Eccentricity.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

8 declarations, 0 contain sorry, 0 declared as axioms

Imports (3)

Imported by 3 files

def SimpleGraph.maxEccentricityVertices The set of vertices of maximum eccentricity line 29 source
Source documentation
The set of vertices of maximum eccentricity.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, Set
Proof uses: Set.ofPred, Eq, ENat, SimpleGraph.eccent, SimpleGraph.ediam
```
noncomputable def maxEccentricityVertices (G : SimpleGraph α) : Set α :=
…
```
def SimpleGraph.averageEccentricity The average eccentricity of a graph G: the mean of G.eccent v over all vertices, converted to a real number line 34 source
Source documentation
The average eccentricity of a graph G: the mean of G.eccent v over all vertices, converted to a real number. Returns 0 if the graph has no vertices.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, SimpleGraph, Real
Proof uses: HDiv.hDiv, Finset.sum, Nat, Finset.univ, ENat.toNat, SimpleGraph.eccent, Fintype.card
6 instances, coercions and auxiliary constants left out.
```
noncomputable def averageEccentricity (G : SimpleGraph α) : ℝ :=
…
```
def SimpleGraph.computable_ediam The diameter of a finite graph as a computable natural number: the maximum BFS distance (computable_dist) over all ordered pairs of vertices line 39 source
Source documentation
The diameter of a finite graph as a computable natural number: the maximum BFS distance (computable_dist) over all ordered pairs of vertices.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, SimpleGraph, DecidableRel, SimpleGraph.Adj, Nat
Proof uses: Finset.sup, Prod, Finset.univ, SimpleGraph.computable_dist, Prod.fst, Prod.snd
4 instances, coercions and auxiliary constants left out.
```
def computable_ediam (G : SimpleGraph α) [DecidableRel G.Adj] : ℕ :=
…
```
theorem SimpleGraph.ediam_eq_computable For a connected finite graph, the extended diameter ediam equals the computable diameter line 43 source
Source documentation
For a connected finite graph, the extended diameter ediam equals the computable diameter.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, SimpleGraph, DecidableRel, SimpleGraph.Adj, Nonempty, SimpleGraph.Connected, Eq, ENat, SimpleGraph.ediam, SimpleGraph.computable_ediam
Proof uses: le_antisymm, Eq.mpr, LE.le, SimpleGraph.edist, id, congrArg, propext, SimpleGraph.ediam_le_iff, SimpleGraph.dist, Eq.symm, SimpleGraph.Reachable.coe_dist_eq_edist, SimpleGraph.Connected.preconnected, SimpleGraph.computable_dist, Nat, SimpleGraph.dist_eq_computable, cast, Finset.le_sup, Prod, Finset.univ, Prod.fst and 11 more
38 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
```
theorem ediam_eq_computable (G : SimpleGraph α) [DecidableRel G.Adj] [Nonempty α]
    (hc : G.Connected) : G.ediam = (computable_ediam G : ℕ∞) := by
…
```
Mentions: SimpleGraph.computable_ediam
def SimpleGraph.computable_eccent The eccentricity of a vertex u in a finite graph as a computable natural number: the maximum BFS distance (computable_dist) from u to any vertex line 57 source
Source documentation
The eccentricity of a vertex u in a finite graph as a computable natural number: the maximum BFS distance (computable_dist) from u to any vertex.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, SimpleGraph, DecidableRel, SimpleGraph.Adj, Nat
Proof uses: Finset.sup, Finset.univ, SimpleGraph.computable_dist
3 instances, coercions and auxiliary constants left out.
```
def computable_eccent (G : SimpleGraph α) [DecidableRel G.Adj] (u : α) : ℕ :=
…
```
def SimpleGraph.computable_radius The radius of a finite graph as a computable natural number: the minimum, over all vertices, of the computable eccentricity line 62 source
Source documentation
The radius of a finite graph as a computable natural number: the minimum, over all vertices, of the computable eccentricity.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, SimpleGraph, DecidableRel, SimpleGraph.Adj, Nonempty, Nat
Proof uses: Finset.inf', Finset.univ, Finset.univ_nonempty, SimpleGraph.computable_eccent
2 instances, coercions and auxiliary constants left out.
```
def computable_radius (G : SimpleGraph α) [DecidableRel G.Adj] [Nonempty α] : ℕ :=
…
```
theorem SimpleGraph.eccent_eq_computable For a connected finite graph, the eccentricity eccent u equals the computable eccentricity line 66 source
Source documentation
For a connected finite graph, the eccentricity eccent u equals the computable eccentricity.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, SimpleGraph, DecidableRel, SimpleGraph.Adj, Nonempty, SimpleGraph.Connected, Eq, ENat, SimpleGraph.eccent, SimpleGraph.computable_eccent
Proof uses: le_antisymm, Eq.mpr, LE.le, SimpleGraph.edist, id, congrArg, propext, SimpleGraph.eccent_le_iff, SimpleGraph.dist, Eq.symm, SimpleGraph.Reachable.coe_dist_eq_edist, SimpleGraph.Connected.preconnected, SimpleGraph.computable_dist, Nat, SimpleGraph.dist_eq_computable, cast, Finset.le_sup, Finset.univ, Finset.mem_univ, And and 7 more
36 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
```
theorem eccent_eq_computable (G : SimpleGraph α) [DecidableRel G.Adj] [Nonempty α]
    (hc : G.Connected) (u : α) : G.eccent u = (computable_eccent G u : ℕ∞) := by
…
```
Mentions: SimpleGraph.computable_eccent
theorem SimpleGraph.radius_eq_computable For a connected finite graph, the radius radius equals the computable radius line 79 source
Source documentation
For a connected finite graph, the radius radius equals the computable radius.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, SimpleGraph, DecidableRel, SimpleGraph.Adj, Nonempty, SimpleGraph.Connected, Eq, ENat, SimpleGraph.radius, SimpleGraph.computable_radius
Proof uses: le_antisymm, And, Membership.mem, Finset, Finset.univ, Nat, Finset.inf', Finset.univ_nonempty, SimpleGraph.computable_eccent, Exists, LE.le, Finset.exists_mem_eq_inf', Eq.mpr, id, congrArg, SimpleGraph.eccent, Eq.symm, SimpleGraph.eccent_eq_computable, SimpleGraph.radius_le_eccent, SimpleGraph.exists_eccent_eq_radius and 3 more
34 instances, coercions and auxiliary constants left out.

Tactic trace: 10 steps
Open to load the checked before-and-after goal states.
```
theorem radius_eq_computable (G : SimpleGraph α) [DecidableRel G.Adj] [Nonempty α]
    (hc : G.Connected) : G.radius = (computable_radius G : ℕ∞) := by
…
```
Mentions: SimpleGraph.computable_radius