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Verification of the Completeness of Unification Algorithms à la Robinson [chapter]

Andréia B. Avelar, Flávio L. C. de Moura, André Luiz Galdino, Mauricio Ayala-Rincón
2010 Lecture Notes in Computer Science  
This work presents a general methodology for verification of the completeness of firstorder unification algorithmsà la Robinson developed in the higher-order proof assistant PVS.  ...  Termination and soundness proofs of any unification algorithm are proved by reusing the formalization of this theorem and completeness should be proved according to the specific way in that non unifiable  ...  Introduction In a previous development, done in the PVS proof assistant [ORS92] , a formalization of the theorem of existence of most general unifiers (mgu's) for unifiable terms over first-order theories  ... 
doi:10.1007/978-3-642-13824-9_10 fatcat:ne54gywxnre4jpqld5mosc5fkm

A Formalization of the Theorem of Existence of First-Order Most General Unifiers

Andréia B Avelar, André L Galdino, Flávio LC de Moura, Mauricio Ayala-Rincón
2012 Electronic Proceedings in Theoretical Computer Science  
This work presents a formalization of the theorem of existence of most general unifiers in first-order signatures in the higher-order proof assistant PVS.  ...  The distinguishing feature of this formalization is that it remains close to the textbook proofs that are based on proving the correctness of the well-known Robinson's first-order unification algorithm  ...  Introduction A formalization in the proof assistant PVS of the theorem of existence of most general unifiers (mgu's) in first-order theories is presented.  ... 
doi:10.4204/eptcs.81.5 fatcat:uuj3bilz2nbgrjavwy2zvzmlwi

Completeness in PVS of a Nominal Unification Algorithm

Mauricio Ayala-Rincón, Maribel Fernández, Ana Cristina Rocha-Oliveira
2016 Electronical Notes in Theoretical Computer Science  
Nominal unification was initially studied by Urban, Pitts and Gabbay and then formalised by Urban in the proof assistant Isabelle/HOL and by Kumar and Norrish in HOL4.  ...  In the nominal approach variable bindings are represented using techniques that are close to first-order logical techniques, instead of using a higher-order metalanguage.  ...  Contribution In this paper, we present a functional specification of a new nominal unification algorithm and formalise its correctness and completeness in the language of the higher-order proof assistant  ... 
doi:10.1016/j.entcs.2016.06.005 fatcat:pgsndhnpbfd5nlpbjht7qo3fge

JProver: Integrating Connection-Based Theorem Proving into Interactive Proof Assistants [chapter]

Stephan Schmitt, Lori Lorigo, Christoph Kreitz, Aleksey Nogin
2001 Lecture Notes in Computer Science  
: multiple provers with different formalisms = expressive power of proof assistants for complex proofs / verifications + efficient proof techniques for first-order subproblems -Dealing with type information  ...  Motivation • Interactive Proof Assistants -Large scale applications of automated reasoning -Expressive logics vs. higher degree of automation -Coq, HOL, Isabelle, Nuprl, OMEGA, PVS • Improving Proof Automation  ... 
doi:10.1007/3-540-45744-5_34 fatcat:3bm75w7mbnbefd5zb4safxl5bq

Twenty Years Later [chapter]

Jean-Pierre Jouannaud
2005 Lecture Notes in Computer Science  
What has happened is that many research areas, such as programming languages, constraint solving, first-order provers, proof assistants, security theory, and verification have all been fertilized by ideas  ...  The first RTA conference took place in Dijon, in 1985. This year, 2005, it takes place in Nara.  ...  Proof assistants Many will agree with me when saying that Isabelle, Coq and PVS are three among the most important proof assistants. Isabelle is based on Nipkow's higher-order rewriting [44] .  ... 
doi:10.1007/978-3-540-32033-3_27 fatcat:6xggpx3i7ve7rfewekdb4vfiv4

A Tool for Automated Theorem Proving in Agda [chapter]

Fredrik Lindblad, Marcin Benke
2006 Lecture Notes in Computer Science  
The proof search is conducted directly in type theory and produces proof terms. Any proof term is verified by the Agda type-checker, which ensures soundness of the tool.  ...  The tool is intended to facilitate interactive proving by relieving the user from filling in simple but tedious parts of a proof.  ...  ACL2, PVS and Isabelle are other major proof assistants. ACL2 and PVS do have automation for induction, but none of the systems produces proof objects.  ... 
doi:10.1007/11617990_10 fatcat:v6tkdrelhjdvbi6gwgk2vftopm

Efficient Full Higher-Order Unification

Petar Vukmirović, Alexander Bentkamp, Visa Nummelin, Zena M. Ariola
2020 International Conference on Formal Structures for Computation and Deduction  
Our unification procedure is implemented in the Zipperposition theorem prover. Experimental evaluation shows a clear advantage over Jensen and Pietrzykowski's procedure.  ...  We developed a procedure to enumerate complete sets of higher-order unifiers based on work by Jensen and Pietrzykowski.  ...  Acknowledgements We are grateful to the maintainers of StarExec for letting us use their service.  ... 
doi:10.4230/lipics.fscd.2020.5 dblp:conf/fscd/VukmirovicBN20 fatcat:quwuu37yrnfl5jyd5c52kwliti

