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title = An Axiomatic Characterization of the Single-Source Shortest Path Problem
author = Christine Rizkallah <https://www.mpi-inf.mpg.de/~crizkall/>
date = 2013-05-22
topic = Mathematics/Graph Theory
abstract = This theory is split into two sections. In the first section, we give a formal proof that a well-known axiomatic characterization of the single-source shortest path problem is correct. Namely, we prove that in a directed graph with a non-negative cost function on the edges the single-source shortest path function is the only function that satisfies a set of four axioms. In the second section, we give a formal proof of the correctness of an axiomatic characterization of the single-source shortest path problem for directed graphs with general cost functions. The axioms here are more involved because we have to account for potential negative cycles in the graph. The axioms are summarized in three Isabelle locales.

title = The Correctness of Launchbury's Natural Semantics for Lazy Evaluation
author = Joachim Breitner <http://pp.ipd.kit.edu/~breitner>
date = 2013-01-31
topic = Computer Science/Programming Languages/Lambda Calculi
abstract = In his seminal paper "Natural Semantics for Lazy Evaluation", John Launchbury proves his semantics correct with respect to a denotational semantics. We have formalized both semantics and identified an ambiguity in what the join operator should do: If it is understood as the least upper bound, as hinted at in the paper, the original proof fails. We fix the proof by taking a detour via a modified natural semantics with an explicit stack. In addition, we show that the original proof goes through when join is understood to be right-sided update.

title = CCS in nominal logic
author = Jesper Bengtson <http://www.itu.dk/people/jebe>
date = 2012-05-29
topic = Computer Science/Process Calculi
abstract = We formalise a large portion of CCS as described in Milner's book 'Communication and Concurrency' using the nominal datatype package in Isabelle. Our results include many of the standard theorems of bisimulation equivalence and congruence, for both weak and strong versions. One main goal of this formalisation is to keep the machine-checked proofs as close to their pen-and-paper counterpart as possible.
 This entry is described in detail in <a href="http://www.itu.dk/people/jebe/files/thesis.pdf">Bengtson's thesis</a>.

title = The pi-calculus in nominal logic
author = Jesper Bengtson <http://www.itu.dk/people/jebe>
date = 2012-05-29
topic = Computer Science/Process Calculi
abstract = We formalise the pi-calculus using the nominal datatype package, based on ideas from the nominal logic by Pitts et al., and demonstrate an implementation in Isabelle/HOL. The purpose is to derive powerful induction rules for the semantics in order to conduct machine checkable proofs, closely following the intuitive arguments found in manual proofs. In this way we have covered many of the standard theorems of bisimulation equivalence and congruence, both late and early, and both strong and weak in a uniform manner. We thus provide one of the most extensive formalisations of a the pi-calculus ever done inside a theorem prover.
 A significant gain in our formulation is that agents are identified up to alpha-equivalence, thereby greatly reducing the arguments about bound names. This is a normal strategy for manual proofs about the pi-calculus, but that kind of hand waving has previously been difficult to incorporate smoothly in an interactive theorem prover. We show how the nominal logic formalism and its support in Isabelle accomplishes this and thus significantly reduces the tedium of conducting completely formal proofs. This improves on previous work using weak higher order abstract syntax since we do not need extra assumptions to filter out exotic terms and can keep all arguments within a familiar first-order logic.
 This entry is described in detail in <a href="http://www.itu.dk/people/jebe/files/thesis.pdf">Bengtson's thesis</a>.

title = Psi-calculi in Isabelle
author = Jesper Bengtson <http://www.itu.dk/people/jebe>
date = 2012-05-29
topic = Computer Science/Process Calculi
abstract = Psi-calculi are extensions of the pi-calculus, accommodating arbitrary nominal datatypes to represent not only data but also communication channels, assertions and conditions, giving it an expressive power beyond the applied pi-calculus and the concurrent constraint pi-calculus.
 We have formalised psi-calculi in the interactive theorem prover Isabelle using its nominal datatype package. One distinctive feature is that the framework needs to treat binding sequences, as opposed to single binders, in an efficient way. While different methods for formalising single binder calculi have been proposed over the last decades, representations for such binding sequences are not very well explored.
 The main effort in the formalisation is to keep the machine checked proofs as close to their pen-and-paper counterparts as possible. This includes treating all binding sequences as atomic elements, and creating custom induction and inversion rules that to remove the bulk of manual alpha-conversions.
 This entry is described in detail in <a href="http://www.itu.dk/people/jebe/files/thesis.pdf">Bengtson's thesis</a>.

title = Isabelle/Circus
author = Abderrahmane Feliachi <mailto:abderrahmane.feliachi@lri.fr>, Burkhart Wolff <mailto:wolff@lri.fr>, Marie-Claude Gaudel <mailto:mcg@lri.fr>
date = 2012-05-27
topic = Computer Science/Process Calculi, Computer Science/Programming Languages/Logics
abstract = The Circus specification language combines elements for complex data and behavior specifications, using an integration of Z and CSP with a refinement calculus. Its semantics is based on Hoare and He's Unifying Theories of Programming (UTP). Isabelle/Circus is a formalization of the UTP and the Circus language in Isabelle/HOL. It contains proof rules and tactic support that allows for proofs of refinement for Circus processes (involving both data and behavioral aspects).
 The Isabelle/Circus environment supports a syntax for the semantic definitions which is close to textbook presentations of Circus. This article contains an extended version of corresponding VSTTE Paper together with the complete formal development of its underlying commented theories.

title = Dijkstra's Shortest Path Algorithm
author = Benedikt Nordhoff <mailto:b.n@wwu.de>, Peter Lammich <http://cs.uni-muenster.de/sev/staff/lammich/>
topic = Computer Science/Algorithms
date = 2012-01-30
depends-on = Refine_Monadic, Collections
abstract = We implement and prove correct Dijkstra's algorithm for the
 single source shortest path problem, conceived in 1956 by E. Dijkstra.
 The algorithm is implemented using the data refinement framework for monadic,
 nondeterministic programs. An efficient implementation is derived using data
 structures from the Isabelle Collection Framework.

title = Refinement for Monadic Programs
author = Peter Lammich <http://cs.uni-muenster.de/sev/staff/lammich/>
topic = Computer Science/Programming Languages/Logics
date = 2012-01-30
depends-on = Collections
abstract = We provide a framework for program and data refinement in Isabelle/HOL.
  The framework is based on a nondeterminism-monad with assertions, i.e.,
  the monad carries a set of results or an assertion failure.
  Recursion is expressed by fixed points. For convenience, we also provide
  while and foreach combinators.
  The framework provides tools to automatize canonical tasks, such as
  verification condition generation, finding appropriate data refinement relations,
  and refine an executable program to a form that is accepted by the
  Isabelle/HOL code generator.
  This submission comes with a collection of examples and a user-guide,
  illustrating the usage of the framework.

title = Automatic Data Refinement
author = Peter Lammich <mailto:lammich@in.tum.de>
topic = Computer Science/Programming Languages/Logics
date = 2013-10-02
abstract = We present the Autoref tool for Isabelle/HOL, which automatically
  refines algorithms specified over abstract concepts like maps
  and sets to algorithms over concrete implementations like red-black-trees,
  and produces a refinement theorem. It is based on ideas borrowed from
  relational parametricity due to Reynolds and Wadler.
  The tool allows for rapid prototyping of verified, executable algorithms.
  Moreover, it can be configured to fine-tune the result to the user~s needs.
  Our tool is able to automatically instantiate generic algorithms, which
  greatly simplifies the implementation of executable data structures.
  This AFP-entry provides the basic tool, which is then used by the
  Refinement and Collection Framework to provide automatic data refinement for
  the nondeterminism monad and various collection datastructures.

title = Pseudo Hoops
author = George Georgescu, Laurentiu Leustean, Viorel Preoteasa <http://users.abo.fi/vpreotea/>
topic = Mathematics/Algebra
date = 2011-09-22
abstract = Pseudo-hoops are algebraic structures introduced by B. Bosbach under the name of complementary semigroups. In this formalization we prove some properties of pseudo-hoops and we define the basic concepts of filter and normal filter. The lattice of normal filters is isomorphic with the lattice of congruences of a pseudo-hoop. We also study some important classes of pseudo-hoops. Bounded Wajsberg pseudo-hoops are equivalent to pseudo-Wajsberg algebras and bounded basic pseudo-hoops are equivalent to pseudo-BL algebras. Some examples of pseudo-hoops are given in the last section of the formalization.

title = Algebra of Monotonic Boolean Transformers
author = Viorel Preoteasa <http://users.abo.fi/vpreotea/>
topic = Computer Science/Programming Languages/Logics
date = 2011-09-22
abstract = Algebras of imperative programming languages have been successful in reasoning about programs. In general an algebra of programs is an algebraic structure with programs as elements and with program compositions (sequential composition, choice, skip) as algebra operations. Various versions of these algebras were introduced to model partial correctness, total correctness, refinement, demonic choice, and other aspects. We formalize here an algebra which can be used to model total correctness, refinement, demonic and angelic choice. The basic model of this algebra are monotonic Boolean transformers (monotonic functions from a Boolean algebra to itself).

title = Lattice Properties
author = Viorel Preoteasa <http://users.abo.fi/vpreotea/>
topic = Mathematics/Algebra
date = 2011-09-22
abstract = This formalization introduces and collects some algebraic structures based on lattices and complete lattices for use in other developments. The structures introduced are modular, and lattice ordered groups. In addition to the results proved for the new lattices, this formalization also introduces theorems about latices and complete lattices in general.
extra-history =
  Change history:
  [2012-01-05]: Removed the theory about distributive complete lattices which is in the standard library now.
      Added a theory about well founded and transitive relations and a result about fixpoints in complete lattices and well founded relations.
      Moved the results about conjunctive and disjunctive functions to a new theory.
      Removed the syntactic classes for inf and sup which are in the standard library now.

title = Proving the Impossibility of Trisecting an Angle and Doubling the Cube
author = Ralph Romanos <mailto:ralph.romanos@student.ecp.fr>, Lawrence Paulson <http://www.cl.cam.ac.uk/~lp15/>
topic = Mathematics/Algebra, Mathematics/Geometry
date = 2012-08-05
abstract = Squaring the circle, doubling the cube and trisecting an angle, using a compass and straightedge alone, are classic unsolved problems first posed by the ancient Greeks. All three problems were proved to be impossible in the 19th century. The following document presents the proof of the impossibility of solving the latter two problems using Isabelle/HOL, following a proof by Carrega. The proof uses elementary methods: no Galois theory or field extensions. The set of points constructible using a compass and straightedge is defined inductively. Radical expressions, which involve only square roots and arithmetic of rational numbers, are defined, and we find that all constructive points have radical coordinates. Finally, doubling the cube and trisecting certain angles requires solving certain cubic equations that can be proved to have no rational roots. The Isabelle proofs require a great many detailed calculations.

title = Knowledge-based programs
author = Peter Gammie <http://peteg.org/>
topic = Computer Science/Automata and Formal Languages
date = 2011-05-17
abstract = Knowledge-based programs (KBPs) are a formalism for directly relating agents' knowledge and behaviour. Here we present a general scheme for compiling KBPs to executable automata with a proof of correctness in Isabelle/HOL. We develop the algorithm top-down, using Isabelle's locale mechanism to structure these proofs, and show that two classic examples can be synthesised using Isabelle's code generator.
extra-history =
  Change history:
  [2012-03-06]: Add some more views and revive the code generation.

title = The independence of Tarski's Euclidean axiom
author = T. J. M. Makarios <mailto:tjm1983@gmail.com>
topic = Mathematics/Geometry
date = 2012-10-30
abstract =
  Tarski's axioms of plane geometry are formalized and, using the standard
  real Cartesian model, shown to be consistent. A substantial theory of
  the projective plane is developed. Building on this theory, the
  Klein-Beltrami model of the hyperbolic plane is defined and shown to
  satisfy all of Tarski's axioms except his Euclidean axiom; thus Tarski's
  Euclidean axiom is shown to be independent of his other axioms of plane
  An earlier version of this work was the subject of the author's
  <a href="http://researcharchive.vuw.ac.nz/handle/10063/2315">MSc thesis</a>,
  which contains natural-language explanations of some of the
  more interesting proofs.

title = The General Triangle Is Unique
author = Joachim Breitner <mailto:mail@joachim-breitner.de>
topic = Mathematics/Geometry
date = 2011-04-01
abstract = Some acute-angled triangles are special, e.g. right-angled or isoscele triangles. Some are not of this kind, but, without measuring angles, look as if they were. In that sense, there is exactly one general triangle. This well-known fact is proven here formally.

title = Lightweight Java
author = Rok Strniša <http://rok.strnisa.com/lj/>, Matthew Parkinson <http://research.microsoft.com/people/mattpark/>
topic = Computer Science/Programming Languages/Language Definitions
date = 2011-02-07
abstract = A fully-formalized and extensible minimal imperative fragment of Java.

title = Lower Semicontinuous Functions
author = Bogdan Grechuk <mailto:grechukbogdan@yandex.ru>
topic = Mathematics/Analysis
date = 2011-01-08
abstract = We define the notions of lower and upper semicontinuity for functions from a metric space to the extended real line. We prove that a function is both lower and upper semicontinuous if and only if it is continuous. We also give several equivalent characterizations of lower semicontinuity. In particular, we prove that a function is lower semicontinuous if and only if its epigraph is a closed set. Also, we introduce the notion of the lower semicontinuous hull of an arbitrary function and prove its basic properties.

