# Theorem Proving in Lean Release 3.23.0 Jeremy Avigad, Leonardo de Moura, and Soonho Kong Apr 25, 2023 Powered by TCPDF (www.tcpdf.org) ## CONTENTS 1 Introduction 1.1 Computers and Theorem Proving the Excluded Middle ..... 163 Bibliography ..... 167 ## I NTRODUCTION ### 1.1 Computers and Theorem Proving Formal verification involves the use of logical and computational methods to establish claims correctness becomes a form of theorem proving. Conversely, the proof of a mathematical theorem may require a lengthy computation, in which case verifying the truth of the theorem requires verifying that the

Game Nim 4 3.1 Rules and How to Play It 4 3.2 Nimrod 5 3.3 Impartial Games 5 3.4 The Sprague-Grundy Theorem 6 4 Machine Learning 6 4.1 Reinforcement learning 7 4.1.1 The Principle 7 4.1.2 The It is also a pretty simple game, and it has a complete mathematical theorem, see "Nim, a game with a complete mathematical theorem". $ ^{1} $ There is a fairly simple algorithm for making winning deck is a type of random element, if done correctly $ ^{6} $ . ### 3.4 The Sprague-Grundy Theorem The Sprague-Grundy Theorem states that all impartial games are equivalent with a number. A number is defined

for the project. ### 1.4 Using the Package Manager leanpkg is the package manager for the Lean theorem prover. It downloads dependencies and manages what modules you can import in your Lean files. This or structure declaration. Similarly, objects can be defined in various ways, such as using def, theorem, or the equation compiler. See Chapter 4 for more information. Writing an expression $ (t : \alpha) Implicit Arguments When declaring arguments to defined objects in Lean (for example, with def, theorem, constant, inductive, or structure; see Chapter 4) or when declaring variables and parameters in

The Idris Community May 18, 2016 Powered by TCPDF (www.tcpdf.org) Contents 2 The Interactive Theorem Prover 1 Type Providers in Idris Tutorials submitted by community members. Note: The documentation DevTBitWidth ## The Interactive Theorem Prover This short guide contributed by a community member illustrates how to prove associativity of addition on Nat using the interactive theorem prover. First we define

assistants, or interactive theorem provers, but a mischievous student coined the phase "proof-preventing beasts," and dictation software occasionally misunderstands "theorem prover" as "fear "fear improver." Consider yourself warned. Rigorous and Formal Proofs Interactive theorem proving has its own terminology, already starting with the notion of “proof.” A formal proof is a logical argument formalization of mathematics have been the proof of the four-color theorem by Gonthier et al. $ [8] $ , the proof of the odd-order theorem by Gonthier et al. [9], and the proof of the Kepler conjecture by

The Idris Community July 26, 2016 Powered by TCPDF (www.tcpdf.org) Contents 2 The Interactive Theorem Prover 1 Type Providers in Idris Tutorials submitted by community members. Note: The documentation DevTBitWidth ## The Interactive Theorem Prover This short guide contributed by a community member illustrates how to prove associativity of addition on Nat using the interactive theorem prover. First we define

The Idris Community July 26, 2016 Powered by TCPDF (www.tcpdf.org) Contents 2 The Interactive Theorem Prover 1 Type Providers in Idris Tutorials submitted by community members. Note: The documentation DevTBitWidth ## The Interactive Theorem Prover This short guide contributed by a community member illustrates how to prove associativity of addition on Nat using the interactive theorem prover. First we define

loop repeat a sequence of events that happened before the loop, the loop must end. void counting_theorem( const int b, const int e ) { extend_stability b, e; claim b <= e; claim implementation; while ( i != e ) { claim i <= e; ++i; } } if ( b <= e ) { counting_theorem( b, e ); int i = b; while ( i != e ) ++i; } { auto i = b; while ( i != e } } void counting_theorem( const int b, const int e ) interface { extend_stability b, e; } if ( b <= e ) { counting_theorem( b, e ); int i = b;

Types 32 3.4 Monads 34 3.5 Input and Output 35 3.6 An Example: Abstract Syntax 36 4 Theorem Proving in Lean 38 4.1 Assertions in Dependent Type Theory 38 4.2 Propositions as Types programs can be written in Lean and run by the bytecode interpreter. In fact, a full-blown resolution theorem prover for Lean has been written in Lean itself. You can profile your code by setting the relevant expressed, and any theorem that can be proved using conventional mathematical means can be carried out formally, with enough effort. Here is a proof that the sum of two even numbers is even: theorem even_add :

Classical Propositional Logic 43 # INTRODUCTION This tutorial can be viewed as a companion to Theorem Proving in Lean, which presents Lean as a system for building mathematical libraries and stating ” or ff, for “false.” This provides another perspective on Lean: instead of thinking of it as a theorem prover whose language just happens to have a computational interpretation, think of it as a programming tour of some of the terms we can write in Lean. For a more detailed and exhaustive account, see Theorem Proving in Lean. ### 2.1 Some Basic Types In Lean: • #check can be used a check the type of an