Lemma Functions for Frama-C: C Programs As Proofs
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Better SMT Proofs for Easier Reconstruction Haniel Barbosa, Jasmin Blanchette, Mathias Fleury, Pascal Fontaine, Hans-Jörg Schurr
Better SMT Proofs for Easier Reconstruction Haniel Barbosa, Jasmin Blanchette, Mathias Fleury, Pascal Fontaine, Hans-Jörg Schurr To cite this version: Haniel Barbosa, Jasmin Blanchette, Mathias Fleury, Pascal Fontaine, Hans-Jörg Schurr. Better SMT Proofs for Easier Reconstruction. AITP 2019 - 4th Conference on Artificial Intelligence and Theorem Proving, Apr 2019, Obergurgl, Austria. hal-02381819 HAL Id: hal-02381819 https://hal.archives-ouvertes.fr/hal-02381819 Submitted on 26 Nov 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Better SMT Proofs for Easier Reconstruction Haniel Barbosa1, Jasmin Christian Blanchette2;3;4, Mathias Fleury4;5, Pascal Fontaine3, and Hans-J¨orgSchurr3 1 University of Iowa, 52240 Iowa City, USA [email protected] 2 Vrije Universiteit Amsterdam, 1081 HV Amsterdam, the Netherlands [email protected] 3 Universit´ede Lorraine, CNRS, Inria, LORIA, 54000 Nancy, France fjasmin.blanchette,pascal.fontaine,[email protected] 4 Max-Planck-Institut f¨urInformatik, Saarland Informatics Campus, Saarbr¨ucken, Germany fjasmin.blanchette,[email protected] 5 Saarbr¨ucken Graduate School of Computer Science, Saarland Informatics Campus, Saarbr¨ucken, Germany [email protected] Proof assistants are used in verification, formal mathematics, and other areas to provide trust- worthy, machine-checkable formal proofs of theorems. -
A Program Logic for First-Order Encapsulated Webassembly
A Program Logic for First-Order Encapsulated WebAssembly Conrad Watt University of Cambridge, UK [email protected] Petar Maksimović Imperial College London, UK; Mathematical Institute SASA, Serbia [email protected] Neelakantan R. Krishnaswami University of Cambridge, UK [email protected] Philippa Gardner Imperial College London, UK [email protected] Abstract We introduce Wasm Logic, a sound program logic for first-order, encapsulated WebAssembly. We design a novel assertion syntax, tailored to WebAssembly’s stack-based semantics and the strong guarantees given by WebAssembly’s type system, and show how to adapt the standard separation logic triple and proof rules in a principled way to capture WebAssembly’s uncommon structured control flow. Using Wasm Logic, we specify and verify a simple WebAssembly B-tree library, giving abstract specifications independent of the underlying implementation. We mechanise Wasm Logic and its soundness proof in full in Isabelle/HOL. As part of the soundness proof, we formalise and fully mechanise a novel, big-step semantics of WebAssembly, which we prove equivalent, up to transitive closure, to the original WebAssembly small-step semantics. Wasm Logic is the first program logic for WebAssembly, and represents a first step towards the creation of static analysis tools for WebAssembly. 2012 ACM Subject Classification Theory of computation Ñ separation logic Keywords and phrases WebAssembly, program logic, separation logic, soundness, mechanisation Acknowledgements We would like to thank the reviewers, whose comments were valuable in improving the paper. All authors were supported by the EPSRC Programme Grant ‘REMS: Rigorous Engineering for Mainstream Systems’ (EP/K008528/1). -
Proof-Assistants Using Dependent Type Systems
