Lean and physics

Lean theorem prover: map
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Related The move to Lean

A proof checker can do one thing for physics: check that a conclusion follows from stated assumptions. It cannot check that the assumptions describe the world. This page says what that division means in practice, how real numbers, units and approximations enter a formal statement, and what has been done in Lean on physics so far. It closes with this wiki's own physics.

A derivation, as Lean sees it

A derivation in physics starts from laws and assumptions and arrives at a result. In Lean the laws and assumptions become hypotheses, the result becomes the statement, and the derivation becomes the proof. Here a body is assumed to move at a constant speed v, and the theorem says how far it moves between two times:

theorem displacement (x : Int → Int) (x₀ v : Int)
    (law : ∀ t, x t = x₀ + v * t) (t₁ t₂ : Int) :
    x t₂ - x t₁ = v * (t₂ - t₁) := by
  rw [law t₂, law t₁]
  grind

Lean accepts it. What it has checked is a conditional: if the position x follows the law named law, then the displacement is speed times elapsed time. It has not checked that any body moves that way. Whether one does is a question for measurement, and no proof can answer it. (The example uses whole numbers so that it runs in Lean without any library; the same statement over the real numbers needs Mathlib.)

The project this wiki records puts the point in one line: Lean proves the model, not the world (see the move to Lean).

What Lean can and cannot check

Lean can check that every step of a derivation is valid, that no case is skipped, and that the assumptions used are exactly the ones stated. An assumption that a written derivation uses without saying so becomes, in Lean, a hypothesis that must be written down, or else the proof fails.

Lean cannot check three things. It cannot check that the assumptions are true of nature; that is what experiments are for. It cannot check that the symbols stand for the things they are meant to, that x really is a position measured from a chosen point; that link between the mathematics and the world is made by people. And it cannot check that the statement proved is the one that was wanted (see stating the right thing).

Real numbers

Physics is written over the real numbers, and core Lean has none (see a library of mathematics). Mathlib defines them as classes of Cauchy sequences of rational numbers, sequences whose terms get and stay arbitrarily close to one another. These are the exact real numbers of mathematics, not the rounded numbers a computer calculates with. A theorem about real numbers therefore says nothing by itself about the rounding errors of a numerical simulation. Connecting the two takes a further statement, one that bounds the error.

Units

On paper, units are checked by hand. In Lean they can be made part of the types, so that adding a length to a time is refused before any proof begins. A small version:

structure Metres where
  val : Int

structure Seconds where
  val : Int

instance : Add Metres := ⟨fun a b => ⟨a.val + b.val⟩⟩

def d : Metres := ⟨3⟩
def t : Seconds := ⟨2⟩

#eval (d + d).val

Lean prints 6: two lengths add. A length and a time do not:

#eval (d + t).val

Lean answers: failed to synthesize instance of type class, naming the missing addition of Metres and Seconds. A fuller treatment exists. A 2025 preprint formalised dimensional analysis in Lean: a dimension is a map from base dimensions to their exponents, dimensions form a group under multiplication, physical quantities pair a value with a dimension, and the framework covers SI units, fundamental constants and the Buckingham Pi theorem, with the Lennard-Jones potential as a worked case.

Approximations

Physics leans on approximations: small angles, weak fields, slow speeds. The sign "≈" means nothing to Lean. An approximation has to be stated as an exact claim, usually an inequality with an explicit bound on the error, or a statement about a limit. Writing it out that way shows what the approximation actually assumes: how small the angle must be, and how large the error can then get. That can be harder than the derivation it serves, and it is part of what formal proof adds.

Physics in Lean so far

In 2024 Joseph Tooby-Smith introduced HepLean, a project to formalise high energy physics in Lean 4, with first examples from flavour physics (CKM matrices), local anomaly cancellation and Higgs physics. Its stated aims include making results easier to find, helping new results to be found by artificial intelligence and automated methods, making the mathematical checking of papers easier, and new ways of teaching. HepLean was renamed PhysLean and later merged with Lean-QuantumInfo, a library for quantum information, to form Physlib, an open-source community project under the Apache 2.0 licence that depends on Mathlib. Its core library holds reviewed formalisations; a separate area with lighter review accepts larger and faster contributions, including some generated by AI.

Formalisation has also found a mistake. In 2026 Tooby-Smith formalised a widely cited paper of 2006, by Maniatis, von Manteuffel, Nachtmann and Nagel, on the stability of the two-Higgs-doublet potential, and found an error in its arguments that invalidates its main theorem on stability. He suggests that this may be the first non-trivial error in a physics paper found through formalisation. Physics in Lean describes this work further.

This wiki's physics

The physics editions on this wiki, Hamilton's principle, Noether's theorems and On Physical Lines of Force, are written in ordinary mathematics and prose, not in Lean. The framework this wiki records is moving to Lean (see Principia Machina). Its own account notes that physics over the real numbers needs Mathlib, which it has not yet chosen to adopt (see the move to Lean).

Next

Terms, types and judgments begins the technical pages, on the type theory that Lean checks. Physics in Lean describes the formal physics done in Lean so far.