`Compfiles/Imo1998P2.lean`

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24 declarations, 0 contain sorry, 0 declared as axioms

International Mathematical Olympiad 1998, Problem 2

In a competition, there are a contestants and b judges, where b ≥ 3 is an odd integer. Each judge rates each contestant as either "pass" or "fail". Suppose k is a number such that, for any two judges, their ratings coincide for at most k contestants.

Prove that k / a ≥ (b - 1) / (2b).

Imports (2)

Imported by 1 file

Compfiles.lean

abbrev Imo1998P2.JudgePair An ordered pair of judges line 40 source
Source documentation
An ordered pair of judges.

no axioms what this means
Proof uses: Prod
```
abbrev JudgePair (J : Type*) :=
…
```
abbrev Imo1998P2.JudgePair.judge₁ The first judge from an ordered pair of judges line 44 source
Source documentation
The first judge from an ordered pair of judges.

no axioms what this means
Statement uses: Imo1998P2.JudgePair
Proof uses: Prod.fst
```
abbrev JudgePair.judge₁ : JudgePair J → J :=
…
```
Mentions: Imo1998P2.JudgePair
abbrev Imo1998P2.JudgePair.judge₂ The second judge from an ordered pair of judges line 48 source
Source documentation
The second judge from an ordered pair of judges.

no axioms what this means
Statement uses: Imo1998P2.JudgePair
Proof uses: Prod.snd
```
abbrev JudgePair.judge₂ : JudgePair J → J :=
…
```
Mentions: Imo1998P2.JudgePair
abbrev Imo1998P2.JudgePair.Distinct The proposition that the judges in an ordered pair are distinct line 52 source
Source documentation
The proposition that the judges in an ordered pair are distinct.

no axioms what this means
Statement uses: Imo1998P2.JudgePair
Proof uses: Ne, Imo1998P2.JudgePair.judge₁, Imo1998P2.JudgePair.judge₂
```
abbrev JudgePair.Distinct (p : JudgePair J) :=
…
```
Mentions: Imo1998P2.JudgePair
abbrev Imo1998P2.JudgePair.Agree The proposition that the judges in an ordered pair agree about a contestant's rating line 56 source
Source documentation
The proposition that the judges in an ordered pair agree about a contestant's rating.

no axioms what this means
Statement uses: Imo1998P2.JudgePair
Proof uses: Iff, Imo1998P2.JudgePair.judge₁, Imo1998P2.JudgePair.judge₂
```
abbrev JudgePair.Agree (p : JudgePair J) (c : C) :=
…
```
Mentions: Imo1998P2.JudgePair
def Imo1998P2.agreedContestants The set of contestants on which two judges agree line 60 source
Source documentation
The set of contestants on which two judges agree.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, Imo1998P2.JudgePair, Finset
Proof uses: Finset.filter, Imo1998P2.JudgePair.Agree, Imo1998P2.JudgePair.judge₁, Imo1998P2.JudgePair.judge₂, Finset.univ
2 instances, coercions and auxiliary constants left out.
```
def agreedContestants [Fintype C] (p : JudgePair J) : Finset C :=
…
```
Mentions: Imo1998P2.JudgePair
abbrev Imo1998P2.AgreedTriple A triple consisting of contestant together with an ordered pair of judges line 66 source
Source documentation
A triple consisting of contestant together with an ordered pair of judges.

no axioms what this means
Proof uses: Prod, Imo1998P2.JudgePair
```
abbrev AgreedTriple (C J : Type*) :=
…
```
abbrev Imo1998P2.AgreedTriple.contestant The contestant from the triple consisting of a contestant and an ordered pair of judges line 70 source
Source documentation
The contestant from the triple consisting of a contestant and an ordered pair of judges.

no axioms what this means
Statement uses: Imo1998P2.AgreedTriple
Proof uses: Prod.fst, Imo1998P2.JudgePair
```
abbrev AgreedTriple.contestant : AgreedTriple C J → C :=
…
```
Mentions: Imo1998P2.AgreedTriple
abbrev Imo1998P2.AgreedTriple.judgePair The ordered pair of judges from the triple consisting of a contestant and an ordered pair of judges line 75 source
Source documentation
The ordered pair of judges from the triple consisting of a contestant and an ordered pair of judges.

