Formalising mathematics in Lean and Mathlib

Se basa en

Requisitos previos obligatorios, hasta el fondo: 16

Aprenderás a

• 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.

Definición

• 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.

Teorema

• 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.

Errores frecuentes

• 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.

Conduce a

Descubrimientos opcionales

• Type theory and propositions as types The type theory beneath Lean explains how types, universes and definitional equality behave.
• ← Proof: direct, contrapositive, contradiction and induction
• ← Real analysis

Itinerarios de aprendizaje

• Computer science and formal methods

Lecturas complementarias

• Jeremy Avigad and Patrick Massot, Mathematics in Lean
• Jeremy Avigad, Leonardo de Moura, Soonho Kong and Sebastian Ullrich, Theorem Proving in Lean 4