locales ISAR IS BASED ON CONTEXTS CONTENT Slide 3 Slide 1 proof - fix x assume Ass: A. x and Ass are visible Slide 2 Slide 4 inside this context
|
|
- Suzanna Lucas
- 6 years ago
- Views:
Transcription
1 LAST TIME Syntax and semantics of IMP Hoare logic rules NICTA Advanced Course Soundness of Hoare logic Slide 1 Theorem Proving Principles, Techniques, Applications Slide 3 Verification conditions Example program proofs locales CONTENT ISAR IS BASED ON CONTEXTS Slide 2 Intro & motivation, getting started with Isabelle Foundations & Principles Lambda Calculus Higher Order Logic, natural deduction Term rewriting Proof & Specification Techniques Inductively defined sets, rule induction Datatypes, recursion, induction Slide 4 theorem x. A = C proof - fix x assume Ass: A. x and Ass are visible qed from Ass show C... inside this context More recursion, Calculational reasoning Hoare logic, proofs about programs Locales, Presentation LAST TIME 1 BEYOND ISAR CONTEXTS 2
2 BEYOND ISAR CONTEXTS DECLARING LOCALES Locales are extended contexts Declaring locale (named context) loc: Slide 5 Locales are named Fixed variables may have syntax It is possible to add and export theorems Locale expression: combine and modify locales Slide 7 locale loc = loc1 + fixes... assumes... Import Context elements CONTEXT ELEMENTS Locales consist of context elements. DECLARING LOCALES Theorems may be stated relative to a named locale. Slide 6 fixes assumes defines notes includes Parameter, with syntax Assumption Definition Record a theorem Import other locales (locale expressions) Slide 8 lemma (in loc) P [simp]: proposition proof Adds theorem P to context loc. Theorem P is in the simpset in context loc. Exported theorem loc.p visible in the entire theory. DECLARING LOCALES 3 4
3 LOCALE EXPRESSIONS Locale name: Rename: n e q 1... q n Change names of parameters in e. Merge: e 1 + e 2 Slide 9 DEMO: LOCALES 1 Slide 11 Context elements of e 1, then e 2. Syntax is lost after rename (currently). PARAMETERS MUST BE CONSISTENT! Parameters in fixes are distinct. Free variables in assumes and defines occur in preceding fixes. Defined parameters cannot occur in preceding assumes nor defines. Slide 10 Slide 12 DEMO: LOCALES 2 LOCALE EXPRESSIONS 5 NORMAL FORM OF LOCALE EXPRESSIONS 6
4 NORMAL FORM OF LOCALE EXPRESSIONS Locale expressions are converted to flattened lists of locale names. With full parameter lists Duplicates removed Slide 13 Allows for multiple inheritance! Slide 15 DEMO: LOCALES 3 INSTANTIATION Move from abstract to concrete. instantiate label: loc Slide 14 From chained fact loc t 1... t n instantiate locale loc. Imports all theorems of loc into current context. Slide 16 PRESENTATION Instantiates the parameters with t 1... t n. Interprets attributes of theorems. Prefixes theorem names with label Currently only works inside Isar contexts. 7 ISABELLE S BATCH MODE 8
5 ISABELLE S BATCH MODE GENERATING L A T E X FROM ISABELLE <..>/isatool usedir -d pdf HOL <session> used to process and check larger number of theories no interactive niceties (no sorry, no quick and dirty) controlled by file ROOT.ML and script set isatool <..>/<session>/root.ml <..>/<session>/mytheory.thy <..>/<session>/document/root.tex Slide 17 can save state for later use (images) can generate HTML and L A T E X documentation Slide 19 In ROOT.ML: no\_document use_thy "MyLibrary"; use_thy "MyTheory"; In document/root.tex: include Isabelle style packages (isabelle.sty, isabellesym.sty) include generated files session.tex (for all theories) or MyTheory.tex ISATOOL isatool <tool> <options> Get help with: isatool shows available tools Slide 18 isatool <tool> -? Interesting tools: isatool mkdir shows options for <tool> create session directory Slide 20 DEMO: EXAMPLE make/makeall run make for directory/all logics usedir batch session (documents, HTML, session graph) document/latex run L A T E X for generated sources GENERATING L A T E X FROM ISABELLE 9 LARGE DEVELOPMENTS 10