no axioms what this means
Statement uses: Imo1998P2.AgreedTriple, Imo1998P2.JudgePair
Proof uses: Prod.snd
```
abbrev AgreedTriple.judgePair : AgreedTriple C J → JudgePair J :=
…
```
Mentions: Imo1998P2.AgreedTriple, Imo1998P2.JudgePair
theorem Imo1998P2.JudgePair.agree_iff_same_rating line 104 source
no axioms what this means
Statement uses: Imo1998P2.JudgePair, Iff, Imo1998P2.JudgePair.Agree, Imo1998P2.JudgePair.judge₁, Imo1998P2.JudgePair.judge₂
Proof uses: Iff.rfl
def Imo1998P2.A All incidences of agreement line 111 source
Source documentation
All incidences of agreement.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, Finset, Imo1998P2.AgreedTriple
Proof uses: Finset.filter, And, Imo1998P2.JudgePair.Agree, Imo1998P2.AgreedTriple.judgePair, Imo1998P2.AgreedTriple.contestant, Imo1998P2.JudgePair.Distinct, Imo1998P2.JudgePair.judge₁, Imo1998P2.JudgePair.judge₂, Eq, Finset.univ, Imo1998P2.JudgePair
5 instances, coercions and auxiliary constants left out.
```
def A : Finset (AgreedTriple C J) :=
…
```
Mentions: Imo1998P2.AgreedTriple
theorem Imo1998P2.A_maps_to_offDiag_judgePair line 115 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, Imo1998P2.AgreedTriple, Membership.mem, Finset, Imo1998P2.A, Imo1998P2.JudgePair, Prod, Finset.offDiag, Finset.univ, Imo1998P2.AgreedTriple.judgePair
Proof uses: of_eq_true, Eq.trans, Iff, Imo1998P2.AgreedTriple.contestant, Imo1998P2.JudgePair.judge₁, Imo1998P2.JudgePair.judge₂, Not, Eq, Prod.fst, Prod.snd, True, And, implies_congr, Finset.filter, congrFun', congrArg, Finset.filter_congr, Imo1998P2.JudgePair.Agree, Imo1998P2.JudgePair.Distinct, Iff.of_eq and 5 more
15 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Imo1998P2.A_fibre_over_contestant line 118 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, Eq, Finset, Imo1998P2.JudgePair, Finset.filter, And, Imo1998P2.JudgePair.Agree, Imo1998P2.JudgePair.Distinct, Imo1998P2.JudgePair.judge₁, Imo1998P2.JudgePair.judge₂, Finset.univ, Finset.image, Prod, Prod.snd, Imo1998P2.AgreedTriple, Imo1998P2.AgreedTriple.contestant, Imo1998P2.A
Proof uses: Finset.ext, of_eq_true, Iff, Membership.mem, Eq.trans, Not, Prod.mk, Imo1998P2.AgreedTriple.judgePair, True, congr, congrArg, congrFun', Finset.filter_congr, Iff.of_eq, true_and, Exists, Finset.filter.congr_simp, Eq.refl, funext, iff_self
20 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem Imo1998P2.A_fibre_over_contestant_card line 124 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, Eq, Nat, Finset.card, Imo1998P2.JudgePair, Finset.filter, And, Imo1998P2.JudgePair.Agree, Imo1998P2.JudgePair.Distinct, Imo1998P2.JudgePair.judge₁, Imo1998P2.JudgePair.judge₂, Finset.univ, Imo1998P2.AgreedTriple, Imo1998P2.AgreedTriple.contestant, Imo1998P2.A
Proof uses: Eq.mpr, Finset.image, Prod, Prod.snd, id, congrArg, Finset, Imo1998P2.A_fibre_over_contestant, Finset.card_image_of_injOn, Set.InjOn, Membership.mem, Set, Prod.fst, Prod.mk, of_eq_true, Eq.trans, True, congrFun', congr, Eq.mp and 4 more
18 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
theorem Imo1998P2.A_fibre_over_judgePair line 132 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, Imo1998P2.JudgePair, Imo1998P2.JudgePair.Distinct, Eq, Finset, Imo1998P2.agreedContestants, Finset.image, Imo1998P2.AgreedTriple, Imo1998P2.AgreedTriple.contestant, Finset.filter, Imo1998P2.AgreedTriple.judgePair, Imo1998P2.A
Proof uses: id, Finset.ext, Imo1998P2.JudgePair.Agree, Imo1998P2.JudgePair.judge₁, Imo1998P2.JudgePair.judge₂, Finset.univ, And, Iff.intro, Membership.mem, Eq.mpr, Exists, congrArg, propext, Finset.mem_image, Exists.intro, Prod.mk, of_eq_true, Eq.trans, True, congr and 17 more
19 instances, coercions and auxiliary constants left out.

