# 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 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

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

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

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 for a ghci style interface which allows evaluation of, as well as type checking of, expressions; theorem proving, compilation; editing; and various other operations. The command :? gives a list of supported complete the definition of parity. 1.8.3 With and proofs To use a dependent pattern match for theorem proving, it is sometimes necessary to explicitly construct the proof resulting from the pattern match

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 and proving mathematical theorems. From that perspective ” 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 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 system are quite powerful

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 a ghci style interface which allows evaluation of, as well as type checking of, expressions; theorem proving, compilation; editing; and various other operations. The command :? gives a list of supported the definition of parity. #### 1.9.3 With and proofs To use a dependent pattern match for theorem proving, it is sometimes necessary to explicitly construct the proof resulting from the pattern match

Implementing State-aware Systems in Idris: The ST Tutorial 72 4 The Effects Tutorial 107 5 Theorem Proving 141 6 Language Reference 152 7 Tutorials on the Idris Language 220 Note: The documentation a ghci style interface which allows evaluation of, as well as type checking of, expressions; theorem proving, compilation; editing; and various other operations. The command :? gives a list of supported the definition of parity. #### 1.8.3 With and proofs To use a dependent pattern match for theorem proving, it is sometimes necessary to explicitly construct the proof resulting from the pattern match

Implementing State-aware Systems in Idris: The ST Tutorial 73 4 The Effects Tutorial 108 5 Theorem Proving 142 6 Language Reference 153 7 Tutorials on the Idris Language 221 Note: The documentation a ghci style interface which allows evaluation of, as well as type checking of, expressions; theorem proving, compilation; editing; and various other operations. The command :? gives a list of supported the definition of parity. #### 1.8.3 With and proofs To use a dependent pattern match for theorem proving, it is sometimes necessary to explicitly construct the proof resulting from the pattern match

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 a ghci style interface which allows evaluation of, as well as type checking of, expressions; theorem proving, compilation; editing; and various other operations. The command :? gives a list of supported complete the definition of parity. ## With and proofs To use a dependent pattern match for theorem proving, it is sometimes necessary to explicitly construct the proof resulting from the pattern match

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 a ghci style interface which allows evaluation of, as well as type checking of, expressions; theorem proving, compilation; editing; and various other operations. The command :? gives a list of supported complete the definition of parity. ## With and proofs To use a dependent pattern match for theorem proving, it is sometimes necessary to explicitly construct the proof resulting from the pattern match