Graphon.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

9 declarations, 0 contain sorry, 0 declared as axioms

Imports (7)

Imported by 2 files

abbrev LimitObjects.UnitInterval The unit interval [0, 1] line 33 source
Source documentation
The unit interval [0, 1].

standard axioms: Classical.choice, Quot.sound, propext what this means
Proof uses: Real, Set.Icc, OfNat.ofNat
6 instances, coercions and auxiliary constants left out.
```
abbrev UnitInterval : Type := Set.Icc (0 : ℝ) 1
```
structure LimitObjects.Graphon A graphon is a symmetric, measurable function W : [0, 1] × [0, 1] → [0, 1] line 39 source
Source documentation
A graphon is a symmetric, measurable function W : [0, 1] × [0, 1] → [0, 1]. The domain is the unit interval squared, and the image is contained in the unit interval.

standard axioms: Classical.choice, Quot.sound, propext what this means
```
structure Graphon where
…
```
instance LimitObjects.(instance at line 44) line 44 source
theorem LimitObjects.Graphon.ext Extensionality principle for Graphon line 50 source
Source documentation
Extensionality principle for Graphon.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: LimitObjects.Graphon, Prod, LimitObjects.UnitInterval, Eq, LimitObjects.Graphon.toFun
Proof uses: Measurable, Real, Membership.mem, Set, Set.Icc, OfNat.ofNat, Prod.mk, LimitObjects.Graphon.mk, Eq.rec, Eq.refl, funext, Eq.symm
11 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
```
theorem ext {W₁ W₂ : Graphon} (h : ∀ p, W₁ p = W₂ p) : W₁ = W₂ := by
…
```
Mentions: LimitObjects.Graphon
def LimitObjects.Graphon.evalSym2 Evaluate a graphon on an unordered pair of vertices in Sym2 UnitInterval as a real number line 54 source
Source documentation
Evaluate a graphon on an unordered pair of vertices in Sym2 UnitInterval as a real number.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: LimitObjects.Graphon, Sym2, LimitObjects.UnitInterval, Real
Proof uses: Equiv, Subtype, Eq, Sym2.lift, Subtype.mk, Membership.mem, Set, Set.Icc, OfNat.ofNat, LimitObjects.Graphon.toFun, Prod.mk
11 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
def evalSym2 (W : Graphon) (e : Sym2 UnitInterval) : ℝ :=
…
```
Mentions: LimitObjects.Graphon, LimitObjects.UnitInterval
def LimitObjects.Graphon.evalEdge Given a configuration x : V → UnitInterval and an edge e : Sym2 V, evaluate W on the endpoints as a real number line 58 source
Source documentation
Given a configuration x : V → UnitInterval and an edge e : Sym2 V, evaluate W on the endpoints as a real number.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: LimitObjects.Graphon, LimitObjects.UnitInterval, Sym2, Real
Proof uses: Equiv, Subtype, Eq, Sym2.lift, Subtype.mk, Membership.mem, Set, Set.Icc, OfNat.ofNat, LimitObjects.Graphon.toFun, Prod.mk
11 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
def evalEdge {V : Type*} (W : Graphon) (x : V → UnitInterval) (e : Sym2 V) : ℝ :=
…
```
Mentions: LimitObjects.Graphon, LimitObjects.UnitInterval
def LimitObjects.graphonHomIntegrand The integrand for the homomorphism density $t(H, W)$ of a simple graph $H$ in a graphon $W$ given a configuration x : V → [0, 1] line 69 source
Source documentation
The integrand for the homomorphism density $t(H, W)$ of a simple graph $H$ in a graphon $W$ given a configuration x : V → [0, 1].

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, SimpleGraph, DecidableRel, SimpleGraph.Adj, LimitObjects.Graphon, LimitObjects.UnitInterval, Real
Proof uses: Finset.prod, Sym2, SimpleGraph.edgeFinset, LimitObjects.Graphon.evalEdge
3 instances, coercions and auxiliary constants left out.
```
def graphonHomIntegrand (H : SimpleGraph V) [DecidableRel H.Adj]
    (W : Graphon) (x : V → UnitInterval) : ℝ :=
…
```
Mentions: LimitObjects.Graphon, LimitObjects.UnitInterval
def LimitObjects.graphonHomDensity The homomorphism density $t(H, W)$ of a finite simple graph $H$ in a graphon $W$ on $[0, 1]$ equipped with the Lebesgue measure volume line 77 source
Source documentation
The homomorphism density $t(H, W)$ of a finite simple graph $H$ in a graphon $W$ on $[0, 1]$ equipped with the Lebesgue measure volume.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, SimpleGraph, DecidableRel, SimpleGraph.Adj, LimitObjects.Graphon, Real
Proof uses: MeasureTheory.integral, LimitObjects.UnitInterval, MeasureTheory.Measure.pi, MeasureTheory.MeasureSpace.volume, LimitObjects.graphonHomIntegrand
8 instances, coercions and auxiliary constants left out.
```
noncomputable def graphonHomDensity (H : SimpleGraph V) [DecidableRel H.Adj]
    (W : Graphon) : ℝ :=
…
```
Mentions: LimitObjects.Graphon
def LimitObjects.graphonEdgeDensity The edge density of a graphon $W$ on $[0, 1]^2 \to [0, 1]$ equipped with the Lebesgue measure volume line 84 source
Source documentation
The edge density of a graphon $W$ on $[0, 1]^2 \to [0, 1]$ equipped with the Lebesgue measure volume.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: LimitObjects.Graphon, Real
Proof uses: MeasureTheory.integral, Prod, LimitObjects.UnitInterval, MeasureTheory.Measure.prod, MeasureTheory.MeasureSpace.volume, Membership.mem, Set, Set.Icc, OfNat.ofNat, LimitObjects.Graphon.toFun
15 instances, coercions and auxiliary constants left out.
```
noncomputable def graphonEdgeDensity (W : Graphon) : ℝ :=
…
```
Mentions: LimitObjects.Graphon