title = RIPEMD-160
author = Fabian Immler <mailto:immler@in.tum.de>
topic = Computer Science/Programming Languages/Static Analysis
date = 2011-01-10
abstract = This work presents a verification of an implementation in SPARK/ADA of the cryptographic hash-function RIPEMD-160. A functional specification of RIPEMD-160 is given in Isabelle/HOL. Proofs for the verification conditions generated by the static-analysis toolset of SPARK certify the functional correctness of the implementation.

title = Regular Sets and Expressions
author = Alexander Krauss <http://www.in.tum.de/~krauss>, Tobias Nipkow <http://www.in.tum.de/~nipkow>
topic = Computer Science/Automata and Formal Languages
date = 2010-05-12
abstract = This is a library of constructions on regular expressions and languages. It provides the operations of concatenation, Kleene star and derivative on languages. Regular expressions and their meaning are defined. An executable equivalence checker for regular expressions is verified; it does not need automata but works directly on regular expressions. <i>By mapping regular expressions to binary relations, an automatic and complete proof method for (in)equalities of binary relations over union, concatenation and (reflexive) transitive closure is obtained.</i> <P> Extended regular expressions with complement and intersection are also defined and an equivalence checker is provided.
extra-history =
  Change history:
  [2011-08-26]: Christian Urban added a theory about derivatives and partial derivatives of regular expressions<br>
  [2012-05-10]: Tobias Nipkow added extended regular expressions<br>
  [2012-05-10]: Tobias Nipkow added equivalence checking with partial derivatives

title = The Myhill-Nerode Theorem Based on Regular Expressions
author = Chunhan Wu, Xingyuan Zhang, Christian Urban <http://www.inf.kcl.ac.uk/staff/urbanc/>
topic = Computer Science/Automata and Formal Languages
date = 2011-08-26
abstract = There are many proofs of the Myhill-Nerode theorem using automata. In this library we give a proof entirely based on regular expressions, since regularity of languages can be conveniently defined using regular expressions (it is more painful in HOL to define regularity in terms of automata).  We prove the first direction of the Myhill-Nerode theorem by solving equational systems that involve regular expressions.  For the second direction we give two proofs: one using tagging-functions and another using partial derivatives. We also establish various closure properties of regular languages. Most details of the theories are described in our ITP 2011 paper.

title = Formalizing the Logic-Automaton Connection
author = Stefan Berghofer <http://www.in.tum.de/~berghofe>, Markus Reiter
date = 2009-12-03
topic = Computer Science/Automata and Formal Languages, Logic
abstract = This work presents a formalization of a library for automata on bit strings. It forms the basis of a reflection-based decision procedure for Presburger arithmetic, which is efficiently executable thanks to Isabelle's code generator. With this work, we therefore provide a mechanized proof of a well-known connection between logic and automata theory. The formalization is also described in a publication [TPHOLs 2009].

title = Functional Automata
author = Tobias Nipkow <http://www.in.tum.de/~nipkow/>
date = 2004-03-30
topic = Computer Science/Automata and Formal Languages
abstract = This theory defines deterministic and nondeterministic automata in a functional representation: the transition function/relation and the finality predicate are just functions. Hence the state space may be infinite. It is shown how to convert regular expressions into such automata. A scanner (generator) is implemented with the help of functional automata: the scanner chops the input up into longest recognized substrings. Finally we also show how to convert a certain subclass of functional automata (essentially the finite deterministic ones) into regular sets.
depends-on = Regular-Sets

title = Formalizing Statecharts using Hierarchical Automata
author = Steffen Helke <mailto:helke@cs.tu-berlin.de>, Florian Kammüller <mailto:flokam@cs.tu-berlin.de>
topic = Computer Science/Automata and Formal Languages
date = 2010-08-08
abstract = We formalize in Isabelle/HOL the abtract syntax and a synchronous
 step semantics for the specification language Statecharts. The formalization
 is based on Hierarchical Automata which allow a structural decomposition of
 Statecharts into Sequential Automata. To support the composition of
 Statecharts, we introduce calculating operators to construct a Hierarchical
 Automaton in a stepwise manner. Furthermore, we present a complete semantics
 of Statecharts including a theory of data spaces, which enables the modelling
 of racing effects. We also adapt CTL for
 Statecharts to build a bridge for future combinations with model
 checking. However the main motivation of this work is to provide a sound and
 complete basis for reasoning on Statecharts. As a central meta theorem we
 prove that the well-formedness of a Statechart is preserved by the semantics.

title = Stuttering Equivalence
author = Stephan Merz <http://www.loria.fr/~merz/>
topic = Computer Science/Automata and Formal Languages
date = 2012-05-07
abstract = <p>Two omega-sequences are stuttering equivalent if they differ only by finite repetitions of elements. Stuttering equivalence is a fundamental concept in the theory of concurrent and distributed systems. Notably, Lamport argues that refinement notions for such systems should be insensitive to finite stuttering. Peled and Wilke showed that all PLTL (propositional linear-time temporal logic) properties that are insensitive to stuttering equivalence can be expressed without the next-time operator. Stuttering equivalence is also important for certain verification techniques such as partial-order reduction for model checking.</p> <p>We formalize stuttering equivalence in Isabelle/HOL. Our development relies on the notion of stuttering sampling functions that may skip blocks of identical sequence elements. We also encode PLTL and prove the theorem due to Peled and Wilke.</p>
extra-history =
  Change history:
  [2013-01-31]: Added encoding of PLTL and proved Peled and Wilke's theorem. Adjusted abstract accordingly.

title = A Codatatype of Formal Languages
author = Dmitriy Traytel <mailto:traytel@in.tum.de>
topic = Computer Science/Automata and Formal Languages
date = 2013-11-15
abstract = <p>We define formal languages as a codataype of infinite trees
 branching over the alphabet. Each node in such a tree indicates whether the
 path to this node constitutes a word inside or outside of the language. This
 codatatype is isormorphic to the set of lists representation of languages,
 but caters for definitions by corecursion and proofs by coinduction.</p>
 <p>Regular operations on languages are then defined by primitive corecursion.
 A difficulty arises here, since the standard definitions of concatenation and
 iteration from the coalgebraic literature are not primitively
 corecursive-they require guardedness up-to union/concatenation.
 Without support for up-to corecursion, these operation must be defined as a
 composition of primitive ones (and proved being equal to the standard
 definitions). As an exercise in coinduction we also prove the axioms of
 Kleene algebra for the defined regular operations.</p>
 <p>Furthermore, a language for context-free grammars given by productions in
 Greibach normal form and an initial nonterminal is constructed by primitive
 corecursion, yielding an executable decision procedure for the word problem
 without further ado.</p>

title = Tree Automata
author = Peter Lammich <http://cs.uni-muenster.de/sev/staff/lammich/>
date = 2009-11-25
topic = Computer Science/Automata and Formal Languages
abstract = This work presents a machine-checked tree automata library for Standard-ML, OCaml and Haskell. The algorithms are efficient by using appropriate data structures like RB-trees. The available algorithms for non-deterministic automata include membership query, reduction, intersection, union, and emptiness check with computation of a witness for non-emptiness. The executable algorithms are derived from less-concrete, non-executable algorithms using data-refinement techniques. The concrete data structures are from the Isabelle Collections Framework. Moreover, this work contains a formalization of the class of tree-regular languages and its closure properties under set operations.
depends-on = Collections

title = A Fast SAT Solver for Isabelle in Standard ML
topic = Computer Science/Algorithms
author = Armin Heller
date = 2009-12-09
abstract = This contribution contains a fast SAT solver for Isabelle written in Standard ML. By loading the theory <tt>DPT_SAT_Solver</tt>, the SAT solver installs itself (under the name ``dptsat'') and certain Isabelle tools like Refute will start using it automatically. This is a port of the DPT (Decision Procedure Toolkit) SAT Solver written in OCaml.

title = Depth First Search
author = Toshiaki Nishihara, Yasuhiko Minamide <http://www.score.is.tsukuba.ac.jp/~minamide/>
date = 2004-06-24
topic = Computer Science/Algorithms
abstract = Depth-first search of a graph is formalized with recdef. It is shown that it visits all of the reachable nodes from a given list of nodes. Executable ML code of depth-first search is obtained using the code generation feature of Isabelle/HOL.

title = Fast Fourier Transform
author = Clemens Ballarin <http://www4.in.tum.de/~ballarin/>
date = 2005-10-12
topic = Computer Science/Algorithms
abstract = We formalise a functional implementation of the FFT algorithm over the complex numbers, and its inverse. Both are shown equivalent to the usual definitions of these operations through Vandermonde matrices. They are also shown to be inverse to each other, more precisely, that composition of the inverse and the transformation yield the identity up to a scalar.

title = Gauss-Jordan Elimination for Matrices Represented as Functions
author = Tobias Nipkow <http://www.in.tum.de/~nipkow/>
date = 2011-08-19
topic = Computer Science/Algorithms, Mathematics/Algebra
abstract = This theory provides a compact formulation of Gauss-Jordan elimination for matrices represented as functions. Its distinctive feature is succinctness. It is not meant for large computations.

title = Verification of the Deutsch-Schorr-Waite Graph Marking Algorithm using Data Refinement
author = Viorel Preoteasa <http://users.abo.fi/vpreotea/>, Ralph-Johan Back <http://users.abo.fi/Ralph-Johan.Back/>
date = 2010-05-28
topic = Computer Science/Algorithms
abstract = The verification of the Deutsch-Schorr-Waite graph marking algorithm is used as a benchmark in many formalizations of pointer programs. The main purpose of this mechanization is to show how data refinement of invariant based programs can be used in verifying practical algorithms. The verification starts with an abstract algorithm working on a graph given by a relation <i>next</i> on nodes. Gradually the abstract program is refined into Deutsch-Schorr-Waite graph marking algorithm where only one bit per graph node of additional memory is used for marking.
depends-on = DataRefinementIBP
extra-history =
  Change history:
  [2012-01-05]: Updated for the new definition of data refinement and the new syntax for demonic and angelic update statements

title = Efficient Mergesort
topic = Computer Science/Algorithms
date = 2011-11-09
author = Christian Sternagel <mailto:c.sternagel@gmail.com>
abstract = We provide a formalization of the mergesort algorithm as used in GHC's Data.List module, proving correctness and stability. Furthermore, experimental data suggests that generated (Haskell-)code for this algorithm is much faster than for previous algorithms available in the Isabelle distribution.
extra-history =
  Change history:
  [2012-10-24]: Added reference to journal article

title = Formal Verification of Modern SAT Solvers
author = Filip Maric <http://poincare.matf.bg.ac.rs/~filip/>
date = 2008-07-23
topic = Computer Science/Algorithms
abstract = This document contains formal correctness proofs of modern SAT solvers. Following (Krstic et al, 2007) and (Nieuwenhuis et al., 2006), solvers are described using state-transition systems. Several different SAT solver descriptions are given and their partial correctness and termination is proved. These include: <ul> <li> a solver based on classical DPLL procedure (using only a backtrack-search with unit propagation),</li> <li> a very general solver with backjumping and learning (similar to the description given in (Nieuwenhuis et al., 2006)), and</li> <li> a solver with a specific conflict analysis algorithm (similar to the description given in (Krstic et al., 2007)).</li> </ul> Within the SAT solver correctness proofs, a large number of lemmas about propositional logic and CNF formulae are proved. This theory is self-contained and could be used for further exploring of properties of CNF based SAT algorithms.

title = Executable Transitive Closures of Finite Relations
topic = Computer Science/Algorithms
date = 2011-03-14
author = Christian Sternagel <mailto:christian.sternagel@uibk.ac.at>, René Thiemann <mailto:rene.thiemann@uibk.ac.at>
license = LGPL
abstract = We provide a generic work-list algorithm to compute the transitive closure of finite relations where only successors of newly detected states are generated. This algorithm is then instantiated for lists over arbitrary carriers and red black trees (which are faster but require a linear order on the carrier), respectively.  Our formalization was performed as part of the IsaFoR/CeTA project where reflexive transitive closures of large tree automata have to be computed.
depends-on = Collections, Abstract-Rewriting

title = Executable Transitive Closures
topic = Computer Science/Algorithms
date = 2012-02-29
author = René Thiemann <mailto:rene.thiemann@uibk.ac.at>
license = LGPL
abstract =
  We provide a generic work-list algorithm to compute the
  (reflexive-)transitive closure of relations where only successors of newly
  detected states are generated.
  In contrast to our previous work, the relations do not have to be finite,
  but each element must only have finitely many (indirect) successors.
  Moreover, a subsumption relation can be used instead of pure equality.
  An executable variant of the algorithm is available where the generic operations
  are instantiated with list operations.

