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LCF Examples in HOL

Sten Agerholm
1994 BRICS Report Series  
In this paper a number of examples are used to demonstrate that an extension of HOL with domain theory combines the benefits of both systems.  ...  The examples illustrate reasoning about infinite values and non-terminating functions and show how mixing domain and set theoretic reasoning eases reasoning about finite LCF types and strict functions.  ...  I am grateful to Larry Paulson for digging up the LCF proof of correctness of the uni cation algorithm.  ... 
doi:10.7146/brics.v1i18.21649 fatcat:wchog7xakjbfrk7eavij4txiiu

Computer Assisted Reasoning

Richard Boulton, Joe Hurd, Konrad Slind
2009 Journal of automated reasoning  
Mike abandoned LCF LSM in favour of higher order logic (HOL) following discussions with Ben Moszkowski; the benefits of higher-order functions for hardware are discussed in [19] .  ...  The techniques developed by them were applied to more sophisticated examples; for example, by Graham Birtwistle and Brian Graham at Calgary [28] .  ... 
doi:10.1007/s10817-009-9144-7 fatcat:rd5iyn6gw5canbyhgz4gfxt2yi

Tactics for mechanized reasoning: a commentary on Milner (1984) 'The use of machines to assist in rigorous proof'

M. J. C. Gordon
2015 Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences  
For example, in the Flyspeck project, parts of the proof of the Kepler conjecture were checked using HOL and other parts checked using Isabelle.  ...  Besides standard ML, the LCF metalanguage also led to OCaml [42] , which is the metalanguage for Coq and one of the HOL implementations (HOL Light), and is growing in popularity in industry.  ... 
doi:10.1098/rsta.2014.0234 pmid:25750147 pmcid:PMC4360087 fatcat:bskjoaarmjc7fofocrwgfqjzsm

HOL Light: An Overview [chapter]

John Harrison
2009 Lecture Notes in Computer Science  
Theorem provers in this family use a version of ML as both the implementation and interaction language; in HOL Light's case this is Objective CAML (OCaml).  ...  Thanks to its adherence to the so-called 'LCF approach', the system can be extended with new inference rules without compromising soundness.  ...  LCF, HOL and HOL Light Both HOL Light and its implementation language OCaml can trace their origins back to Edinburgh LCF, developed by Milner and his research assistants in the 1970s [6] .  ... 
doi:10.1007/978-3-642-03359-9_4 fatcat:hmqwv5qmyjajxbz52xatepemo4

From LCF to Isabelle/HOL

Lawrence C. Paulson, Tobias Nipkow, Makarius Wenzel
2019 Formal Aspects of Computing  
For example, simplification by rewriting-coupled with recursive simplification to handle conditional rewrite rules-was already realised in both the Boyer/Moore theorem prover [BM79] and Edinburgh LCF by  ...  The resulting type, thm, From LCF to Isabelle/HOL 677  ...  distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in  ... 
doi:10.1007/s00165-019-00492-1 fatcat:rv5zmo22fjedvjxjz4lm22v6de

HOLCF: Higher order logic of computable functions [chapter]

Franz Regensburger
1995 Lecture Notes in Computer Science  
Therefore techniques from higher order logic and LCF can be combined in a fruitful manner avoiding drawbacks of both logics.  ...  With the help of type classes the extension can be formulated in a way s u c h t h a t the logic LCF constitutes a proper sublanguage of HOLCF.  ...  The theory Fun1 is an example for an extension by a new constant which corresponds to the same notion in Gordon's HOL.  ... 
doi:10.1007/3-540-60275-5_72 fatcat:mlrfpmxhtzh2zfz5jtoicd7mxm

An LCF-Style Interface between HOL and First-Order Logic [chapter]

Joe Hurd
2002 Lecture Notes in Computer Science  
In addition to MESON in HOL, there have been many other examples of automatic first-order provers being used to prove problems in an interactive theorem prover, including: FAUST in HOL [5] ; Paulson's  ...  For example, the HOL boolean term m ≤ n can be translated to the first-order formula B(@(@(≤, m), n)).  ... 
doi:10.1007/3-540-45620-1_10 fatcat:au5jgjzldnadld7ntuxjvvfkhe

Fast LCF-Style Proof Reconstruction for Z3 [chapter]

Sascha Böhme, Tjark Weber
2010 Lecture Notes in Computer Science  
We present independent reconstruction of these proofs in the theorem provers Isabelle/HOL and HOL4 with particular focus on efficiency.  ...  Detailed performance data shows that LCF-style proof reconstruction can be faster than proof search in Z3.  ...  Conclusions We have presented LCF-style proof reconstruction for Z3 in the theorem provers Isabelle/HOL and HOL4.  ... 
doi:10.1007/978-3-642-14052-5_14 fatcat:yvi5yovfijayhmc6nnkj34mfju

