`Compfiles/Usa1995P5.lean`

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5 declarations, 0 contain sorry, 0 declared as axioms

USA Mathematical Olympiad 1995, Problem 5

A graph with n points and k edges has no triangles. Show that it has a point P such that there are at most k(1 - 4k/n²) edges between points not joined to P (by an edge).

Imports (4)

Imported by 1 file

Compfiles.lean

lemma Usa1995P5.not_adj_and_adj_of_mem_edgeFinset In a triangle-free graph, no edge can have two distinct endpoints that are both adjacent to P: together with P they would form a triangle line 36 source
Source documentation
In a triangle-free graph, no edge can have two distinct endpoints that are both adjacent to P: together with P they would form a triangle.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, SimpleGraph, Fin, DecidableRel, SimpleGraph.Adj, SimpleGraph.CliqueFree, OfNat.ofNat, Ne, Sym2, Membership.mem, Finset, SimpleGraph.edgeFinset, False
Proof uses: Sym2.ind, Sym2.mk, Eq, Or, Eq.mp, congrArg, propext, Sym2.mem_iff, absurd, rfl, Set, SimpleGraph.edgeSet, SimpleGraph.mem_edgeSet, SimpleGraph.mem_edgeFinset, SimpleGraph.Adj.symm, Insert.insert, Singleton.singleton, Iff.mpr, SimpleGraph.IsNClique, And and 2 more
20 instances, coercions and auxiliary constants left out.

Tactic trace: 15 steps
Open to load the checked before-and-after goal states.
```
lemma not_adj_and_adj_of_mem_edgeFinset (hG : G.CliqueFree 3) {P Q₁ Q₂ : Fin n}
    (hPQ₁ : G.Adj P Q₁) (hPQ₂ : G.Adj P Q₂) (hne : Q₁ ≠ Q₂)
    {e : Sym2 (Fin n)} (he : e ∈ G.edgeFinset) (h₁ : Q₁ ∈ e) (h₂ : Q₂ ∈ e) :
    False := by
…
```
Mentions: P
lemma Usa1995P5.sum_degree_neighborFinset_eq_card Double counting: the sum of the degrees of the neighbors of P equals the number of edges of G having an endpoint adjacent to P line 56 source
Source documentation
Double counting: the sum of the degrees of the neighbors of P equals the number of edges of G having an endpoint adjacent to P. Indeed, every edge incident to a neighbor Q of P has Q as its unique endpoint adjacent to P, since the graph is triangle-free.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, SimpleGraph, Fin, DecidableRel, SimpleGraph.Adj, SimpleGraph.CliqueFree, OfNat.ofNat, Eq, Finset.sum, SimpleGraph.neighborFinset, Membership.mem, Set, SimpleGraph.neighborSet, SimpleGraph.degree, Finset.card, Sym2, Finset.filter, Exists, And, SimpleGraph.edgeFinset
Proof uses: Eq.mpr, SimpleGraph.incidenceFinset, id, congrArg, Eq.symm, SimpleGraph.card_incidenceFinset_eq_degree, Finset, SimpleGraph.incidenceFinset_eq_filter, Eq.refl, congrFun', Finset.sum_congr, Set.PairwiseDisjoint, Ne, Function.onFun, Disjoint, Not, propext, Finset.disjoint_left, Usa1995P5.not_adj_and_adj_of_mem_edgeFinset, Iff.mp and 17 more
39 instances, coercions and auxiliary constants left out.

Tactic trace: 22 steps
Open to load the checked before-and-after goal states.
```
lemma sum_degree_neighborFinset_eq_card (hG : G.CliqueFree 3) (P : Fin n) :
    ∑ Q ∈ G.neighborFinset P, G.degree Q
      = #(G.edgeFinset.filter fun e => ∃ v ∈ e, G.Adj P v) := by
…
```
Mentions: P
lemma Usa1995P5.card_far_add_sum_degree The edges of G split into those with no endpoint adjacent to P and those with a (unique) endpoint adjacent to P, counted in sum_degree_neighborFinset_eq_card line 86 source
Source documentation
The edges of G split into those with no endpoint adjacent to P and those with a (unique) endpoint adjacent to P, counted in sum_degree_neighborFinset_eq_card.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, SimpleGraph, Fin, DecidableRel, SimpleGraph.Adj, SimpleGraph.CliqueFree, OfNat.ofNat, Eq, HAdd.hAdd, Finset.card, Sym2, Finset.filter, Membership.mem, Not, SimpleGraph.edgeFinset, Finset.sum, SimpleGraph.neighborFinset, Set, SimpleGraph.neighborSet, SimpleGraph.degree
Proof uses: Finset, Exists, And, Finset.filter_congr, of_eq_true, Iff, Eq.trans, True, congrArg, forall_congr, iff_self, Finset.card_filter_add_card_filter_not, Eq.mpr, id, Usa1995P5.sum_degree_neighborFinset_eq_card, add_comm
42 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
```
lemma card_far_add_sum_degree (hG : G.CliqueFree 3) (P : Fin n) :
    #(G.edgeFinset.filter fun e => ∀ v ∈ e, ¬G.Adj P v)
      + ∑ Q ∈ G.neighborFinset P, G.degree Q = #G.edgeFinset := by
…
```
Mentions: P
lemma Usa1995P5.sum_sum_degree_neighborFinset Summing ∑ Q ∈ N(P), deg Q over all vertices P counts deg Q once for each neighbor of Q, giving the sum of the squares of the degrees line 99 source
Source documentation
Summing ∑ Q ∈ N(P), deg Q over all vertices P counts deg Q once for each neighbor of Q, giving the sum of the squares of the degrees.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, SimpleGraph, Fin, DecidableRel, SimpleGraph.Adj, Eq, Finset.sum, Finset.univ, SimpleGraph.neighborFinset, Membership.mem, Set, SimpleGraph.neighborSet, SimpleGraph.degree, HPow.hPow, OfNat.ofNat
Proof uses: ite, Finset, Finset.filter, Finset.ext, Eq.mpr, Iff, id, congr, congrArg, Eq.trans, And, True, congrFun', true_and, SimpleGraph.adj_comm, Eq.symm, Finset.sum_filter, HSMul.hSMul, Finset.card, Finset.sum_const and 10 more
47 instances, coercions and auxiliary constants left out.

Tactic trace: 10 steps
Open to load the checked before-and-after goal states.
```
lemma sum_sum_degree_neighborFinset :
    ∑ P : Fin n, ∑ Q ∈ G.neighborFinset P, G.degree Q
      = ∑ Q : Fin n, (G.degree Q) ^ 2 := by
…
```
Mentions: P
problem Usa1995P5.usa1995_p5 line 117 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, NeZero, SimpleGraph, Fin, DecidableRel, SimpleGraph.Adj, SimpleGraph.CliqueFree, OfNat.ofNat, Exists, LE.le, Real, Finset.card, Sym2, Finset.filter, Membership.mem, Not, SimpleGraph.edgeFinset, HMul.hMul, HSub.hSub, HDiv.hDiv and 1 more
Proof uses: Ne, Iff.mpr, Nat.cast_ne_zero, NeZero.ne, LT.lt, Nat.cast_pos, NeZero.pos, Eq.mpr, id, Eq, congrArg, rfl, Finset.sum, Finset.univ, SimpleGraph.neighborFinset, Set, SimpleGraph.neighborSet, SimpleGraph.degree, cast, Eq.symm and 152 more
203 instances, coercions and auxiliary constants left out.

Tactic trace: 33 steps
Open to load the checked before-and-after goal states.