  This formalization was performed as part of the IsaFoR/CeTA project,
  and it has been used to certify size-change
  termination proofs where large transitive closures have to be computed.
depends-on = Regular-Sets

title = Much Ado About Two
author = Sascha Böhme <http://www4.in.tum.de/~boehmes/>
date = 2007-11-06
topic = Computer Science/Algorithms
abstract = This article is an Isabelle formalisation of a paper with the same title. In a similar way as Knuth's 0-1-principle for sorting algorithms, that paper develops a 0-1-2-principle for parallel prefix computations.

title = Proving the Correctness of Disk Paxos
date = 2005-06-22
author = Mauro Jaskelioff <http://www.fceia.unr.edu.ar/~mauro/>, Stephan Merz <http://www.loria.fr/~merz/>
topic = Computer Science/Algorithms/Distributed
abstract = Disk Paxos is an algorithm for building arbitrary fault-tolerant distributed systems. The specification of Disk Paxos has been proved correct informally and tested using the TLC model checker, but up to now, it has never been fully formally verified. In this work we have formally verified its correctness using the Isabelle theorem prover and the HOL logic system, showing that Isabelle is a practical tool for verifying properties of TLA+ specifications.

title = Formalization of a Generalized Protocol for Clock Synchronization
author = Alwen Tiu <http://users.cecs.anu.edu.au/~tiu/>
date = 2005-06-24
topic = Computer Science/Algorithms/Distributed
abstract = We formalize the generalized Byzantine fault-tolerant clock synchronization protocol of Schneider. This protocol abstracts from particular algorithms or implementations for clock synchronization. This abstraction includes several assumptions on the behaviors of physical clocks and on general properties of concrete algorithms/implementations. Based on these assumptions the correctness of the protocol is proved by Schneider. His proof was later verified by Shankar using the theorem prover EHDM (precursor to PVS). Our formalization in Isabelle/HOL is based on Shankar's formalization.

title = Instances of Schneider's generalized protocol of clock synchronization
author = Damián Barsotti <http://www.cs.famaf.unc.edu.ar/~damian/>
date = 2006-03-15
topic = Computer Science/Algorithms/Distributed
abstract = F. B. Schneider ("Understanding protocols for Byzantine clock synchronization") generalizes a number of protocols for Byzantine fault-tolerant clock synchronization and presents a uniform proof for their correctness. In Schneider's schema, each processor maintains a local clock by periodically adjusting each value to one computed by a convergence function applied to the readings of all the clocks. Then, correctness of an algorithm, i.e. that the readings of two clocks at any time are within a fixed bound of each other, is based upon some conditions on the convergence function. To prove that a particular clock synchronization algorithm is correct it suffices to show that the convergence function used by the algorithm meets Schneider's conditions. Using the theorem prover Isabelle, we formalize the proofs that the convergence functions of two algorithms, namely, the Interactive Convergence Algorithm (ICA) of Lamport and Melliar-Smith and the Fault-tolerant Midpoint algorithm of Lundelius-Lynch, meet Schneider's conditions. Furthermore, we experiment on handling some parts of the proofs with fully automatic tools like ICS and CVC-lite. These theories are part of a joint work with Alwen Tiu and Leonor P. Nieto <a href="http://users.rsise.anu.edu.au/~tiu/clocksync.pdf">"Verification of Clock Synchronization Algorithms: Experiments on a combination of deductive tools"</a> in proceedings of AVOCS 2005. In this work the correctness of Schneider schema was also verified using Isabelle (entry <a href="GenClock.shtml">GenClock</a> in AFP).

title = Verifying Fault-Tolerant Distributed Algorithms in the Heard-Of Model
date = 2012-07-27
author = Henri Debrat <mailto:henri.debrat@loria.fr>, Stephan Merz <http://www.loria.fr/~merz/>
topic = Computer Science/Algorithms/Distributed
abstract =
 Distributed computing is inherently based on replication, promising
 increased tolerance to failures of individual computing nodes or
 communication channels. Realizing this promise, however, involves
 quite subtle algorithmic mechanisms, and requires precise statements
 about the kinds and numbers of faults that an algorithm tolerates (such
 as process crashes, communication faults or corrupted values).  The
 landmark theorem due to Fischer, Lynch, and Paterson shows that it is
 impossible to achieve Consensus among N asynchronously communicating
 nodes in the presence of even a single permanent failure. Existing
 solutions must rely on assumptions of "partial synchrony".
 Indeed, there have been numerous misunderstandings on what exactly a given
 algorithm is supposed to realize in what kinds of environments. Moreover, the
 abundance of subtly different computational models complicates comparisons
 between different algorithms. Charron-Bost and Schiper introduced the Heard-Of
 model for representing algorithms and failure assumptions in a uniform
 framework, simplifying comparisons between algorithms.
 In this contribution, we represent the Heard-Of model in Isabelle/HOL. We define
 two semantics of runs of algorithms with different unit of atomicity and relate
 these through a reduction theorem that allows us to verify algorithms in the
 coarse-grained semantics (where proofs are easier) and infer their correctness
 for the fine-grained one (which corresponds to actual executions). We
 instantiate the framework by verifying six Consensus algorithms that differ in
 the underlying algorithmic mechanisms and the kinds of faults they tolerate.

title = Abortable Linearizable Modules
author = Rachid Guerraoui <mailto:rachid.guerraoui@epfl.ch>, Viktor Kuncak <http://lara.epfl.ch/~kuncak/>, Giuliano Losa <mailto:giuliano.losa@epfl.ch>
date = 2012-03-1
topic = Computer Science/Algorithms/Distributed
abstract =
  We define the Abortable Linearizable Module automaton (ALM for short)
  and prove its key composition property using the IOA theory of
  HOLCF. The ALM is at the heart of the Speculative Linearizability
  framework. This framework simplifies devising correct speculative
  algorithms by enabling their decomposition into independent modules
  that can be analyzed and proved correct in isolation. It is
  particularly useful when working in a distributed environment, where
  the need to tolerate faults and asynchrony has made current
  monolithic protocols so intricate that it is no longer tractable to
  check their correctness. Our theory contains a typical example of a
  refinement proof in the I/O-automata framework of Lynch and Tuttle.

title = AVL Trees
author = Tobias Nipkow <http://www.in.tum.de/~nipkow/>, Cornelia Pusch
date = 2004-03-19
topic = Computer Science/Data Structures
abstract = Two formalizations of AVL trees with room for extensions. The first formalization is monolithic and shorter, the second one in two stages, longer and a bit simpler. The final implementation is the same. If you are interested in developing this further, please contact <tt>gerwin.klein@nicta.com.au</tt>.
extra-history =
  Change history:
  [2011-04-11]: Ondrej Kuncar added delete function

title = BDD Normalisation
author = Veronika Ortner, Norbert Schirmer
date = 2008-02-29
topic = Computer Science/Data Structures
abstract = We present the verification of the normalisation of a binary decision diagram (BDD). The normalisation follows the original algorithm presented by Bryant in 1986 and transforms an ordered BDD in a reduced, ordered and shared BDD. The verification is based on Hoare logics.
depends-on = Simpl

title = Binary Search Trees
author = Viktor Kuncak <http://lara.epfl.ch/~kuncak/>
date = 2004-04-05
topic = Computer Science/Data Structures
abstract = The correctness is shown of binary search tree operations (lookup, insert and remove) implementing a set. Two versions are given, for both structured and linear (tactic-style) proofs. An implementation of integer-indexed maps is also verified.

title = Functional Binomial Queues
author = René Neumann <mailto:neumannr@in.tum.de>
date = 2010-10-28
topic = Computer Science/Data Structures
abstract = Priority queues are an important data structure and efficient implementations of them are crucial. We implement a functional variant of binomial queues in Isabelle/HOL and show its functional correctness. A verification against an abstract reference specification of priority queues has also been attempted, but could not be achieved to the full extent.

title = Binomial Heaps and Skew Binomial Heaps
author = Rene Meis <mailto:rene.meis@uni-muenster.de>, Finn Nielsen <mailto:finn.nielsen@uni-muenster.de>, Peter Lammich <http://cs.uni-muenster.de/sev/staff/lammich/>
date = 2010-10-28
topic = Computer Science/Data Structures
abstract =
 We implement and prove correct binomial heaps and skew binomial heaps.
 Both are data-structures for priority queues.
 While binomial heaps have logarithmic <em>findMin</em>, <em>deleteMin</em>,
 <em>insert</em>, and <em>meld</em> operations,
 skew binomial heaps have constant time <em>findMin</em>, <em>insert</em>,
 and <em>meld</em> operations, and only the <em>deleteMin</em>-operation is
 logarithmic. This is achieved by using <em>skew links</em> to avoid
 cascading linking on <em>insert</em>-operations, and <em>data-structural
 bootstrapping</em> to get constant-time <em>findMin</em> and <em>meld</em>
 operations.  Our implementation follows the paper by Brodal and Okasaki.

title = Finger Trees
author = Benedikt Nordhoff <mailto:b_nord01@uni-muenster.de>, Stefan Körner <mailto:s_koer03@uni-muenster.de>, Peter Lammich <http://cs.uni-muenster.de/sev/staff/lammich/>
date = 2010-10-28
topic = Computer Science/Data Structures
abstract =
 We implement and prove correct 2-3 finger trees.
 Finger trees are a general purpose data structure, that can be used to
 efficiently implement other data structures, such as priority queues.
 Intuitively, a finger tree is an annotated sequence, where the annotations are
 elements of a monoid. Apart from operations to access the ends of the sequence,
 the main operation is to split the sequence at the point where a
 <em>monotone predicate</em> over the sum of the left part of the sequence
 becomes true for the first time.
 The implementation follows the paper of Hintze and Paterson.
 The code generator can be used to get efficient, verified code.

title = Code Generation for Functions as Data
author = Andreas Lochbihler <http://www.infsec.ethz.ch/people/andreloc>
date = 2009-05-06
topic = Computer Science/Data Structures
abstract = <b>FinFun is now part of the Isabelle distribution. The entry is kept for archival; maintained but not developed further. Use the Isabelle distribution version instead.</b><p>FinFuns are total functions that are constant except for a finite set of points, i.e. a generalisation of finite maps. They are formalised as a new type in Isabelle/HOL such that the code generator can handle equality tests and quantification on FinFuns. On the code output level, FinFuns are explicitly represented by constant functions and pointwise updates, similarly to associative lists. Inside the logic, they behave like ordinary functions with extensionality. Via the update/constant pattern, a recursion combinator and an induction rule for FinFuns allow for defining and reasoning about operators on FinFun that are also executable.</p>
extra-history =
  Change history:
        new concept domain of a FinFun as a FinFun
        (revision 34b3517cbc09)<br>
        new conversion function from FinFun to list of elements in the domain
        (revision 0c167102e6ed)<br>
        replace sets as FinFuns by predicates as FinFuns because the set type constructor has been reintroduced
        (revision b7aa87989f3a)

title = Collections Framework
author = Peter Lammich <http://cs.uni-muenster.de/sev/staff/lammich/>, Andreas Lochbihler <http://www.infsec.ethz.ch/people/andreloc>, Thomas Tuerk
date = 2009-11-25
topic = Computer Science/Data Structures
abstract = This development provides an efficient, extensible, machine checked collections framework. The library adopts the concepts of interface, implementation and generic algorithm from object-oriented programming and implements them in Isabelle/HOL. The framework features the use of data refinement techniques to refine an abstract specification (using high-level concepts like sets) to a more concrete implementation (using collection datastructures, like red-black-trees). The code-generator of Isabelle/HOL can be used to generate efficient code.
depends-on = Binomial-Heaps, Finger-Trees
extra-history =
  Change history:
  [2010-10-08]: New Interfaces: OrderedSet, OrderedMap, List.
    Fifo now implements list-interface: Function names changed: put/get --> enqueue/dequeue.
    New Implementations: ArrayList, ArrayHashMap, ArrayHashSet, TrieMap, TrieSet.
    Invariant-free datastructures: Invariant implicitely hidden in typedef.
    Record-interfaces: All operations of an interface encapsulated as record.
    Examples moved to examples subdirectory.<br>
  [2010-12-01]: New Interfaces: Priority Queues, Annotated Lists. Implemented by finger trees, (skew) binomial queues.
  [2011-10-10]: SetSpec: Added operations: sng, isSng, bexists, size_abort, diff, filter, iterate_rule_insertP
                MapSpec: Added operations: sng, isSng, iterate_rule_insertP, bexists, size, size_abort, restrict,
                          map_image_filter, map_value_image_filter
                Some maintenance changes
  [2012-04-25]: New iterator foundation by Tuerk. Various maintenance changes.

title = Light-weight Containers
author = Andreas Lochbihler <http://www.infsec.ethz.ch/people/andreloc>
date = 2013-04-15
topic = Computer Science/Data Structures
abstract =
  This development provides a framework for container types like sets and maps such that generated code implements these containers with different (efficient) data structures.
  Thanks to type classes and refinement during code generation, this light-weight approach can seamlessly replace Isabelle's default setup for code generation.
  Heuristics automatically pick one of the available data structures depending on the type of elements to be stored, but users can also choose on their own.
  The extensible design permits to add more implementations at any time.
  To support arbitrary nesting of sets, we define a linear order on sets based on a linear order of the elements and provide efficient implementations.
  It even allows to compare complements with non-complements.
depends-on = Datatype_Order_Generator
extra-history =
  Change history:
  [2013-07-11]: add pretty printing for sets (revision 7f3f52c5f5fa)<br>
        provide generators for canonical type class instantiations
        (revision 159f4401f4a8 by René Thiemann)

title = File Refinement
author = Karen Zee <http://www.mit.edu/~kkz/>, Viktor Kuncak <http://lara.epfl.ch/~kuncak/>
date = 2004-12-09
topic = Computer Science/Data Structures
abstract = These theories illustrates the verification of basic file operations (file creation, file read and file write) in the Isabelle theorem prover. We describe a file at two levels of abstraction: an abstract file represented as a resizable array, and a concrete file represented using data blocks.

