← problemsAN010openMathematics/ Analysis

Schanuel's conjecture

A transcendence statement that implies almost everything else in the subject.

▸Motivation

Stephen Schanuel, 1960s, communicated by Lang. Given n complex numbers linearly independent over ℚ, the field they generate together with their exponentials should have transcendence degree at least n.

It implies the algebraic independence of e and π, the Lindemann–Weierstrass theorem, the four exponentials conjecture, and much of transcendence theory at once. That is the reason to doubt a proof is near: it is too strong to fall to any single technique.

Stephen Schanuel, 1960s; recorded by Serge Lang.

▸Lean API

import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.RingTheory.Algebraic.Defs

namespace Conjectura.AN010

/-- Given complex numbers linearly independent over ℚ, the field generated by them
together with their exponentials has transcendence degree at least `n`. It implies
almost every known and conjectured transcendence result at once, including the
algebraic independence of `e` and `π`. -/
def goal : Prop :=
  ∀ (n : ℕ) (z : Fin n → ℂ),
    LinearIndependent ℚ z →
      ¬ ∃ (m : ℕ), m < n ∧ ∃ B : Fin m → ℂ,
        ∀ i : Fin n, IsAlgebraic (Algebra.adjoin ℚ (Set.range B)) (z i) ∧
                     IsAlgebraic (Algebra.adjoin ℚ (Set.range B)) (Complex.exp (z i))

end Conjectura.AN010

▸Definition5

LinearIndependentLinearIndependent: def
IsAlgebraicIsAlgebraic: abbrev
adjoinAlgebra.adjoin: def
rangeSet.range: def
expComplex.exp: def

Download the Lean definitions

▸Related work0

▸For your AI

Download as .md

I am proving a theorem in Lean 4 and submitting it to Conjectura.

## Problem AN010 — Schanuel's conjecture

A transcendence statement that implies almost everything else in the subject.

## Environment (fixed — do not assume anything newer)

- Lean toolchain: `leanprover/lean4:v4.33.0-rc1`
- Mathlib: `v4.33.0-rc1`

If a lemma you want does not exist in that Mathlib, prove it inline instead of
importing something newer.

## The exact statement I must prove

```lean
theorem solution : ∀ (n : ℕ) (z : Fin n → ℂ), LinearIndependent ℚ z → ... := by
  sorry
```

## The file I submit

```lean
import Conjectura.Problems.AN010.Statement

namespace Submission

theorem solution : Conjectura.AN010.goal := by
  sorry

end Submission

```

## The Lean definitions of every term in this problem

These are the actual definitions your proof will be checked against. Do not
substitute your own version of any of them.

### LinearIndependent

def

```lean
LinearIndependent
```
Defined in Mathlib.

### IsAlgebraic

abbrev

```lean
IsAlgebraic
```
Defined in Mathlib.

### adjoin

def

```lean
Algebra.adjoin
```
Defined in Mathlib.

### range

def

```lean
Set.range
```
Defined in Mathlib.

### exp

def

```lean
Complex.exp
```
Defined in Mathlib.

## Rules — submissions violating these are rejected automatically

1. **Do not change the name or type of `solution`.** It must satisfy the
   statement above exactly.
2. **Do not redefine or shadow anything from the problem's Statement module.**
   Declaring your own `goal`, or redefining a name it depends on, produces a
   proof of a *different* statement and is rejected. This is the single most
   common rejection.
3. **No `sorry`** anywhere, including in helper lemmas. It surfaces as the
   axiom `sorryAx` and is detected transitively through imports.
4. **No `native_decide`** — it surfaces as `Lean.ofReduceBool` and is not
   accepted, because it trusts compiled code rather than the kernel.
5. Only these axioms are permitted: `propext`, `Classical.choice`,
   `Quot.sound`.
6. Follow Mathlib style: hypotheses left of the colon, explicit types,
   `snake_case` theorem names, `UpperCamelCase` types.

## What I want from you

Here is my argument in informal mathematics:

> [PASTE YOUR PROOF SKETCH HERE]

Turn it into Lean 4 that compiles under the environment above and satisfies the
statement exactly. Where you are unsure a lemma exists in this Mathlib version,
say so explicitly rather than guessing a name.

Submissions are not open yet

Conjectura is in beta. You can read every statement, every definition and the Lean behind them, and download the exact files the checker uses — but proofs are not being accepted yet.

The reason is a deliberate order of operations. Accepting a proof means running a stranger’s code and standing behind a verdict, and no statement here yet carries a researcher’s name. A machine-checked answer to a question nobody has vouched for is worth very little, so the vouching comes first.

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The English write-ups are also switched off during the beta. Nothing on this page is generated by a model.

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