6 LARGE DEVELOPMENTS ANTIQUOTATIONS Creating Images: Inside L A T E X you can go back to Isabelle commands and syntax. <..>/<session>/isatool usedir -b HOL <session> Useful Antiquotations: <..>/<session>/root.ml t} print type τ print term t Slide 21 Slide φ} print proposition φ Processes [display] φ} print proposition φ with linebreaks Saves state after processing in [source] a} check proposition φ, print its input print fact a Makes HOL-<session> available as logic in menu Isabelle Logics Direct start of Isabelle with new logic: Isabelle -l a [no [source] s} print fact a, fixing schematic variables check availability of a, print its name print uninterpreted text s MARKUP COMMANDS WRITING ABOUT ISABELLE THEORIES document structure commands: normal text header section subsection subsubsection (meaning defined in isabelle.sty) To document definitions and proofs: put comments explanations directly in original theory keep explanations short and to the point formal definitions, lemmas, syntax should speak for themself text {... } text raw {... } Slide 22 text inside proofs Slide 24 To write a paper/thesis about a formal development txt {... } txt raw {... } use a separate theory/document on top of the development formal comments only talk about the interesting parts -- {... } use antiquoations for theorems and definitions make text invisible: ( < )... ( > ) use extra locales, definitions, syntax for polish make full proof document available separately ANTIQUOTATIONS 11 POLISH 12
7 POLISH Know your audience. Use the right notation. Change L A T E X symbol interpretations \renewcommand{\isasymlongrightarrow} {\isamath{\longrightarrow}} Slide 25 Declare special L A T E X output syntax: syntax (latex) Cons :: a a list a list ( / [66,65] 65) Slide 27 DEMO Use translations to change output syntax: syntax (latex) notex :: ( a bool) bool (binder \<notex> 10) translations \<notex>x. P <= ( x. P) in document/root.tex: \newcommand{\isasymnotex}{\isamath{\neg\exists}} USING LOCALES WE HAVE SEEN TODAY... making large developments more accessible Math textbook: Locale Declarations + Theorems in Locales Let (A,, 0) in the following be a group with x y = y x Locale Expressions + Inheritance Isabelle: Locale Instantiation Slide 26 Use locales to formalize contexts Slide 28 Generating L A T E X Antiquotations are sensitive to current locale context Writing a thesis/paper in Isabelle Example: locale agroup = group + assumes com: x y = y x... ( < ) lemma (in agroup) True ( > ) txt {... } ( < ) oops ( > ) 13 EXERCISES 14
8 EXERCISES No Exercise Today Slide 29 Theorem Proving Principles, Techniques, Applications The End EXERCISES 15
type classes & locales
Content Rough timeline Intro & motivation, getting started [1] COMP 4161 NICTA Advanced Course Advanced Topics in Software Verification Gerwin Klein, June Andronick, Toby Murray type classes & locales
More informationHOL DEFINING HIGHER ORDER LOGIC LAST TIME ON HOL CONTENT. Slide 3. Slide 1. Slide 4. Slide 2 WHAT IS HIGHER ORDER LOGIC? 2 LAST TIME ON HOL 1
LAST TIME ON HOL Proof rules for propositional and predicate logic Safe and unsafe rules NICTA Advanced Course Forward Proof Slide 1 Theorem Proving Principles, Techniques, Applications Slide 3 The Epsilon
More informationA CRASH COURSE IN SEMANTICS
LAST TIME Recdef More induction NICTA Advanced Course Well founded orders Slide 1 Theorem Proving Principles, Techniques, Applications Slide 3 Well founded recursion Calculations: also/finally {P}... {Q}
More informationλ calculus is inconsistent
Content Rough timeline COMP 4161 NICTA Advanced Course Advanced Topics in Software Verification Gerwin Klein, June Andronick, Toby Murray λ Intro & motivation, getting started [1] Foundations & Principles
More informationTheorem Proving Principles, Techniques, Applications Recursion
NICTA Advanced Course Theorem Proving Principles, Techniques, Applications Recursion 1 CONTENT Intro & motivation, getting started with Isabelle Foundations & Principles Lambda Calculus Higher Order Logic,
More informationCOMP 4161 NICTA Advanced Course. Advanced Topics in Software Verification. Toby Murray, June Andronick, Gerwin Klein
COMP 4161 NICTA Advanced Course Advanced Topics in Software Verification Toby Murray, June Andronick, Gerwin Klein λ 1 Last time... λ calculus syntax free variables, substitution β reduction α and η conversion
More informationPreuves Interactives et Applications
Preuves Interactives et Applications Christine Paulin & Burkhart Wolff http://www.lri.fr/ paulin/preuvesinteractives Université Paris-Saclay HOL and its Specification Constructs 10/12/16 B. Wolff - M2
More informationTheorem proving. PVS theorem prover. Hoare style verification PVS. More on embeddings. What if. Abhik Roychoudhury CS 6214
Theorem proving PVS theorem prover Abhik Roychoudhury National University of Singapore Both specification and implementation can be formalized in a suitable logic. Proof rules for proving statements in
More informationIsabelle/HOL:Selected Features and Recent Improvements
/: Selected Features and Recent Improvements webertj@in.tum.de Security of Systems Group, Radboud University Nijmegen February 20, 2007 /:Selected Features and Recent Improvements 1 2 Logic User Interface
More informationThe Isar Proof Language in 2016
The Isar Proof Language in 2016 Makarius Wenzel sketis.net August 2016 = Isabelle λ β Isar α Introduction History of Isar 1999: first usable version primary notion of proof document (not proof script )
More informationIsabelle/jEdit as IDE for domain-specific formal languages and informal text documents
Isabelle/jEdit as IDE for domain-specific formal languages and informal text documents Makarius Wenzel http://sketis.net June 2018 λ = Isabelle β PIDE α Isabelle/jEdit as Formal IDE Abstract Isabelle/jEdit
More informationCOMP4161: Advanced Topics in Software Verification. fun. Gerwin Klein, June Andronick, Ramana Kumar S2/2016. data61.csiro.au
COMP4161: Advanced Topics in Software Verification fun Gerwin Klein, June Andronick, Ramana Kumar S2/2016 data61.csiro.au Content Intro & motivation, getting started [1] Foundations & Principles Lambda
More informationOrganisatorials. About us. Binary Search (java.util.arrays) When Tue 9:00 10:30 Thu 9:00 10:30. COMP 4161 NICTA Advanced Course
Organisatorials COMP 4161 NICTA Advanced Course When Tue 9:00 10:30 Thu 9:00 10:30 Where Tue: Law 163 (F8-163) Thu: Australian School Business 205 (E12-205) Advanced Topics in Software Verification Rafal
More informationIntegration of SMT Solvers with ITPs There and Back Again
Integration of SMT Solvers with ITPs There and Back Again Sascha Böhme and University of Sheffield 7 May 2010 1 2 Features: SMT-LIB vs. Yices Translation Techniques Caveats 3 4 Motivation Motivation System
More informationOverview. A Compact Introduction to Isabelle/HOL. Tobias Nipkow. System Architecture. Overview of Isabelle/HOL
Overview A Compact Introduction to Isabelle/HOL Tobias Nipkow TU München 1. Introduction 2. Datatypes 3. Logic 4. Sets p.1 p.2 System Architecture Overview of Isabelle/HOL ProofGeneral Isabelle/HOL Isabelle
More informationThe Formal Semantics of Programming Languages An Introduction. Glynn Winskel. The MIT Press Cambridge, Massachusetts London, England
The Formal Semantics of Programming Languages An Introduction Glynn Winskel The MIT Press Cambridge, Massachusetts London, England Series foreword Preface xiii xv 1 Basic set theory 1 1.1 Logical notation
More informationPackaging Theories of Higher Order Logic
Packaging Theories of Higher Order Logic Joe Hurd Galois, Inc. joe@galois.com Theory Engineering Workshop Tuesday 9 February 2010 Joe Hurd Packaging Theories of Higher Order Logic 1 / 26 Talk Plan 1 Introduction
More informationBasic Foundations of Isabelle/HOL
Basic Foundations of Isabelle/HOL Peter Wullinger May 16th 2007 1 / 29 1 Introduction into Isabelle s HOL Why Type Theory Basic Type Syntax 2 More HOL Typed λ Calculus HOL Rules 3 Example proof 2 / 29
More informationNote that in this definition, n + m denotes the syntactic expression with three symbols n, +, and m, not to the number that is the sum of n and m.
CS 6110 S18 Lecture 8 Structural Operational Semantics and IMP Today we introduce a very simple imperative language, IMP, along with two systems of rules for evaluation called small-step and big-step semantics.
