Conway99Graph.lean`

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

9 declarations, 1 contain sorry, 0 declared as axioms

Conway's 99-graph problem

Reference: Wikipedia

Imports (1)

FormalConjecturesUtil

lemma Conway99Graph.completeGraphIsClique A finset of vertices in a complete graph is always a clique line 38 source
Source documentation
A finset of vertices in a complete graph is always a clique.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finset, SimpleGraph.IsClique, Top.top, SimpleGraph
Proof uses: Pairwise.set_pairwise, Ne
8 instances, coercions and auxiliary constants left out.
```
lemma completeGraphIsClique (s : Finset V) : (⊤ : SimpleGraph V).IsClique s :=
…
```
lemma Conway99Graph.completeGraph_cliqueSet The only clique of size n in a complete graph on n vertices is the entire set of vertices line 45 source
Source documentation
The only clique of size n in a complete graph on n vertices is the entire set of vertices.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, Eq, Set, Finset, SimpleGraph.cliqueSet, Top.top, SimpleGraph, Fintype.card, Singleton.singleton, Set.toFinset, Set.univ
Proof uses: Eq.mpr, Set.ofPred, Nat, Finset.card, Finset.univ, id, congr, congrArg, SimpleGraph.IsNClique, funext, Eq.trans, And, SimpleGraph.IsClique, propext, SimpleGraph.isNClique_iff, True, congrFun', true_and, Set.toFinset_univ, Iff.mp and 3 more
16 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
lemma completeGraph_cliqueSet :
    (⊤ : SimpleGraph V).cliqueSet (Fintype.card V) = {Set.univ.toFinset} := by
…
```
def Conway99Graph.NonEdgesAreDiagonals Each two non-adjacent vertices have exactly two common neighbors line 55 source
Source documentation
Each two non-adjacent vertices have exactly two common neighbors.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph
Proof uses: Pairwise, Not, SimpleGraph.Adj, Eq, Nat, Set.ncard, Inter.inter, Set, SimpleGraph.neighborSet, OfNat.ofNat
2 instances, coercions and auxiliary constants left out.
```
def NonEdgesAreDiagonals (G : SimpleGraph V) : Prop :=
…
```
theorem Conway99Graph.conway99Graph Documented declaration: conway99Graph sorry line 67 source
Source documentation
Does there exist an undirected graph with 99 vertices, in which each two adjacent vertices have exactly one common neighbor, and in which each two non-adjacent vertices have exactly two common neighbors? Equivalently, every edge should be part of a unique triangle and every non-adjacent pair should be one of the two diagonals of a unique 4-cycle. The first condition is equivalent to being locally linear.

depends on sorry what this means
Statement uses: Iff, True, Exists, SimpleGraph, Fin, OfNat.ofNat, Nat, And, SimpleGraph.LocallyLinear, Conway99Graph.NonEdgesAreDiagonals
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conway99Graph : answer(sorry) ↔ ∃ G : SimpleGraph (Fin 99),
    G.LocallyLinear ∧ NonEdgesAreDiagonals G := by
…
```
Mentions: Conway99Graph.NonEdgesAreDiagonals
theorem Conway99Graph.triangle_locallyLinear_and_nonEdgesAreDiagonals The triangle is an example with 3 vertices satisfying the condition line 75 source
Source documentation
The triangle is an example with 3 vertices satisfying the condition.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: And, SimpleGraph.LocallyLinear, Fin, OfNat.ofNat, Nat, SimpleGraph.completeGraph, Conway99Graph.NonEdgesAreDiagonals
Proof uses: And.intro, Eq.mpr, SimpleGraph.EdgeDisjointTriangles, Top.top, SimpleGraph, Not, Eq, Exists, Finset, SimpleGraph.IsNClique, Membership.mem, id, congrArg, SimpleGraph.Adj, forall_congr, Ne, implies_congr, Eq.refl, Set, SimpleGraph.cliqueSet and 60 more
54 instances, coercions and auxiliary constants left out.

Tactic trace: 22 steps
Open to load the checked before-and-after goal states.
```
theorem triangle_locallyLinear_and_nonEdgesAreDiagonals : (completeGraph (Fin 3)).LocallyLinear ∧
    NonEdgesAreDiagonals (completeGraph (Fin 3)) := by
…
```
Mentions: Conway99Graph.NonEdgesAreDiagonals
abbrev Conway99Graph.Conway9 The box product of two triangles is an example with 9 vertices satisfying the condition line 98 source
Source documentation
The box product of two triangles is an example with 9 vertices satisfying the condition. (This graph is the complement of the one described in https://vimeo.com/109815595 and it is also isomorphic to it and to the Paley graph and the graph of the 3-3 duoprism)

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, Prod, Fin, OfNat.ofNat, Nat
Proof uses: SimpleGraph.boxProd, SimpleGraph.completeGraph
2 instances, coercions and auxiliary constants left out.
```
abbrev Conway9 := completeGraph (Fin 3) □ completeGraph (Fin 3)
…
```
theorem Conway99Graph.conway9_nonEdgesAreDiagonals line 101 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Conway99Graph.NonEdgesAreDiagonals, Prod, Fin, OfNat.ofNat, Nat, Conway99Graph.Conway9
Proof uses: id, Eq, Finset.card, Inter.inter, Finset, SimpleGraph.neighborFinset, SimpleGraph.completeGraph, Membership.mem, Set, SimpleGraph.neighborSet, Prod.fst, Prod.snd, Set.ncard, Eq.mpr, Set.toFinset, congrArg, Eq.symm, Set.toFinset_inter, Set.ncard_eq_toFinset_card', Eq.refl and 46 more
35 instances, coercions and auxiliary constants left out.

Tactic trace: 15 steps
Open to load the checked before-and-after goal states.
lemma Conway99Graph.completeGraph_boxProd_completeGraph_cliqueSet line 116 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Set, Finset, Prod, Fin, OfNat.ofNat, Nat, SimpleGraph.cliqueSet, SimpleGraph.boxProd, SimpleGraph.completeGraph, Union.union, Set.ofPred, Exists, Finset.filter, Prod.mk, Finset.univ
Proof uses: Set.ext, Eq.mpr, Iff, Membership.mem, SimpleGraph.IsNClique, Top.top, SimpleGraph, Or, id, congr, congrArg
22 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
theorem Conway99Graph.conway9_locallyLinear line 126 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph.LocallyLinear, Prod, Fin, OfNat.ofNat, Nat, Conway99Graph.Conway9
Proof uses: id, And.intro, SimpleGraph.EdgeDisjointTriangles, SimpleGraph.Adj, Exists, Finset, And, SimpleGraph.IsNClique, Membership.mem, Set, SimpleGraph.cliqueSet, Ne, Set.ofPred, Eq, Finset.filter, Prod.mk, Finset.univ, Union.union, Set.Subsingleton, Inter.inter and 67 more
49 instances, coercions and auxiliary constants left out.

Tactic trace: 39 steps
Open to load the checked before-and-after goal states.