`Compfiles/Usa1979P3.lean`

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13 declarations, 0 contain sorry, 0 declared as axioms

USA Mathematical Olympiad 1979, Problem 3

$a_1, a_2, \dots, a_n$ is an arbitrary sequence of positive integers. A member of the sequence is picked at random. Its value is $a$. Another member is picked at random, independently of the first. Its value is $b$. Then a third, value $c$. Show that the probability that $a + b + c$ is divisible by $3$ is at least $1 / 4$.

Imports (9)

Imported by 1 file

Compfiles.lean

def Usa1979P3.res The residue of a i modulo 3 line 37 source
Source documentation
The residue of a i modulo 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, ZMod, OfNat.ofNat
10 instances, coercions and auxiliary constants left out.
```
def res {n : ℕ} (a : Fin n → ℕ) (i : Fin n) : ZMod 3 := (a i : ZMod 3)
```
def Usa1979P3.cnt The number of entries of a having residue x modulo 3 line 40 source
Source documentation
The number of entries of a having residue x modulo 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, ZMod, OfNat.ofNat
Proof uses: Finset.card, Finset.filter, Eq, Usa1979P3.res, Finset.univ
4 instances, coercions and auxiliary constants left out.
```
def cnt {n : ℕ} (a : Fin n → ℕ) (x : ZMod 3) : ℕ :=
…
```
def Usa1979P3.patterns The "good" residue patterns: triples of residues that sum to 0 line 44 source
Source documentation
The "good" residue patterns: triples of residues that sum to 0.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finset, Fin, OfNat.ofNat, Nat, ZMod
Proof uses: Finset.filter, Eq, HAdd.hAdd, Finset.univ
20 instances, coercions and auxiliary constants left out.
```
def patterns : Finset (Fin 3 → ZMod 3) := univ.filter fun u ↦ u 0 + u 1 + u 2 = 0
…
```
lemma Usa1979P3.dvd_iff_res Divisibility by 3 in terms of residues in ZMod 3 line 49 source
Source documentation
Divisibility by 3 in terms of residues in ZMod 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Iff, Dvd.dvd, OfNat.ofNat, HAdd.hAdd, Eq, ZMod
Proof uses: Eq.mpr, id, congrArg, Eq.symm, propext, ZMod.natCast_eq_zero_iff, Nat.cast_add, Iff.rfl
46 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
lemma dvd_iff_res {x y z : ℕ} :
    3 ∣ x + y + z ↔ (x : ZMod 3) + (y : ZMod 3) + (z : ZMod 3) = 0 := by
…
```
lemma Usa1979P3.sum_univ_zmod3 Summing over ZMod 3 gives three terms line 54 source
Source documentation
Summing over ZMod 3 gives three terms.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: ZMod, OfNat.ofNat, Nat, Eq, Finset.sum, Finset.univ, HAdd.hAdd
Proof uses: Fin.sum_univ_three
25 instances, coercions and auxiliary constants left out.
```
lemma sum_univ_zmod3 (f : ZMod 3 → ℕ) : ∑ x : ZMod 3, f x = f 0 + f 1 + f 2 :=
…
```
Mentions: f
lemma Usa1979P3.cnt_sum The residues split Fin n into the three fibers of res a line 58 source
Source documentation
The residues split Fin n into the three fibers of res a.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, Eq, HAdd.hAdd, Usa1979P3.cnt, OfNat.ofNat, ZMod
Proof uses: Finset.card, Finset.univ, Finset.sum, Finset.filter, Usa1979P3.res, Finset.card_eq_sum_card_fiberwise, Membership.mem, Set, Finset, Finset.mem_univ, Eq.symm, Eq.mp, Fintype.card, congrArg, Fintype.card_fin, Finset.card_univ, Usa1979P3.sum_univ_zmod3