More informationCOMP 4161 Data61 Advanced Course. Advanced Topics in Software Verification. Gerwin Klein, June Andronick, Christine Rizkallah, Miki Tanaka
COMP 4161 Data61 Advanced Course Advanced Topics in Software Verification Gerwin Klein, June Andronick, Christine Rizkallah, Miki Tanaka 1 COMP4161 c Data61, CSIRO: provided under Creative Commons Attribution
More informationPrograms and Proofs in Isabelle/HOL
Programs and Proofs in Isabelle/HOL Makarius Wenzel http://sketis.net March 2016 = Isabelle λ β α Introduction What is Isabelle? Hanabusa Itcho : Blind monks examining an elephant Introduction 2 History:
More informationAn Introduction to ProofPower
An Introduction to ProofPower Roger Bishop Jones Date: 2006/10/21 16:53:33 Abstract An introductory illustrated description of ProofPower (not progressed far enough to be useful). Contents http://www.rbjones.com/rbjpub/pp/doc/t015.pdf
More informationFunctional Programming with Isabelle/HOL
Functional Programming with Isabelle/HOL = Isabelle λ β HOL α Florian Haftmann Technische Universität München January 2009 Overview Viewing Isabelle/HOL as a functional programming language: 1. Isabelle/HOL
More informationThe Prototype Verification System PVS
The Prototype Verification System PVS Wolfgang Schreiner Wolfgang.Schreiner@risc.uni-linz.ac.at Research Institute for Symbolic Computation (RISC) Johannes Kepler University, Linz, Austria http://www.risc.uni-linz.ac.at
More informationIsabelle Tutorial: System, HOL and Proofs
Isabelle Tutorial: System, HOL and Proofs Burkhart Wolff, Makarius Wenzel Université Paris-Sud What we will talk about What we will talk about Isabelle with: its System Framework the Logical Framework
More informationFoundations of AI. 9. Predicate Logic. Syntax and Semantics, Normal Forms, Herbrand Expansion, Resolution
Foundations of AI 9. Predicate Logic Syntax and Semantics, Normal Forms, Herbrand Expansion, Resolution Wolfram Burgard, Andreas Karwath, Bernhard Nebel, and Martin Riedmiller 09/1 Contents Motivation
More informationComputing Fundamentals 2 Introduction to CafeOBJ
Computing Fundamentals 2 Introduction to CafeOBJ Lecturer: Patrick Browne Lecture Room: K408 Lab Room: A308 Based on work by: Nakamura Masaki, João Pascoal Faria, Prof. Heinrich Hußmann. See notes on slides
More informationVerification Condition Generation via Theorem Proving
Verification Condition Generation via Theorem Proving John Matthews Galois Connections Inc. J Strother Moore University of Texas at Austin Sandip Ray University of Texas at Austin Daron Vroon Georgia Institute
More information15 212: Principles of Programming. Some Notes on Continuations
15 212: Principles of Programming Some Notes on Continuations Michael Erdmann Spring 2011 These notes provide a brief introduction to continuations as a programming technique. Continuations have a rich
More information1.3. Conditional expressions To express case distinctions like
Introduction Much of the theory developed in the underlying course Logic II can be implemented in a proof assistant. In the present setting this is interesting, since we can then machine extract from a
More informationTyped Lambda Calculus for Syntacticians
Department of Linguistics Ohio State University January 12, 2012 The Two Sides of Typed Lambda Calculus A typed lambda calculus (TLC) can be viewed in two complementary ways: model-theoretically, as a
More informationFormal editing: jedit-mmt. Narrative editing: LaTeX-MMT. Browsing: MMT web server. Building: MMT scripting language. The MMT API: A Generic MKM System
The MMT API: A Generic MKM System Florian Rabe MMT is foundation-independent: 1. Developer defines new logic 2. MMT yields complete MKM system for it MMT is application-independent: No single MMT application
More informationLee Pike. June 3, 2005
Proof NASA Langley Formal Methods Group lee.s.pike@nasa.gov June 3, 2005 Proof Proof Quantification Quantified formulas are declared by quantifying free variables in the formula. For example, lem1: LEMMA
More informationThe Isabelle System Manual
= Isabelle λ β α The Isabelle System Manual Markus Wenzel and Stefan Berghofer TU München June 8, 2008 Contents 1 The Isabelle system environment 1 1.1 Isabelle settings.......................... 1 Building
More informationCS152: Programming Languages. Lecture 11 STLC Extensions and Related Topics. Dan Grossman Spring 2011
CS152: Programming Languages Lecture 11 STLC Extensions and Related Topics Dan Grossman Spring 2011 Review e ::= λx. e x e e c v ::= λx. e c τ ::= int τ τ Γ ::= Γ, x : τ (λx. e) v e[v/x] e 1 e 1 e 1 e
More informationSpecification, Verification, and Interactive Proof
Specification, Verification, and Interactive Proof SRI International May 23, 2016 PVS PVS - Prototype Verification System PVS is a verification system combining language expressiveness with automated tools.
