Lean 4 Leonardo de Moura - MSR - USA Sebastian Ullrich - KIT - Germany Programming Language http://leanprover.github.io • Goals • Extensibility, Expressivity, Scalability, Proof stability io/theorem_proving_in_lean/ • Zulip channel: https://leanprover.zulipchat.com/ • Community website: https://leanprover-community.github.io/ • Maintainers of the official release (Lean 3) • Mathlib: https://leanprover-community.github.io/mathlib-overview.html • Lean 4 repository: https://github.com/leanprover/lean4 Programming Language Mathlib The Lean mathematical library, mathlib, is a community-driven

Lean in Lean Leonardo de Moura - MSR - USA Workshop Programming Language http://leanprover.github.io Lean • Goals • Extensibility, Expressivity, Scalability, Proof stability • Functional Programming Language Lean Timeline • Lean 1 (2013) Leo and Soonho Kong • Almost useless • Brave (crazy?) users in 2014: Jeremy Avigad, Cody Roux and Floris van Doorn • Lean 2 (2015) Leo and Soonho the HoTT library for Lean • First Math library (Jeremy Avigad, Rob Lewis, and many others) • Lean 3 (2016) Leo, Daniel Selsam, Gabriel Ebner, Jared Roesch, Sebastian Ullrich • Lean is now a programming

An Introduction to Lean Jeremy Avigad Leonardo de Moura Gabriel Ebner and Sebastian Ullrich Version 1fc176a, updated at 2017-01-09 14:16:26 -0500 2 Contents Contents 3 1 Overview 5 1.1 Perspectives Perspectives on Lean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Where To Go From Here . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2 Defining Objects in Lean 13 2.1 Nonconstructive Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3 Programming in Lean 27 3.1 Evaluating Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.2

Programming in Lean Release 3.4.2 Jeremy Avigad and Simon Hudon Apr 02, 2019 CONTENTS 1 Introduction 1 2 Types and Terms 3 2.1 Some Basic Types . . . . . . . . . . . . . . . . . . . . . . . . . and Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4 Monads 23 4.1 The option monad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theorem Proving in Lean, which presents Lean as a system for building mathematical libraries and stating and proving mathematical theorems. From that perspective, the point of Lean is to implement a formal

Lean 2 Quick Reference Jeremy Avigad, Leonardo de Moura, Soonho Kong Version d0dd6d0, updated at 2017-01-30 19:53:44 -0500 Quick Reference Note that this quick reference guide describes Lean 2 only intros : same as intro intro : let Lean choose a name intros : introduce variables as long as the goal reduces to a function type and let Lean choose the names rename : rename a variable can combine rewrite rules from different groups in the following order, starting with the innermost: 4 e : match left-hand-side of equation e to a goal subterm, then replace every occurence with right-hand-side

The Lean Reference Manual Release 3.3.0 Jeremy Avigad, Gabriel Ebner, and Sebastian Ullrich Sep 06, 2018 CONTENTS 1 Using Lean 1 1.1 Using Lean Online . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Using Lean with VSCode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.3 Using Lean with Emacs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4 Declarations 21 4.1 Declaration Names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Theorem Proving in Lean Release 3.23.0 Jeremy Avigad, Leonardo de Moura, and Soonho Kong Apr 25, 2023 CONTENTS 1 Introduction 1 1.1 Computers and Theorem Proving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 About Lean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 About this Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4 Quantifiers and Equality 37 4.1 The Universal Quantifier . . . . . . . . . . . . . . . . . . . . .

turn(‘page’, 4); API DOCUMENTATION turn.js 4th release www.turnjs.com ©2012 Emmanuel Garcia – All rights reserved The turn.js API was conveniently built as a jQuery plugin, it provides iPod) • Safari 5 • Chrome 11 • Chrome for Android • Firefox 9 • IE 9 • IE 8 Using turn.html4.js 4th Release - Commercial Version + Added option autoCenter + Added option zoom + Added property own-size + Added class .sheet + Added class .sheet + Added the ignore attribute + New turn.html4.js + New scissors.js + Changed the class .turn-page to .page + Improved the animation frame generator

TIME_START SECOND MINUTE HOUR DAY WEEK MONTH YEAR Chronos Debug Kit Migrations ElasticSearch Appendices 4.x Migration Guide 4.0 Migration Guide 4.1 Migration Guide 4.2 Migration Guide 4.3 Migration Guide Application is loaded and bound to an HttpServer. 3. Your application’s middleware is initialized. 4. A request and response is dispatched through the PSR-7 Middleware that your application uses. Typically File and Class Name Conventions In general, filenames match the class names, and follow the PSR-4 standard for autoloading. The following are some examples of class names and their filenames: The Controller

CakePHP Cookbook Documentation Release 4.x Cake Software Foundation Dec 07, 2023 Contents 1 CakePHP at a Glance 1 Conventions Over Configuration . . . . . . . . . . . . . . . . . . . . . . . . . Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4 Tutorials & Examples 69 Content Management Tutorial . . . . . . . . . . . . . . . . . . . . . . . 930 42 Chronos 931 43 Debug Kit 933 44 Migrations 935 45 ElasticSearch 937 46 Appendices 939 4.x Migration Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .