`Compfiles/Imo2007P3.lean`

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14 declarations, 0 contain sorry, 0 declared as axioms

International Mathematical Olympiad 2007, Problem 3

In a mathematical competition some competitors are (mutual) friends. Call a group of competitors a clique if each two of them are friends. Given that the largest size of a clique is even, prove that the competitors can be arranged into two rooms such that the largest size of a clique contained in one room is the same as the largest size of a clique contained in the other room.

Imports (2)

Imported by 1 file

Compfiles.lean

def Imo2007P3.maxCliqueCard The largest size of a clique of the graph G all of whose vertices lie in A line 32 source
Source documentation
The largest size of a clique of the graph G all of whose vertices lie in A.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, Finset, Nat
Proof uses: SupSet.sSup, Set.ofPred, Exists, And, LE.le, SimpleGraph.IsClique, Eq, Finset.card
6 instances, coercions and auxiliary constants left out.
```
noncomputable def maxCliqueCard {V : Type*} (G : SimpleGraph V) (A : Finset V) : ℕ :=
…
```
lemma Imo2007P3.bddAbove_cliqueCard line 41 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, Finset, BddAbove, Nat, Set.ofPred, Exists, And, LE.le, SimpleGraph.IsClique, Eq, Finset.card
Proof uses: Exists.intro, Membership.mem, Set, upperBounds, Finset.card_le_card
16 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Imo2007P3.nonempty_cliqueCard line 47 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, Finset, Set.Nonempty, Nat, Set.ofPred, Exists, And, LE.le, SimpleGraph.IsClique, Eq, Finset.card
Proof uses: Exists.intro, Membership.mem, Set, OfNat.ofNat, EmptyCollection.emptyCollection, And.intro, Finset.empty_subset, Eq.mpr, id, congrArg, Finset.coe_empty, Set.pairwise_empty, SimpleGraph.Adj, Finset.card_empty
14 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Imo2007P3.card_le_maxCliqueCard line 52 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, Finset, LE.le, SimpleGraph.IsClique, Nat, Finset.card, Imo2007P3.maxCliqueCard
Proof uses: le_csSup, Set.ofPred, Exists, And, Eq, Imo2007P3.bddAbove_cliqueCard, Exists.intro, And.intro, rfl
14 instances, coercions and auxiliary constants left out.
lemma Imo2007P3.maxCliqueCard_le line 56 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, Finset, Nat, LE.le, SimpleGraph.IsClique, Finset.card, Imo2007P3.maxCliqueCard
Proof uses: csSup_le, Set.ofPred, Exists, And, Eq, Imo2007P3.nonempty_cliqueCard, Membership.mem, Set
22 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma Imo2007P3.maxCliqueCard_le_card line 63 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, Finset, LE.le, Nat, Imo2007P3.maxCliqueCard, Finset.card
Proof uses: Imo2007P3.maxCliqueCard_le, SimpleGraph.IsClique, Finset.card_le_card
6 instances, coercions and auxiliary constants left out.
lemma Imo2007P3.maxCliqueCard_eq_card_of_isClique line 66 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, Finset, SimpleGraph.IsClique, Eq, Nat, Imo2007P3.maxCliqueCard, Finset.card
Proof uses: le_antisymm, Imo2007P3.maxCliqueCard_le_card, Imo2007P3.card_le_maxCliqueCard, Finset.Subset.refl
5 instances, coercions and auxiliary constants left out.
lemma Imo2007P3.maxCliqueCard_mono line 70 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, Finset, LE.le, Nat, Imo2007P3.maxCliqueCard
Proof uses: Imo2007P3.maxCliqueCard_le, SimpleGraph.IsClique, Imo2007P3.card_le_maxCliqueCard, LE.le.trans
9 instances, coercions and auxiliary constants left out.
lemma Imo2007P3.isClique_of_subset line 74 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, Finset, LE.le, SimpleGraph.IsClique
Proof uses: SimpleGraph.IsClique.subset, Iff.mpr, Set, Finset.coe_subset
11 instances, coercions and auxiliary constants left out.
lemma Imo2007P3.isClique_union line 78 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, Finset, SimpleGraph.IsClique, Membership.mem, Ne, SimpleGraph.Adj, Union.union, Set
Proof uses: Or, Eq.mp, congrArg, propext, Set.mem_union, SimpleGraph.Adj.symm, Ne.symm
10 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.
lemma Imo2007P3.maxCliqueCard_insert_le line 88 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, DecidableEq, Finset, LE.le, Nat, Imo2007P3.maxCliqueCard, Insert.insert, HAdd.hAdd, OfNat.ofNat
Proof uses: Imo2007P3.maxCliqueCard_le, SimpleGraph.IsClique, dite, Finset.card, Membership.mem, Finset.erase, Iff.mp, Finset.subset_insert_iff, Imo2007P3.card_le_maxCliqueCard, Imo2007P3.isClique_of_subset, Finset.erase_subset, Eq, Eq.symm, Finset.card_erase_add_one, Not, Finset.subset_insert_iff_of_notMem, LE.le.trans
20 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.
lemma Imo2007P3.maxCliqueCard_erase_le line 100 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, DecidableEq, Finset, LE.le, Nat, Imo2007P3.maxCliqueCard, HAdd.hAdd, Finset.erase, OfNat.ofNat
Proof uses: dite, Membership.mem, Insert.insert, Imo2007P3.maxCliqueCard_insert_le, Eq.mp, congrArg, Finset.insert_erase, Not, Eq.mpr, id, Eq, Finset.erase_eq_of_notMem
13 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
lemma Imo2007P3.exists_isClique_card_eq line 108 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, Finset, Exists, And, LE.le, SimpleGraph.IsClique, Eq, Nat, Finset.card, Imo2007P3.maxCliqueCard
Proof uses: SupSet.sSup, Set.ofPred, Membership.mem, Set, Nat.sSup_mem, Imo2007P3.nonempty_cliqueCard, Imo2007P3.bddAbove_cliqueCard, Exists.intro, And.intro
14 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
problem Imo2007P3.imo2007_p3 line 117 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, SimpleGraph, Even, Nat, Imo2007P3.maxCliqueCard, Finset.univ, Exists, Finset, And, Disjoint, Eq, Union.union
Proof uses: HAdd.hAdd, LE.le, SimpleGraph.IsClique, Finset.card, Imo2007P3.exists_isClique_card_eq, HMul.hMul, OfNat.ofNat, Eq.trans, Eq.mp, congrArg, Eq.symm, Nat.two_mul, SDiff.sdiff, Compl.compl, Finset.exists_subset_card_eq, Exists.intro, And.intro, Imo2007P3.maxCliqueCard_eq_card_of_isClique, Imo2007P3.isClique_of_subset, Finset.sdiff_subset and 91 more
80 instances, coercions and auxiliary constants left out.

Tactic trace: 290 steps (first 200 of 290)
Open to load the checked before-and-after goal states.