More informationImporting HOL-Light into Coq
Outlines Importing HOL-Light into Coq Deep and shallow embeddings of the higher order logic into Coq Work in progress Chantal Keller chantal.keller@ens-lyon.fr Bejamin Werner benjamin.werner@inria.fr 2009
More informationProgramming and Proving in
Programming and Proving in = λ β Isabelle HOL α Tobias Nipkow Fakultät für Informatik Technische Universität München 1 Notation Implication associates to the right: A = B = C means A = (B = C) Similarly
More informationContents. Chapter 1 SPECIFYING SYNTAX 1
Contents Chapter 1 SPECIFYING SYNTAX 1 1.1 GRAMMARS AND BNF 2 Context-Free Grammars 4 Context-Sensitive Grammars 8 Exercises 8 1.2 THE PROGRAMMING LANGUAGE WREN 10 Ambiguity 12 Context Constraints in Wren
More informationVerification of an LCF-Style First-Order Prover with Equality
Verification of an LCF-Style First-Order Prover with Equality Alexander Birch Jensen, Anders Schlichtkrull, and Jørgen Villadsen DTU Compute, Technical University of Denmark, 2800 Kongens Lyngby, Denmark
More informationInduction Schemes. Math Foundations of Computer Science
Induction Schemes Math Foundations of Computer Science Topics Induction Example Induction scheme over the naturals Termination Reduction to equational reasoning ACL2 proof General Induction Schemes Induction
More informationAbstract Interpretation
Abstract Interpretation Ranjit Jhala, UC San Diego April 22, 2013 Fundamental Challenge of Program Analysis How to infer (loop) invariants? Fundamental Challenge of Program Analysis Key issue for any analysis
More informationA Natural Deduction Environment for Matita
A Natural Deduction Environment for Matita Claudio Sacerdoti Coen University of Bologna 12/07/2009 Outline 1 The Problem 2 The (MKM) Technologies at Hand 3 Demo 4 Conclusion Outline
More informationCoq, a formal proof development environment combining logic and programming. Hugo Herbelin
Coq, a formal proof development environment combining logic and programming Hugo Herbelin 1 Coq in a nutshell (http://coq.inria.fr) A logical formalism that embeds an executable typed programming language:
More informationProgramming Languages Fall 2014
Programming Languages Fall 2014 Lecture 7: Simple Types and Simply-Typed Lambda Calculus Prof. Liang Huang huang@qc.cs.cuny.edu 1 Types stuck terms? how to fix it? 2 Plan First I For today, we ll go back
More informationFormal Systems and their Applications
Formal Systems and their Applications Dave Clarke (Dave.Clarke@cs.kuleuven.be) Acknowledgment: these slides are based in part on slides from Benjamin Pierce and Frank Piessens 1 Course Overview Introduction
More informationFunctional Programming and Modeling
Chapter 2 2. Functional Programming and Modeling 2.0 2. Functional Programming and Modeling 2.0 Overview of Chapter Functional Programming and Modeling 2. Functional Programming and Modeling 2.1 Overview
More informationAutomatic Proof and Disproof in Isabelle/HOL
Automatic Proof and Disproof in Isabelle/HOL Jasmin Blanchette, Lukas Bulwahn, Tobias Nipkow Fakultät für Informatik TU München 1 Introduction 2 Isabelle s Standard Proof Methods 3 Sledgehammer 4 Quickcheck:
More informationEmbedding logics in Dedukti
1 INRIA, 2 Ecole Polytechnique, 3 ENSIIE/Cedric Embedding logics in Dedukti Ali Assaf 12, Guillaume Burel 3 April 12, 2013 Ali Assaf, Guillaume Burel: Embedding logics in Dedukti, 1 Outline Introduction
More informationChapter 3: Propositional Languages
Chapter 3: Propositional Languages We define here a general notion of a propositional language. We show how to obtain, as specific cases, various languages for propositional classical logic and some non-classical
More informationProgramming and Proving in Isabelle/HOL
Programming and Proving in Isabelle/HOL Tobias Nipkow Fakultät für Informatik Technische Universität München 2013 MOD Summer School 1 Notation Implication associates to the right: A = B = C means A = (B
More informationPart II. Hoare Logic and Program Verification. Why specify programs? Specification and Verification. Code Verification. Why verify programs?
Part II. Hoare Logic and Program Verification Part II. Hoare Logic and Program Verification Dilian Gurov Props: Models: Specs: Method: Tool: safety of data manipulation source code logic assertions Hoare
More informationLogik für Informatiker Logic for computer scientists
Logik für Informatiker for computer scientists WiSe 2011/12 Overview Motivation Why is logic needed in computer science? The LPL book and software Scheinkriterien Why is logic needed in computer science?