title = Generating linear orders for datatypes
author = René Thiemann <mailto:rene.thiemann@uibk.ac.at>
date = 2012-08-07
topic = Computer Science/Data Structures
abstract =
  We provide a framework for registering automatic methods to derive
  class instances of datatypes, as it is possible using Haskell's ``deriving Ord, Show, ...'' feature.
  We further implemented such automatic methods to derive (linear) orders or hash-functions which are
  required in the Isabelle Collection Framework. Moreover, for the tactic of Huffman and Krauss to show that a
  datatype is countable, we implemented a wrapper so that this tactic becomes accessible in our framework.
  Our formalization was performed as part of the <a href="http://cl-informatik.uibk.ac.at/software/ceta">IsaFoR/CeTA</a> project.
  With our new tactic we could completely remove
  tedious proofs for linear orders of two datatypes.

title = List Index
date = 2010-02-20
author = Tobias Nipkow <http://www.in.tum.de/~nipkow>
topic = Computer Science/Data Structures
abstract = This theory provides functions for finding the index of an element in a list, by predicate and by value.

title = Infinite Lists
date = 2011-02-23
author = David Trachtenherz <http://www4.in.tum.de/~trachten>
topic = Computer Science/Data Structures
abstract = We introduce a theory of infinite lists in HOL formalized as functions over naturals (folder ListInf, theories ListInf and ListInf_Prefix). It also provides additional results for finite lists (theory ListInf/List2), natural numbers (folder CommonArith, esp. division/modulo, naturals with infinity), sets (folder CommonSet, esp. cutting/truncating sets, traversing sets of naturals).

title = Executable Matrix Operations on Matrices of Arbitrary Dimensions
topic = Computer Science/Data Structures
date = 2010-06-17
author = Christian Sternagel <http://cl-informatik.uibk.ac.at/~griff>, René Thiemann <http://cl-informatik.uibk.ac.at/~thiemann>
license = LGPL
abstract =
  We provide the operations of matrix addition, multiplication,
  transposition, and matrix comparisons as executable functions over
  ordered semirings. Moreover, it is proven that strongly normalizing
  (monotone) orders can be lifted to strongly normalizing (monotone) orders
  over matrices. We further show that the standard semirings over the
  naturals, integers, and rationals, as well as the arctic semirings
  satisfy the axioms that are required by our matrix theory. Our
  formalization is part of the <a
  href="http://cl-informatik.uibk.ac.at/software/ceta">CeTA</a> system
  which contains several termination techniques. The provided theories have
  been essential to formalize matrix-interpretations and arctic
depends-on = Abstract-Rewriting
extra-history =
  Change history:
  [2010-09-17]: Moved theory on arbitrary (ordered) semirings to Abstract Rewriting.

title = The Textbook Proof of Huffman's Algorithm
author = Jasmin Christian Blanchette <http://www4.in.tum.de/~blanchet>
date = 2008-10-15
topic = Computer Science/Data Structures
abstract = Huffman's algorithm is a procedure for constructing a binary tree with minimum weighted path length. This report presents a formal proof of the correctness of Huffman's algorithm written using Isabelle/HOL. Our proof closely follows the sketches found in standard algorithms textbooks, uncovering a few snags in the process. Another distinguishing feature of our formalization is the use of custom induction rules to help Isabelle's automatic tactics, leading to very short proofs for most of the lemmas.

title = Coinductive
topic = Computer Science/Functional Programming
author = Andreas Lochbihler <http://www.infsec.ethz.ch/people/andreloc>
date = 2010-02-12
abstract = This article collects formalisations of general-purpose coinductive data types and sets. Currently, it contains coinductive natural numbers, coinductive lists, i.e. lazy lists or streams, infinite streams, coinductive terminated lists, coinductive resumptions, a library of operations on coinductive lists, and a version of König's lemma as an application for coinductive lists.<br>The initial theory was contributed by Paulson and Wenzel. Extensions and other coinductive formalisations of general interest are welcome.
extra-history =
  Change history:
        coinductive lists: setup for quotient package
        (revision 015574f3bf3c)<br>
        new codatatype terminated lazy lists
        (revision e12de475c558)<br>
        terminated lazy lists: setup for quotient package;
        more lemmas
        (revision 6ead626f1d01)<br>
        Koenig's lemma as an example application for coinductive lists
        (revision f81ce373fa96)<br>
        lazy implementation of coinductive (terminated) lists for the code generator
        (revision 6034973dce83)<br>
        new codatatype resumption
        (revision 811364c776c7)<br>
        new codatatype stream with operations (with contributions by Peter Gammie)
        (revision dd789a56473c)<br>
        construct codatatypes with the BNF package and adjust the definitions and proofs,
        setup for lifting and transfer packages
        (revision f593eda5b2c0)<br>
        stream theory uses type and operations from HOL/BNF/Examples/Stream
        (revision 692809b2b262)<br>

title = Stream Fusion
author = Brian Huffman <http://cs.pdx.edu/~brianh>
topic = Computer Science/Functional Programming
date = 2009-04-29
abstract = Stream Fusion is a system for removing intermediate list structures from Haskell programs; it consists of a Haskell library along with several compiler rewrite rules. (The library is available <a href="http://hackage.haskell.org/package/stream-fusion">online</a>.)<br><br>These theories contain a formalization of much of the Stream Fusion library in HOLCF. Lazy list and stream types are defined, along with coercions between the two types, as well as an equivalence relation for streams that generate the same list. List and stream versions of map, filter, foldr, enumFromTo, append, zipWith, and concatMap are defined, and the stream versions are shown to respect stream equivalence.

title = Type Constructor Classes and Monad Transformers
author = Brian Huffman <mailto:huffman@in.tum.de>
date = 2012-06-26
topic = Computer Science/Functional Programming
abstract =
 These theories contain a formalization of first class type constructors
 and axiomatic constructor classes for HOLCF. This work is described
 in detail in the ICFP 2012 paper <i>Formal Verification of Monad
 Transformers</i> by the author. The formalization is a revised and
 updated version of earlier joint work with Matthews and White.
 Based on the hierarchy of type classes in Haskell, we define classes
 for functors, monads, monad-plus, etc. Each one includes all the
 standard laws as axioms. We also provide a new user command,
 tycondef, for defining new type constructors in HOLCF. Using tycondef,
 we instantiate the type class hierarchy with various monads and monad

title = CoreC++
author = Daniel Wasserrab <http://pp.info.uni-karlsruhe.de/personhp/daniel_wasserrab.php>
date = 2006-05-15
topic = Computer Science/Programming Languages/Language Definitions
abstract = We present an operational semantics and type safety proof for multiple inheritance in C++. The semantics models the behavior of method calls, field accesses, and two forms of casts in C++ class hierarchies. For explanations see the OOPSLA 2006 paper by Wasserrab, Nipkow, Snelting and Tip.

title = A Theory of Featherweight Java in Isabelle/HOL
author = J. Nathan Foster <http://www.cs.cornell.edu/~jnfoster/>, Dimitrios Vytiniotis <http://research.microsoft.com/en-us/people/dimitris/>
date = 2006-03-31
topic = Computer Science/Programming Languages/Language Definitions
abstract = We formalize the type system, small-step operational semantics, and type soundness proof for Featherweight Java, a simple object calculus, in Isabelle/HOL.

title = Jinja is not Java
author = Gerwin Klein <http://www.cse.unsw.edu.au/~kleing/>, Tobias Nipkow <http://www.in.tum.de/~nipkow/>
date = 2005-06-01
topic = Computer Science/Programming Languages/Language Definitions
abstract = We introduce Jinja, a Java-like programming language with a formal semantics designed to exhibit core features of the Java language architecture. Jinja is a compromise between realism of the language and tractability and clarity of the formal semantics. The following aspects are formalised: a big and a small step operational semantics for Jinja and a proof of their equivalence; a type system and a definite initialisation analysis; a type safety proof of the small step semantics; a virtual machine (JVM), its operational semantics and its type system; a type safety proof for the JVM; a bytecode verifier, i.e. data flow analyser for the JVM; a correctness proof of the bytecode verifier w.r.t. the type system; a compiler and a proof that it preserves semantics and well-typedness. The emphasis of this work is not on particular language features but on providing a unified model of the source language, the virtual machine and the compiler. The whole development has been carried out in the theorem prover Isabelle/HOL.

title = Jinja with Threads
author = Andreas Lochbihler <http://www.infsec.ethz.ch/people/andreloc>
date = 2007-12-03
topic = Computer Science/Programming Languages/Language Definitions
abstract = We extend the Jinja source code semantics by Klein and Nipkow with Java-style arrays and threads. Concurrency is captured in a generic framework semantics for adding concurrency through interleaving to a sequential semantics, which features dynamic thread creation, inter-thread communication via shared memory, lock synchronisation and joins. Also, threads can suspend themselves and be notified by others. We instantiate the framework with the adapted versions of both Jinja source and byte code and show type safety for the multithreaded case. Equally, the compiler from source to byte code is extended, for which we prove weak bisimilarity between the source code small step semantics and the defensive Jinja virtual machine. On top of this, we formalise the JMM and show the DRF guarantee and consistency. For description of the different parts, see Lochbihler's papers at FOOL 2008, ESOP 2010, ITP 2011, and ESOP 2012.
depends-on = Collections, Coinductive
extra-history =
  Change history:
        added bytecode formalisation with arrays and threads, added thread joins
        (revision f74a8be156a7)<br>
        added verified compiler from source code to bytecode;
        encapsulate native methods in separate semantics
        (revision e4f26541e58a)<br>
        extended compiler correctness proof to infinite and deadlocking computations
        (revision e50282397435)<br>
        added thread interruption;
        new abstract memory model with sequential consistency as implementation
        (revision 0cb9e8dbd78d)<br>
        new thread interruption model
        (revision c0440d0a1177)<br>
        preliminary version of the Java memory model for source code
        (revision 02fee0ef3ca2)<br>
        improved version of the Java memory model, also for bytecode
        executable scheduler for source code semantics
        (revision 1f41c1842f5a)<br>
        simplified code generator setup
        new random scheduler
        (revision 3059dafd013f)<br>
        new interruption model,
        generalized JMM proof of DRF guarantee,
        allow class Object to declare methods and fields,
        simplified subtyping relation,
        corrected division and modulo implementation
        (revision 46e4181ed142)<br>
        added example programs
        (revision bf0b06c8913d)<br>
        type safety proof for the Java memory model,
        allow spurious wake-ups
        (revision 76063d860ae0)<br>
        support for non-deterministic memory allocators
        (revision cc3344a49ced)

title = Locally Nameless Sigma Calculus
author = Ludovic Henrio <mailto:Ludovic.Henrio@sophia.inria.fr>, Florian Kammüller <mailto:flokam@cs.tu-berlin.de>, Bianca Lutz <mailto:sowilo@cs.tu-berlin.de>, Henry Sudhof <mailto:hsudhof@cs.tu-berlin.de>
date = 2010-04-30
topic = Computer Science/Programming Languages/Language Definitions
abstract = We present a Theory of Objects based on the original functional sigma-calculus by Abadi and Cardelli but with an additional parameter to methods. We prove confluence of the operational semantics following the outline of Nipkow's proof of confluence for the lambda-calculus reusing his theory Commutation, a generic diamond lemma reduction. We furthermore formalize a simple type system for our sigma-calculus including a proof of type safety. The entire development uses the concept of Locally Nameless representation for binders. We reuse an earlier proof of confluence for a simpler sigma-calculus based on de Bruijn indices and lists to represent objects.

title = AutoFocus Stream Processing for Single-Clocking and Multi-Clocking Semantics
author = David Trachtenherz <http://www4.in.tum.de/~trachten/>
date = 2011-02-23
topic = Computer Science/Programming Languages/Language Definitions
abstract = We formalize the AutoFocus Semantics (a time-synchronous subset of the Focus formalism) as stream processing functions on finite and infinite message streams represented as finite/infinite lists. The formalization comprises both the conventional single-clocking semantics (uniform global clock for all components and communications channels) and its extension to multi-clocking semantics (internal execution clocking of a component may be a multiple of the external communication clocking). The semantics is defined by generic stream processing functions making it suitable for simulation/code generation in Isabelle/HOL. Furthermore, a number of AutoFocus semantics properties are formalized using definitions from the IntervalLogic theories.

title = Logical Relations for PCF
author = Peter Gammie <mailto:peteg42@gmail.com>
date = 2012-07-01
topic = Computer Science/Programming Languages/Lambda Calculi
abstract = We apply Andy Pitts's methods of defining relations over domains to
  several classical results in the literature. We show that the Y
  combinator coincides with the domain-theoretic fixpoint operator,
  that parallel-or and the Plotkin existential are not definable in
  PCF, that the continuation semantics for PCF coincides with the
  direct semantics, and that our domain-theoretic semantics for PCF is
  adequate for reasoning about contextual equivalence in an
  operational semantics. Our version of PCF is untyped and has both
  strict and non-strict function abstractions. The development is
  carried out in HOLCF.

title = POPLmark Challenge Via de Bruijn Indices
author = Stefan Berghofer <http://www.in.tum.de/~berghofe>
date = 2007-08-02
topic = Computer Science/Programming Languages/Lambda Calculi
abstract = We present a solution to the POPLmark challenge designed by Aydemir et al., which has as a goal the formalization of the meta-theory of System F<sub>&lt;:</sub>. The formalization is carried out in the theorem prover Isabelle/HOL using an encoding based on de Bruijn indices. We start with a relatively simple formalization covering only the basic features of System F<sub>&lt;:</sub>, and explain how it can be extended to also cover records and more advanced binding constructs.

