Machine-Assisted Theorem Proving

Machine-assisted theorem proving represents a revolution in mathematical rigor, using proof assistants like Lean, Coq, and Isabelle/HOL to verify complex mathematical proofs and formalize entire theories with absolute certainty.

Interactive Lean 4 Proof Assistant

Try Proving: √2 is Irrational

import Mathlib.Data.Real.Irrational theorem sqrt_two_irrational : Irrational (Real.sqrt 2) := by -- Your proof here sorry
Current Goal:
⊢ Irrational (Real.sqrt 2)
Ready to check your proof...

Proof Tree Visualization

Major Proof Assistants

System Foundation Language Key Features Major Projects
Lean 4 Dependent Type Theory Lean Modern, fast, great automation Liquid Tensor Experiment, Mathlib
Coq Calculus of Constructions Gallina Mature, extraction to code Four Color Theorem, CompCert
Isabelle/HOL Higher-Order Logic Isar Readable proofs, sledgehammer seL4, Archive of Formal Proofs
Agda Dependent Types Agda Programming focus, Unicode HoTT Book, Cubical Agda

The State of Formalized Mathematics

100k+
Theorems in Mathlib
3,000+
Contributors
500+
Formalized Papers
15+
Active Systems

🏆 The Liquid Tensor Experiment

Peter Scholze challenged the formalization community to verify his proof of a key theorem about liquid vector spaces. The Lean community completed this monumental task in 2022, marking a watershed moment for formalized mathematics.

Explore the Project

Common Proof Tactics

intro
apply
exact
rw
simp
ring
linarith
norm_num

Example: Proving Commutativity

theorem add_comm (a b : ℕ) : a + b = b + a := by induction a with | zero => rw [Nat.zero_add, Nat.add_zero] | succ n ih => rw [Nat.succ_add, Nat.add_succ, ih]

Example: Using Classical Logic

open Classical theorem not_not (p : Prop) : ¬¬p → p := by intro h by_cases hp : p · exact hp · contradiction

Type Theory Foundations

Landmark Formalizations

A Machine-Checked Proof of the Four Color Theorem
Georges Gonthier (2008)
AMS Notices

First major theorem formalized in Coq, revolutionizing computer-assisted proofs.

The Lean Mathematical Library
The mathlib Community (2020)
arXiv:1910.09336

Describes the largest coherent library of formalized mathematics.

Homotopy Type Theory: Univalent Foundations
Univalent Foundations Program (2013)
HoTT Book

Revolutionary connection between type theory, homotopy theory, and foundations.

Formalising Perfectoid Spaces
Kevin Buzzard, et al. (2022)
arXiv:2102.06572

Formalization of Scholze's perfectoid spaces in Lean.

Getting Started with Proof Assistants

Your First Lean Proof

  1. Install Lean 4: curl -sSf https://raw.githubusercontent.com/leanprover/elan/master/elan-init.sh | sh
  2. Install VS Code with Lean 4 extension
  3. Create a new file with .lean extension
  4. Try this simple proof:
example (p q : Prop) (hp : p) (hq : q) : p ∧ q := by constructor · exact hp · exact hq

Leading Groups & Initiatives

  • Microsoft Research - Lean development team
  • Carnegie Mellon University - HoTT and formal verification
  • Imperial College London - Kevin Buzzard's formalization group
  • INRIA - Coq development
  • TU Munich - Isabelle development
  • Formal Abstracts Project - Thomas Hales