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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

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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

10 declarations, 0 contain sorry, 0 declared as axioms

Uniformity and weak colorings of hypergraphs

This file extends Mathlib's Hypergraph with finite-edge constructors, uniformity, and weak proper colorings. Uniformity uses extended cardinality, so every edge is finite, including at uniformity zero. Empty and singleton edges admit no weak proper coloring.

Imports (1)

Mathlib.Combinatorics.Hypergraph.Basic

Imported by 2 files

def Hypergraph.IsUniform Every edge has exactly k vertices line 33 source
Source documentation
Every edge has exactly k vertices.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Hypergraph, Nat
Proof uses: Set, Membership.mem, Hypergraph.edgeSet, Eq, ENat, Set.encard
3 instances, coercions and auxiliary constants left out.
```
def IsUniform {V : Type*} (H : Hypergraph V) (k : ℕ) : Prop :=
…
```
def Hypergraph.IsProperColoring A weak proper coloring gives two vertices different colors in every edge line 37 source
Source documentation
A weak proper coloring gives two vertices different colors in every edge.

no axioms what this means
Statement uses: Hypergraph
Proof uses: Set, Membership.mem, Hypergraph.edgeSet, Exists, And, Ne
1 instances, coercions and auxiliary constants left out.
```
def IsProperColoring {V C : Type*} (H : Hypergraph V) (f : V → C) : Prop :=
…
```
def Hypergraph.IsNColorable A weak proper coloring using at most n colors line 41 source
Source documentation
A weak proper coloring using at most n colors.

no axioms what this means
Statement uses: Hypergraph, Nat
Proof uses: Exists, Fin, Hypergraph.IsProperColoring
```
def IsNColorable {V : Type*} (H : Hypergraph V) (n : ℕ) : Prop :=
…
```
def Hypergraph.ofEdgeFamily A family of finite edges on an explicit vertex set, retaining isolated vertices line 45 source
Source documentation
A family of finite edges on an explicit vertex set, retaining isolated vertices.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Finset, Membership.mem, LE.le, Hypergraph
Proof uses: Hypergraph.mk, Set.ofPred, Exists, And, Eq
9 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
def ofEdgeFamily {V : Type*} (F : Set (Finset V)) (S : Set V)
    (h : ∀ e ∈ F, (e : Set V) ⊆ S) : Hypergraph V where
…
```
theorem Hypergraph.mem_edgeSet_ofEdgeFamily line 54 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Finset, Membership.mem, LE.le, Iff, Hypergraph.edgeSet, Hypergraph.ofEdgeFamily
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, Exists, And, Eq, funext, iff_self
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Hypergraph.isUniform_ofEdgeFamily_iff line 60 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Finset, Membership.mem, LE.le, Nat, Iff, Hypergraph.IsUniform, Hypergraph.ofEdgeFamily, Eq, Finset.card
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, Exists, And, ENat, Set.encard, forall_congr, implies_congr, Eq.refl, Set.encard_coe_eq_coe_finsetCard, iff_self
16 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Hypergraph.isProperColoring_ofEdgeFamily_iff line 66 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Finset, Membership.mem, LE.le, Iff, Hypergraph.IsProperColoring, Hypergraph.ofEdgeFamily, Exists, And, Ne
Proof uses: of_eq_true, Not, Eq, Eq.trans, True, congrFun', congrArg, forall_congr, implies_congr, Eq.refl, funext, congr, iff_self
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Hypergraph.IsUniform.finite_edge Every edge of a uniform hypergraph is finite line 73 source
Source documentation
Every edge of a uniform hypergraph is finite.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Hypergraph, Nat, Hypergraph.IsUniform, Set, Membership.mem, Hypergraph.edgeSet, Set.Finite
Proof uses: Set.finite_of_encard_eq_coe
2 instances, coercions and auxiliary constants left out.
```
theorem IsUniform.finite_edge {V : Type*} {H : Hypergraph V} {k : ℕ}
    (hH : H.IsUniform k) {e : Set V} (he : e ∈ H.edgeSet) : e.Finite :=
…
```
theorem Hypergraph.isProperColoring_of_injective An injective coloring is proper when every edge has at least two vertices line 78 source
Source documentation
An injective coloring is proper when every edge has at least two vertices.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Hypergraph, Set, Membership.mem, Hypergraph.edgeSet, LE.le, ENat, OfNat.ofNat, Nat, Set.encard, Function.Injective, Hypergraph.IsProperColoring
Proof uses: Exists, And, Ne, Iff.mp, LT.lt, Set.one_lt_encard_iff, lt_of_lt_of_le, of_decide_eq_true, id, Eq, Bool, Decidable.decide, Bool.true, Eq.refl, Exists.intro, And.intro
23 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
```
theorem isProperColoring_of_injective {V C : Type*} {H : Hypergraph V}
    (hH : ∀ e ∈ H.edgeSet, 2 ≤ e.encard) {f : V → C}
    (hf : Function.Injective f) : H.IsProperColoring f := by
…
```
theorem Hypergraph.IsUniform.isProperColoring_of_injective An injective coloring is proper when uniformity is at least two line 87 source
Source documentation
An injective coloring is proper when uniformity is at least two.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Hypergraph, Nat, Hypergraph.IsUniform, LE.le, OfNat.ofNat, Function.Injective, Hypergraph.IsProperColoring
Proof uses: Hypergraph.isProperColoring_of_injective, Set, Membership.mem, Hypergraph.edgeSet, Eq.mpr, ENat, Set.encard, id, Eq, congrArg, cast, Eq.symm
30 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
```
theorem IsUniform.isProperColoring_of_injective {V C : Type*} {H : Hypergraph V}
    {k : ℕ} (hH : H.IsUniform k) (hk : 2 ≤ k) {f : V → C}
    (hf : Function.Injective f) : H.IsProperColoring f := by
…
```