-- Mathlib/Combinatorics/Additive/ErdosGinzburgZiv.lean
theorem Int.erdos_ginzburg_ziv (a : ι → ℤ) (hs : 2 * n - 1 ≤ #s) :
    ∃ t ⊆ s, #t = n ∧ ↑n ∣ ∑ i ∈ t, a i

Proof state:
m : ℕ
hm : 2 ≤ m
ihm : ∀ {ι : Type u_1} {s : Finset ι} (a : ι → ℤ), 2 * m - 1 ≤ #s → ∃ t ⊆ s, #t = m ∧ ↑m ∣ ∑ i ∈ t, a i
n : ℕ
hn : 2 ≤ n
ihn : ∀ {ι : Type u_1} {s : Finset ι} (a : ι → ℤ), 2 * n - 1 ≤ #s → ∃ t ⊆ s, #t = n ∧ ↑n ∣ ∑ i ∈ t, a i
ι : Type u_1
s : Finset ι
a : ι → ℤ
hs : 2 * (m * n) - 1 ≤ #s
k : ℕ
ih : k ≤ 2 * m - 1 → ∃ 𝒜, #𝒜 = k ∧ (↑𝒜).Pairwise Disjoint ∧ ∀ ⦃t : Finset ι⦄, t ∈ 𝒜 → t ⊆ s ∧ #t = n ∧ ↑n ∣ ∑ i ∈ t, a i
hk : k + 1 ≤ 2 * m - 1
𝒜 : Finset (Finset ι)
h𝒜card : #𝒜 = k
h𝒜disj : (↑𝒜).Pairwise Disjoint
h𝒜 : ∀ ⦃t : Finset ι⦄, t ∈ 𝒜 → t ⊆ s ∧ #t = n ∧ ↑n ∣ ∑ i ∈ t, a i
this✝ : 2 * n - 1 ≤ #(s \ 𝒜.biUnion id)
t₀ : Finset ι
ht₀ : t₀ ⊆ s \ 𝒜.biUnion id
ht₀card : #t₀ = n
ht₀sum : ↑n ∣ ∑ i ∈ t₀, a i
this : t₀ ∉ 𝒜
⊢ #(cons t₀ 𝒜 this) = k + 1

Tactic:
rw [card_cons, h𝒜card]

Result:
no goals

Proof state:
case composite
m : ℕ
hm : 2 ≤ m
ihm : ∀ {ι : Type u_1} {s : Finset ι} (a : ι → ℤ), 2 * m - 1 ≤ #s → ∃ t ⊆ s, #t = m ∧ ↑m ∣ ∑ i ∈ t, a i
n : ℕ
hn : 2 ≤ n
ihn : ∀ {ι : Type u_1} {s : Finset ι} (a : ι → ℤ), 2 * n - 1 ≤ #s → ∃ t ⊆ s, #t = n ∧ ↑n ∣ ∑ i ∈ t, a i
ι : Type u_1
s : Finset ι
a : ι → ℤ
hs : 2 * (m * n) - 1 ≤ #s
⊢ ∀ k ≤ 2 * m - 1, ∃ 𝒜, #𝒜 = k ∧ (↑𝒜).Pairwise Disjoint ∧ ∀ ⦃t : Finset ι⦄, t ∈ 𝒜 → t ⊆ s ∧ #t = n ∧ ↑n ∣ ∑ i ∈ t, a i

