Independence.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

15 declarations, 0 contain sorry, 0 declared as axioms

Imports (2)

Imported by 3 files

def SimpleGraph.aprime The maximum cardinality among all independent sets s that maximize the quantity |s| - |N(s)|, where N(s) is the neighborhood of the set s line 31 source
Source documentation
The maximum cardinality among all independent sets s that maximize the quantity |s| - |N(s)|, where N(s) is the neighborhood of the set s.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, SimpleGraph, DecidableRel, SimpleGraph.Adj, Real
Proof uses: Option.getD, Nat, Finset.max, Finset.image, Finset, Finset.card, Finset.filter, Eq, Int, HSub.hSub, Set.toFinset, Set.iUnion, Membership.mem, Set, SimpleGraph.neighborSet, SimpleGraph.IsIndepSet, Finset.powerset, Finset.univ, OfNat.ofNat
19 instances, coercions and auxiliary constants left out.
```
noncomputable def aprime (G : SimpleGraph α) [DecidableRel G.Adj] : ℝ :=
…
```
def SimpleGraph.indepNeighborsCard Independence number of the neighbourhood of v line 40 source
Source documentation
Independence number of the neighbourhood of v.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, Nat
Proof uses: SimpleGraph.indepNum, SimpleGraph.neighborSet, SimpleGraph.induce
1 instances, coercions and auxiliary constants left out.
```
noncomputable def indepNeighborsCard (G : SimpleGraph α) (v : α) : ℕ :=
…
```
def SimpleGraph.indepNeighbors The same quantity as a real number line 44 source
Source documentation
The same quantity as a real number.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, Real
Proof uses: SimpleGraph.indepNeighborsCard
2 instances, coercions and auxiliary constants left out.
```
noncomputable def indepNeighbors (G : SimpleGraph α) (v : α) : ℝ :=
…
```
def SimpleGraph.averageIndepNeighbors Average of indepNeighbors over all vertices line 48 source
Source documentation
Average of indepNeighbors over all vertices.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, SimpleGraph, Real
Proof uses: HDiv.hDiv, Finset.sum, Finset.univ, SimpleGraph.indepNeighbors, Fintype.card
6 instances, coercions and auxiliary constants left out.
```
noncomputable def averageIndepNeighbors (G : SimpleGraph α) : ℝ :=
…
```
abbrev SimpleGraph.l Abbreviation for the average independence number of the neighborhoods line 53 source
Source documentation
Abbreviation for the average independence number of the neighborhoods.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, SimpleGraph, Real
Proof uses: SimpleGraph.averageIndepNeighbors
```
noncomputable abbrev l (G : SimpleGraph α) : ℝ := averageIndepNeighbors G
```
Mentions: SimpleGraph.averageIndepNeighbors
abbrev SimpleGraph.l_avg The same quantity under a different name, used in some conjectures line 57 source
Source documentation
The same quantity under a different name, used in some conjectures.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, SimpleGraph, Real
Proof uses: SimpleGraph.averageIndepNeighbors
```
noncomputable abbrev l_avg (G : SimpleGraph α) : ℝ := averageIndepNeighbors G
```
Mentions: SimpleGraph.averageIndepNeighbors
def SimpleGraph.computable_indep_num Computable independence number via powerset enumeration line 60 source
Source documentation
Computable independence number via powerset enumeration.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, SimpleGraph, DecidableRel, SimpleGraph.Adj, Nat
Proof uses: Finset.sup, Finset, Finset.filter, Membership.mem, Ne, Not, Eq, Finset.powerset, Finset.univ, Finset.card
9 instances, coercions and auxiliary constants left out.
```
def computable_indep_num (G : SimpleGraph α) [DecidableRel G.Adj] : ℕ :=
…
```
theorem SimpleGraph.indep_num_eq_computable Equivalence between computable and noncomputable independence number line 65 source
Source documentation
Equivalence between computable and noncomputable independence number.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, SimpleGraph, DecidableRel, SimpleGraph.Adj, Eq, Nat, SimpleGraph.indepNum, SimpleGraph.computable_indep_num
Proof uses: id, SupSet.sSup, Set.ofPred, Exists, Finset, SimpleGraph.IsNIndepSet, le_antisymm, Finset.sup, Finset.filter, Membership.mem, Ne, Not, Finset.powerset, Finset.univ, Finset.card, csSup_le, Exists.intro, Set, OfNat.ofNat, EmptyCollection.emptyCollection and 30 more
32 instances, coercions and auxiliary constants left out.