More informationSoftware Verification : Introduction
Software Verification : Introduction Ranjit Jhala, UC San Diego April 4, 2013 What is Algorithmic Verification? Algorithms, Techniques and Tools to ensure that Programs Don t Have Bugs (What does that
More informationAdam Chlipala University of California, Berkeley ICFP 2006
Modular Development of Certified Program Verifiers with a Proof Assistant Adam Chlipala University of California, Berkeley ICFP 2006 1 Who Watches the Watcher? Program Verifier Might want to ensure: Memory
More informationIsabelle Tutorial: System, HOL and Proofs
Isabelle Tutorial: System, HOL and Proofs Burkhart Wolff Université Paris-Sud What we will talk about What we will talk about Isabelle with: Brief Revision Advanced Automated Proof Techniques Structured
More informationDeductive Program Verification with Why3, Past and Future
Deductive Program Verification with Why3, Past and Future Claude Marché ProofInUse Kick-Off Day February 2nd, 2015 A bit of history 1999: Jean-Christophe Filliâtre s PhD Thesis Proof of imperative programs,
More informationNegations in Refinement Type Systems
Negations in Refinement Type Systems T. Tsukada (U. Tokyo) 14th March 2016 Shonan, JAPAN This Talk About refinement intersection type systems that refute judgements of other type systems. Background Refinement
More informationTyped Lambda Calculus
Department of Linguistics Ohio State University Sept. 8, 2016 The Two Sides of A typed lambda calculus (TLC) can be viewed in two complementary ways: model-theoretically, as a system of notation for functions
More informationThe Isabelle/Isar Reference Manual
= Isabelle λ β Isar α The Isabelle/Isar Reference Manual Makarius Wenzel With Contributions by Clemens Ballarin, Stefan Berghofer, Jasmin Blanchette, Timothy Bourke, Lukas Bulwahn, Amine Chaieb, Lucas
More informationFormally Certified Satisfiability Solving
SAT/SMT Proof Checking Verifying SAT Solver Code Future Work Computer Science, The University of Iowa, USA April 23, 2012 Seoul National University SAT/SMT Proof Checking Verifying SAT Solver Code Future
More informationThe Isabelle/HOL type-class hierarchy
= Isabelle λ β Isar α The Isabelle/HOL type-class hierarchy Florian Haftmann 8 October 2017 Abstract This primer introduces corner stones of the Isabelle/HOL type-class hierarchy and gives some insights
More informationc constructor P, Q terms used as propositions G, H hypotheses scope identifier for a notation scope M, module identifiers t, u arbitrary terms
Coq quick reference Meta variables Usage Meta variables Usage c constructor P, Q terms used as propositions db identifier for a hint database s string G, H hypotheses scope identifier for a notation scope
More informationCoq quick reference. Category Example Description. Inductive type with instances defined by constructors, including y of type Y. Inductive X : Univ :=
Coq quick reference Category Example Description Meta variables Usage Meta variables Usage c constructor P, Q terms used as propositions db identifier for a hint database s string G, H hypotheses scope
More informationDynamic Logic David Harel, The Weizmann Institute Dexter Kozen, Cornell University Jerzy Tiuryn, University of Warsaw The MIT Press, Cambridge, Massac
Dynamic Logic David Harel, The Weizmann Institute Dexter Kozen, Cornell University Jerzy Tiuryn, University of Warsaw The MIT Press, Cambridge, Massachusetts, 2000 Among the many approaches to formal reasoning
More informationAppendix G: Some questions concerning the representation of theorems
Appendix G: Some questions concerning the representation of theorems Specific discussion points 1. What should the meta-structure to represent mathematics, in which theorems naturally fall, be? There obviously
More informationVerification of Selection and Heap Sort Using Locales
Verification of Selection and Heap Sort Using Locales Danijela Petrović September 19, 2015 Abstract Stepwise program refinement techniques can be used to simplify program verification. Programs are better
More informationPropositional Calculus. Math Foundations of Computer Science
Propositional Calculus Math Foundations of Computer Science Propositional Calculus Objective: To provide students with the concepts and techniques from propositional calculus so that they can use it to
More informationCoq projects for type theory 2018
Coq projects for type theory 2018 Herman Geuvers, James McKinna, Freek Wiedijk February 6, 2018 Here are five projects for the type theory course to choose from. Each student has to choose one of these
More informationAn Industrially Useful Prover
An Industrially Useful Prover J Strother Moore Department of Computer Science University of Texas at Austin July, 2017 1 Recap Yesterday s Talk: ACL2 is used routinely in the microprocessor industry to
More informationGoals: Define the syntax of a simple imperative language Define a semantics using natural deduction 1
Natural Semantics Goals: Define the syntax of a simple imperative language Define a semantics using natural deduction 1 1 Natural deduction is an instance of first-order logic; that is, it is the formal
More informationThe Proposition. Roger Bishop Jones. Abstract. A place to play formally with the concept of proposition. Created 2009/11/22
The Proposition Roger Bishop Jones Abstract A place to play formally with the concept of proposition. Created 2009/11/22 Last Change Date: 2010-08-05 12:04:21 http://www.rbjones.com/rbjpub/pp/doc/t033.pdf
More informationLecture Notes on Static and Dynamic Semantics
Lecture Notes on Static and Dynamic Semantics 15-312: Foundations of Programming Languages Frank Pfenning Lecture 4 September 9, 2004 In this lecture we illustrate the basic concepts underlying the static
More informationProving Properties on Programs From the Coq Tutorial at ITP 2015
Proving Properties on Programs From the Coq Tutorial at ITP 2015 Reynald Affeldt August 29, 2015 Hoare logic is a proof system to verify imperative programs. It consists of a language of Hoare triples
More informationFrom Types to Sets in Isabelle/HOL
From Types to Sets in Isabelle/HOL Extented Abstract Ondřej Kunčar 1 and Andrei Popescu 1,2 1 Fakultät für Informatik, Technische Universität München, Germany 2 Institute of Mathematics Simion Stoilow
More informationLectures 20, 21: Axiomatic Semantics
Lectures 20, 21: Axiomatic Semantics Polyvios Pratikakis Computer Science Department, University of Crete Type Systems and Static Analysis Based on slides by George Necula Pratikakis (CSD) Axiomatic Semantics
More informationProgram Design in PVS. Eindhoven University of Technology. Abstract. Hoare triples (precondition, program, postcondition) have
Program Design in PVS Jozef Hooman Dept. of Computing Science Eindhoven University of Technology P.O. Box 513, 5600 MB Eindhoven, The Netherlands e-mail: wsinjh@win.tue.nl Abstract. Hoare triples (precondition,
More informationNatural deduction environment for Matita
Natural deduction environment for Matita Claudio Sacerdoti Coen and Enrico Tassi Department of Computer Science, University of Bologna Mura Anteo Zamboni, 7 40127 Bologna, ITALY {sacerdot,tassi}@cs.unibo.it
More informationLOGIC AND DISCRETE MATHEMATICS
LOGIC AND DISCRETE MATHEMATICS A Computer Science Perspective WINFRIED KARL GRASSMANN Department of Computer Science University of Saskatchewan JEAN-PAUL TREMBLAY Department of Computer Science University
More informationCSE505, Fall 2012, Midterm Examination October 30, 2012
CSE505, Fall 2012, Midterm Examination October 30, 2012 Rules: The exam is closed-book, closed-notes, except for one side of one 8.5x11in piece of paper. Please stop promptly at Noon. You can rip apart
More informationStatic and User-Extensible Proof Checking. Antonis Stampoulis Zhong Shao Yale University POPL 2012
Static and User-Extensible Proof Checking Antonis Stampoulis Zhong Shao Yale University POPL 2012 Proof assistants are becoming popular in our community CompCert [Leroy et al.] sel4 [Klein et al.] Four-color
More informationHIGHER-ORDER ABSTRACT SYNTAX IN TYPE THEORY
HIGHER-ORDER ABSTRACT SYNTAX IN TYPE THEORY VENANZIO CAPRETTA AND AMY P. FELTY Abstract. We develop a general tool to formalize and reason about languages expressed using higher-order abstract syntax in
More informationReasoning About Programs Panagiotis Manolios
Reasoning About Programs Panagiotis Manolios Northeastern University February 26, 2017 Version: 100 Copyright c 2017 by Panagiotis Manolios All rights reserved. We hereby grant permission for this publication
More informationComputer-supported Modeling and Reasoning. First-Order Logic. 1 More on Isabelle. 1.1 Isabelle System Architecture
Dipl-Inf Achim D Brucker Dr Burkhart Wolff Computer-supported Modeling and easoning http://wwwinfsecethzch/ education/permanent/csmr/ (rev 16814) Submission date: First-Order Logic In this lecture you
More informationCS 242. Fundamentals. Reading: See last slide
