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Proof Pearl: Defining Functions over Finite Sets
[chapter]

2005
*
Lecture Notes in Computer Science
*

This paper outlines a theory of

doi:10.1007/11541868_25
fatcat:q6qjaqckzzhopirl3ixeygus44
*function*definition for*finite**sets*, based on the fold*functionals*often used with lists. ... Structural recursion*over**sets*is meaningful only if the result is independent of the order in which the set's elements are enumerated. ... The sum of the*function*f*over*the*set*A is*defined*in terms of fold. Cardinality of a*Finite**Set*. ...##
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Proof Pearl: Mechanizing the Textbook Proof of Huffman's Algorithm

2009
*
Journal of automated reasoning
*

Our Isabelle/HOL

doi:10.1007/s10817-009-9116-y
fatcat:p4samcta4zfoxnspkrdefn23fe
*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. ... Alexander Krauss answered all my Isabelle questions and helped me with the trickier*proofs*. ...##
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Proof Pearl: A Probabilistic Proof for the Girth-Chromatic Number Theorem
[chapter]

2012
*
Lecture Notes in Computer Science
*

We formalize a probabilistic

doi:10.1007/978-3-642-32347-8_27
fatcat:mhfexq2shnblnkg34zhumcfybm
*proof*of this theorem in the Isabelle/HOL theorem prover, closely following a standard textbook*proof*and use this to explore the use of the probabilistic method in a theorem ... In the future, an automated reasoner for inequalities*over*real-value*functions*like MetiTarski [1] might be useful. However, the*set*of a few example inequalities from our*proof*which L. ... However, an (underspecified)*function*with this equation can easily*defined*by the partial*function*command of Isabelle. ...##
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Proof Pearl: Regular Expression Equivalence and Relation Algebra

2011
*
Journal of automated reasoning
*

By mapping regular expressions to binary relations, an automatic and complete

doi:10.1007/s10817-011-9223-4
fatcat:j4ovv2ddkzgd5bx6nb4i5ljtv4
*proof*method for (in)equalities of binary relations*over*union, composition and (reflexive) transitive closure is obtained. ... We*define*the*set*D(r ) inductively by the following rules. norm r ∈ D(r ) s ∈ D(r ) =⇒ Da(s) ∈ D(r ) By induction on the structure of r one can prove that D(r ) is always*finite*. ... as (hd ws) . p / ∈*set*ps ∪*set*ws] in (new @ tl ws, ps ))*Functions*hd and tl take the head and tail of a list. ...##
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A Proof Pearl with the Fan Theorem and Bar Induction
[chapter]

2011
*
Lecture Notes in Computer Science
*

We study temporal properties

doi:10.1007/978-3-642-25318-8_26
fatcat:ow4mbnbhlncbrngod4422usszi
*over*infinite binary red-blue trees in the*setting*of constructive type theory. ... We have fully formalized our development with the Coq*proof*assistant. ... Lists : A **over*a*set*A are*finite*sequences*over*A*defined*inductively by : A * a : A : A * a : A * The notation a denotes singletons, i.e., a = a . ...##
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A String of Pearls: Proofs of Fermat's Little Theorem
[chapter]

2012
*
Lecture Notes in Computer Science
*

We discuss mechanised

doi:10.1007/978-3-642-35308-6_16
fatcat:oiswy7gzmzfmdlw5bbvqqwae7i
*proofs*of Fermat's Little Theorem in a variety of styles, focusing in particular on an elegant combinatorial "necklace"*proof*that has not been mechanised previously. ... This has pleasant consequences both for the necklace*proof*, and also for some of the direct number-theoretic approaches. ... We write IMAGE f s for the image of*set*s under*function*f , and BIJ f s 1 s 2 means that*function*f is a bijection between*sets*s 1 and s 2 . ...##
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EDUCATIONAL PEARL: 'Proof-directed debugging' revisited for a first-order version

2006
*
Journal of functional programming
*

Some 10 years ago, Harper illustrated the powerful method of

doi:10.1017/s0956796806006149
fatcat:4rv6iq7v5zhwzkvocovzo2bc6u
*proof*-directed debugging for developing programs with an article in this journal. ... Unfortunately, his example uses both higher-order*functions*and continuation-passing style, which is too difficult for students in an introductory programming course. ... Background Let Σ be an alphabet, that is, a*finite**set*of letters. We use c to denote a letter. Σ * is the*set*of*finite*strings*over*the alphabet Σ. We use s to denote a string. ...##
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A Formalisation of the Myhill-Nerode Theorem Based on Regular Expressions (Proof Pearl)
[chapter]

