SzegedIndex.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

7 declarations, 0 contain sorry, 0 declared as axioms

Imports (4)

Imported by 1 file

FormalConjecturesForMathlib.lean

def SimpleGraph.szeged_aux Auxiliary function for Szeged index: counts vertices closer to u than v line 30 source
Source documentation
Auxiliary function for Szeged index: counts vertices closer to u than v.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, SimpleGraph, Nat
Proof uses: Finset.card, Finset.filter, LT.lt, ENat, SimpleGraph.edist, Finset.univ
3 instances, coercions and auxiliary constants left out.
```
noncomputable def szeged_aux (G : SimpleGraph α) (u v : α) : ℕ :=
…
```
def SimpleGraph.szegedIndex The Szeged index of G line 38 source
Source documentation
The Szeged index of G.
This is define as the sum ∑_{uv ∈ E(G)} n_u(u,v) * n_v(u,v) where n_u(uv) is the number of vertices closer to u than v.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, SimpleGraph, DecidableRel, SimpleGraph.Adj, Nat
Proof uses: Finset.sum, Sym2, SimpleGraph.edgeFinset, Equiv, Subtype, Eq, Sym2.lift, Subtype.mk, HMul.hMul, SimpleGraph.szeged_aux
9 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
noncomputable def szegedIndex (G : SimpleGraph α) [DecidableRel G.Adj] : ℕ :=
…
```
def SimpleGraph.computable_szeged_aux Computable Szeged auxiliary: count vertices closer to u than v line 43 source
Source documentation
Computable Szeged auxiliary: count vertices closer to u than v.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, SimpleGraph, DecidableRel, SimpleGraph.Adj, Nat
Proof uses: Finset.card, Finset.filter, LT.lt, SimpleGraph.computable_dist, Finset.univ
2 instances, coercions and auxiliary constants left out.
```
def computable_szeged_aux (G : SimpleGraph α) [DecidableRel G.Adj] (u v : α) : ℕ :=
…
```
def SimpleGraph.computable_szeged_index Computable Szeged index line 47 source
Source documentation
Computable Szeged index.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, SimpleGraph, DecidableRel, SimpleGraph.Adj, Nat
Proof uses: Finset.sum, Sym2, SimpleGraph.edgeFinset, Equiv, Subtype, Eq, Sym2.lift, Subtype.mk, HMul.hMul, SimpleGraph.computable_szeged_aux
9 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
def computable_szeged_index (G : SimpleGraph α) [DecidableRel G.Adj] : ℕ :=
…
```
lemma SimpleGraph.edist_lt_iff_dist_lt line 54 source
lemma SimpleGraph.szeged_aux_eq_computable line 64 source
theorem SimpleGraph.szeged_eq_computable line 72 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, SimpleGraph, DecidableRel, SimpleGraph.Adj, Eq, Nat, SimpleGraph.szegedIndex, SimpleGraph.computable_szeged_index
Proof uses: id, Finset.sum, Sym2, SimpleGraph.edgeFinset, Equiv, Subtype, Sym2.lift, Subtype.mk, HMul.hMul, SimpleGraph.szeged_aux, Finset.sum_congr, SimpleGraph.computable_szeged_aux, rfl, Finset, Membership.mem, Sym2.ind, Sym2.mk, Eq.mp, Set, SimpleGraph.edgeSet and 8 more
14 instances, coercions and auxiliary constants left out.

Tactic trace: 14 steps
Open to load the checked before-and-after goal states.