A HOL Basis for Reasoning about Functional Programs

Sten Agerholm
1994 BRICS Report Series  
<br /> <br />The advantages of combining the best of the domain and set theoretic worlds in the same system are demonstrated in a larger example, showing the correctness of a unification algorithm.  ...  A thorough comparison with LCF is provided.  ...  Note that types of LCF correspond to pointed cpos in HOL-CPO.  ... 
doi:10.7146/brics.v1i44.21598 fatcat:5aowedoqnbdlxnjcfplcsygyum

Standalone Tactics Using OpenTheory [chapter]

Ramana Kumar, Joe Hurd
2012 Lecture Notes in Computer Science  
This enables, for example, LCF-style proof reconstruction efforts to be shared by users of different interactive theorem provers and removes the need for each user to install the external tool being integrated  ...  Proof tools in interactive theorem provers are usually developed within and tied to a specific system, which leads to a duplication of effort to make the functionality available in different systems.  ...  Introduction There are many LCF-style systems for interactively developing machine-checked formal theories, including HOL4 [1] , HOL Light [2] , ProofPower [3] and Isabelle/HOL [4] .  ... 
doi:10.1007/978-3-642-32347-8_28 fatcat:jas6ssyzhjd6ddqgb6p3zbp5cq

Michael John Caldwell Gordon. 28 February 1948—22 August 2017

Lawrence C. Paulson
2018 Biographical Memoirs of Fellows of the Royal Society  
His achievements include his work at Edinburgh University helping to create Edinburgh LCF, the first interactive theorem prover of its kind, and the ML family of functional programming languages.  ...  Michael Gordon was a pioneer in the field of interactive theorem proving and hardware verification.  ...  In the 1980s, he worked with Mike Gordon on further development of the LCF proof assistant, which became the foundation of Gordon's LCF LSM and HOL systems.  ... 
doi:10.1098/rsbm.2018.0019 fatcat:lionn75yjnddfeupt7dpwmvdcu

Integrating a SAT Solver with an LCF-style Theorem Prover

Tjark Weber
2006 Electronical Notes in Theoretical Computer Science  
This paper describes the integration of a leading SAT solver with Isabelle/HOL, a popular interactive theorem prover.  ...  The LCF-approach instead demands that we verify zChaff's claim of unsatisfiability within Isabelle/HOL.  ...  A Simple Example In this section we illustrate the proof reconstruction using a small example. Consider the following input formula φ ≡ (¬v1 ∨ v2) ∧ (¬v2 ∨ ¬v3) ∧ (v1 ∨ v2) ∧ (¬v2 ∨ v3).  ... 
doi:10.1016/j.entcs.2005.12.007 fatcat:52nuhgbxs5efxkfzqwmztuy5qy

History of Interactive Theorem Proving [chapter]

John Harrison, Josef Urban, Freek Wiedijk
2014 Handbook of the History of Logic  
ACKNOWLEDGEMENTS The authors are grateful to Jörg Siekmann for inviting them to prepare this chapter and for his patience in the face of our lengthy delays.  ...  HOL As emphasized by Gordon [1982] , despite the name 'LCF', nothing in the Edinburgh LCF methodology is tied to the Logic of Computable Functions.  ...  A good example of such methods at work is the implementation by Slind [1991] of Knuth-Bendix completion in HOL.  ... 
doi:10.1016/b978-0-444-51624-4.50004-6 fatcat:sngnxnl3ivectj5taj3dgsaa7m

Reconstruction of Z3's Bit-Vector Proofs in HOL4 and Isabelle/HOL [chapter]

Sascha Böhme, Anthony C. J. Fox, Thomas Sewell, Tjark Weber
2011 Lecture Notes in Computer Science  
We present independent reconstruction of unsatisfiability proofs for bit-vector theories in the theorem provers HOL4 and Isabelle/HOL.  ...  We thereby obtain high correctness assurances for Z3's results, and increase the degree of proof automation for bit-vector problems in HOL4 and Isabelle/HOL.  ...  The LCF-style systems that we consider here, HOL4 and Isabelle/HOL, are popular theorem provers for polymorphic higher-order logic (HOL) [21] , based on the simply-typed λ-calculus.  ... 
doi:10.1007/978-3-642-25379-9_15 fatcat:gesxkp2cm5girliosyngzyuwmi

Possibly infinite sequences in theorem provers: A comparative study [chapter]

Marco Devillers, David Griffioen, Olaf Müller
1997 Lecture Notes in Computer Science  
The formalizations have been carried out in di erent proof tools, namely in Gordon's HOL, in Isabelle and in PVS.  ...  We compare four di erent formalizations of possibly in nite sequences in theorem provers based on higher-order logic.  ...  For example, suppose one wants to lter all symbols a in the sequence x = (b a a b a : : : 6 HOL-LCF: Domain Theory in Isabelle/HOL De nition 12 (Type of Sequences).  ... 
doi:10.1007/bfb0028388 fatcat:jkh254z3tvfrhkn72ndcnknq24
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