Tactic trace: 21 steps
Open to load the checked before-and-after goal states.
```
theorem indep_num_eq_computable (G : SimpleGraph α) [DecidableRel G.Adj] :
    G.indepNum = computable_indep_num G := by
…
```
Mentions: SimpleGraph.computable_indep_num
def SimpleGraph.maxLocalIndependence Maximum local independence number over all vertices line 90 source
Source documentation
Maximum local independence number over all vertices. For each vertex, we compute the independence number of the subgraph induced by its neighbors, and take the maximum over all vertices.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, SimpleGraph, Nat
Proof uses: Finset, Finset.image, SimpleGraph.indepNum, SimpleGraph.neighborSet, SimpleGraph.induce, Finset.univ, Option.getD, Finset.max, OfNat.ofNat
4 instances, coercions and auxiliary constants left out.
```
noncomputable def maxLocalIndependence (G : SimpleGraph α) : ℕ :=
…
```
def SimpleGraph.minLocalIndependence Minimum local independence number over all vertices line 97 source
Source documentation
Minimum local independence number over all vertices. For each vertex, we compute the independence number of the subgraph induced by its neighbors, and take the minimum over all vertices.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, SimpleGraph, Nat
Proof uses: Finset, Finset.image, SimpleGraph.indepNum, SimpleGraph.neighborSet, SimpleGraph.induce, Finset.univ, Option.getD, Finset.min, OfNat.ofNat
4 instances, coercions and auxiliary constants left out.
```
noncomputable def minLocalIndependence (G : SimpleGraph α) : ℕ :=
…
```
def SimpleGraph.internalEdges Edges of G whose two endpoints lie in S line 102 source
Source documentation
Edges of G whose two endpoints lie in S.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, SimpleGraph, DecidableRel, SimpleGraph.Adj, Finset, Sym2
Proof uses: Finset.filter, LE.le, Sym2.toFinset, SimpleGraph.edgeFinset
6 instances, coercions and auxiliary constants left out.
```
def internalEdges (G : SimpleGraph α) [DecidableRel G.Adj] (S : Finset α) :
    Finset (Sym2 α) :=
…
```
Mentions: SimpleGraph.S
def SimpleGraph.IsTwoIndependent A vertex set is 2-independent when its induced graph has maximum degree at most one line 108 source
Source documentation
A vertex set is 2-independent when its induced graph has maximum degree at most one.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, DecidableRel, SimpleGraph.Adj, Finset
Proof uses: Membership.mem, LE.le, Nat, Finset.card, Finset.filter, OfNat.ofNat
4 instances, coercions and auxiliary constants left out.
```
def IsTwoIndependent (G : SimpleGraph α) [DecidableRel G.Adj] (S : Finset α) : Prop :=
…
```
Mentions: SimpleGraph.S
def SimpleGraph.indepNumOn Maximum cardinality of an independent subset contained in B line 112 source
Source documentation
Maximum cardinality of an independent subset contained in B.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: DecidableEq, SimpleGraph, DecidableRel, SimpleGraph.Adj, Finset, Nat
Proof uses: Finset.sup, Finset.filter, SimpleGraph.IsIndepSet, Finset.powerset, Finset.card
6 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
noncomputable def indepNumOn (G : SimpleGraph α) [DecidableRel G.Adj]
    (B : Finset α) : ℕ :=
…
```
def SimpleGraph.alphaTwo Maximum cardinality of a 2-independent vertex set line 119 source
Source documentation
Maximum cardinality of a 2-independent vertex set.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, SimpleGraph, DecidableRel, SimpleGraph.Adj, Nat
Proof uses: Finset.sup, Finset, Finset.filter, SimpleGraph.IsTwoIndependent, Classical.propDecidable, Finset.powerset, Finset.univ, Finset.card
3 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
noncomputable def alphaTwo (G : SimpleGraph α) [DecidableRel G.Adj] : ℕ :=
…
```
def SimpleGraph.lowDegreeLayer The set of vertices of degree at most two line 125 source
Source documentation
The set of vertices of degree at most two.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, SimpleGraph, DecidableRel, SimpleGraph.Adj, Finset
Proof uses: Finset.filter, LE.le, Nat, SimpleGraph.degree, Membership.mem, Set, SimpleGraph.neighborSet, OfNat.ofNat, Finset.univ
6 instances, coercions and auxiliary constants left out.
```
def lowDegreeLayer (G : SimpleGraph α) [DecidableRel G.Adj] : Finset α :=
…
```