possible to bring gradual certified programming to Coq in a straightforward manner.  ...  venues for future work.  ... 
doi:10.1145/2936313.2816710 fatcat:5bkpblagy5a6rcf6z7oosrsyza

Gradual certified programming in coq

Éric Tanter, Nicolas Tabareau
2015 Proceedings of the 11th Symposium on Dynamic Languages - DLS 2015  
Casts can be integrated with the implicit coercion mechanism of Coq to support implicit cast insertion a la gradual typing.  ...  A simple mix of type classes and axioms makes it possible to bring gradual certified programming to Coq in a straightforward manner.  ...  venues for future work.  ... 
doi:10.1145/2816707.2816710 dblp:conf/dls/TanterT15 fatcat:ujuwbzy2krh3jeeskgw3ghhwx4

Embracing a mechanized formalization gap [article]

Antal Spector-Zabusky, Joachim Breitner, Yao Li, Stephanie Weirich
2019 arXiv   pre-print
Acknowledgments Thanks to Leonidas Lampropoulos and William Mansky for their valuable comments on a draft of this paper.  ...  Thanks to Josh Cohen, Christine Rizkallah, and John Wiegley for assistance with the development of hs-to-coq and the theory of the containers library.  ...  As in the Coercion and Type_ axiomatization, we justify all axioms that we depend on for our reasoning. A complete list of these axioms appears in Appendix B.  ... 
arXiv:1910.11724v1 fatcat:iancabvl3nbrxi2coyol7c4iie

Program Analysis and Verification Based on Kleene Algebra in Isabelle/HOL [chapter]

Alasdair Armstrong, Georg Struth, Tjark Weber
2013 Lecture Notes in Computer Science  
All the machinery for these coercions, such as abs, is thereby hidden from the user. A simple application example is given by the following lemma.  ...  It is therefore relatively simple, concise and well suited for automation [15, 12, 13, 3] .  ...  We have implemented a similar tactic for our SKAT-based implementation, called hoare-auto.  ... 
doi:10.1007/978-3-642-39634-2_16 fatcat:bzxg2fl7i5gvfevgx444jsuwgy
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