-- Mathlib/Combinatorics/Schnirelmann.lean
/-- The Schnirelmann density of any finset is `0`. -/
lemma schnirelmannDensity_finset (A : Finset ℕ) : schnirelmannDensity A = 0

Proof state:
A : Finset ℕ
ε : ℝ
hε : 0 < ε
hε₁ : ε ≤ 1
n : ℕ := ⌊↑(#A) / ε⌋₊ + 1
hn : 0 < n
⊢ ↑(#(filter (fun a => a ∈ ↑A) (Ioc 0 n))) / ε ≤ ↑(#A) / ε

Tactic:
gcongr

Result:
case hab.h.a
A : Finset ℕ
ε : ℝ
hε : 0 < ε
hε₁ : ε ≤ 1
n : ℕ := ⌊↑(#A) / ε⌋₊ + 1
hn : 0 < n
⊢ filter (fun a => a ∈ ↑A) (Ioc 0 n) ⊆ A

Proof state:
case inr
A : Finset ℕ
ε : ℝ
hε : 0 < ε
this : ∀ (ε : ℝ), 0 < ε → ε ≤ 1 → ∃ n, 0 < n ∧ ↑(#(filter (fun a => a ∈ ↑A) (Ioc 0 n))) / ↑n < ε
hε₁ : ¬ε ≤ 1
⊢ ∃ n, 0 < n ∧ ↑(#(filter (fun a => a ∈ ↑A) (Ioc 0 n))) / ↑n < ε

Tactic:
obtain ⟨n, hn, hn'⟩ := this 1 zero_lt_one le_rfl

Result:
case inr.intro.intro
A : Finset ℕ
ε : ℝ
hε : 0 < ε
this : ∀ (ε : ℝ), 0 < ε → ε ≤ 1 → ∃ n, 0 < n ∧ ↑(#(filter (fun a => a ∈ ↑A) (Ioc 0 n))) / ↑n < ε
hε₁ : ¬ε ≤ 1
n : ℕ
hn : 0 < n
hn' : ↑(#(filter (fun a => a ∈ ↑A) (Ioc 0 n))) / ↑n < 1
⊢ ∃ n, 0 < n ∧ ↑(#(filter (fun a => a ∈ ↑A) (Ioc 0 n))) / ↑n < ε

Proof state:
A : Finset ℕ
ε : ℝ
hε : 0 < ε
hε₁ : ε ≤ 1
n : ℕ := ⌊↑(#A) / ε⌋₊ + 1
⊢ ∃ n, 0 < n ∧ ↑(#(filter (fun a => a ∈ ↑A) (Ioc 0 n))) / ↑n < ε

Tactic:
have hn : 0 < n := Nat.succ_pos _

Result:
A : Finset ℕ
ε : ℝ
hε : 0 < ε
hε₁ : ε ≤ 1
n : ℕ := ⌊↑(#A) / ε⌋₊ + 1
hn : 0 < n
⊢ ∃ n, 0 < n ∧ ↑(#(filter (fun a => a ∈ ↑A) (Ioc 0 n))) / ↑n < ε

Proof state:
A : Finset ℕ
ε : ℝ
hε : 0 < ε
hε₁ : ε ≤ 1
n : ℕ := ⌊↑(#A) / ε⌋₊ + 1
hn : 0 < n
⊢ ∃ n, 0 < n ∧ ↑(#(filter (fun a => a ∈ ↑A) (Ioc 0 n))) / ↑n < ε

Tactic:
use n, hn

Result:
case right
A : Finset ℕ
ε : ℝ
hε : 0 < ε
hε₁ : ε ≤ 1
n : ℕ := ⌊↑(#A) / ε⌋₊ + 1
hn : 0 < n
⊢ ↑(#(filter (fun a => a ∈ ↑A) (Ioc 0 n))) / ↑n < ε

Proof state:
A : Finset ℕ
ε : ℝ
hε : 0 < ε
⊢ ∃ n, 0 < n ∧ ↑(#(filter (fun a => a ∈ ↑A) (Ioc 0 n))) / ↑n < ε

Tactic:
wlog hε₁ : ε ≤ 1 generalizing ε

Result:
case inr
A : Finset ℕ
ε : ℝ
hε : 0 < ε
this : ∀ (ε : ℝ), 0 < ε → ε ≤ 1 → ∃ n, 0 < n ∧ ↑(#(filter (fun a => a ∈ ↑A) (Ioc 0 n))) / ↑n < ε
hε₁ : ¬ε ≤ 1
⊢ ∃ n, 0 < n ∧ ↑(#(filter (fun a => a ∈ ↑A) (Ioc 0 n))) / ↑n < ε