49 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
```
lemma cnt_sum {n : ℕ} (a : Fin n → ℕ) :
    cnt a 0 + cnt a 1 + cnt a 2 = n := by
…
```
Mentions: Usa1979P3.cnt
lemma Usa1979P3.fiber_card The fiber of triples with prescribed residues is a product of fibers line 67 source
Source documentation
The fiber of triples with prescribed residues is a product of fibers.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, OfNat.ofNat, ZMod, Eq, Finset.card, Finset.filter, Usa1979P3.res, Finset.univ, Finset.prod, Usa1979P3.cnt
Proof uses: Finset, Fintype.piFinset, Finset.ext, of_eq_true, Iff, Membership.mem, Eq.trans, True, congr, congrArg, And, congrFun', true_and, forall_congr, iff_self, Eq.mpr, id, Fintype.card_piFinset, Eq.refl
19 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
```
lemma fiber_card {n : ℕ} (a : Fin n → ℕ) (u : Fin 3 → ZMod 3) :
    (univ.filter fun t : Fin 3 → Fin n ↦ ∀ j, res a (t j) = u j).card
      = ∏ j, cnt a (u j) := by
…
```
Mentions: Usa1979P3.res, Usa1979P3.cnt
lemma Usa1979P3.good_card The number of good triples, grouped by their residue pattern line 78 source
Source documentation
The number of good triples, grouped by their residue pattern.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, Eq, Fintype.card, Subtype, OfNat.ofNat, Dvd.dvd, HAdd.hAdd, Finset.sum, ZMod, Usa1979P3.patterns, Finset.prod, Finset.univ, Usa1979P3.cnt
Proof uses: Membership.mem, Finset, Finset.filter, Usa1979P3.res, id, Eq.mpr, congrArg, Eq.symm, propext, Usa1979P3.dvd_iff_res, Eq.mp, Eq.trans, And, True, congrFun', true_and, of_eq_true, congr, eq_self, and_self and 16 more
54 instances, coercions and auxiliary constants left out.

Tactic trace: 27 steps
Open to load the checked before-and-after goal states.
```
lemma good_card {n : ℕ} (a : Fin n → ℕ) :
    Fintype.card {t : Fin 3 → Fin n // 3 ∣ a (t 0) + a (t 1) + a (t 2)}
      = ∑ u ∈ patterns, ∏ j, cnt a (u j) := by
…
```
Mentions: Usa1979P3.patterns, Usa1979P3.cnt
lemma Usa1979P3.sum_over_patterns Summing over the good patterns by reindexing with the first two entries line 111 source
Source documentation
Summing over the good patterns by reindexing with the first two entries.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fin, OfNat.ofNat, Nat, ZMod, Eq, Finset.sum, Usa1979P3.patterns, Prod, Finset.univ, Matrix.vecCons, Prod.fst, Prod.snd, Neg.neg, HAdd.hAdd, Matrix.vecEmpty
Proof uses: Finset.sum_bij, Membership.mem, Finset, Prod.mk, Finset.mem_univ, funext, And.right, Iff.mp, Finset.filter, And, Finset.mem_filter, congrArg, List, List.Mem, List.finRange, HEq, Fintype.elems, Fintype.complete, List.cons, List.Mem.head and 71 more
93 instances, coercions and auxiliary constants left out.

Tactic trace: 47 steps
Open to load the checked before-and-after goal states.
```
lemma sum_over_patterns (F : (Fin 3 → ZMod 3) → ℕ) :
    ∑ u ∈ patterns, F u
      = ∑ p : ZMod 3 × ZMod 3, F ![p.1, p.2, -(p.1 + p.2)] := by
…
```
Mentions: Usa1979P3.patterns
lemma Usa1979P3.pattern_sum The good triple count in terms of the residue counts line 150 source
Source documentation