CHAPTER 18 Proof-Assistants Using Dependent Type Systems Henk Barendregt Herman Geuvers Contents I Proof checking 1151 2 Type-theoretic notions for proof checking 1153 2.1 Proof checking mathematical statements 1153 2.2 Propositions as types 1156 2.3 Examples of proofs as terms 1157 2.4 Intermezzo: Logical frameworks. 1160 2.5 Functions: algorithms versus graphs 1164 2.6 Subject Reduction . 1166 2.7 Conversion and Computation 1166 2.8 Equality . 1168 2.9 Connection between logic and type theory 1175 3 Type systems for proof checking 1180 3. l Higher order predicate logic . 1181 3.2 Higher order typed A-calculus . 1185 3.3 Pure Type Systems 1196 3.4 Properties of P ure Type Systems . 1199 3.5 Extensions of Pure Type Systems 1202 3.6 Products and Sums 1202 3.7 E-typcs 1204 3.8 Inductive Types 1206 4 Proof-development in type systems 1211 4.1 Tactics 1212 4.2 Examples of Proof Development 1214 4.3 Autarkic Computations 1220 5 P roof assistants 1223 5.1 Comparing proof-assistants . 1224 5.2 Applications of proof-assistants 1228 Bibliography 1230 Index 1235 Name index 1238 HANDBOOK OF AUTOMAT8D REASONING Edited by Alan Robinson and Andrei Voronkov © 2001 Elsevier Science Publishers 8.V. All rights reserved PROOF-ASSISTANTS USING DEPENDENT TYPE SYSTEMS 1151 I. Proof checking Proof checking consists of the automated verification of mathematical theories by first fully formalizing the underlying primitive notions, the definitions, the axioms and the proofs. Then the definitions are checked for their well-formedness and the proofs for their correctness, all this within a given logic. -
Can the Computer Really Help Us to Prove Theorems?
Can the computer really help us to prove theorems? Herman Geuvers1 Radboud University Nijmegen and Eindhoven University of Technology The Netherlands ICT.Open 2011 Veldhoven, November 2011 1Thanks to Freek Wiedijk & Foundations group, RU Nijmegen Can the computer really help us to prove theorems? Can the computer really help us to prove theorems? Yes it can Can the computer really help us to prove theorems? Yes it can But it’s hard ... ◮ How does it work? ◮ Some state of the art ◮ What needs to be done Overview ◮ What are Proof Assistants? ◮ How can a computer program guarantee correctness? ◮ Challenges What are Proof Assistants – History John McCarthy (1927 – 2011) 1961, Computer Programs for Checking Mathematical Proofs What are Proof Assistants – History John McCarthy (1927 – 2011) 1961, Computer Programs for Checking Mathematical Proofs Proof-checking by computer may be as important as proof generation. It is part of the definition of formal system that proofs be machine checkable. For example, instead of trying out computer programs on test cases until they are debugged, one should prove that they have the desired properties. What are Proof Assistants – History Around 1970 five new systems / projects / ideas ◮ Automath De Bruijn (Eindhoven) ◮ Nqthm Boyer, Moore (Austin, Texas) ◮ LCF Milner (Stanford; Edinburgh) What are Proof Assistants – History Around 1970 five new systems / projects / ideas ◮ Automath De Bruijn (Eindhoven) ◮ Nqthm Boyer, Moore (Austin, Texas) ◮ LCF Milner (Stanford; Edinburgh) ◮ Mizar Trybulec (Bia lystok, Poland) -
Reconstructing Verit Proofs in Isabelle/HOL Mathias Fleury, Hans-Jörg Schurr
Reconstructing veriT Proofs in Isabelle/HOL Mathias Fleury, Hans-Jörg Schurr To cite this version: Mathias Fleury, Hans-Jörg Schurr. Reconstructing veriT Proofs in Isabelle/HOL. PxTP 2019 - Sixth Workshop on Proof eXchange for Theorem Proving, Aug 2019, Natal, Brazil. pp.36-50, 10.4204/EPTCS.301.6. hal-02276530 HAL Id: hal-02276530 https://hal.inria.fr/hal-02276530 Submitted on 2 Sep 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Reconstructing veriT Proofs in Isabelle/HOL Mathias Fleury Hans-Jorg¨ Schurr Max-Planck-Institut fur¨ Informatik, University of Lorraine, CNRS, Inria, and Saarland Informatics Campus, Saarbrucken,¨ Germany LORIA, Nancy, France [email protected] [email protected] Graduate School of Computer Science, Saarland Informatics Campus, Saarbrucken,¨ Germany [email protected] Automated theorem provers are now commonly used within interactive theorem provers to discharge an increasingly large number of proof obligations. To maintain the trustworthiness of a proof, the automatically found proof must be verified inside the proof assistant. We present here a reconstruc- tion procedure in the proof assistant Isabelle/HOL for proofs generated by the satisfiability modulo theories solver veriT which is part of the smt tactic. -
Mathematics in the Computer
