本科低年级Computation计划中

Formalising mathematics in Lean and Mathlib

Formalising undergraduate mathematics in Lean 4 with Mathlib: structures and type classes, sets and functions, quantifier reasoning, algebraic structures, and searching and contributing to a large library.

math.comp.formal-proof-lean

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前置知识

First steps in Lean

16 项必需的前置知识（一直追溯到底）

你将学会

Formalise definitions and proofs about sets, functions and quantifiers, such as the fact that a composite of injections is injective.
Use Mathlib's algebraic hierarchy to state and prove results about groups and rings.
Search Mathlib for existing results and explain the trade-off between using library lemmas and proving from scratch.

定义

Mathlib is the community-maintained mathematical library for Lean 4.
A type class packages structure, such as a group operation, which Lean then finds automatically by instance search.
A structure in Lean is a type whose elements bundle several fields, such as the carrier operations and laws of a monoid.

定理

Mathlib's Function.Injective.comp states that a composite of injective functions is injective.
Lean's #print axioms command lists the axioms on which a proof depends.
A formal library is useful only if its definitions match standard mathematical usage, which people must check.

常见误解

Thinking formalisation is mere translation; it often exposes hidden assumptions and edge cases.
Believing Lean's convention x / 0 = 0 changes mathematics; it is a definitional convenience, and theorems carry hypotheses such as x ≠ 0 where needed.