# 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

11 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 future releases the version string will differ: / ### 2.1 Explore the Context We start the interactive session by asking Idris to prove the hole $ r_{hs} $ using the command: p $ r_{hs} $ . Idris

12 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 future releases the version string will differ: / ### 2.1 Explore the Context We start the interactive session by asking Idris to prove the hole $ r_{hs} $ using the command: p $ r_{hs} $ . Idris

12 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 future releases the version string will differ: / ### 2.1 Explore the Context We start the interactive session by asking Idris to prove the hole $ r_{hs} $ using the command: p $ r_{hs} $ . Idris

6 Writing Automation 43 6.1 A Tableau Prover for Classical Propositional Logic 43 # INTRODUCTION This tutorial can be viewed as a companion to Theorem Proving in Lean, which presents Lean as a system 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 programming language that just happens to come equipped with a rich specification language and an interactive environment for proving that programs meet their specifications. The specification language and proof

Version 0.99 1 The Idris Tutorial 2 2 Frequently Asked Questions 61 3 The Effects Tutorial 66 4 Theorem Proving 98 5 Language Reference 109 6 Tutorials on the Idris Language 173 Note: The documentation executable called prog. • --check type check the file and its dependencies without starting the interactive environment. --package pkg add package as dependency, e.g. --package contrib to make use of the display usage summary and command line options. #### 1.2.3 The Interactive Environment Entering idris at the shell prompt starts up the interactive environment. You should see something like the following:

66 3 Implementing State-aware Systems in Idris: The ST Tutorial 71 4 The Effects Tutorial 106 5 Theorem Proving 140 6 Language Reference 151 7 Tutorials on the Idris Language 219 Note: The documentation software. For a more in-depth introduction to Idris, which proceeds at a much slower pace, covering interactive program development, with many more examples, see Type-Driven Development with Idris by Edwin Brady executable called prog. • --check type check the file and its dependencies without starting the interactive environment. • --package pkg add package as dependency, e.g. --package contrib to make use of the

3 Implementing State-aware Systems in Idris: The ST Tutorial 67 4 The Effects Tutorial 102 5 Theorem Proving 136 6 Language Reference 147 7 Tutorials on the Idris Language 214 Note: The documentation software. For a more in-depth introduction to Idris, which proceeds at a much slower pace, covering interactive program development, with many more examples, see Type-Driven Development with Idris by Edwin Brady executable called prog. • --check type check the file and its dependencies without starting the interactive environment. --package pkg add package as dependency, e.g. --package contrib to make use of the

3 Implementing State-aware Systems in Idris: The ST Tutorial 67 4 The Effects Tutorial 102 5 Theorem Proving 136 6 Language Reference 147 7 Tutorials on the Idris Language 214 Note: The documentation software. For a more in-depth introduction to Idris, which proceeds at a much slower pace, covering interactive program development, with many more examples, see Type-Driven Development with Idris by Edwin Brady executable called prog. • --check type check the file and its dependencies without starting the interactive environment. --package pkg add package as dependency, e.g. --package contrib to make use of the

3 Implementing State-aware Systems in Idris: The ST Tutorial 67 4 The Effects Tutorial 102 5 Theorem Proving 136 6 Language Reference 147 7 Tutorials on the Idris Language 214 Note: The documentation software. For a more in-depth introduction to Idris, which proceeds at a much slower pace, covering interactive program development, with many more examples, see Type-Driven Development with Idris by Edwin Brady executable called prog. • --check type check the file and its dependencies without starting the interactive environment. --package pkg add package as dependency, e.g. --package contrib to make use of the