title = Strong Normalization of Moggis's Computational Metalanguage
author = Christian Doczkal <mailto:doczkal@ps.uni-saarland.de>
date = 2010-08-29
topic = Computer Science/Programming Languages/Lambda Calculi
abstract = Handling variable binding is one of the main difficulties in formal proofs. In this context, Moggi's computational metalanguage serves as an interesting case study. It features monadic types and a commuting conversion rule that rearranges the binding structure. Lindley and Stark have given an elegant proof of strong normalization for this calculus. The key construction in their proof is a notion of relational TT-lifting, using stacks of elimination contexts to obtain a Girard-Tait style logical relation. I give a formalization of their proof in Isabelle/HOL-Nominal with a particular emphasis on the treatment of bound variables.

title = Mini ML
author = Wolfgang Naraschewski, Tobias Nipkow <http://www.in.tum.de/~nipkow/>
date = 2004-03-19
topic = Computer Science/Programming Languages/Type Systems
abstract = This theory defines the type inference rules and the type inference algorithm <i>W</i> for MiniML (simply-typed lambda terms with <tt>let</tt>) due to Milner. It proves the soundness and completeness of <i>W</i> w.r.t. the rules.

title = A Sequential Imperative Programming Language Syntax, Semantics, Hoare Logics and Verification Environment
author = Norbert Schirmer
date = 2008-02-29
topic = Computer Science/Programming Languages/Language Definitions, Computer Science/Programming Languages/Logics
license = LGPL
abstract = We present the theory of Simpl, a sequential imperative programming language. We introduce its syntax, its semantics (big and small-step operational semantics) and Hoare logics for both partial as well as total correctness. We prove soundness and completeness of the Hoare logic. We integrate and automate the Hoare logic in Isabelle/HOL to obtain a practically usable verification environment for imperative programs. Simpl is independent of a concrete programming language but expressive enough to cover all common language features: mutually recursive procedures, abrupt termination and exceptions, runtime faults, local and global variables, pointers and heap, expressions with side effects, pointers to procedures, partial application and closures, dynamic method invocation and also unbounded nondeterminism.

title = Separation Algebra
author = Gerwin Klein <mailto:kleing@cse.unsw.edu.au>, Rafal Kolanski <mailto:rafal.kolanski@nicta.com.au>, Andrew Boyton <mailto:andrew.boyton@nicta.com.au>
date = 2012-05-11
topic = Computer Science/Programming Languages/Logics
license = BSD
abstract = We present a generic type class implementation of separation algebra for Isabelle/HOL as well as lemmas and generic tactics which can be used directly for any instantiation of the type class. <P> The ex directory contains example instantiations that include structures such as a heap or virtual memory. <P> The abstract separation algebra is based upon "Abstract Separation Logic" by Calcagno et al. These theories are also the basis of the ITP 2012 rough diamond "Mechanised Separation Algebra" by the authors. <P> The aim of this work is to support and significantly reduce the effort for future separation logic developments in Isabelle/HOL by factoring out the part of separation logic that can be treated abstractly once and for all. This includes developing typical default rule sets for reasoning as well as automated tactic support for separation logic.

title = A Separation Logic Framework for Imperative HOL
author = Peter Lammich <http://cs.uni-muenster.de/sev/staff/lammich/>, Rene Meis <mailto:rene.meis@uni-due.de>
date = 2012-11-14
topic = Computer Science/Programming Languages/Logics
license = BSD
abstract =
  We provide a framework for separation-logic based correctness proofs of
  Imperative HOL programs. Our framework comes with a set of proof methods to
  automate canonical tasks such as verification condition generation and
  frame inference. Moreover, we provide a set of examples that show the
  applicability of our framework. The examples include algorithms on lists,
  hash-tables, and union-find trees. We also provide abstract interfaces for
  lists, maps, and sets, that allow to develop generic imperative algorithms
  and use data-refinement techniques.
  As we target Imperative HOL, our programs can be translated to
  efficiently executable code in various target languages, including
  ML, OCaml, Haskell, and Scala.

title = Inductive Study of Confidentiality
author = Giampaolo Bella <http://www.dmi.unict.it/~giamp/>
date = 2012-05-02
topic = Computer Science/Security
abstract = This document contains the full theory files accompanying article <i>Inductive Study of Confidentiality --- for Everyone</i> in <i>Formal Aspects of Computing</i>. They aim at an illustrative and didactic presentation of the Inductive Method of protocol analysis, focusing on the treatment of one of the main goals of security protocols: confidentiality against a threat model. The treatment of confidentiality, which in fact forms a key aspect of all protocol analysis tools, has been found cryptic by many learners of the Inductive Method, hence the motivation for this work. The theory files in this document guide the reader step by step towards design and proof of significant confidentiality theorems. These are developed against two threat models, the standard Dolev-Yao and a more audacious one, the General Attacker, which turns out to be particularly useful also for teaching purposes.

title = Possibilistic Noninterference
author = Andrei Popescu <mailto:uuomul@yahoo.com>, Johannes Hölzl <mailto:hoelzl@in.tum.de>
date = 2012-09-10
topic = Computer Science/Security, Computer Science/Programming Languages/Type Systems
abstract = We formalize a wide variety of Volpano/Smith-style  noninterference
  notions for a while language with parallel composition.
  We systematize and classify these notions according to
  compositionality w.r.t. the language constructs. Compositionality
  yields sound syntactic criteria (a.k.a. type systems) in a uniform way.
  An <a href="http://www.in.tum.de/~nipkow/pubs/cpp12.html">article</a>
  about these proofs is published in the proceedings
  of the conference Certified Programs and Proofs 2012.

title = A Correctness Proof for the Volpano/Smith Security Typing System
author = Gregor Snelting <http://pp.info.uni-karlsruhe.de/personhp/gregor_snelting.php>, Daniel Wasserrab <http://pp.info.uni-karlsruhe.de/personhp/daniel_wasserrab.php>
date = 2008-09-02
topic = Computer Science/Programming Languages/Type Systems, Computer Science/Security
abstract = The Volpano/Smith/Irvine security type systems requires that variables are annotated as high (secret) or low (public), and provides typing rules which guarantee that secret values cannot leak to public output ports. This property of a program is called confidentiality. For a simple while-language without threads, our proof shows that typeability in the Volpano/Smith system guarantees noninterference. Noninterference means that if two initial states for program execution are low-equivalent, then the final states are low-equivalent as well. This indeed implies that secret values cannot leak to public ports. The proof defines an abstract syntax and operational semantics for programs, formalizes noninterference, and then proceeds by rule induction on the operational semantics. The mathematically most intricate part is the treatment of implicit flows. Note that the Volpano/Smith system is not flow-sensitive and thus quite unprecise, resulting in false alarms. However, due to the correctness property, all potential breaks of confidentiality are discovered.

title = Abstract Hoare Logics
author = Tobias Nipkow <http://www.in.tum.de/~nipkow>
date = 2006-08-08
topic = Computer Science/Programming Languages/Logics
abstract = These therories describe Hoare logics for a number of imperative language constructs, from while-loops to mutually recursive procedures. Both partial and total correctness are treated. In particular a proof system for total correctness of recursive procedures in the presence of unbounded nondeterminism is presented.

title = Kleene Algebra
author = Alasdair Armstrong <http://staffwww.dcs.shef.ac.uk/people/A.Armstrong/>, Georg Struth <http://staffwww.dcs.shef.ac.uk/people/G.Struth/>, Tjark Weber <http://user.it.uu.se/~tjawe125/>
date = 2013-01-15
topic = Computer Science/Programming Languages/Logics, Computer Science/Automata and Formal Languages, Mathematics/Algebra
abstract =
 These files contain a formalisation of variants of Kleene algebras and
 their most important models as axiomatic type classes in Isabelle/HOL.
 Kleene algebras are foundational structures in computing with
 applications ranging from automata and language theory to computational
 modeling, program construction and verification.
 We start with formalising dioids, which are additively idempotent
 semirings, and expand them by axiomatisations of the Kleene star for
 finite iteration and an omega operation for infinite iteration. We
 show that powersets over a given monoid, (regular) languages, sets of
 paths in a graph, sets of computation traces, binary relations and
 formal power series form Kleene algebras, and consider further models
 based on lattices, max-plus semirings and min-plus semirings. We also
 demonstrate that dioids are closed under the formation of matrices
 (proofs for Kleene algebras remain to be completed).
 On the one hand we have aimed at a reference formalisation of variants
 of Kleene algebras that covers a wide range of variants and the core
 theorems in a structured and modular way and provides readable proofs
 at text book level. On the other hand, we intend to use this algebraic
 hierarchy and its models as a generic algebraic middle-layer from which
 programming applications can quickly be explored, implemented and verified.

title = A Bytecode Logic for JML and Types
author = Lennart Beringer <http://www.tcs.informatik.uni-muenchen.de/~beringer>, Martin Hofmann <http://www.tcs.informatik.uni-muenchen.de/~mhofmann>
date = 2008-12-12
topic = Computer Science/Programming Languages/Logics
abstract = This document contains the Isabelle/HOL sources underlying the paper <i>A bytecode logic for JML and types</i> by Beringer and Hofmann, updated to Isabelle 2008. We present a program logic for a subset of sequential Java bytecode that is suitable for representing both, features found in high-level specification language JML as well as interpretations of high-level type systems. To this end, we introduce a fine-grained collection of assertions, including strong invariants, local annotations and VDM-reminiscent partial-correctness specifications. Thanks to a goal-oriented structure and interpretation of judgements, verification may proceed without recourse to an additional control flow analysis. The suitability for interpreting intensional type systems is illustrated by the proof-carrying-code style encoding of a type system for a first-order functional language which guarantees a constant upper bound on the number of objects allocated throughout an execution, be the execution terminating or non-terminating. Like the published paper, the formal development is restricted to a comparatively small subset of the JVML, lacking (among other features) exceptions, arrays, virtual methods, and static fields. This shortcoming has been overcome meanwhile, as our paper has formed the basis of the Mobius base logic, a program logic for the full sequential fragment of the JVML. Indeed, the present formalisation formed the basis of a subsequent formalisation of the Mobius base logic in the proof assistant Coq, which includes a proof of soundness with respect to the Bicolano operational semantics by Pichardie.

title = Semantics and Data Refinement of Invariant Based Programs
author = Viorel Preoteasa <http://users.abo.fi/vpreotea/>, Ralph-Johan Back <http://users.abo.fi/Ralph-Johan.Back/>
date = 2010-05-28
topic = Computer Science/Programming Languages/Logics
abstract = The invariant based programming is a technique of constructing correct programs by first identifying the basic situations (pre- and post-conditions and invariants) that can occur during the execution of the program, and then defining the transitions and proving that they preserve the invariants. Data refinement is a technique of building correct programs working on concrete datatypes as refinements of more abstract programs. In the theories presented here we formalize the predicate transformer semantics for invariant based programs and their data refinement.
depends-on = LatticeProperties
extra-history =
  Change history:
  [2012-01-05]: Moved some general complete lattice properties to the AFP entry Lattice Properties.
      Changed the definition of the data refinement relation to be more general and updated all corresponding theorems.
      Added new syntax for demonic and angelic update statements.

title = Secure information flow and program logics
author = Lennart Beringer <http://www.tcs.informatik.uni-muenchen.de/~beringer>, Martin Hofmann <http://www.tcs.informatik.uni-muenchen.de/~mhofmann>
date = 2008-11-10
topic = Computer Science/Programming Languages/Logics, Computer Science/Security
abstract = We present interpretations of type systems for secure information flow in Hoare logic, complementing previous encodings in relational program logics. We first treat the imperative language IMP, extended by a simple procedure call mechanism. For this language we consider base-line non-interference in the style of Volpano et al. and the flow-sensitive type system by Hunt and Sands. In both cases, we show how typing derivations may be used to automatically generate proofs in the program logic that certify the absence of illicit flows. We then add instructions for object creation and manipulation, and derive appropriate proof rules for base-line non-interference. As a consequence of our work, standard verification technology may be used for verifying that a concrete program satisfies the non-interference property.<br><br>The present proof development represents an update of the formalisation underlying our paper [CSF 2007] and is intended to resolve any ambiguities that may be present in the paper.

title = A Definitional Encoding of TLA* in Isabelle/HOL
author = Gudmund Grov <http://homepages.inf.ed.ac.uk/ggrov>, Stephan Merz <http://www.loria.fr/~merz>
date = 2011-11-19
topic = Computer Science/Programming Languages/Logics
abstract = We mechanise the logic TLA*
 <a href="http://www.springerlink.com/content/ax3qk557qkdyt7n6/">[Merz 1999]</a>,
 an extension of Lamport's  Temporal Logic of Actions (TLA)
 <a href="http://dl.acm.org/citation.cfm?doid=177492.177726">[Lamport 1994]</a>
 for specifying and reasoning
 about concurrent and reactive systems. Aiming at a framework for mechanising]  the verification of TLA (or TLA*) specifications, this contribution reuses
 some elements from a previous axiomatic encoding of TLA in Isabelle/HOL
 by the second author [Merz 1998], which has been part of the Isabelle
 distribution. In contrast to that previous work, we give here a shallow,
 definitional embedding, with the following highlights:
 <li>a theory of infinite sequences, including a formalisation of the concepts of stuttering invariance central to TLA and TLA*;
 <li>a definition of the semantics of TLA*, which extends TLA by a mutually-recursive definition of formulas and pre-formulas, generalising TLA action formulas;
 <li>a substantial set of derived proof rules, including the TLA* axioms and Lamport's proof rules for system verification;
 <li>a set of examples illustrating the usage of Isabelle/TLA* for reasoning about systems.
 Note that this work is unrelated to the ongoing development of a proof system
 for the specification language TLA+, which includes an encoding of TLA+ as a
 new Isabelle object logic <a href="http://www.springerlink.com/content/354026160p14j175/">[Chaudhuri et al 2010]</a>.