Tactic trace: 10 steps
Open to load the checked before-and-after goal states.
theorem Imo1998P2.A_fibre_over_judgePair_card line 139 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, Imo1998P2.JudgePair, Imo1998P2.JudgePair.Distinct, Eq, Nat, Finset.card, Imo1998P2.agreedContestants, Imo1998P2.AgreedTriple, Finset.filter, Imo1998P2.AgreedTriple.judgePair, Imo1998P2.A
Proof uses: Eq.mpr, Finset.image, Imo1998P2.AgreedTriple.contestant, id, congrArg, Finset, Imo1998P2.A_fibre_over_judgePair, Finset.card_image_of_injOn, Set.InjOn, Membership.mem, Set, Prod.snd, Not, Imo1998P2.JudgePair.judge₁, Imo1998P2.JudgePair.judge₂, And, Prod.mk, of_eq_true, Eq.trans, True and 8 more
11 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
theorem Imo1998P2.A_card_upper_bound line 147 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, Nat, Imo1998P2.JudgePair, Imo1998P2.JudgePair.Distinct, LE.le, Finset.card, Imo1998P2.agreedContestants, Imo1998P2.AgreedTriple, Imo1998P2.A, HMul.hMul, HSub.hSub, Fintype.card
Proof uses: id, Eq.mpr, Finset.univ, Prod, Finset.offDiag, Eq, congrArg, Eq.symm, Finset.offDiag_card, Finset.card_le_mul_card_image_of_maps_to, Imo1998P2.AgreedTriple.judgePair, Imo1998P2.A_maps_to_offDiag_judgePair, Membership.mem, Finset, Eq.mp, Not, Prod.fst, Prod.snd, Eq.trans, And and 7 more
16 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
theorem Imo1998P2.add_sq_add_sq_sub line 159 source
standard axioms: propext what this means
Statement uses: Ring, Eq, HAdd.hAdd, HMul.hMul, HSub.hSub, OfNat.ofNat, Nat
Proof uses: Eq.mpr, Neg.neg, HSMul.hSMul, id, congr, congrArg, Eq.trans, mul_add, add_mul, sub_eq_add_neg, neg_mul, mul_neg, congrFun', Mathlib.Tactic.NoncommRing.nat_lit_mul_eq_nsmul, smul_mul_assoc
59 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Imo1998P2.norm_bound_of_odd_sum line 162 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Int, Eq, HAdd.hAdd, HMul.hMul, OfNat.ofNat, LE.le
Proof uses: Eq.mpr, id, congrArg, Mathlib.Tactic.Ring.of_eq, Nat.rawCast, HPow.hPow, Nat, Mathlib.Tactic.Ring.Common.mul_congr, Mathlib.Tactic.Ring.Common.add_congr, Mathlib.Tactic.Ring.cast_pos, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Tactic.Ring.Common.atom_pf, rfl, Eq.symm, Mathlib.Tactic.Ring.Common.add_mul, Mathlib.Tactic.Ring.Common.mul_add, Mathlib.Tactic.Ring.Common.mul_pf_right, Mathlib.Meta.NormNum.IsNat.to_raw_eq, Mathlib.Meta.NormNum.isNat_mul and 33 more
88 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.
theorem Imo1998P2.judge_pairs_card_lower_bound line 175 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, Nat, Eq, Fintype.card, HAdd.hAdd, HMul.hMul, OfNat.ofNat, LE.le, Finset.card, Imo1998P2.JudgePair, Finset.filter, Imo1998P2.JudgePair.Agree, Imo1998P2.JudgePair.judge₁, Imo1998P2.JudgePair.judge₂, Finset.univ
Proof uses: Not, of_eq_true, Eq.trans, Iff, Prod, True, congr, congrArg, Finset, Finset.filter_congr, Membership.mem, Iff.of_eq, Prod.fst, Prod.snd, SProd.sprod, eq_self, Eq.mpr, id, Int.le_of_ofNat_le_ofNat, Int and 8 more
51 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.
theorem Imo1998P2.distinct_judge_pairs_card_lower_bound line 186 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, Nat, Eq, Fintype.card, HAdd.hAdd, HMul.hMul, OfNat.ofNat, LE.le, Finset.card, Imo1998P2.JudgePair, Finset.filter, And, Imo1998P2.JudgePair.Agree, Imo1998P2.JudgePair.Distinct, Imo1998P2.JudgePair.judge₁, Imo1998P2.JudgePair.judge₂, Finset.univ
Proof uses: Finset, Imo1998P2.judge_pairs_card_lower_bound, SDiff.sdiff, Finset.diag, Finset.ext, Iff.intro, Membership.mem, of_eq_true, Prod, Eq.trans, Prod.fst, Prod.snd, True, congr, congrArg, id, Iff, Eq.mp, Not, congrFun' and 21 more
35 instances, coercions and auxiliary constants left out.

