UnitDistancePlaneGraph.lean`

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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, 0 contain sorry, 0 declared as axioms

Imports (3)

Imported by 1 file

FormalConjecturesForMathlib.lean

def SimpleGraph.UnitDistancePlaneGraph A unit distance graph in ℝ²: A graph where the vertices V are a collection of points in ℝ² and there is an edge between two points if and only if the distance between them is 1 line 32 source
Source documentation
A unit distance graph in ℝ²: A graph where the vertices V are a collection of points in ℝ² and there is an edge between two points if and only if the distance between them is 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, EuclideanSpace, Real, Fin, OfNat.ofNat, Nat, SimpleGraph
Proof uses: SimpleGraph.mk, Eq, Dist.dist, ENNReal, Membership.mem
20 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
def UnitDistancePlaneGraph (V : Set (EuclideanSpace ℝ (Fin 2))) : SimpleGraph V where
…
```
def SimpleGraph.IntegerDistancePlaneGraph Documented declaration: IntegerDistancePlaneGraph line 39 source
Source documentation
An integer distance graph in ℝ²: the same construction with 1 replaced by an arbitrary positive integer, so two distinct points are adjacent exactly when the distance between them is a positive whole number.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, EuclideanSpace, Real, Fin, OfNat.ofNat, Nat, SimpleGraph
Proof uses: SimpleGraph.mk, Exists, And, LT.lt, Eq, Dist.dist, ENNReal, Membership.mem
21 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
```
def IntegerDistancePlaneGraph (V : Set (EuclideanSpace ℝ (Fin 2))) : SimpleGraph V where
…
```
def SimpleGraph.UnitDistanceEmbeddable Documented declaration: UnitDistanceEmbeddable line 51 source
Source documentation
G can be embedded in ℝ^n with every edge a unit line segment if there is an injective map from the vertices of G to ℝ^n sending any two adjacent vertices to points at distance 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, Nat
Proof uses: Exists, EuclideanSpace, Real, Fin, And, Function.Injective, SimpleGraph.Adj, Eq, Dist.dist, OfNat.ofNat, ENNReal
11 instances, coercions and auxiliary constants left out.
```
def UnitDistanceEmbeddable {V : Type*} (G : SimpleGraph V) (n : ℕ) : Prop :=
…
```
def SimpleGraph.HasDimension G has dimension n: the least m for which G embeds in ℝ^m with every edge a unit line segment line 56 source
Source documentation
G has dimension n: the least m for which G embeds in ℝ^m with every edge a unit line segment.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, Nat
Proof uses: IsLeast, Set.ofPred, SimpleGraph.UnitDistanceEmbeddable
1 instances, coercions and auxiliary constants left out.
```
def HasDimension {V : Type*} (G : SimpleGraph V) (n : ℕ) : Prop :=
…
```