Mathematics in the Computer Mario Carneiro Carnegie Mellon University April 26, 2021 1 / 31 Who am I? I PhD student in Logic at CMU I Proof engineering since 2013 I Metamath (maintainer) I Lean 3 (maintainer) I Dabbled in Isabelle, HOL Light, Coq, Mizar I Metamath Zero (author) Github: digama0 I Proved 37 of Freek’s 100 theorems list in Zulip: Mario Carneiro Metamath I Lots of library code in set.mm and mathlib I Say hi at https://leanprover.zulipchat.com 2 / 31 I Found Metamath via a random internet search I they already formalized half of the book! I .! . and there is some stuff on cofinality they don’t have yet, maybe I can help I Got involved, did it as a hobby for a few years I Got a job as an android developer, kept on the hobby I Norm Megill suggested that I submit to a (Mizar) conference, it went well I Met Leo de Moura (Lean author) at a conference, he got me in touch with Jeremy Avigad (my current advisor) I Now I’m a PhD at CMU philosophy! How I got involved in formalization I Undergraduate at Ohio State University I Math, CS, Physics I Reading Takeuti & Zaring, Axiomatic Set Theory 3 / 31 I they already formalized half of the book! I .! . and there is some stuff on cofinality they don’t have yet, maybe I can help I Got involved, did it as a hobby for a few years I Got a job as an android developer, kept on the hobby I Norm Megill suggested that I submit to a (Mizar) conference, it went well I Met Leo de Moura (Lean author) at a conference, he got me in touch with Jeremy Avigad (my current advisor) I Now I’m a PhD at CMU philosophy! How I got involved in formalization I Undergraduate at Ohio State University I Math, CS, Physics I Reading Takeuti & Zaring, Axiomatic Set Theory I Found Metamath via a random internet search 3 / 31 I . -
Formalized Mathematics in the Lean Proof Assistant
Formalized mathematics in the Lean proof assistant Robert Y. Lewis Vrije Universiteit Amsterdam CARMA Workshop on Computer-Aided Proof Newcastle, NSW June 6, 2019 Credits Thanks to the following people for some of the contents of this talk: Leonardo de Moura Jeremy Avigad Mario Carneiro Johannes Hölzl 1 38 Table of Contents 1 Proof assistants in mathematics 2 Dependent type theory 3 The Lean theorem prover 4 Lean’s mathematical libraries 5 Automation in Lean 6 The good and the bad 2 38 bfseries Proof assistants in mathematics Computers in mathematics Mathematicians use computers in various ways. typesetting numerical calculations symbolic calculations visualization exploration Let’s add to this list: checking proofs. 3 38 Interactive theorem proving We want a language (and an implementation of this language) for: Defining mathematical objects Stating properties of these objects Showing that these properties hold Checking that these proofs are correct Automatically generating these proofs This language should be: Expressive User-friendly Computationally eicient 4 38 Interactive theorem proving Working with a proof assistant, users construct definitions, theorems, and proofs. The proof assistant makes sure that definitions are well-formed and unambiguous, theorem statements make sense, and proofs actually establish what they claim. In many systems, this proof object can be extracted and verified independently. Much of the system is “untrusted.” Only a small core has to be relied on. 5 38 Interactive theorem proving Some systems with large -
Dafny: an Automatic Program Verifier for Functional Correctness K
