`Compfiles/Usa1989P2.lean`

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3 declarations, 0 contain sorry, 0 declared as axioms

USA Mathematical Olympiad 1989, Problem 2

The 20 members of a local tennis club have scheduled exactly 14 two-person games among themselves, with each member playing in at least one game. Prove that within this schedule can be found a set of six games with 12 distinct players.

Imports (3)

Imported by 1 file

Compfiles.lean

lemma Usa1989P2.incidenceFinset_nonempty_of_degree_pos Every vertex of positive degree is incident to at least one edge line 48 source
Source documentation
Every vertex of positive degree is incident to at least one edge.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, SimpleGraph, DecidableRel, SimpleGraph.Adj, LE.le, Nat, OfNat.ofNat, SimpleGraph.degree, Membership.mem, Set, SimpleGraph.neighborSet, Finset.Nonempty, Sym2, SimpleGraph.incidenceFinset
Proof uses: Eq.mpr, LT.lt, Finset.card, id, Eq, congrArg, Eq.symm, propext, Finset.card_pos, SimpleGraph.card_incidenceFinset_eq_degree
13 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
lemma incidenceFinset_nonempty_of_degree_pos (G : SimpleGraph V) [DecidableRel G.Adj]
    {v : V} (hv : 1 ≤ G.degree v) : (G.incidenceFinset v).Nonempty := by
…
```
theorem Usa1989P2.exists_pairwise_disjoint_edges In a finite simple graph with minimum degree at least one, there is a set of pairwise vertex-disjoint edges of size at least |V| - |E| line 55 source
Source documentation
In a finite simple graph with minimum degree at least one, there is a set of pairwise vertex-disjoint edges of size at least |V| - |E|.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, SimpleGraph, DecidableRel, SimpleGraph.Adj, LE.le, Nat, OfNat.ofNat, SimpleGraph.degree, Membership.mem, Set, SimpleGraph.neighborSet, Exists, Finset, Sym2, And, SimpleGraph.edgeFinset, HSub.hSub, Fintype.card, Finset.card and 2 more
Proof uses: Finset.Nonempty, SimpleGraph.incidenceFinset, Usa1989P2.incidenceFinset_nonempty_of_degree_pos, Exists.choose, Exists.choose_spec, Finset.biUnion, Finset.univ, Finset.erase, Eq, rfl, Eq.mp, congrArg, propext, Finset.mem_biUnion, SimpleGraph.incidenceFinset_subset, And.right, Iff.mp, Finset.mem_erase, HAdd.hAdd, HMul.hMul and 36 more
52 instances, coercions and auxiliary constants left out.

Tactic trace: 36 steps
Open to load the checked before-and-after goal states.
```
theorem exists_pairwise_disjoint_edges (G : SimpleGraph V) [DecidableRel G.Adj]
    (hdeg : ∀ v : V, 1 ≤ G.degree v) :
    ∃ s : Finset (Sym2 V), s ⊆ G.edgeFinset ∧
      Fintype.card V - #G.edgeFinset ≤ #s ∧
      ∀ e₁ ∈ s, ∀ e₂ ∈ s, e₁ ≠ e₂ → ∀ v : V, ¬ (v ∈ e₁ ∧ v ∈ e₂) := by
…
```
problem Usa1989P2.usa1989_p2 line 111 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, Fin, OfNat.ofNat, Nat, DecidableRel, SimpleGraph.Adj, Eq, Finset.card, Sym2, SimpleGraph.edgeFinset, LE.le, SimpleGraph.degree, Membership.mem, Set, SimpleGraph.neighborSet, Exists, Finset, And, Finset.biUnion, Sym2.toFinset
Proof uses: HSub.hSub, Fintype.card, Ne, Not, Usa1989P2.exists_pairwise_disjoint_edges, Finset.exists_subset_card_eq, Eq.mp, congrArg, Fintype.card_fin, Exists.intro, And.intro, LE.le.trans, Set.PairwiseDisjoint, Iff.mpr, Disjoint, Finset.disjoint_left, Iff.mp, Finset.mem_coe, propext, Sym2.mem_toFinset and 16 more
40 instances, coercions and auxiliary constants left out.

Tactic trace: 16 steps
Open to load the checked before-and-after goal states.