`Compfiles/Usa2009P2.lean`

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8 declarations, 0 contain sorry, 0 declared as axioms

USA Mathematical Olympiad 2009, Problem 2

Let n be a positive integer. Determine the size of the largest subset of {−n, −n+1, ..., n−1, n} which does not contain three elements a, b, c (not necessarily distinct) satisfying a + b + c = 0.

Imports (5)

Imported by 1 file

Compfiles.lean

abbrev Usa2009P2.IsGood A set of integers is called good if it does not contain three elements a, b, c (not necessarily distinct) satisfying a + b + c = 0 line 31 source
Source documentation
A set of integers is called good if it does not contain three elements a, b, c (not necessarily distinct) satisfying a + b + c = 0.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finset, Int
Proof uses: Membership.mem, Ne, HAdd.hAdd, OfNat.ofNat
5 instances, coercions and auxiliary constants left out.
```
abbrev IsGood (A : Finset ℤ) : Prop :=
…
```
determine Usa2009P2.answer line 34 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: ite, Even, HAdd.hAdd, OfNat.ofNat
4 instances, coercions and auxiliary constants left out.
lemma Usa2009P2.injOn_min_sub The pairing map x ↦ min x (c - x) is injective on any set containing no two elements summing to c: equality of the images forces x = y or x + y = c line 40 source
Source documentation
The pairing map x ↦ min x (c - x) is injective on any set containing no two elements summing to c: equality of the images forces x = y or x + y = c.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finset, Int, Membership.mem, Ne, HAdd.hAdd, Set.InjOn, Min.min, HSub.hSub
Proof uses: Set, Eq, Or, min_choice, Eq.mp, congrArg, absurd
26 instances, coercions and auxiliary constants left out.

Tactic trace: 15 steps
Open to load the checked before-and-after goal states.
```
lemma injOn_min_sub {B : Finset ℤ} {c : ℤ} (hB : ∀ x ∈ B, ∀ y ∈ B, x + y ≠ c) :
    Set.InjOn (fun x ↦ min x (c - x)) B := by
…
```
lemma Usa2009P2.card_le_of_pair Pairing counting lemma line 55 source
Source documentation
Pairing counting lemma. If B ⊆ [lo, hi] contains no two elements (not necessarily distinct) summing to c = lo + hi, then pairing x with c - x shows that B has at most as many elements as the interval [lo, (c-1)/2].

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finset, Int, LE.le, Finset.Icc, Eq, HAdd.hAdd, Membership.mem, Ne, Nat, Finset.card, Int.toNat, HSub.hSub, HDiv.hDiv, OfNat.ofNat
Proof uses: Min.min, And.left, Iff.mp, And, Finset.mem_Icc, And.right, Eq.mpr, id, congrArg, propext, And.intro, Iff.mpr, le_min_iff, Decidable.byContradiction, Not, LT.lt, lt_of_lt_of_le, Eq.mp, min_le_left, min_le_right and 4 more
42 instances, coercions and auxiliary constants left out.

Tactic trace: 14 steps
Open to load the checked before-and-after goal states.
```
lemma card_le_of_pair {B : Finset ℤ} {lo hi c : ℤ} (hsub : B ⊆ Finset.Icc lo hi)
    (hc : c = lo + hi) (hB : ∀ x ∈ B, ∀ y ∈ B, x + y ≠ c) :
    B.card ≤ ((c - 1) / 2 + 1 - lo).toNat := by
…
```
lemma Usa2009P2.card_le_case2 Case (ii) of the induction step: if both 2m and -(2m) lie in A (with m ≥ 1), then A has at most 2m elements line 76 source
Source documentation
Case (ii) of the induction step: if both 2m and -(2m) lie in A (with m ≥ 1), then A has at most 2m elements.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Finset, Int, Finset.Icc, Neg.neg, HMul.hMul, Usa2009P2.IsGood, Membership.mem, Finset.card
Proof uses: Not, Mathlib.Tactic.Ring.of_eq, HAdd.hAdd, Mathlib.Tactic.Ring.Common.add_congr, HPow.hPow, Nat.rawCast, Int.rawCast, inferInstance, Ring, Int.negOfNat, Mathlib.Tactic.Ring.Common.mul_congr, Mathlib.Tactic.Ring.cast_pos, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Tactic.Ring.Common.atom_pf, rfl, Eq.mpr, Eq, id, congrArg and 71 more
94 instances, coercions and auxiliary constants left out.