Dafny: An Automatic Program Verifier for Functional Correctness K. Rustan M. Leino Microsoft Research [email protected] Abstract Traditionally, the full verification of a program’s functional correctness has been obtained with pen and paper or with interactive proof assistants, whereas only reduced verification tasks, such as ex- tended static checking, have enjoyed the automation offered by satisfiability-modulo-theories (SMT) solvers. More recently, powerful SMT solvers and well-designed program verifiers are starting to break that tradition, thus reducing the effort involved in doing full verification. This paper gives a tour of the language and verifier Dafny, which has been used to verify the functional correctness of a number of challenging pointer-based programs. The paper describes the features incorporated in Dafny, illustrating their use by small examples and giving a taste of how they are coded for an SMT solver. As a larger case study, the paper shows the full functional specification of the Schorr-Waite algorithm in Dafny. 0 Introduction Applications of program verification technology fall into a spectrum of assurance levels, at one extreme proving that the program lives up to its functional specification (e.g., [8, 23, 28]), at the other extreme just finding some likely bugs (e.g., [19, 24]). Traditionally, program verifiers at the high end of the spectrum have used interactive proof assistants, which require the user to have a high degree of prover expertise, invoking tactics or guiding the tool through its various symbolic manipulations. Because they limit which program properties they reason about, tools at the low end of the spectrum have been able to take advantage of satisfiability-modulo-theories (SMT) solvers, which offer some fixed set of automatic decision procedures [18, 5]. -
Separation Logic: a Logic for Shared Mutable Data Structures
Separation Logic: A Logic for Shared Mutable Data Structures John C. Reynolds∗ Computer Science Department Carnegie Mellon University [email protected] Abstract depends upon complex restrictions on the sharing in these structures. To illustrate this problem,and our approach to In joint work with Peter O’Hearn and others, based on its solution,consider a simple example. The following pro- early ideas of Burstall, we have developed an extension of gram performs an in-place reversal of a list: Hoare logic that permits reasoning about low-level impera- tive programs that use shared mutable data structure. j := nil ; while i = nil do The simple imperative programming language is ex- (k := [i +1];[i +1]:=j ; j := i ; i := k). tended with commands (not expressions) for accessing and modifying shared structures, and for explicit allocation and (Here the notation [e] denotes the contents of the storage at deallocation of storage. Assertions are extended by intro- address e.) ducing a “separating conjunction” that asserts that its sub- The invariant of this program must state that i and j are formulas hold for disjoint parts of the heap, and a closely lists representing two sequences α and β such that the re- related “separating implication”. Coupled with the induc- flection of the initial value α0 can be obtained by concate- tive definition of predicates on abstract data structures, this nating the reflection of α onto β: extension permits the concise and flexible description of ∃ ∧ ∧ † †· structures with controlled sharing. α, β. list α i list β j α0 = α β, In this paper, we will survey the current development of this program logic, including extensions that permit unre- where the predicate list α i is defined by induction on the stricted address arithmetic, dynamically allocated arrays, length of α: and recursive procedures. -
Pointer Programs, Separation Logic
Chapter 6 Pointer Programs, Separation Logic The goal of this section is to drop the hypothesis that we have until now, that is, references are not values of the language, and in particular there is nothing like a reference to a reference, and there is no way to program with data structures that can be modified in-place, such as records where fields can be assigned directly. There is a fundamental difference of semantics between in-place modification of a record field and the way we modeled records in Chapter 4. The small piece of code below, in the syntax of the C programming language, is a typical example of the kind of programs we want to deal with in this chapter. typedef struct L i s t { i n t data; list next; } ∗ l i s t ; list create( i n t d , l i s t n ) { list l = (list)malloc( sizeof ( struct L i s t ) ) ; l −>data = d ; l −>next = n ; return l ; } void incr_list(list p) { while ( p <> NULL) { p−>data++; p = p−>next ; } } Like call by reference, in-place assignment of fields is another source of aliasing issues. Consider this small test program void t e s t ( ) { list l1 = create(4,NULL); list l2 = create(7,l1); list l3 = create(12,l1); assert ( l3 −>next−>data == 4 ) ; incr_list(l2); assert ( l3 −>next−>data == 5 ) ; } which builds the following structure: 63 7 4 null 12 the list node l1 is shared among l2 and l3, hence the call to incr_list(l2) modifies the second node of list l3. -
A Survey of Engineering of Formally Verified Software
The version of record is available at: http://dx.doi.org/10.1561/2500000045 QED at Large: A Survey of Engineering of Formally Verified Software Talia Ringer Karl Palmskog University of Washington University of Texas at Austin [email protected] [email protected] Ilya Sergey Milos Gligoric Yale-NUS College University of Texas at Austin and [email protected] National University of Singapore [email protected] Zachary Tatlock University of Washington [email protected] arXiv:2003.06458v1 [cs.LO] 13 Mar 2020 Errata for the present version of this paper may be found online at https://proofengineering.org/qed_errata.html The version of record is available at: http://dx.doi.org/10.1561/2500000045 Contents 1 Introduction 103 1.1 Challenges at Scale . 104 1.2 Scope: Domain and Literature . 105 1.3 Overview . 106 1.4 Reading Guide . 106 2 Proof Engineering by Example 108 3 Why Proof Engineering Matters 111 3.1 Proof Engineering for Program Verification . 112 3.2 Proof Engineering for Other Domains . 117 3.3 Practical Impact . 124 4 Foundations and Trusted Bases 126 4.1 Proof Assistant Pre-History . 126 4.2 Proof Assistant Early History . 129 4.3 Proof Assistant Foundations . 130 4.4 Trusted Computing Bases of Proofs and Programs . 137 5 Between the Engineer and the Kernel: Languages and Automation 141 5.1 Styles of Automation . 142 5.2 Automation in Practice . 156 The version of record is available at: http://dx.doi.org/10.1561/2500000045 6 Proof Organization and Scalability 162 6.1 Property Specification and Encodings . -
Translating Scala Programs to Isabelle/HOL (System
Translating Scala Programs to Isabelle/HOL System Description Lars Hupel1 and Viktor Kuncak2 1 Technische Universität München 2 École Polytechnique Fédérale de Lausanne (EPFL) Abstract. We present a trustworthy connection between the Leon verification system and the Isabelle proof assistant. Leon is a system for verifying functional Scala programs. It uses a variety of automated theorem provers (ATPs) to check verification conditions (VCs) stemming from the input program. Isabelle, on the other hand, is an interactive theorem prover used to verify mathematical spec- ifications using its own input language Isabelle/Isar. Users specify (inductive) definitions and write proofs about them manually, albeit with the help of semi- automated tactics. The integration of these two systems allows us to exploit Is- abelle’s rich standard library and give greater confidence guarantees in the cor- rectness of analysed programs. Keywords: Isabelle, HOL, Scala, Leon, compiler 1 Introduction This system description presents a new tool that aims to connect two important worlds: the world of interactive proof assistant users who create a body of verified theorems, and the world of professional programmers who increasingly adopt functional programming to develop important applications. The Scala language (www.scala-lang.org) en- joys a prominent role today for its adoption in industry, a trend most recently driven by the Apache Spark data analysis framework (to which, e.g., IBM committed 3500 researchers recently [16]). We hope to introduce some of the many Scala users to formal methods by providing tools they can use directly on Scala code. Leon sys- tem (http://leon.epfl.ch) is a verification and synthesis system for a subset of Scala [2, 10].