The good triple count in terms of the residue counts.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: ZMod, OfNat.ofNat, Nat, Eq, Finset.sum, Fin, Usa1979P3.patterns, Finset.prod, Finset.univ, HAdd.hAdd, HPow.hPow, HMul.hMul
Proof uses: Eq.mpr, Prod, Matrix.vecCons, Prod.fst, Prod.snd, Neg.neg, Matrix.vecEmpty, id, congrArg, Usa1979P3.sum_over_patterns, Prod.mk, Fintype.sum_prod_type, congrFun', Finset.sum_congr, Eq.refl, Finset, Membership.mem, Eq.trans, Finset.prod_congr, Fin.prod_univ_three and 43 more
85 instances, coercions and auxiliary constants left out.

Tactic trace: 24 steps
Open to load the checked before-and-after goal states.
```
lemma pattern_sum (c : ZMod 3 → ℕ) :
    ∑ u ∈ patterns, ∏ j, c (u j)
      = c 0 ^ 3 + c 1 ^ 3 + c 2 ^ 3 + 6 * (c 0 * c 1 * c 2) := by
…
```
Mentions: Usa1979P3.patterns
lemma Usa1979P3.ineq_max The key inequality in the case where x is a maximum of x, y, z line 171 source
Source documentation
The key inequality in the case where x is a maximum of x, y, z.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, LE.le, OfNat.ofNat, HPow.hPow, Nat, HAdd.hAdd, HMul.hMul
Proof uses: add_nonneg, HSub.hSub, le_of_not_gt, LT.lt, Mathlib.Tactic.Linarith.lt_irrefl, Eq.mp, congrArg, Mathlib.Tactic.Ring.of_eq, Mathlib.Tactic.Ring.Common.add_congr, Nat.rawCast, Int.rawCast, inferInstance, Ring, Int.negOfNat, Mathlib.Meta.NormNum.instAddMonoidWithOne, Mathlib.Tactic.Ring.Common.sub_congr, Mathlib.Tactic.Ring.Common.atom_pf, rfl, Eq.mpr, Eq and 87 more
112 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
```
lemma ineq_max {x y z : ℝ} (hx : 0 ≤ x) (hy : 0 ≤ y) (hz : 0 ≤ z)
    (hyx : y ≤ x) (hzx : z ≤ x) :
    (x + y + z) ^ 3 ≤ 4 * (x ^ 3 + y ^ 3 + z ^ 3 + 6 * (x * y * z)) := by
…
```
lemma Usa1979P3.ineq3 The key inequality: 4 * (p³ + q³ + r³ + 6pqr) ≥ (p + q + r)³ for p q r ≥ 0 line 180 source
Source documentation
The key inequality: 4 * (p³ + q³ + r³ + 6pqr) ≥ (p + q + r)³ for p q r ≥ 0.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, LE.le, OfNat.ofNat, HPow.hPow, Nat, HAdd.hAdd, HMul.hMul
Proof uses: Or, le_total, Usa1979P3.ineq_max, le_of_not_gt, LT.lt, Mathlib.Tactic.Linarith.lt_irrefl, Eq.mp, HSub.hSub, congrArg, Mathlib.Tactic.Ring.of_eq, Mathlib.Tactic.Ring.Common.add_congr, Nat.rawCast, Int.rawCast, inferInstance, Ring, Int.negOfNat, Mathlib.Tactic.Ring.Common.sub_congr, Mathlib.Tactic.Ring.Common.atom_pf, rfl, Eq.mpr and 62 more
78 instances, coercions and auxiliary constants left out.

Tactic trace: 23 steps
Open to load the checked before-and-after goal states.
```
lemma ineq3 {x y z : ℝ} (hx : 0 ≤ x) (hy : 0 ≤ y) (hz : 0 ≤ z) :
    (x + y + z) ^ 3 ≤ 4 * (x ^ 3 + y ^ 3 + z ^ 3 + 6 * (x * y * z)) := by
…
```
problem Usa1979P3.usa1979_p3 line 195 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, NeZero, Fin, LT.lt, OfNat.ofNat, LE.le, ENNReal, HDiv.hDiv, MeasureTheory.OuterMeasure, Set, PMF.toOuterMeasure, PMF.uniformOfFintype, Set.ofPred, Dvd.dvd, HAdd.hAdd
Proof uses: Usa1979P3.cnt, ZMod, Eq, rfl, HPow.hPow, HMul.hMul, Fintype.card, of_eq_true, Eq.trans, True, congrFun', congrArg, Finset.prod, Finset.univ, Fintype.card_pi, Finset.card, Finset.prod_congr, Eq.refl, Finset, Membership.mem and 47 more
162 instances, coercions and auxiliary constants left out.

Tactic trace: 31 steps
Open to load the checked before-and-after goal states.