CS 242 Fundamentals Reading: See last slide Syntax and Semantics of Programs Syntax The symbols used to write a program Semantics The actions that occur when a program is executed Programming language
More informationFrom natural numbers to the lambda calculus
From natural numbers to the lambda calculus Benedikt Ahrens joint work with Ralph Matthes and Anders Mörtberg Outline 1 About UniMath 2 Signatures and associated syntax Outline 1 About UniMath 2 Signatures
More informationAssistant for Language Theory. SASyLF: An Educational Proof. Corporation. Microsoft. Key Shin. Workshop on Mechanizing Metatheory
SASyLF: An Educational Proof Assistant for Language Theory Jonathan Aldrich Robert J. Simmons Key Shin School of Computer Science Carnegie Mellon University Microsoft Corporation Workshop on Mechanizing
More informationComputer-Aided Program Design
Computer-Aided Program Design Spring 2015, Rice University Unit 1 Swarat Chaudhuri January 22, 2015 Reasoning about programs A program is a mathematical object with rigorous meaning. It should be possible
More informationThe design of a programming language for provably correct programs: success and failure
The design of a programming language for provably correct programs: success and failure Don Sannella Laboratory for Foundations of Computer Science School of Informatics, University of Edinburgh http://homepages.inf.ed.ac.uk/dts
More informationReview. CS152: Programming Languages. Lecture 11 STLC Extensions and Related Topics. Let bindings (CBV) Adding Stuff. Booleans and Conditionals
Review CS152: Programming Languages Lecture 11 STLC Extensions and Related Topics e ::= λx. e x ee c v ::= λx. e c (λx. e) v e[v/x] e 1 e 2 e 1 e 2 τ ::= int τ τ Γ ::= Γ,x : τ e 2 e 2 ve 2 ve 2 e[e /x]:
More informationContext aware Calculation and Deduction
Context aware Calculation and Deduction Ring Equalities via Gröbner Bases in Isabelle Amine Chaieb and Makarius Wenzel Technische Universität München Institut für Informatik, Boltzmannstraße 3, 85748 Garching,
More informationFormal semantics of loosely typed languages. Joep Verkoelen Vincent Driessen
Formal semantics of loosely typed languages Joep Verkoelen Vincent Driessen June, 2004 ii Contents 1 Introduction 3 2 Syntax 5 2.1 Formalities.............................. 5 2.2 Example language LooselyWhile.................
More informationFirst-Class Type Classes
First-Class Type Classes Matthieu Sozeau Joint work with Nicolas Oury LRI, Univ. Paris-Sud - Démons Team & INRIA Saclay - ProVal Project Gallium Seminar November 3rd 2008 INRIA Rocquencourt Solutions for
More informationIntroduction to Axiomatic Semantics
Introduction to Axiomatic Semantics Meeting 10, CSCI 5535, Spring 2009 Announcements Homework 3 due tonight Homework 2 is graded 13 (mean), 14 (median), out of 21 total, but Graduate class: final project
More informationCSCI.6962/4962 Software Verification Fundamental Proof Methods in Computer Science (Arkoudas and Musser) Chapter p. 1/27
CSCI.6962/4962 Software Verification Fundamental Proof Methods in Computer Science (Arkoudas and Musser) Chapter 2.1-2.7 p. 1/27 CSCI.6962/4962 Software Verification Fundamental Proof Methods in Computer
More informationAutosubst Manual. May 20, 2016
Autosubst Manual May 20, 2016 Formalizing syntactic theories with variable binders is not easy. We present Autosubst, a library for the Coq proof assistant to automate this process. Given an inductive
More informationIsabelle s meta-logic. p.1
Isabelle s meta-logic p.1 Basic constructs Implication = (==>) For separating premises and conclusion of theorems p.2 Basic constructs Implication = (==>) For separating premises and conclusion of theorems
More informationCombining Programming with Theorem Proving
Combining Programming with Theorem Proving Chiyan Chen and Hongwei Xi Boston University Programming with Theorem Proving p.1/27 Motivation for the Research To support advanced type systems for practical
More informationCSCI.6962/4962 Software Verification Fundamental Proof Methods in Computer Science (Arkoudas and Musser) Sections p.
CSCI.6962/4962 Software Verification Fundamental Proof Methods in Computer Science (Arkoudas and Musser) Sections 10.1-10.3 p. 1/106 CSCI.6962/4962 Software Verification Fundamental Proof Methods in Computer
More informationInteractive Theorem Proving (ITP) Course Parts VII, VIII
Interactive Theorem Proving (ITP) Course Parts VII, VIII Thomas Tuerk (tuerk@kth.se) KTH Academic Year 2016/17, Period 4 version 5056611 of Wed May 3 09:55:18 2017 65 / 123 Part VII Backward Proofs 66
More information