Tactic:
rintro k hk

Result:
case composite
m : ℕ
hm : 2 ≤ m
ihm : ∀ {ι : Type u_1} {s : Finset ι} (a : ι → ℤ), 2 * m - 1 ≤ #s → ∃ t ⊆ s, #t = m ∧ ↑m ∣ ∑ i ∈ t, a i
n : ℕ
hn : 2 ≤ n
ihn : ∀ {ι : Type u_1} {s : Finset ι} (a : ι → ℤ), 2 * n - 1 ≤ #s → ∃ t ⊆ s, #t = n ∧ ↑n ∣ ∑ i ∈ t, a i
ι : Type u_1
s : Finset ι
a : ι → ℤ
hs : 2 * (m * n) - 1 ≤ #s
k : ℕ
hk : k ≤ 2 * m - 1
⊢ ∃ 𝒜, #𝒜 = k ∧ (↑𝒜).Pairwise Disjoint ∧ ∀ ⦃t : Finset ι⦄, t ∈ 𝒜 → t ⊆ s ∧ #t = n ∧ ↑n ∣ ∑ i ∈ t, a i

Proof state:
case intro.intro.intro.intro.intro.intro.refine_2
m : ℕ
hm : 2 ≤ m
ihm : ∀ {ι : Type u_1} {s : Finset ι} (a : ι → ℤ), 2 * m - 1 ≤ #s → ∃ t ⊆ s, #t = m ∧ ↑m ∣ ∑ i ∈ t, a i
n : ℕ
hn : 2 ≤ n
ihn : ∀ {ι : Type u_1} {s : Finset ι} (a : ι → ℤ), 2 * n - 1 ≤ #s → ∃ t ⊆ s, #t = n ∧ ↑n ∣ ∑ i ∈ t, a i
ι : Type u_1
s : Finset ι
a : ι → ℤ
hs : 2 * (m * n) - 1 ≤ #s
this :
  ∀ k ≤ 2 * m - 1, ∃ 𝒜, #𝒜 = k ∧ (↑𝒜).Pairwise Disjoint ∧ ∀ ⦃t : Finset ι⦄, t ∈ 𝒜 → t ⊆ s ∧ #t = n ∧ ↑n ∣ ∑ i ∈ t, a i
𝒜 : Finset (Finset ι)
h𝒜card : #𝒜 = 2 * m - 1
h𝒜disj : (↑𝒜).Pairwise Disjoint
h𝒜 : ∀ ⦃t : Finset ι⦄, t ∈ 𝒜 → t ⊆ s ∧ #t = n ∧ ↑n ∣ ∑ i ∈ t, a i
ℬ : Finset (Finset ι)
hℬ𝒜 : ℬ ⊆ 𝒜
hℬcard : #ℬ = m
hℬ : ↑m ∣ ∑ i ∈ ℬ, (∑ i ∈ i, a i) / ↑n
⊢ (↑ℬ).PairwiseDisjoint fun x => x

Tactic:
exact h𝒜disj.mono hℬ𝒜

Result:
no goals

Proof state:
case composite.succ.intro.intro.intro.intro.intro.intro.refine_1
m : ℕ
hm : 2 ≤ m
ihm : ∀ {ι : Type u_1} {s : Finset ι} (a : ι → ℤ), 2 * m - 1 ≤ #s → ∃ t ⊆ s, #t = m ∧ ↑m ∣ ∑ i ∈ t, a i
n : ℕ
hn : 2 ≤ n
ihn : ∀ {ι : Type u_1} {s : Finset ι} (a : ι → ℤ), 2 * n - 1 ≤ #s → ∃ t ⊆ s, #t = n ∧ ↑n ∣ ∑ i ∈ t, a i
ι : Type u_1
s : Finset ι
a : ι → ℤ
hs : 2 * (m * n) - 1 ≤ #s
k : ℕ
ih : k ≤ 2 * m - 1 → ∃ 𝒜, #𝒜 = k ∧ (↑𝒜).Pairwise Disjoint ∧ ∀ ⦃t : Finset ι⦄, t ∈ 𝒜 → t ⊆ s ∧ #t = n ∧ ↑n ∣ ∑ i ∈ t, a i
hk : k + 1 ≤ 2 * m - 1
𝒜 : Finset (Finset ι)
h𝒜card : #𝒜 = k
h𝒜disj : (↑𝒜).Pairwise Disjoint
h𝒜 : ∀ ⦃t : Finset ι⦄, t ∈ 𝒜 → t ⊆ s ∧ #t = n ∧ ↑n ∣ ∑ i ∈ t, a i
this✝ : 2 * n - 1 ≤ #(s \ 𝒜.biUnion id)
t₀ : Finset ι
ht₀ : t₀ ⊆ s \ 𝒜.biUnion id
ht₀card : #t₀ = n
ht₀sum : ↑n ∣ ∑ i ∈ t₀, a i
this : t₀ ∉ 𝒜
⊢ (↑(cons t₀ 𝒜 this)).Pairwise Disjoint