2011
*
Lecture Notes in Computer Science
*

Nearly all of them introduce the subject by describing

doi:10.1007/978-3-642-22863-6_25
fatcat:bceexo2b5nd4nkbrep5dcjasyq
*finite*automata and only mentioning on the side a connection with regular expressions. ... In contrast, regular expressions can be*defined*conveniently as a datatype and a corresponding reasoning infrastructure comes for free. ... Our*proof*will rely on some tagging-*functions**defined**over*strings. ...##
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POSIX Lexing with Derivatives of Regular Expressions (Proof Pearl)
[chapter]

2016
*
Lecture Notes in Computer Science
*

Our definitions and

doi:10.1007/978-3-319-43144-4_5
fatcat:opnx4er5mnelzdztmnyhnnqe34
*proof*are much simpler than those by Sulzmann and Lu and can be easily formalised in Isabelle/HOL. ... Sulzmann and Lu have made available on-line what they call a "rigorous*proof*" of the correctness of their algorithm w.r.t. their specification; regrettably, it appears to us to have unfillable gaps. ... By using the type char for characters we have a supply of*finitely*many characters roughly corresponding to the ASCII character*set*. ...##
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Proof Pearl: A Verified Bignum Implementation in x86-64 Machine Code
[chapter]

2013
*
Lecture Notes in Computer Science
*

This paper presents a case study which shows that machine-code verification does not necessarily require ghastly low-level

doi:10.1007/978-3-319-03545-1_5
fatcat:vwdjj3wqfrdy5lwusekyndhkiu
*proofs*. ... We achieve this improvement as a result of using previously developed tools, namely, a*proof*-producing decompiler and compiler. ... We start by*defining*the algorithms involved as*functions*in logic. The*functions*operate*over*lists of binary words. ...##
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Proof Pearl: Purely Functional, Simple and Efficient Priority Search Trees and Applications to Prim and Dijkstra

2019
*
International Conference on Interactive Theorem Proving
*

As two applications of this data structure we verify purely

doi:10.4230/lipics.itp.2019.23
dblp:conf/itp/LammichN19
fatcat:cqyyx4x2czdadgh6fhhku4kdi4
*functional*, simple and efficient implementations of Prim's and Dijkstra's algorithms. ... The starting point of this paper is a new, purely*functional*, simple and efficient data structure combining a search tree and a priority queue, which we call a priority search tree. ... E ⊆ V ×V ∧*finite*V ∧ sym E ∧ irrefl E } Next, we*define*accessor*functions*to obtain the nodes and edges of a graph: nodes:: v ugraph ⇒ v*set*edges:: v ugraph ⇒ ( v × v)*set*We also*define**functions*to ...##
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Functional pearl

2008
*
SIGPLAN notices
*

In fact, we redevelop the theory of recurrences,

doi:10.1145/1411203.1411232
fatcat:5qyktmskjjfstb3epkmvj2khym
*finite*calculus and generating*functions*using streams and stream operators building on the cornerstone of unique solutions. ... Finally, we rephrase the*proof*of uniqueness using generalised algebraic data types. ... Sigma denotes summation and, traditionally, summation is a binder introducing an index variable that ranges*over*some*set*. ...##
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Functional pearl

2008
*
Proceeding of the 13th ACM SIGPLAN international conference on Functional programming - ICFP '08
*

In fact, we redevelop the theory of recurrences,

doi:10.1145/1411204.1411232
dblp:conf/icfp/Hinze08
fatcat:oo35hmry2fflvmzjm7oszr7vyy
*finite*calculus and generating*functions*using streams and stream operators building on the cornerstone of unique solutions. ... Finally, we rephrase the*proof*of uniqueness using generalised algebraic data types. ... Sigma denotes summation and, traditionally, summation is a binder introducing an index variable that ranges*over*some*set*. ...##
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Functional pearl

2010
*
Proceedings of the 15th ACM SIGPLAN international conference on Functional programming - ICFP '10
*

The

doi:10.1145/1863543.1863548
dblp:conf/icfp/VytiniotisK10
fatcat:z7jv2las6fdtpfvnu4mpqgvjdm
*functions*are parameterized on subtraction and addition operations, and are instantiated with appropriate concrete operations to obtain games for*finite*multisets and*sets*of natural numbers, as follows ... Proper*finite*filtering. ...##
###
Functional pearl

2010
*
SIGPLAN notices
*

The

doi:10.1145/1932681.1863548
fatcat:rlokws6ddzhfpoeylsirgzlawy
*functions*are parameterized on subtraction and addition operations, and are instantiated with appropriate concrete operations to obtain games for*finite*multisets and*sets*of natural numbers, as follows ... Proper*finite*filtering. ...
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