`Compfiles/Usa1978P5.lean`

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4 declarations, 0 contain sorry, 0 declared as axioms

USA Mathematical Olympiad 1978, Problem 5

There are 9 delegates at a conference, each speaking at most three languages. Given any three delegates, at least 2 speak a common language. Show that there are three delegates with a common language.

Imports (2)

Imported by 1 file

Compfiles.lean

abbrev Usa1978P5.Share Delegate a and delegate b speak a common language line 27 source
Source documentation
Delegate a and delegate b speak a common language.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fin, OfNat.ofNat, Nat, Finset
Proof uses: Exists, And, Membership.mem
4 instances, coercions and auxiliary constants left out.
```
abbrev Share {L : Type*} (speaks : Fin 9 → Finset L) (a b : Fin 9) : Prop :=
…
```
abbrev Usa1978P5.SharedWith The delegates other than d that share a language with d line 35 source
Source documentation
The delegates other than d that share a language with d.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fin, OfNat.ofNat, Nat, Finset
Proof uses: Finset.filter, And, Ne, Usa1978P5.Share, Eq, Membership.mem, Classical.propDecidable, Finset.univ
10 instances, coercions and auxiliary constants left out.
```
noncomputable abbrev SharedWith {L : Type*} (speaks : Fin 9 → Finset L) (d : Fin 9) :
    Finset (Fin 9) :=
…
```
theorem Usa1978P5.card_sharedWith_le Documented declaration: card_sharedWith_le line 44 source
Source documentation
Assuming that no three delegates share a common language, a delegate d can share a language with at most three other delegates: among any four delegates sharing a language with d, two of them would share the same language with d by the pigeonhole principle, since d speaks at most three languages, and those two together with d would be three delegates with a common language.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fin, OfNat.ofNat, Nat, Finset, LE.le, Finset.card, Not, Exists, And, Ne, Membership.mem, Usa1978P5.SharedWith
Proof uses: Decidable.byContradiction, Subtype, And.right, Usa1978P5.Share, Finset.univ, Iff.mp, Finset.filter, Eq, Classical.propDecidable, Finset.mem_filter, Subtype.property, Finset.attach, False, Finset.exists_ne_map_eq_of_card_lt_of_maps_to, Eq.mpr, LT.lt, id, congrArg, Finset.card_attach, lt_of_le_of_lt and 9 more
27 instances, coercions and auxiliary constants left out.

Tactic trace: 14 steps
Open to load the checked before-and-after goal states.
```
theorem card_sharedWith_le {L : Type*} (speaks : Fin 9 → Finset L)
    (hthree : ∀ d : Fin 9, (speaks d).card ≤ 3)
    (hnothree : ¬∃ a b c : Fin 9, a ≠ b ∧ a ≠ c ∧ b ≠ c ∧
      ∃ l : L, l ∈ speaks a ∧ l ∈ speaks b ∧ l ∈ speaks c)
    (d : Fin 9) : (SharedWith speaks d).card ≤ 3 := by
…
```
Mentions: Usa1978P5.SharedWith
problem Usa1978P5.usa1978_p5 line 67 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fin, OfNat.ofNat, Nat, Finset, LE.le, Finset.card, Ne, Or, Usa1978P5.Share, Exists, And, Membership.mem
Proof uses: Decidable.byContradiction, Eq, Classical.propDecidable, Not, Usa1978P5.SharedWith, Usa1978P5.card_sharedWith_le, Nonempty, False, Nonempty.intro, Union.union, Singleton.singleton, HAdd.hAdd, Finset.card_union_le, Eq.mp, congrArg, Finset.card_singleton, Finset.univ, Finset.card_le_card, Eq.trans, binderNameHint and 21 more
32 instances, coercions and auxiliary constants left out.

Tactic trace: 41 steps
Open to load the checked before-and-after goal states.