Tactic trace: 59 steps
Open to load the checked before-and-after goal states.
```
lemma card_le_case2 {m : ℕ} (hm : 1 ≤ m) {A : Finset ℤ}
    (hsub : A ⊆ Finset.Icc (-(2 * (m : ℤ))) (2 * (m : ℤ))) (hgood : IsGood A)
    (hN : (2 * (m : ℤ)) ∈ A) (hnegN : -(2 * (m : ℤ)) ∈ A) :
    A.card ≤ 2 * m := by
…
```
Mentions: Usa2009P2.IsGood
lemma Usa2009P2.card_le_case3 Case (iii) of the induction step: if 2m, 2m-1 and -(2m-1) lie in A but -(2m) does not (with m ≥ 1), then A has at most 2m elements line 144 source
Source documentation
Case (iii) of the induction step: if 2m, 2m-1 and -(2m-1) lie in A but -(2m) does not (with m ≥ 1), then A has at most 2m elements.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Finset, Int, Finset.Icc, Neg.neg, HMul.hMul, Usa2009P2.IsGood, Membership.mem, HSub.hSub, Not, Finset.card
Proof uses: dite, Eq, False.elim, Eq.mp, congrArg, Mathlib.Meta.NormNum.IsNat.to_eq, Mathlib.Meta.NormNum.isNat_mul, Eq.refl, Mathlib.Meta.NormNum.isNat_ofNat, Mathlib.Meta.NormNum.isNat_natCast, Mathlib.Meta.NormNum.IsInt.neg_to_eq, Mathlib.Meta.NormNum.isInt_neg, Int.ofNat, Int.negOfNat, Mathlib.Meta.NormNum.IsNat.to_isInt, Mathlib.Meta.NormNum.IsInt.to_isNat, Mathlib.Meta.NormNum.isInt_sub, of_eq_true, HAdd.hAdd, Eq.trans and 76 more
105 instances, coercions and auxiliary constants left out.

Tactic trace: 78 steps
Open to load the checked before-and-after goal states.
```
lemma card_le_case3 {m : ℕ} (hm : 1 ≤ m) {A : Finset ℤ}
    (hsub : A ⊆ Finset.Icc (-(2 * (m : ℤ))) (2 * (m : ℤ))) (hgood : IsGood A)
    (hN : (2 * (m : ℤ)) ∈ A) (hN1 : (2 * (m : ℤ)) - 1 ∈ A)
    (hnegN1 : -((2 * (m : ℤ)) - 1) ∈ A) (hnegN : -(2 * (m : ℤ)) ∉ A) :
    A.card ≤ 2 * m := by
…
```
Mentions: Usa2009P2.IsGood
lemma Usa2009P2.card_le_even The upper bound for even n: any good subset of [-2m, 2m] has at most 2m elements, by strong induction on m line 226 source
Source documentation
The upper bound for even n: any good subset of [-2m, 2m] has at most 2m elements, by strong induction on m.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Finset, Int, LE.le, Finset.Icc, Neg.neg, HMul.hMul, OfNat.ofNat, Usa2009P2.IsGood, Finset.card
Proof uses: Nat.strong_induction_on, LT.lt, Eq, GT.gt, Or, Nat.eq_zero_or_pos, Not, Membership.mem, of_eq_true, HAdd.hAdd, Eq.trans, True, congrFun', congrArg, Mathlib.Meta.NormNum.IsNat.to_eq, Mathlib.Meta.NormNum.isNat_add, Eq.refl, Mathlib.Meta.NormNum.isNat_ofNat, eq_self, eq_true and 52 more
99 instances, coercions and auxiliary constants left out.

Tactic trace: 141 steps
Open to load the checked before-and-after goal states.
```
lemma card_le_even (m : ℕ) (A : Finset ℤ) :
    A ⊆ Finset.Icc (-(2 * (m : ℤ))) (2 * (m : ℤ)) → IsGood A → A.card ≤ 2 * m := by
…
```
Mentions: Usa2009P2.IsGood
problem Usa2009P2.usa2009_p2 line 360 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, IsGreatest, Set.ofPred, Exists, Finset, Int, And, LE.le, Finset.Icc, Neg.neg, Usa2009P2.IsGood, Eq, Finset.card, Usa2009P2.answer
Proof uses: And.intro, Membership.mem, Set, upperBounds, Exists.intro, Union.union, HAdd.hAdd, HDiv.hDiv, Or, Eq.mp, congrArg, propext, Finset.mem_union, Eq.mpr, id, Finset.mem_Icc, GE.ge, HMul.hMul, False, Disjoint and 47 more
92 instances, coercions and auxiliary constants left out.

Tactic trace: 69 steps
Open to load the checked before-and-after goal states.