`Compfiles/Usa2002P1.lean`

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7 declarations, 0 contain sorry, 0 declared as axioms

USA Mathematical Olympiad 2002, Problem 1

Let S be a set with 2002 elements, and let N be an integer with

0 ≤ N ≤ 2^2002.

Prove that it is possible to color every subset of S either blue or red so that the following conditions hold:

a) the union of any two red subsets is red; b) the union of any two blue subsets is blue; c) there are exactly N red subsets.

Imports (5)

Imported by 1 file

Compfiles.lean

inductive Usa2002P1.Color line 38 source
no axioms what this means

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
instance Usa2002P1.(instance at line 43) line 43 source
lemma Usa2002P1.Finset.subtype_union Seems like this maybe belongs in mathlib? line 50 source
Source comment
Seems like this maybe belongs in mathlib?

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: DecidableEq, DecidablePred, Finset, Eq, Subtype, Finset.subtype, Union.union
Proof uses: Finset.ext, of_eq_true, Iff, Membership.mem, Eq.trans, Or, True, congr, congrArg, iff_self
9 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
lemma Finset.subtype_union {α} (p : α → Prop) [DecidableEq α] [DecidablePred p] (s1 s2 : Finset α) :
    Finset.subtype p (s1 ∪ s2) = Finset.subtype p s1 ∪ Finset.subtype p s2 := by
…
```
lemma Usa2002P1.Finset.subtype_map_subtype line 54 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: DecidablePred, Finset, Subtype, Eq, Finset.subtype, Finset.map, Function.Embedding.subtype
Proof uses: Finset.ext, Iff, Membership.mem, of_eq_true, Subtype.mk, Eq.trans, Iff.mpr, Iff.of_eq, Eq.refl, True, eq_true, True.intro, congrFun', congrArg, Exists, And, Function.Embedding, funext, propext, exists_prop_congr and 1 more
12 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Usa2002P1.lemma1 line 60 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, Eq, Nat, Fintype.card, Subtype, Finset, Membership.mem, Not
Proof uses: Subtype.mk, Finset.cons, Finset.mem_cons_self, Function.Bijective, And.intro, Function.Injective, Function.Surjective, Eq.mpr, id, congrArg, Subtype.mk.injEq, Eq.mp, Finset.erase, Subtype.property, congr, Eq.trans, Insert.insert, congrFun', Finset.cons_eq_insert, Finset.erase_insert_eq_erase and 16 more
18 instances, coercions and auxiliary constants left out.

Tactic trace: 15 steps
Open to load the checked before-and-after goal states.
lemma Usa2002P1.usa2002_p1_generalized line 77 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: DecidableEq, Fintype, Nat, Eq, Fintype.card, LE.le, HPow.hPow, OfNat.ofNat, Exists, Finset, Usa2002P1.Color, And, Union.union, Subtype, Usa2002P1.Color.red
Proof uses: Nat.recAux, dite, Exists.intro, of_eq_true, Eq.trans, True, congr, congrArg, forall_congr, implies_congr, eq_self, implies_true, congrFun', Finset.card, Finset.filter, Finset.univ, Fintype.card_congr', funext, Fintype.card_subtype, Finset.filter_true and 112 more
116 instances, coercions and auxiliary constants left out.

Tactic trace: 198 steps
Open to load the checked before-and-after goal states.
problem Usa2002P1.usa2002_p1 line 279 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: DecidableEq, Fintype, Eq, Nat, Fintype.card, OfNat.ofNat, LE.le, HPow.hPow, Exists, Finset, Usa2002P1.Color, And, Union.union, Subtype, Usa2002P1.Color.red
Proof uses: Usa2002P1.usa2002_p1_generalized
16 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.