Tactic trace: 23 steps
Open to load the checked before-and-after goal states.
theorem Imo1998P2.A_card_lower_bound line 202 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, Nat, Eq, Fintype.card, HAdd.hAdd, HMul.hMul, OfNat.ofNat, LE.le, Finset.card, Imo1998P2.AgreedTriple, Imo1998P2.A
Proof uses: Membership.mem, Finset, Finset.univ, Prod.fst, Imo1998P2.JudgePair, Finset.mem_univ, Finset.mul_card_image_le_card_of_maps_to, Eq.mpr, Finset.filter, And, Imo1998P2.JudgePair.Agree, Imo1998P2.JudgePair.Distinct, Imo1998P2.JudgePair.judge₁, Imo1998P2.JudgePair.judge₂, id, congrArg, Imo1998P2.AgreedTriple.contestant, Eq.symm, Imo1998P2.A_fibre_over_contestant_card, Imo1998P2.distinct_judge_pairs_card_lower_bound
19 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
theorem Imo1998P2.clear_denominators line 212 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Iff, LE.le, Rat, HDiv.hDiv, HSub.hSub, HMul.hMul
Proof uses: Eq.mpr, id, Eq, congrArg, propext, div_le_div_iff₀, of_eq_true, Eq.trans, True, eq_true, eq_of_heq, HEq, Eq.refl, HEq.refl, Eq.symm, LE, Nat.cast_mul, congrFun', Nat.cast_pred, eq_self and 1 more
86 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
problem Imo1998P2.imo1998_p2 line 225 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, Nat, Eq, Fintype.card, LT.lt, OfNat.ofNat, Odd, Imo1998P2.JudgePair, Imo1998P2.JudgePair.Distinct, LE.le, Finset.card, Imo1998P2.agreedContestants, Rat, HDiv.hDiv, HSub.hSub, HMul.hMul
Proof uses: Eq.mpr, id, congrArg, propext, Imo1998P2.clear_denominators, Odd.pos, HAdd.hAdd, le_trans, Imo1998P2.AgreedTriple, Imo1998P2.A, Imo1998P2.A_card_lower_bound, Eq.mp, Imo1998P2.A_card_upper_bound, congr, Eq.trans, congrFun', mul_comm, add_mul, one_mul, add_tsub_cancel_right and 36 more
95 instances, coercions and auxiliary constants left out.

Tactic trace: 15 steps
Open to load the checked before-and-after goal states.