title = Compiling Exceptions Correctly
author = Tobias Nipkow <http://www.in.tum.de/~nipkow/>
date = 2004-07-09
topic = Computer Science/Programming Languages/Compiling
abstract = An exception compilation scheme that dynamically creates and removes exception handler entries on the stack. A formalization of an article of the same name by <a href="http://www.cs.nott.ac.uk/~gmh/">Hutton</a> and Wright.

title = Normalization by Evaluation
author = Klaus Aehlig <http://www.linta.de/~aehlig/>, Tobias Nipkow <http://www.in.tum.de/~nipkow/>
date = 2008-02-18
topic = Computer Science/Programming Languages/Compiling
abstract = This article formalizes normalization by evaluation as implemented in Isabelle. Lambda calculus plus term rewriting is compiled into a functional program with pattern matching. It is proved that the result of a successful evaluation is a) correct, i.e. equivalent to the input, and b) in normal form.

title = Formalization of Conflict Analysis of Programs with Procedures, Thread Creation, and Monitors
topic = Computer Science/Programming Languages/Static Analysis
author = Peter Lammich <http://cs.uni-muenster.de/sev/staff/lammich/>, Markus Müller-Olm <http://cs.uni-muenster.de/u/mmo/>
date = 2007-12-14
abstract = In this work we formally verify the soundness and precision of a static program analysis that detects conflicts (e. g. data races) in programs with procedures, thread creation and monitors with the Isabelle theorem prover. As common in static program analysis, our program model abstracts guarded branching by nondeterministic branching, but completely interprets the call-/return behavior of procedures, synchronization by monitors, and thread creation. The analysis is based on the observation that all conflicts already occur in a class of particularly restricted schedules. These restricted schedules are suited to constraint-system-based program analysis. The formalization is based upon a flowgraph-based program model with an operational semantics as reference point.

title = Shivers' Control Flow Analysis
topic = Computer Science/Programming Languages/Static Analysis
author = Joachim Breitner <mailto:mail@joachim-breitner.de>
date = 2010-11-16
abstract =
 In his dissertation, Olin Shivers introduces a concept of control flow graphs
 for functional languages, provides an algorithm to statically derive a safe
 approximation of the control flow graph and proves this algorithm correct. In
 this research project, Shivers' algorithms and proofs are formalized
 in the HOLCF extension of HOL.

title = Towards Certified Slicing
author = Daniel Wasserrab <http://pp.info.uni-karlsruhe.de/personhp/daniel_wasserrab.php>
date = 2008-09-16
topic = Computer Science/Programming Languages/Static Analysis
abstract = Slicing is a widely-used technique with applications in e.g. compiler technology and software security. Thus verification of algorithms in these areas is often based on the correctness of slicing, which should ideally be proven independent of concrete programming languages and with the help of well-known verifying techniques such as proof assistants. As a first step in this direction, this contribution presents a framework for dynamic and static intraprocedural slicing based on control flow and program dependence graphs. Abstracting from concrete syntax we base the framework on a graph representation of the program fulfilling certain structural and well-formedness properties.<br><br>The formalization consists of the basic framework (in subdirectory Basic/), the correctness proof for dynamic slicing (in subdirectory Dynamic/), the correctness proof for static intraprocedural slicing (in subdirectory StaticIntra/) and instantiations of the framework with a simple While language (in subdirectory While/) and the sophisticated object-oriented bytecode language of Jinja (in subdirectory JinjaVM/). For more information on the framework, see the TPHOLS 2008 paper by Wasserrab and Lochbihler and the PLAS 2009 paper by Wasserrab et al.
depends-on = Jinja

title = Backing up Slicing: Verifying the Interprocedural Two-Phase Horwitz-Reps-Binkley Slicer
author = Daniel Wasserrab <http://pp.info.uni-karlsruhe.de/personhp/daniel_wasserrab.php>
date = 2009-11-13
topic = Computer Science/Programming Languages/Static Analysis
abstract = After verifying <a href="Slicing.shtml">dynamic and static interprocedural slicing</a>, we present a modular framework for static interprocedural slicing. To this end, we formalized the standard two-phase slicer from Horwitz, Reps and Binkley (see their TOPLAS 12(1) 1990 paper) together with summary edges as presented by Reps et al. (see FSE 1994). The framework is again modular in the programming language by using an abstract CFG, defined via structural and well-formedness properties. Using a weak simulation between the original and sliced graph, we were able to prove the correctness of static interprocedural slicing. We also instantiate our framework with a simple While language with procedures. This shows that the chosen abstractions are indeed valid.
depends-on = Jinja

title = The Worker/Wrapper Transformation
author = Peter Gammie <http://peteg.org/>
date = 2009-10-30
topic = Computer Science/Programming Languages/Transformations
abstract = Gill and Hutton formalise the worker/wrapper transformation, building on the work of Launchbury and Peyton-Jones who developed it as a way of changing the type at which a recursive function operates. This development establishes the soundness of the technique and several examples of its use.

title = Jive Data and Store Model
author = Nicole Rauch <mailto:rauch@informatik.uni-kl.de>, Norbert Schirmer
date = 2005-06-20
license = LGPL
topic = Computer Science/Programming Languages/Misc
abstract = This document presents the formalization of an object-oriented data and store model in Isabelle/HOL. This model is being used in the Java Interactive Verification Environment, Jive.

title = Hotel Key Card System
author = Tobias Nipkow <http://www.in.tum.de/~nipkow>
date = 2006-09-09
topic = Computer Science/Security
abstract = Two models of an electronic hotel key card system are contrasted: a state based and a trace based one. Both are defined, verified, and proved equivalent in the theorem prover Isabelle/HOL. It is shown that if a guest follows a certain safety policy regarding her key cards, she can be sure that nobody but her can enter her room.

title = SHA1, RSA, PSS and more
author = Christina Lindenberg, Kai Wirt
date = 2005-05-02
topic = Computer Science/Security
abstract = Formal verification is getting more and more important in computer science. However the state of the art formal verification methods in cryptography are very rudimentary. These theories are one step to provide a tool box allowing the use of formal methods in every aspect of cryptography. Moreover we present a proof of concept for the feasibility of verification techniques to a standard signature algorithm.

; todo: multiple entries with different naming scheme
title = Information Flow Noninterference via Slicing
author = Daniel Wasserrab <http://pp.info.uni-karlsruhe.de/personhp/daniel_wasserrab.php>
date = 2010-03-23
topic = Computer Science/Security
ignore = entry
abstract = In this contribution, we show how correctness proofs for <a href="Slicing.shtml">intra-</a> and <a href="HRB-Slicing.shtml">interprocedural slicing</a> can be used to prove that slicing is able to guarantee information flow noninterference.  Moreover, we also illustrate how to lift the control flow graphs of the respective frameworks such that they fulfil the additional assumptions needed in the noninterference proofs. A detailed description of the intraprocedural proof and its interplay with the slicing framework can be found in the PLAS'09 paper by Wasserrab et al.
depends-on = Slicing, HRB-Slicing

title = A Mechanically Verified, Efficient, Sound and Complete Theorem Prover For First Order Logic
author = Tom Ridge
date = 2004-09-28
topic = Logic
abstract = Soundness and completeness for a system of first order logic are formally proved, building on James Margetson's formalization of work by Wainer and Wallen. The completeness proofs naturally suggest an algorithm to derive proofs. This algorithm, which can be implemented tail recursively, is formalized in Isabelle/HOL. The algorithm can be executed via the rewriting tactics of Isabelle. Alternatively, the definitions can be exported to OCaml, yielding a directly executable program.

title = Completeness theorem
author = James Margetson, Tom Ridge
date = 2004-09-20
topic = Logic
abstract = The completeness of first-order logic is proved, following the first five pages of Wainer and Wallen's chapter of the book <i>Proof Theory</i> by Aczel et al., CUP, 1992. Their presentation of formulas allows the proofs to use symmetry arguments. Margetson formalized this theorem by early 2000. The Isar conversion is thanks to Tom Ridge. A paper describing the formalization is available <a href="Completeness-paper.pdf">[pdf]</a>.

title = Countable Ordinals
author = Brian Huffman <http://web.cecs.pdx.edu/~brianh/>
date = 2005-11-11
topic = Logic
abstract = This development defines a well-ordered type of countable ordinals. It includes notions of continuous and normal functions, recursively defined functions over ordinals, least fixed-points, and derivatives. Much of ordinal arithmetic is formalized, including exponentials and logarithms. The development concludes with formalizations of Cantor Normal Form and Veblen hierarchies over normal functions.

title = Ordinals and Cardinals
author = Andrei Popescu
date = 2009-09-01
topic = Logic
abstract = We develop a basic theory of ordinals and cardinals in Isabelle/HOL, up to the point where some cardinality facts relevant for the ``working mathematician" become available. Unlike in set theory, here we do not have at hand canonical notions of ordinal and cardinal. Therefore, here an ordinal is merely a well-order relation and a cardinal is an ordinal minim w.r.t. order embedding on its field.
extra-history =
  Change history:
  [2012-09-25]: This entry has been discontinued because it is now part of the Isabelle distribution.

title = First-Order Logic According to Fitting
author = Stefan Berghofer <http://www.in.tum.de/~berghofe>
date = 2007-08-02
topic = Logic
abstract = We present a formalization of parts of Melvin Fitting's book ``First-Order Logic and Automated Theorem Proving''. The formalization covers the syntax of first-order logic, its semantics, the model existence theorem, a natural deduction proof calculus together with a proof of correctness and completeness, as well as the Löwenheim-Skolem theorem.

title = Invertibility in Sequent Calculi
author = Peter Chapman <http://www.cs.st-andrews.ac.uk/~pc>
date = 2009-08-28
topic = Logic
license = LGPL
abstract = The invertibility of the rules of a sequent calculus is important for guiding proof search and can be used in some formalised proofs of Cut admissibility. We present sufficient conditions for when a rule is invertible with respect to a calculus. We illustrate the conditions with examples. It must be noted we give purely syntactic criteria; no guarantees are given as to the suitability of the rules.

title = Quantifier Elimination for Linear Arithmetic
author = Tobias Nipkow <http://www.in.tum.de/~nipkow/>
date = 2008-01-11
topic = Logic
abstract = This article formalizes quantifier elimination procedures for dense linear orders, linear real arithmetic and Presburger arithmetic. In each case both a DNF-based non-elementary algorithm and one or more (doubly) exponential NNF-based algorithms are formalized, including the well-known algorithms by Ferrante and Rackoff and by Cooper. The NNF-based algorithms for dense linear orders are new but based on Ferrante and Rackoff and on an algorithm by Loos and Weisspfenning which simulates infenitesimals. All algorithms are directly executable. In particular, they yield reflective quantifier elimination procedures for HOL itself. The formalization makes heavy use of locales and is therefore highly modular.

title = Interval Temporal Logic on Natural Numbers
author = David Trachtenherz <http://www4.in.tum.de/~trachten/>
date = 2011-02-23
topic = Logic
abstract = We introduce a theory of temporal logic operators using sets of natural numbers as time domain, formalized in a shallow embedding manner. The theory comprises special natural intervals (theory IL_Interval: open and closed intervals, continuous and modulo intervals, interval traversing results), operators for shifting intervals to left/right on the number axis as well as expanding/contracting intervals by constant factors (theory IL_IntervalOperators.thy), and ultimately definitions and results for unary and binary temporal operators on arbitrary natural sets (theory IL_TemporalOperators).
depends-on = List-Infinite

title = Recursion Theory I
author = Michael Nedzelsky
date = 2008-04-05
topic = Logic
abstract = This document presents the formalization of introductory material from  recursion theory --- definitions and basic properties of primitive recursive  functions, Cantor pairing function and computably enumerable sets  (including a proof of existence of a one-complete computably enumerable set  and a proof of the Rice's theorem).

topic = Logic
title = Free Boolean Algebra
author = Brian Huffman <http://web.cecs.pdx.edu/~brianh/>
date = 2010-03-29
abstract = This theory defines a type constructor representing the free Boolean algebra over a set of generators. Values of type (α)<i>formula</i> represent propositional formulas with uninterpreted variables from type α, ordered by implication. In addition to all the standard Boolean algebra operations, the library also provides a function for building homomorphisms to any other Boolean algebra type.

title = Sound and Complete Sort Encodings for First-Order Logic
author = Jasmin Christian Blanchette <http://www21.in.tum.de/~blanchet>, Andrei Popescu <http://www21.in.tum.de/~popescua>
date = 2013-06-27
topic = Logic
abstract =
  This is a formalization of the soundness and completeness properties
  for various efficient encodings of sorts in unsorted first-order logic
  used by Isabelle's Sledgehammer tool.
  Essentially, the encodings proceed as follows:
  a many-sorted problem is decorated with (as few as possible) tags or
  guards that make the problem monotonic; then sorts can be soundly
  The development employs a formalization of many-sorted first-order logic
  in clausal form (clauses, structures and the basic properties
  of the satisfaction relation), which could be of interest as the starting
  point for other formalizations of first-order logic metatheory.