A Certifying Square Root and Division Elimination

Pierre Neron
2015 Electronical Notes in Theoretical Computer Science  
proofs become very difficult since any mathematical intuition is lost.  ...  Section 3 describes the embedding of a subset of Pvs to provide the certifying process. Section 4 introduces some of the technical details and features of the transformation.  ...  One approach to produce code that may satisfy this required level of safety is to verify its correctness in a proof assistant such as Pvs.  ... 
doi:10.1016/j.entcs.2015.10.012 fatcat:l2hnyqpftzgjfbeu3jqfhbuo6u

Efficient Full Higher-Order Unification [article]

Petar Vukmirović, Alexander Bentkamp, Visa Nummelin
2021 arXiv   pre-print
Our unification procedure, together with new higher-order term indexing data structures, is implemented in the Zipperposition theorem prover.  ...  We developed a procedure to enumerate complete sets of higher-order unifiers based on work by Jensen and Pietrzykowski.  ...  We are grateful to the maintainers of StarExec for letting us use their service. We thank  ... 
arXiv:2011.09507v3 fatcat:hbdpvf6vg5fb3alsdtxuer67li

Rewriting, Inference, and Proof [chapter]

Natarajan Shankar
2010 Lecture Notes in Computer Science  
We discuss these problems in the context of our ongoing effort to use PVS as a metatheoretic framework to construct a proof kernel for justifying the claims of theorem provers, rewriters, model checkers  ...  Rewriting is a form of inference, and one that interacts in several ways with other forms of inference such as decision procedures and proof search.  ...  In our approach, we use a kernel proof checker as the reference standard. In our case, we use PVS to define a proof checker for first-order logic with the axioms of ZFC.  ... 
doi:10.1007/978-3-642-16310-4_1 fatcat:vqoukc27wnhtrdsvcr2hymspt4

Automation of Higher-Order Logic [chapter]

Christoph Benzmüller, Dale Miller
2014 Handbook of the History of Logic  
We thank Chad Brown for sharing notes that he has written related to the material in this chapter.  ...  The first author has been supported by the German Research Foundation under Heisenberg grant BE2501/9-1 and the second author has been supported by the ERC Advanced Grant ProofCert.  ...  ., 2007) are prominent first-order interactive proof assistants that integrate induction.  ... 
doi:10.1016/b978-0-444-51624-4.50005-8 fatcat:jfcztdvymjfujg3bzb2rq2qyzy

A Tool-Assisted Framework for Certified Bytecode Verification [chapter]

Gilles Barthe, Guillaume Dufay
2004 Lecture Notes in Computer Science  
proofs that the execution function is monotone w.r.t. the order on states and the unification is decreasing and monotone.  ...  The purpose of this paper is to complete the automation process by certifying the different dataflow analyses involved in bytecode verification, using the Coq proof assistant.  ...  We mostly use Jakarta in conjunction with the proof assistant Coq [8] , although prototypes interfaces to Isabelle [18] and PVS [20] exist 2 .  ... 
doi:10.1007/978-3-540-24721-0_7 fatcat:2vgaiikiijakrop26csl7t5dry

A PVS Theory for Term Rewriting Systems

André L. Galdino, Mauricio Ayala-Rincón
2009 Electronical Notes in Theoretical Computer Science  
A theory, called trs, for Term Rewriting Systems in the theorem Prover PVS is described.  ...  This theory is built on the PVS libraries for finite sequences and sets and a previously developed PVS theory named ars for Abstract Reduction Systems which was built on the PVS libraries for sets.  ...  the proof assistant PVS.  ... 
doi:10.1016/j.entcs.2009.07.049 fatcat:soxxpwyy3ng7zcjlzeejjvgvhe

From LCF to Isabelle/HOL

Lawrence C. Paulson, Tobias Nipkow, Makarius Wenzel
2019 Formal Aspects of Computing  
They include automatic proof search, borrowing techniques from the world of first order theorem proving, but also the automatic search for counterexamples.  ...  Like other proof assistants, Isabelle is not directly concerned with program verification, i.e. with verifying code written in a programming language, but it can be used as a back end to prove verification  ...  , and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.  ... 
doi:10.1007/s00165-019-00492-1 fatcat:rv5zmo22fjedvjxjz4lm22v6de

Extensional Higher-Order Paramodulation in Leo-III [article]

Alexander Steen, Christoph Benzmüller
2021 arXiv   pre-print
Leo-III supports reasoning in polymorphic first-order and higher-order logic, in all normal quantified modal logics, as well as in different deontic logics.  ...  proof assistants such as Isabelle/HOL.  ...  Further notable higher-order reasoning systems include proof assistants such as PVS [81] , Isabelle/HOL, the HOL prover family including HOL4 [61] , and the HOL Light system [63] .  ... 
arXiv:1907.11501v2 fatcat:hv3uysew6fdrtixvtv5d7ydgb4
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