sorry
A✝ B✝ : Set ℕ
n✝ : ℕ
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
α : ℝ := schnirelmannDensity A
halpha : α = schnirelmannDensity A
β : ℝ := schnirelmannDensity B
hbeta : β = schnirelmannDensity B
dum : α * (1 - β) + β = α + β - α * β
n : ℕ
n1 : n > 0
lem : ⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n} ⊆ (A + B) ∩ ↑(Icc 1 n)
aux : countelements (⋃ a, {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n}) n ≤ countelements (A + B) n
a : ↑A
⊢ 1 + countelements B (next_elm A a n - ↑a - 1) ≤ countelements {c | c ∈ A + B ∧ 0 < c - ↑a ∧ c ≤ next_elm A a n} n
no goals