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 . . . . . . . . . . . . . . . . . . . . .

OpenShift Dedicated 4 入门 OpenShift Dedicated 入门 Last Updated: 2024-02-23 OpenShift Dedicated 4 入门 OpenShift Dedicated 入门 法律通告 法律通告 Copyright © 2024 Red Hat, Inc. The text of and illustrations in Hat, as the licensor of this document, waives the right to enforce, and agrees not to assert, Section 4d of CC-BY-SA to the fullest extent permitted by applicable law. Red Hat, Red Hat Enterprise Linux, 从用户撤销管理员权限 2.9. 撤销对集群的用户访问权限 2.10. 删除集群 2.11. 后续步骤 2.12. 其他资源 3 3 3 4 4 4 4 6 7 7 9 9 10 11 11 11 目 目录 录 1 OpenShift Dedicated 4 入 入门 门 2 第 1 章 了解您的云部署选项 您可以使用您拥有的云帐户或在 Amazon Web Services (AWS)

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

The Julia Language V1.12.0-beta4 The Julia Project June 6, 2025Contents Contents i I Manual 1 1 Julia 1.12-beta4 Documentation 2 1.1 Important Links . . . . . . . . . . . . . . . . . . . . . . 7 3.3 Alternative installation methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4 Variables 10 4.1 Allowed Variable Names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2042Part I Manual 1Chapter 1 Julia 1.12-beta4 Documentation Welcome to the documentation for Julia 1.12-beta4. Work in progress! This documentation is for an unreleased, in-development