GraphConjecture309.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

5 declarations, 1 contain sorry, 0 declared as axioms

Written on the Wall II - Conjecture 309

References:

The conjecture is false for the clique blow-ups $C_5[K_k]$ for every $k \geq 3$.

Imports (1)

FormalConjecturesUtil

def WrittenOnTheWallII.GraphConjecture309.evenHorizontal The number of edges whose endpoints are at the same even distance from $v$ line 41 source
Source documentation
The number of edges whose endpoints are at the same even distance from $v$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, SimpleGraph, DecidableRel, SimpleGraph.Adj, Nat
Proof uses: Finset.card, Sym2, Finset.filter, Finset, Finset.image, SimpleGraph.computable_dist, Sym2.toFinset, And, Eq, OfNat.ofNat, Exists, Membership.mem, Even, SimpleGraph.edgeFinset
10 instances, coercions and auxiliary constants left out.
```
def evenHorizontal (G : SimpleGraph V) [DecidableRel G.Adj] (v : V) : ℕ :=
…
```
def WrittenOnTheWallII.GraphConjecture309.maxEvenCorrection The maximum of $\operatorname{distEven}(v)-\operatorname{evenHorizontal}(v)$ line 47 source
Source documentation
The maximum of $\operatorname{distEven}(v)-\operatorname{evenHorizontal}(v)$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, SimpleGraph, Nonempty, DecidableRel, SimpleGraph.Adj, Int
Proof uses: Finset.max', Finset.image, HSub.hSub, SimpleGraph.distEven, WrittenOnTheWallII.GraphConjecture309.evenHorizontal, Finset.univ
7 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
noncomputable def maxEvenCorrection (G : SimpleGraph V) [Nonempty V]
    [DecidableRel G.Adj] : ℤ :=
…
```
def WrittenOnTheWallII.GraphConjecture309.minComplementEdgeNeighborhood The minimum complement-edge neighborhood-union order, when a complement edge exists line 53 source
Source documentation
The minimum complement-edge neighborhood-union order, when a complement edge exists.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, SimpleGraph, DecidableRel, SimpleGraph.Adj, Option, Nat
Proof uses: Finset.min, Finset.image, Sym2, Equiv, Subtype, Eq, Sym2.lift, Subtype.mk, Finset.card, Union.union, Finset, SimpleGraph.neighborFinset, Compl.compl, Membership.mem, Set, SimpleGraph.neighborSet, SimpleGraph.edgeFinset
14 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
def minComplementEdgeNeighborhood (G : SimpleGraph V)
    [DecidableRel G.Adj] : Option ℕ :=
…
```
def WrittenOnTheWallII.GraphConjecture309.conjecture309Statement The universal inequality proposed in WOWII Conjecture 309 line 68 source
Source documentation
The universal inequality proposed in WOWII Conjecture 309. The Option-valued minimum makes the statement vacuous for complete graphs, whose complements have no edge; this totalization does not affect the counterexample.

standard axioms: Classical.choice, Quot.sound, propext what this means
Proof uses: Fintype, DecidableEq, Nonempty, SimpleGraph, DecidableRel, SimpleGraph.Adj, SimpleGraph.Connected, LT.lt, Nat, OfNat.ofNat, Fintype.card, Membership.mem, Option, WrittenOnTheWallII.GraphConjecture309.minComplementEdgeNeighborhood, LE.le, Real, SimpleGraph.totalDominationNumber, HDiv.hDiv, HAdd.hAdd, WrittenOnTheWallII.GraphConjecture309.maxEvenCorrection
15 instances, coercions and auxiliary constants left out.
```
def conjecture309Statement : Prop :=
…
```
theorem WrittenOnTheWallII.GraphConjecture309.conjecture309 sorry line 86 source
depends on sorry what this means
Statement uses: Iff, False, WrittenOnTheWallII.GraphConjecture309.conjecture309Statement
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.