GraphConjecture217.lean`

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4 declarations, 1 contain sorry, 0 declared as axioms

Written on the Wall II - Conjecture 217

Per the WOWII definitions popup linked from this conjecture:

$L(G)$ is the maximum number of leaves of a spanning tree of $G$ — i.e. Ls G in our invariant library.
$\chi_{\mathrm{residue}=2}(G)$ is the characteristic function for the predicate $\mathrm{residue}\, G = 2$, i.e. $1$ when $\mathrm{residue}\, G = 2$ and $0$ otherwise. It is not a connected-component count of any 2-core.

Reference: E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc

Imports (1)

FormalConjecturesUtil

def WrittenOnTheWallII.GraphConjecture217.residueEqTwoIndicator The characteristic function for the predicate $\mathrm{residue}\, G = 2$: returns $1$ when $G.\mathrm{residue} = 2$ and $0$ otherwise line 46 source
Source documentation
The characteristic function for the predicate $\mathrm{residue}\, G = 2$: returns $1$ when $G.\mathrm{residue} = 2$ and $0$ otherwise. This is the WOWII $\chi_{\mathrm{residue}=2}(G)$ indicator appearing in Conjecture 217.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, SimpleGraph, DecidableRel, SimpleGraph.Adj, Nat
Proof uses: ite, Eq, SimpleGraph.residue, OfNat.ofNat
2 instances, coercions and auxiliary constants left out.
```
noncomputable def residueEqTwoIndicator (G : SimpleGraph α) [DecidableRel G.Adj] : ℕ :=
…
```
theorem WrittenOnTheWallII.GraphConjecture217.conjecture217 sorry line 61 source
depends on sorry what this means
Statement uses: Fintype, DecidableEq, Nontrivial, SimpleGraph, DecidableRel, SimpleGraph.Adj, SimpleGraph.Connected, LE.le, Real, SimpleGraph.Ls, HAdd.hAdd, HMul.hMul, OfNat.ofNat, Nat, WrittenOnTheWallII.GraphConjecture217.residueEqTwoIndicator, Exists, SimpleGraph.Walk, SimpleGraph.Walk.IsHamiltonian
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
22 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
example WrittenOnTheWallII.GraphConjecture217.(example at line 70) line 70 source
Source documentation
residueEqTwoIndicator is always $0$ or $1$.
```
example (G : SimpleGraph (Fin 4)) [DecidableRel G.Adj] :
    residueEqTwoIndicator G ≤ 1 := by
…
```
Mentions: WrittenOnTheWallII.GraphConjecture217.residueEqTwoIndicator
example WrittenOnTheWallII.GraphConjecture217.(example at line 76) line 76 source
Source documentation
residueEqTwoIndicator is nonneg.
```
example (G : SimpleGraph (Fin 4)) [DecidableRel G.Adj] :
    0 ≤ residueEqTwoIndicator G := Nat.zero_le _
…
```
Mentions: WrittenOnTheWallII.GraphConjecture217.residueEqTwoIndicator