Degeneracy.lean`

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4 declarations, 0 contain sorry, 0 declared as axioms

Graph degeneracy

A graph is $r$-degenerate if each nonempty finite vertex set contains a vertex with at most $r$ neighbours in that set.

Imports (2)

Imported by 1 file

FormalConjecturesForMathlib.lean

def SimpleGraph.neighborsWithin The neighbours of v lying inside s line 35 source
Source documentation
The neighbours of v lying inside s.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, Finset
Proof uses: Finset.filter, SimpleGraph.Adj, Classical.propDecidable
```
noncomputable def neighborsWithin {V : Type*} (H : SimpleGraph V) (s : Finset V) (v : V) :
    Finset V := s.filter (H.Adj v)
```
def SimpleGraph.IsDegenerate H is r-degenerate when every induced subgraph has a vertex of degree at most r, that is, every nonempty vertex set contains a vertex with at most r neighbours inside it line 40 source
Source documentation
H is r-degenerate when every induced subgraph has a vertex of degree at most r, that is, every nonempty vertex set contains a vertex with at most r neighbours inside it.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, Nat
Proof uses: Finset, Finset.Nonempty, Exists, And, Membership.mem, LE.le, Finset.card, SimpleGraph.neighborsWithin
3 instances, coercions and auxiliary constants left out.
```
def IsDegenerate {V : Type*} (H : SimpleGraph V) (r : ℕ) : Prop :=
…
```
lemma SimpleGraph.neighborsWithin_empty line 44 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, Eq, Finset, SimpleGraph.neighborsWithin, EmptyCollection.emptyCollection
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, Finset.filter_empty, SimpleGraph.Adj, Classical.propDecidable, eq_self
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma SimpleGraph.isDegenerate_bot line 49 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, SimpleGraph.IsDegenerate, Bot.bot, SimpleGraph
Proof uses: Finset, Finset.Nonempty, Membership.mem, Exists, And, LE.le, Finset.card, SimpleGraph.neighborsWithin, Exists.intro, And.intro, of_eq_true, Eq.trans, OfNat.ofNat, True, congrFun', congrArg, EmptyCollection.emptyCollection, Finset.filter, SimpleGraph.Adj, Classical.propDecidable and 4 more
17 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.