title = Abstract Rewriting
topic = Logic/Rewriting
date = 2010-06-14
author = Christian Sternagel <http://cl-informatik.uibk.ac.at/~griff>, René Thiemann <http://cl-informatik.uibk.ac.at/~thiemann>
license = LGPL
abstract =
  We present an Isabelle formalization of abstract rewriting (see, e.g.,
  the book by Baader and Nipkow). First, we define standard relations like
  <i>joinability</i>, <i>meetability</i>, <i>conversion</i>, etc. Then, we
  formalize important properties of abstract rewrite systems, e.g.,
  confluence and strong normalization. Our main concern is on strong
  normalization, since this formalization is the basis of <a
  href="http://cl-informatik.uibk.ac.at/software/ceta">CeTA</a> (which is
  mainly about strong normalization of term rewrite systems). Hence lemmas
  involving strong normalization constitute by far the biggest part of this
  theory. One of those is Newman's lemma.
extra-history =
  Change history:
  [2010-09-17]: Added theories defining several (ordered)
  semirings related to strong normalization and giving some standard
  instances. <br>
  [2013-10-16]: Generalized delta-orders from rationals to Archimedean fields.

title = Free Groups
author = Joachim Breitner <mailto:mail@joachim-breitner.de>
date = 2010-06-24
topic = Mathematics/Algebra
abstract =
 Free Groups are, in a sense, the most generic kind of group. They
 are defined over a set of generators with no additional relations in between
 them. They play an important role in the definition of group presentations
 and in other fields. This theory provides the definition of Free Group as
 the set of fully canceled words in the generators. The universal property is
 proven, as well as some isomorphisms results about Free Groups.
depends-on = Ordinals_and_Cardinals
extra-history =
 Change history:
 [2011-12-11]: Added the Ping Pong Lemma.

title = An Example of a Cofinitary Group in Isabelle/HOL
author = Bart Kastermans <http://kasterma.net>
date = 2009-08-04
topic = Mathematics/Algebra
abstract = We formalize the usual proof that the group generated by the function k -> k + 1 on the integers gives rise to a cofinitary group.

title = Groups, Rings and Modules
author = Hidetsune Kobayashi, L. Chen, H. Murao
date = 2004-05-18
topic = Mathematics/Algebra
abstract = The theory of groups, rings and modules is developed to a great depth. Group theory results include Zassenhaus's theorem and the Jordan-Hoelder theorem. The ring theory development includes ideals, quotient rings and the Chinese remainder theorem. The module development includes the Nakayama lemma, exact sequences and Tensor products.

title = A Complete Proof of the Robbins Conjecture
author = Matthew Wampler-Doty
date = 2010-05-22
topic = Mathematics/Algebra
abstract = This document gives a formalization of the proof of the Robbins conjecture, following A. Mann, <i>A Complete Proof of the Robbins Conjecture</i>, 2003.

title = Fundamental Properties of Valuation Theory and Hensel's Lemma
author = Hidetsune Kobayashi
date = 2007-08-08
topic = Mathematics/Algebra
abstract = Convergence with respect to a valuation is discussed as convergence of a Cauchy sequence. Cauchy sequences of polynomials are defined. They are used to formalize Hensel's lemma.
depends-on = Group-Ring-Module

title = Rank-Nullity Theorem in Linear Algebra
author = Jose Divasón <http://www.unirioja.es/cu/jodivaso>, Jesús Aransay <http://www.unirioja.es/cu/jearansa>
topic = Mathematics/Algebra
date = 2013-01-16
abstract = In this article we present a proof of the result known in Linear Algebra as the ``rank nullity Theorem'', which states that, given any linear form f from a finite dimensional vector space V to a vector space W, then the dimension of V is equal to the dimension of the kernel of f (which is a subspace of V) and the dimension of the range of f (which is a subspace of W). The proof presented here is based on the one given by Sheldon Axler in his book <i>Linear Algebra Done Right</i>. It makes use of the HOL-Multivariate-Analysis session of Isabelle, and of several of its results and definitions. As a corollary of the previous theorem, and taking advantage of the relationship between linear forms and matrices, we prove that, for every matrix A (which has associated a linear form between finite dimensional vector spaces), the sum of its null space and its column space (which is equal to the range of the linear form) is equal to the number of columns of A.

title = Cauchy's Mean Theorem and the Cauchy-Schwarz Inequality
author = Benjamin Porter
date = 2006-03-14
topic = Mathematics/Analysis
abstract = This document presents the mechanised proofs of two popular theorems attributed to Augustin Louis Cauchy - Cauchy's Mean Theorem and the Cauchy-Schwarz Inequality.

title = Integration theory and random variables
author = Stefan Richter <http://www-lti.informatik.rwth-aachen.de/~richter/>
date = 2004-11-19
topic = Mathematics/Analysis
abstract = Lebesgue-style integration plays a major role in advanced probability. We formalize concepts of elementary measure theory, real-valued random variables as Borel-measurable functions, and a stepwise inductive definition of the integral itself. All proofs are carried out in human readable style using the Isar language.
extra-note = Note: This article is of historical interest only. Lebesgue-style integration and probability theory are now available as part of the Isabelle/HOL distribution (directory Probability).

title = Ordinary Differential Equations
author = Fabian Immler <http://home.in.tum.de/~immler>, Johannes Hölzl <http://home.in.tum.de/~hoelzl>
topic = Mathematics/Analysis
date = 2012-04-26
abstract = We formalize initial value problems (IVPs) of ODEs and prove the existence of a unique solution, i.e. the Picard-Lindelöf theorem. We introduce general one-step methods for numerical approximation of the solution and provide an analysis regarding the local and global error of one-step methods. We give an executable specification of the Euler method to approximate the solution of IVPs and prove an explicit bound for the global error. We use arbitrary-precision floating-point numbers and also handle rounding errors when we truncate the numbers for efficiency reasons.

title = Executable Multivariate Polynomials
author = Christian Sternagel <http://cl-informatik.uibk.ac.at/~griff>, René Thiemann <http://cl-informatik.uibk.ac.at/~thiemann>
date = 2010-08-10
topic = Mathematics/Analysis
license = LGPL
abstract =
  We define multivariate polynomials over arbitrary (ordered) semirings in
  combination with (executable) operations like addition, multiplication,
  and substitution. We also define (weak) monotonicity of polynomials and
  comparison of polynomials where we provide standard estimations like
  absolute positiveness or the more recent approach of Neurauter, Zankl,
  and Middeldorp. Moreover, it is proven that strongly normalizing
  (monotone) orders can be lifted to strongly normalizing (monotone) orders
  over polynomials. Our formalization was performed as part of the <a
  which contains several termination techniques. The provided theories have
  been essential to  formalize polynomial interpretations.
depends-on = Abstract-Rewriting
extra-history =
  Change history:
  [2010-09-17]: Moved theories on arbitrary (ordered) semirings to Abstract Rewriting.

title = Computing Square Roots using the Babylonian Method
author = René Thiemann <mailto:rene.thiemann@uibk.ac.at>
date = 2013-01-03
topic = Mathematics/Analysis
license = LGPL
abstract =
  We implement the Babylonian method to compute square roots of numbers.
  We provide precise algorithms for naturals, integers and rationals, and
  offer an approximation algorithm for linear ordered fields. Moreover, there
  are precise algorithms to compute the floor and the ceiling of the square root
  of an integer.
extra-history =
  Change history:
  [2013-10-16]: Added algorithms to compute floor and ceiling of sqrt of integers.

title = Markov Models
author = Johannes Hölzl <http://www.in.tum.de/~hoelzl>, Tobias Nipkow <http://www.in.tum.de/~nipkow/>
date = 2012-01-03
topic = Mathematics/Probability Theory, Computer Science/Automata and Formal Languages
abstract = This is a formalization of Markov models in Isabelle/HOL. It
  builds on Isabelle's probability theory. The available models are
  currently Discrete-Time Markov Chains and a extensions of them with
  As application of these models we formalize probabilistic model
  checking of pCTL formulas, analysis of IPv4 address allocation in
  ZeroConf and an analysis of the anonymity of the Crowds protocol.

title = Fermat's Last Theorem for Exponents 3 and 4 and the Parametrisation of Pythagorean Triples
author = Roelof Oosterhuis <http://www.roelofoosterhuis.nl/>
date = 2007-08-12
topic = Mathematics/Number Theory
abstract = This document presents the mechanised proofs of<ul><li>Fermat's Last Theorem for exponents 3 and 4 and</li><li>the parametrisation of Pythagorean Triples.</li></ul>

title = Perfect Number Theorem
author = Mark Ijbema <mailto:ijbema@fmf.nl>
date = 2009-11-22
topic = Mathematics/Number Theory
abstract = These theories present the mechanised proof of the Perfect Number Theorem.

title = Sums of Two and Four Squares
author = Roelof Oosterhuis <http://www.roelofoosterhuis.nl/>
date = 2007-08-12
topic = Mathematics/Number Theory
abstract = This document presents the mechanised proofs of the following results:<ul><li>any prime number of the form 4m+1 can be written as the sum of two squares;</li><li>any natural number can be written as the sum of four squares</li></ul>
depends-on = Fermat3_4

title = Lehmer's Theorem
author = Simon Wimmer <mailto:simon.wimmer@tum.de>, Lars Noschinski <http://www21.in.tum.de/~noschinl/>
date = 2013-07-22
topic = Mathematics/Number Theory
abstract = In 1927, Lehmer presented criterions for primality, based on the converse of Fermat's litte theorem. This work formalizes the second criterion from Lehmer's paper, a necessary and sufficient condition for primality.
  As a side product we formalize some properties of Euler's phi-function,
  the notion of the order of an element of a group, and the cyclicity of the multiplicative group of a finite field.

title = Pratt's Primality Certificates
author = Simon Wimmer <mailto:simon.wimmer@tum.de>, Lars Noschinski <http://www21.in.tum.de/~noschinl/>
date = 2013-07-22
topic = Mathematics/Number Theory
abstract = In 1975, Pratt introduced a proof system for certifying primes. He showed that a number <i>p</i> is prime iff a primality certificate for <i>p</i> exists. By showing a logarithmic upper bound on the length of the certificates in size of the prime number, he concluded that the decision problem for prime numbers is in NP. This work formalizes soundness and completeness of Pratt's proof system as well as an upper bound for the size of the certificate.
depends-on = Lehmer

title = Arrow and Gibbard-Satterthwaite
author = Tobias Nipkow <http://www.in.tum.de/~nipkow/>
date = 2008-09-01
topic = Mathematics/Social Choice Theory
abstract = This article formalizes two proofs of Arrow's impossibility theorem due to Geanakoplos and derives the Gibbard-Satterthwaite theorem as a corollary. One formalization is based on utility functions, the other one on strict partial orders.<br><br>An article about these proofs is found <a href="http://www.in.tum.de/~nipkow/pubs/arrow.html">here</a>.

title = Some classical results in Social Choice Theory
author = Peter Gammie <http://peteg.org/>
date = 2008-11-09
topic = Mathematics/Social Choice Theory
abstract = Drawing on Sen's landmark work "Collective Choice and Social Welfare" (1970), this development proves Arrow's General Possibility Theorem, Sen's Liberal Paradox and May's Theorem in a general setting. The goal was to make precise the classical statements and proofs of these results, and to provide a foundation for more recent results such as the Gibbard-Satterthwaite and Duggan-Schwartz theorems.

title = Topology
author = Stefan Friedrich
date = 2004-04-26
topic = Mathematics/Topology
abstract = This entry contains two theories. The first, <tt>Topology</tt>, develops the basic notions of general topology. The second, which can be viewed as a demonstration of the first, is called <tt>LList_Topology</tt>. It develops the topology of lazy lists.
depends-on = Lazy-Lists-II

title = Graph Theory
author = Lars Noschinski <http://www21.in.tum.de/~noschinl/>
date = 2013-04-28
topic = Mathematics/Graph Theory
abstract = This development provides a formalization of directed graphs, supporting (labelled) multi-edges and infinite graphs. A polymorphic edge type allows edges to be treated as pairs of vertices, if multi-edges are not required. Formalized properties are i.a. walks (and related concepts), connectedness and subgraphs and basic properties of isomorphisms.
 This formalization is used to prove characterizations of Euler Trails, Shortest Paths and Kuratowski subgraphs.

title = Maximum Cardinality Matching
author = Christine Rizkallah <http://www.mpi-inf.mpg.de/homepage/crizkall/>
date = 2011-07-21
topic = Mathematics/Graph Theory
abstract = A <em>matching</em> in a graph <i>G</i> is a subset <i>M</i> of the
 edges of <i>G</i> such that no two share an endpoint. A matching has maximum
 cardinality if its cardinality is at least as large as that of any other
 matching. An <em>odd-set cover</em> <i>OSC</i> of a graph <i>G</i> is a
 labeling of the nodes of <i>G</i> with integers such that every edge of
 <i>G</i> is either incident to a node labeled 1 or connects two nodes
 labeled with the same number <i>i &ge; 2</i>.
 This article proves Edmonds theorem:

 Let <i>M</i> be a matching in a graph <i>G</i> and let <i>OSC</i> be an
 odd-set cover of <i>G</i>.
 For any <i>i &ge; 0</i>, let <var>n(i)</var> be the number of nodes
 labeled <i>i</i>. If <i>|M| = n(1) +
 &sum;<sub>i &ge; 2</sub>(n(i) div 2)</i>,
 then <i>M</i> is a maximum cardinality matching.

title = A Probabilistic Proof of the Girth-Chromatic Number Theorem
author = Lars Noschinski <http://www21.in.tum.de/~noschinl/>
date = 2012-02-06
topic = Mathematics/Graph Theory
abstract = This works presents a formalization of the Girth-Chromatic number theorem in graph theory, stating that graphs with arbitrarily large girth and chromatic number exist. The proof uses the theory of Random Graphs to prove the existence with probabilistic arguments.

title = Flyspeck I: Tame Graphs
author = Gertrud Bauer, Tobias Nipkow <http://www.in.tum.de/~nipkow>
date = 2006-05-22
topic = Mathematics/Graph Theory
abstract = These theories present the verified enumeration of <i>tame</i> plane graphs as defined by Thomas C. Hales in his <i>revised</i> proof of the Kepler Conjecture. The revison concerns the definition of tameness. The <a href="http://www.in.tum.de/~nipkow/pubs/Flyspeck/">IJCAR 2006 paper by Nipkow, Bauer and Schultz</a> refers to the original version of Hales' proof.
extra-history =
  Change history:
  [2010-11-02]: modified theories to reflect the modified
  definition of tameness in Hales' revised proof.

title = Well-Quasi-Orders
author = Christian Sternagel <mailto:c-sterna@jaist.ac.jp>
date = 2012-04-13
topic = Mathematics/Combinatorics
abstract = Based on Isabelle/HOL's type class for preorders,
 we introduce a type class for well-quasi-orders (wqo)
 which is characterized by the absence of "bad" sequences
 (our proofs are along the lines of the proof of Nash-Williams,
 from which we also borrow terminology). Our main results are
 instantiations for the product type, the list type, and a type of finite trees,
 which (almost) directly follow from our proofs of (1) Dickson's Lemma, (2)
 Higman's Lemma, and (3) Kruskal's Tree Theorem. More concretely:
 <li>If the sets A and B are wqo then their Cartesian product is wqo.</li>
 <li>If the set A is wqo then the set of finite lists over A is wqo.</li>
 <li>If the set A is wqo then the set of finite trees over A is wqo.</li>
 The research was funded by the Austrian Science Fund (FWF): J3202.
extra-history =
  Change history:
  [2012-06-11]: Added Kruskal's Tree Theorem.<br>
  [2012-12-19]: New variant of Kruskal's tree theorem for terms (as opposed to
  variadic terms, i.e., trees), plus finite version of the tree theorem as
  [2013-05-16]: Simplified construction of minimal bad sequences.

title = Hall's Marriage Theorem
author = Dongchen Jiang <mailto:dongchenjiang@googlemail.com>, Tobias Nipkow <http://www.in.tum.de/~nipkow/>
date = 2010-12-17
topic = Mathematics/Combinatorics
abstract = Two proofs of Hall's Marriage Theorem: one due to Halmos and Vaughan, one due to Rado.
extra-history =
  Change history:
  [2011-09-09]: Added Rado's proof

title = Bondy's Theorem
author = Jeremy Avigad <http://www.andrew.cmu.edu/user/avigad/>, Stefan Hetzl <http://www.logic.at/people/hetzl/>
date = 2012-10-27
topic = Mathematics/Combinatorics
abstract = A proof of Bondy's theorem following B. Bollabas, Combinatorics, 1986, Cambridge University Press.

title = Ramsey's theorem, infinitary version
author = Tom Ridge
date = 2004-09-20
topic = Mathematics/Combinatorics
abstract = This formalization of Ramsey's theorem (infinitary version) is taken from Boolos and Jeffrey, <i>Computability and Logic</i>, 3rd edition, Chapter 26. It differs slightly from the text by assuming a slightly stronger hypothesis. In particular, the induction hypothesis is stronger, holding for any infinite subset of the naturals. This avoids the rather peculiar mapping argument between kj and aikj on p.263, which is unnecessary and slightly mars this really beautiful result.

title = Open Induction
author = Mizuhito Ogawa, Christian Sternagel <http://www.jaist.ac.jp/~c-sterna/>
date = 2012-11-02
topic = Mathematics/Combinatorics
abstract =
  A proof of the open induction schema based on J.-C. Raoult, Proving open properties by induction, <i>Information Processing Letters</i> 29, 1988, pp.19-23.
  <p>This research was supported by the Austrian Science Fund (FWF): J3202.</p>

title = Category Theory to Yoneda's Lemma
author = Greg O'Keefe <http://users.rsise.anu.edu.au/~okeefe/>
date = 2005-04-21
topic = Mathematics/Category Theory
license = LGPL
abstract = This development proves Yoneda's lemma and aims to be readable by humans. It only defines what is needed for the lemma: categories, functors and natural transformations. Limits, adjunctions and other important concepts are not included.
extra-history =
    Change history:
    [2010-04-23]: The definition of the constant <tt>equinumerous</tt> was slightly too weak in the original submission and has been fixed in revision <a href="http://afp.hg.sourceforge.net/hgweb/afp/afp/rev/8c2b5b3c995f">8c2b5b3c995f</a>.

title = Category Theory
author = Alexander Katovsky <mailto:apk32@cam.ac.uk>
date = 2010-06-20
topic = Mathematics/Category Theory
abstract = This article presents a development of Category Theory in Isabelle/HOL. A Category is defined using records and locales. Functors and Natural Transformations are also defined. The main result that has been formalized is that the Yoneda functor is a full and faithful embedding. We also formalize the completeness of many sorted monadic equational logic. Extensive use is made of the HOLZF theory in both cases. For an informal description see <a href="http://www.srcf.ucam.org/~apk32/Isabelle/Category/Cat.pdf">here [pdf]</a>.

title = Fun With Functions
author = Tobias Nipkow <http://www.in.tum.de/~nipkow/>
date = 2008-08-26
topic = Mathematics/Misc
abstract = This is a collection of cute puzzles of the form ``Show that if a function satisfies the following constraints, it must be ...'' Please add further examples to this collection!

title = Fun With Tilings
author = Tobias Nipkow <http://www.in.tum.de/~nipkow/>, Lawrence Paulson <http://www.cl.cam.ac.uk/users/lcp/>
date = 2008-11-07
topic = Mathematics/Misc
abstract = Tilings are defined inductively. It is shown that one form of mutilated chess board cannot be tiled with dominoes, while another one can be tiled with L-shaped tiles. Please add further fun examples of this kind!

title = Lazy Lists II
author = Stefan Friedrich
date = 2004-04-26
topic = Computer Science/Data Structures
abstract = This theory contains some useful extensions to the LList (lazy list) theory by <a href="http://www.cl.cam.ac.uk/users/lcp/">Larry Paulson</a>, including finite, infinite, and positive llists over an alphabet, as well as the new constants take and drop and the prefix order of llists. Finally, the notions of safety and liveness in the sense of Alpern and Schneider (1985) are defined.
depends-on = Coinductive

title = Ribbon Proofs
author = John Wickerson <http://www.pes.tu-berlin.de/menue/ueber_uns/team/john_wickerson/>
date = 2013-01-19
topic = Computer Science/Programming Languages/Logics
abstract = This document concerns the theory of ribbon proofs: a diagrammatic proof system, based on separation logic, for verifying program correctness. We include the syntax, proof rules, and soundness results for two alternative formalisations of ribbon proofs. <p> Compared to traditional proof outlines, ribbon proofs emphasise the structure of a proof, so are intelligible and pedagogical. Because they contain less redundancy than proof outlines, and allow each proof step to be checked locally, they may be more scalable. Where proof outlines are cumbersome to modify, ribbon proofs can be visually manoeuvred to yield proofs of variant programs.

title = The Königsberg Bridge Problem and the Friendship Theorem
author = Wenda Li <mailto:wl302@cam.ac.uk>
date = 2013-07-19
topic = Mathematics/Graph Theory
depends-on = Dijkstra_Shortest_Path
abstract = This development provides a formalization of undirected graphs and simple graphs, which are based on Benedikt Nordhoff and Peter Lammich's simple formalization of labelled directed graphs in the archive. Then, with our formalization of graphs, we show both necessary and sufficient conditions for Eulerian trails and circuits as well as the fact that the Königsberg Bridge Problem does not have a solution. In addition, we show the Friendship Theorem in simple graphs.

title = A Formal Model of IEEE Floating Point Arithmetic
author = Lei Yu <mailto:ly271@cam.ac.uk>
date = 2013-07-27
topic = Computer Science/Data Structures
abstract = This development provides a formal model of IEEE-754 floating-point arithmetic. This formalization, including formal specification of the standard and proofs of important properties of floating-point arithmetic, forms the foundation for verifying programs with floating-point computation. There is also a code generation setup for floats so that we can execute programs using this formalization in functional programming languages.

title = Native Word
author = Andreas Lochbihler <http://www.infsec.ethz.ch/people/andreloc>
date = 2013-09-17
topic = Computer Science/Data Structures
abstract = This entry makes machine words and machine arithmetic available for code generation from Isabelle/HOL.  It provides a common abstraction that hides the differences between the different target languages.  The code generator maps these operations to the APIs of the target languages.  Apart from that, we extend the available bit operations on types int and integer, and map them to the operations in the target languages.

title = The Hereditarily Finite Sets
author = Lawrence C. Paulson <mailto:lp15@cam.ac.uk>
date = 2013-11-17
topic = Logic
abstract = The theory of hereditarily finite sets is formalised, following
	the <a href="http://journals.impan.gov.pl/dm/Inf/422-0-1.html">development</a> of Swierczkowski.
	An HF set is a finite collection of other HF sets; they enjoy an induction principle
	and satisfy all the axioms of ZF set theory apart from the axiom of infinity, which is negated.
	All constructions that are possible in ZF set theory (Cartesian products, disjoint sums, natural numbers,
	functions) without using infinite sets are possible here.
	The definition of addition for the HF sets follows Kirby.
	This development forms the foundation for the Isabelle proof of Gödel's incompleteness theorems,
	which has been <a href="Incompleteness.shtml">formalised separately</a>.

title = Gödel's Incompleteness Theorems
author = Lawrence C. Paulson <mailto:lp15@cam.ac.uk>
depends-on = HereditarilyFinite
date = 2013-11-17
topic = Logic
abstract = Gödel's two incompleteness theorems are formalised, following a careful  <a href="http://journals.impan.gov.pl/dm/Inf/422-0-1.html">presentation</a> by Swierczkowski, in the theory of <a href="HereditarilyFinite.shtml">hereditarily finite sets</a>. This represents the first ever machine-assisted proof of the second incompleteness theorem. Compared with traditional formalisations using Peano arithmetic (see e.g. Boolos), coding is simpler, with no need to formalise the notion
	of multiplication (let alone that of a prime number)
	in the formalised calculus upon which the theorem is based.
	However, other technical problems had to be solved in order to complete the argument.

title = Decreasing Diagrams
author = Harald Zankl <http://cl-informatik.uibk.ac.at/users/hzankl>
license = LGPL
depends-on = Abstract-Rewriting
date = 2013-11-01
topic = Logic/Rewriting
abstract = This theory contains a formalization of decreasing diagrams showing that any locally decreasing abstract rewrite system is confluent. We consider the valley (van Oostrom, TCS 1994) and the conversion version (van Oostrom, RTA 2008) and closely follow the original proofs. As an application we prove Newman's lemma.

title = Stream Processing Components: Isabelle/HOL Formalisation and Case Studies
author = Maria Spichkova
date = 2013-11-14
topic = Computer Science/Programming Languages/Language Definitions
abstract = This set of theories presents an Isabelle/HOL formalisation of stream processing components introduced
        in Focus,
	a framework for formal specification and development of interactive systems.
	This is an extended and updated version of the formalisation, which was
	elaborated within the methodology "Focus on Isabelle".
	In addition, we also applied the formalisation on three case studies
	that cover different application areas: process control (Steam Boiler System),
	data transmission (FlexRay communication protocol),
	memory and processing components (Automotive-Gateway System).

title = Gödel's God in Isabelle/HOL
author = Christoph Benzmüller <http://page.mi.fu-berlin.de/cbenzmueller/>, Bruno Woltzenlogel Paleo <http://www.logic.at/staff/bruno/>
date = 2013-11-12
topic = Logic
abstract = Dana Scott's version of Gödel's proof of God's existence is formalized in quantified
      modal logic KB (QML KB).
      QML KB is modeled as a fragment of classical higher-order logic (HOL);
      thus, the formalization is essentially a formalization in HOL.

title = A General Method for the Proof of Theorems on Tail-recursive Functions
author = Pasquale Noce <mailto:pasquale.noce@arjowiggins-it.com>
date = 2013-12-01
topic = Computer Science/Functional Programming
abstract = Tail-recursive function definitions are sometimes more straightforward than alternatives, but proving theorems on them may be roundabout because of the peculiar form of the resulting recursion induction rules. This paper describes a proof method that provides a general solution to this problem by means of suitable invariants over inductive sets, and illustrates the application of such method by examining two case studies.

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