Tactic:
simp only [cons_eq_insert, coe_insert, Set.pairwise_insert_of_symmetric symmetric_disjoint,
  mem_coe, ne_eq]

Result:
case composite.succ.intro.intro.intro.intro.intro.intro.refine_1
m : ℕ
hm : 2 ≤ m
ihm : ∀ {ι : Type u_1} {s : Finset ι} (a : ι → ℤ), 2 * m - 1 ≤ #s → ∃ t ⊆ s, #t = m ∧ ↑m ∣ ∑ i ∈ t, a i
n : ℕ
hn : 2 ≤ n
ihn : ∀ {ι : Type u_1} {s : Finset ι} (a : ι → ℤ), 2 * n - 1 ≤ #s → ∃ t ⊆ s, #t = n ∧ ↑n ∣ ∑ i ∈ t, a i
ι : Type u_1
s : Finset ι
a : ι → ℤ
hs : 2 * (m * n) - 1 ≤ #s
k : ℕ
ih : k ≤ 2 * m - 1 → ∃ 𝒜, #𝒜 = k ∧ (↑𝒜).Pairwise Disjoint ∧ ∀ ⦃t : Finset ι⦄, t ∈ 𝒜 → t ⊆ s ∧ #t = n ∧ ↑n ∣ ∑ i ∈ t, a i
hk : k + 1 ≤ 2 * m - 1
𝒜 : Finset (Finset ι)
h𝒜card : #𝒜 = k
h𝒜disj : (↑𝒜).Pairwise Disjoint
h𝒜 : ∀ ⦃t : Finset ι⦄, t ∈ 𝒜 → t ⊆ s ∧ #t = n ∧ ↑n ∣ ∑ i ∈ t, a i
this✝ : 2 * n - 1 ≤ #(s \ 𝒜.biUnion id)
t₀ : Finset ι
ht₀ : t₀ ⊆ s \ 𝒜.biUnion id
ht₀card : #t₀ = n
ht₀sum : ↑n ∣ ∑ i ∈ t₀, a i
this : t₀ ∉ 𝒜
⊢ (↑𝒜).Pairwise Disjoint ∧ ∀ b ∈ 𝒜, ¬t₀ = b → Disjoint t₀ b

Proof state:
case prime.intro.intro.intro.intro
p : ℕ
hp : Nat.Prime p
ι : Type u_1
s : Finset ι
a : ι → ℤ
hs : 2 * p - 1 ≤ #s
this : Fact (Nat.Prime p)
t : Finset ι
hts : t ⊆ s
ht : #t = 2 * p - 1
u : Finset ι
hut : u ⊆ t
hu : #u = p ∧ ↑p ∣ ∑ i ∈ u, a i
⊢ ∃ t ⊆ s, #t = p ∧ ↑p ∣ ∑ i ∈ t, a i

Tactic:
exact ⟨u, hut.trans hts, hu⟩

Result:
no goals

Proof state:
case zero
ι : Type u_1
s : Finset ι
a : ι → ℤ
hs : 2 * 0 - 1 ≤ #s
⊢ ∃ t ⊆ s, #t = 0 ∧ ↑0 ∣ ∑ i ∈ t, a i

Tactic:
exact ⟨∅, by simp⟩

Result:
no goals