`Compfiles/Imo2001P3.lean`

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4 declarations, 0 contain sorry, 0 declared as axioms

International Mathematical Olympiad 2001, Problem 3

Twenty-one girls and twenty-one boys took part in a mathematical competition. It turned out that each contestant solved at most six problems, and for each pair of a girl and a boy, there was at most one problem solved by both the girl and the boy. Show that there was a problem solved by at least three girls and at least three boys.

Imports (6)

Imported by 1 file

Compfiles.lean

def Imo2001P3.Easy A problem is easy for a cohort (boys or girls) if at least three of its members solved it line 40 source
Source documentation
A problem is easy for a cohort (boys or girls) if at least three of its members solved it.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, Finset, Nat
Proof uses: LE.le, OfNat.ofNat, Finset.card, Finset.filter, Membership.mem, Finset.univ
6 instances, coercions and auxiliary constants left out.
```
def Easy {α : Type} [Fintype α] (F : α → Finset ℕ) (p : ℕ) : Prop := 3 ≤ #{i | p ∈ F i}
…
```
lemma Imo2001P3.card_not_easy_le_five Every contestant solved at most five problems that were not easy for the other cohort line 59 source
Source documentation
Every contestant solved at most five problems that were not easy for the other cohort.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, Finset, Nat, Eq, OfNat.ofNat, Fintype.card, LE.le, Finset.card, Not, Disjoint, Finset.filter, Imo2001P3.Easy, Classical.propDecidable
Proof uses: Decidable.byContradiction, le_antisymm, Finset.card_filter_le, LE.le.trans, Eq.mp, LT.lt, Trans.trans, Finset.sum, Membership.mem, Finset.univ, Finset.biUnion, Eq.mpr, id, congrArg, Finset.filter_true_of_mem, Eq.rec, Eq.refl, Finset.ext, of_eq_true, Iff and 27 more
49 instances, coercions and auxiliary constants left out.

Tactic trace: 13 steps
Open to load the checked before-and-after goal states.
```
lemma card_not_easy_le_five {α β : Type} [Fintype α]
    {β_solved : β → Finset ℕ} {α_solved : α → Finset ℕ}
    (hcard : 21 = Fintype.card α)
    {i : β} (hG : #(β_solved i) ≤ 6) (hB : ∀ j, ¬Disjoint (β_solved i) (α_solved j)) :
    #{p ∈ β_solved i | ¬Easy α_solved p} ≤ 5 := by
…
```
Mentions: Imo2001P3.Easy
lemma Imo2001P3.card_not_easy_le_210 There are at most 210 girl-boy pairs who solved some problem in common that was not easy for a fixed cohort line 79 source
Source documentation
There are at most 210 girl-boy pairs who solved some problem in common that was not easy for a fixed cohort.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, Eq, Nat, OfNat.ofNat, Fintype.card, Finset, LE.le, Finset.card, Not, Disjoint, Prod, Finset.filter, Exists, And, Imo2001P3.Easy, Membership.mem, Inter.inter, Prod.fst, Prod.snd, Classical.propDecidable and 1 more
Proof uses: Trans.trans, Finset.sum, HMul.hMul, Finset.biUnion, Eq.mpr, ite, id, congr, congrArg, Finset.card_filter, Finset.sum_congr, Eq.refl, of_eq_true, SProd.sprod, Eq.trans, Prod.mk, True, congrFun', Finset.sum_product, eq_self and 25 more
51 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.
```
lemma card_not_easy_le_210 {α β : Type} [Fintype α] [Fintype β]
    (hcard_α : 21 = Fintype.card α)
    (hcard_β : 21 = Fintype.card β)
    {α_solved : α → Finset ℕ} {β_solved : β → Finset ℕ}
    (hA : ∀ i, #(β_solved i) ≤ 6) (hB : ∀ i j, ¬Disjoint (β_solved i) (α_solved j)) :
    #{ij : β × α | ∃ p, ¬Easy α_solved p ∧ p ∈ β_solved ij.1 ∩ α_solved ij.2} ≤ 210 :=
…
```
Mentions: Imo2001P3.Easy
problem Imo2001P3.imo2001_p3 line 105 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, Finset, Nat, Eq, OfNat.ofNat, Fintype.card, LE.le, Finset.card, Not, Disjoint, Exists, And, Imo2001P3.Easy
Proof uses: Eq.mpr, id, congrArg, propext, disjoint_comm, Prod, Finset.filter, Membership.mem, Inter.inter, Prod.fst, Prod.snd, Classical.propDecidable, Finset.univ, Imo2001P3.card_not_easy_le_210, LT.lt, Union.union, Finset.map, Function.Embedding.mk, Prod.swap, Prod.swap_injective and 40 more
44 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.