193.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

4 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 193

References:

erdosproblems.com/193
[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).
[GeRa79] Gerver, Joseph L. and Ramsey, L. Thomas, "On certain sequences of lattice points." Pacific J. Math. (1979), 357-363.
[CaKa26] Cambie, Stijn and Kalviainen, Erik, "An infinite small-step Z³-walk with no collinear triple." arXiv:2609.01766.

Imports (1)

FormalConjecturesUtil

def Erdos193.IsSWalk An $S$-walk is a sequence where every difference is in $S$ line 40 source
Source documentation
An $S$-walk is a sequence where every difference is in $S$.

no axioms what this means
Statement uses: AddCommGroup, Set, Nat
Proof uses: Membership.mem, HSub.hSub, HAdd.hAdd, OfNat.ofNat
8 instances, coercions and auxiliary constants left out.
```
def IsSWalk {V : Type*} [AddCommGroup V] (S : Set V) (a : ℕ → V) : Prop :=
…
```
def Erdos193.HasCollinearTriple True if set $A$ contains 3 distinct collinear points over $R$ line 44 source
Source documentation
True if set $A$ contains 3 distinct collinear points over $R$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: DivisionRing, AddCommGroup, Module, Set
Proof uses: Exists, And, Membership.mem, Ne, Collinear, Insert.insert, Singleton.singleton
11 instances, coercions and auxiliary constants left out.
```
def HasCollinearTriple (R) {V : Type*} [DivisionRing R] [AddCommGroup V] [Module R V] (A : Set V) : Prop :=
…
```
theorem Erdos193.erdos_193 sorry line 57 source
depends on sorry what this means
Statement uses: Iff, False, Set, Fin, OfNat.ofNat, Nat, Int, Set.Finite, Erdos193.IsSWalk, Set.Infinite, Set.range, Erdos193.HasCollinearTriple, Rat, Function.comp
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
24 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos193.erdos_193_z2 [GeRa79] showed that the answer is yes for $\mathbb{Z}^2$ sorry line 76 source
Source documentation
[GeRa79] showed that the answer is yes for $\mathbb{Z}^2$

depends on sorry what this means
Statement uses: Set, Fin, OfNat.ofNat, Nat, Int, Set.Finite, Erdos193.IsSWalk, Set.Infinite, Set.range, Erdos193.HasCollinearTriple, Rat, Function.comp
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
24 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_193_z2 :
    ∀ S : Set (Fin 2 → ℤ), S.Finite →
      ∀ a : ℕ → Fin 2 → ℤ, IsSWalk S a → (range a).Infinite →
      HasCollinearTriple ℚ (range (fun n ↦ (↑) ∘ a n : ℕ → Fin 2 → ℚ)) := by
…
```
Mentions: Erdos193.IsSWalk, Erdos193.HasCollinearTriple