Sendov.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 10 curated AI-written annotations and 7190 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

5 declarations, 3 contain sorry, 0 declared as axioms

Sendov's conjecture

References:

Wikipedia
Lech Mazur, A Computer-Assisted Proof of Sendov's Conjecture, ProofAtlas, August 5, 2026.
Terence Tao, A digestion of the proof of Sendov's conjecture.
Tao's Lean formalization of the digested proof.

Tags: Sendov Conjecture, Ilieff's Conjecture.

Imports (1)

FormalConjecturesUtil

def Sendov.Polynomial.IsSendov The predicate that a polynomial satisfies the hypotheses of Sendov's conjecture line 46 source
Source documentation
The predicate that a polynomial satisfies the hypotheses of Sendov's conjecture.
f.IsSendov holds if f has degree at least 2 and all roots of f lie in the unit disc of the complex plane.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Polynomial, Complex
Proof uses: And, LE.le, Nat, OfNat.ofNat, Polynomial.natDegree, Set, Polynomial.rootSet, Metric.closedBall, Real
18 instances, coercions and auxiliary constants left out.
```
def Polynomial.IsSendov (f : ℂ[X]) : Prop :=
…
```
def Sendov.Nat.SatisfiesSendovConjecture SatisfiesSendovConjecture n states that Sendov's conjecture is true for every polynomial of degree n line 51 source
Source documentation
SatisfiesSendovConjecture n states that Sendov's conjecture is true for every polynomial of degree n.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Polynomial, Complex, Sendov.Polynomial.IsSendov, Eq, Polynomial.natDegree, Membership.mem, Set, Polynomial.rootSet, LE.le, Real, Metric.infDist, LinearMap, RingHom.id, Polynomial.derivative, OfNat.ofNat
21 instances, coercions and auxiliary constants left out.
```
def Nat.SatisfiesSendovConjecture (n : ℕ) : Prop :=
…
```
theorem Sendov.sendov_conjecture sorry line 62 source
depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Sendov.Nat.SatisfiesSendovConjecture
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Sendov.sendov_conjecture.variants.le_eight Sendov's conjecture states that for a polynomial sorry line 73 source
Source documentation
Sendov's conjecture states that for a polynomial $$f(z)=(z-r_{1})\cdots (z-r_{n}),\qquad (n\geq 2)$$ with all roots $r_1, ..., r_n$ inside the closed unit disk $|z| ≤ 1$, each of the $n$ roots is at a distance no more than $1$ from at least one critical point.
It has been shown that Sendov's conjecture holds when the degree $n$ is at most $8$.

depends on sorry what this means
Statement uses: Nat, Membership.mem, Set, Set.Icc, OfNat.ofNat, Sendov.Nat.SatisfiesSendovConjecture
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
6 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem sendov_conjecture.variants.le_eight (n : ℕ) (hn : n ∈ Set.Icc 2 8) :
    n.SatisfiesSendovConjecture := by
…
```
theorem Sendov.sendov_conjecture.variants.eventually_true Sendov's conjecture states that for a polynomial sorry line 85 source
Source documentation
Sendov's conjecture states that for a polynomial $$f(z)=(z-r_{1})\cdots (z-r_{n}),\qquad (n\geq 2)$$ with all roots $r_1, ..., r_n$ inside the closed unit disk $|z| ≤ 1$, each of the $n$ roots is at a distance no more than $1$ from at least one critical point.
It has been shown that Sendov's conjecture holds for polynomials of sufficiently large degree.

depends on sorry what this means
Statement uses: Filter.Eventually, Nat, Sendov.Nat.SatisfiesSendovConjecture, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem sendov_conjecture.variants.eventually_true :
    ∀ᶠ (n : ℕ) in Filter.atTop, n.SatisfiesSendovConjecture := by
…
```