A : Finset ℕ
ε : ℝ
hε : 0 < ε
hε₁ : ε ≤ 1
⊢ ∃ n, 0 < n ∧ ↑(#(filter (fun a => a ∈ ↑A) (Ioc 0 n))) / ↑n < ε

Proof state:
case right
A : Finset ℕ
ε : ℝ
hε : 0 < ε
hε₁ : ε ≤ 1
n : ℕ := ⌊↑(#A) / ε⌋₊ + 1
hn : 0 < n
⊢ ↑(#(filter (fun a => a ∈ ↑A) (Ioc 0 n))) / ε < ↑⌊↑(#A) / ε⌋₊ + 1

Tactic:
exact (Nat.lt_floor_add_one _).trans_le' <| by gcongr; simp [subset_iff]

Result:
no goals

Proof state:
A : Finset ℕ
⊢ ∀ (ε : ℝ), 0 < ε → ∃ n, 0 < n ∧ ↑(#(filter (fun a => a ∈ ↑A) (Ioc 0 n))) / ↑n < ε

Tactic:
intro ε hε

Result:
A : Finset ℕ
ε : ℝ
hε : 0 < ε
⊢ ∃ n, 0 < n ∧ ↑(#(filter (fun a => a ∈ ↑A) (Ioc 0 n))) / ↑n < ε

Proof state:
case right
A : Finset ℕ
ε : ℝ
hε : 0 < ε
hε₁ : ε ≤ 1
n : ℕ := ⌊↑(#A) / ε⌋₊ + 1
hn : 0 < n
⊢ ↑(#(filter (fun a => a ∈ ↑A) (Ioc 0 n))) / ↑n < ε

Tactic:
rw [div_lt_iff₀ (Nat.cast_pos.2 hn), ← div_lt_iff₀' hε, Nat.cast_add_one]

Result:
case right
A : Finset ℕ
ε : ℝ
hε : 0 < ε
hε₁ : ε ≤ 1
n : ℕ := ⌊↑(#A) / ε⌋₊ + 1
hn : 0 < n
⊢ ↑(#(filter (fun a => a ∈ ↑A) (Ioc 0 n))) / ε < ↑⌊↑(#A) / ε⌋₊ + 1

Proof state:
A : Finset ℕ
⊢ schnirelmannDensity ↑A ≤ 0

Tactic:
simp only [schnirelmannDensity_le_iff_forall, zero_add]

Result:
A : Finset ℕ
⊢ ∀ (ε : ℝ), 0 < ε → ∃ n, 0 < n ∧ ↑(#(filter (fun a => a ∈ ↑A) (Ioc 0 n))) / ↑n < ε

Proof state:
case inr.intro.intro
A : Finset ℕ
ε : ℝ
hε : 0 < ε
this : ∀ (ε : ℝ), 0 < ε → ε ≤ 1 → ∃ n, 0 < n ∧ ↑(#(filter (fun a => a ∈ ↑A) (Ioc 0 n))) / ↑n < ε
hε₁ : ¬ε ≤ 1
n : ℕ
hn : 0 < n
hn' : ↑(#(filter (fun a => a ∈ ↑A) (Ioc 0 n))) / ↑n < 1
⊢ ∃ n, 0 < n ∧ ↑(#(filter (fun a => a ∈ ↑A) (Ioc 0 n))) / ↑n < ε

Tactic:
exact ⟨n, hn, hn'.trans_le (le_of_not_le hε₁)⟩

Result:
no goals

Proof state:
A : Finset ℕ
⊢ schnirelmannDensity ↑A = 0

Tactic:
refine le_antisymm ?_ schnirelmannDensity_nonneg

Result:
A : Finset ℕ
⊢ schnirelmannDensity ↑A ≤ 0

Proof state:
case hab.h.a
A : Finset ℕ
ε : ℝ
hε : 0 < ε
hε₁ : ε ≤ 1
n : ℕ := ⌊↑(#A) / ε⌋₊ + 1
hn : 0 < n
⊢ filter (fun a => a ∈ ↑A) (Ioc 0 n) ⊆ A

Tactic:
simp [subset_iff]

Result:
no goals

Proof state:
A : Finset ℕ
ε : ℝ
hε : 0 < ε
hε₁ : ε ≤ 1
⊢ ∃ n, 0 < n ∧ ↑(#(filter (fun a => a ∈ ↑A) (Ioc 0 n))) / ↑n < ε

Tactic:
let n : ℕ := ⌊#A / ε⌋₊ + 1

Result:
A : Finset ℕ
ε : ℝ
hε : 0 < ε
hε₁ : ε ≤ 1
n : ℕ := ⌊↑(#A) / ε⌋₊ + 1
⊢ ∃ n, 0 < n ∧ ↑(#(filter (fun a => a ∈ ↑A) (Ioc 0 n))) / ↑n < ε