-- Mathlib/NumberTheory/Bertrand.lean
theorem centralBinom_le_of_no_bertrand_prime (n : ℕ) (n_large : 2 < n)
    (no_prime : ¬∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n) :
    centralBinom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3)

Proof state:
case h₁.refine_2
n : ℕ
n_large : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
n_pos : 0 < n
n2_pos : 1 ≤ 2 * n
S : Finset ℕ := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : ℕ → ℕ := fun x => x ^ n.centralBinom.factorization x
this✝ : ∏ x ∈ S, f x = ∏ x ∈ Finset.range (2 * n / 3 + 1), f x
this : (Finset.Icc 1 (2 * n).sqrt).card = (2 * n).sqrt
⊢ ∏ i ∈ Finset.filter (fun x => x ≤ (2 * n).sqrt) S, 2 * n ≤ (2 * n) ^ (2 * n).sqrt

Tactic:
rw [Finset.prod_const]

Result:
case h₁.refine_2
n : ℕ
n_large : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
n_pos : 0 < n
n2_pos : 1 ≤ 2 * n
S : Finset ℕ := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : ℕ → ℕ := fun x => x ^ n.centralBinom.factorization x
this✝ : ∏ x ∈ S, f x = ∏ x ∈ Finset.range (2 * n / 3 + 1), f x
this : (Finset.Icc 1 (2 * n).sqrt).card = (2 * n).sqrt
⊢ (2 * n) ^ (Finset.filter (fun x => x ≤ (2 * n).sqrt) S).card ≤ (2 * n) ^ (2 * n).sqrt

Proof state:
case h₂.refine_1.intro
n : ℕ
n_large : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
n_pos : 0 < n
n2_pos : 1 ≤ 2 * n
S : Finset ℕ := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : ℕ → ℕ := fun x => x ^ n.centralBinom.factorization x
this : ∏ x ∈ S, f x = ∏ x ∈ Finset.range (2 * n / 3 + 1), f x
p : ℕ
hp : p ∈ Finset.filter (fun x => ¬x ≤ (2 * n).sqrt) S
h1 : p ∈ S
h2 : ¬p ≤ (2 * n).sqrt
⊢ n.centralBinom.factorization p ≤ 1

Tactic:
exact Nat.factorization_choose_le_one (sqrt_lt'.mp <| not_le.1 h2)

Result:
no goals

Proof state:
n : ℕ
n_large : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
n_pos : 0 < n
n2_pos : 1 ≤ 2 * n
S : Finset ℕ := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
⊢ n.centralBinom ≤ (2 * n) ^ (2 * n).sqrt * 4 ^ (2 * n / 3)

Tactic:
let f x := x ^ n.centralBinom.factorization x

Result:
n : ℕ
n_large : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
n_pos : 0 < n
n2_pos : 1 ≤ 2 * n
S : Finset ℕ := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : ℕ → ℕ := fun x => x ^ n.centralBinom.factorization x
⊢ n.centralBinom ≤ (2 * n) ^ (2 * n).sqrt * 4 ^ (2 * n / 3)

Proof state:
n : ℕ
n_large : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
n_pos : 0 < n
n2_pos : 1 ≤ 2 * n
S : Finset ℕ := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : ℕ → ℕ := fun x => x ^ n.centralBinom.factorization x
p : ℕ
x✝ : p ∈ Finset.range (2 * n / 3 + 1)
h : ¬Nat.Prime p
⊢ p ^ n.centralBinom.factorization p = 1

Tactic:
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]

Result:
no goals

Proof state:
n : ℕ
n_large : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
n_pos : 0 < n
n2_pos : 1 ≤ 2 * n
S : Finset ℕ := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : ℕ → ℕ := fun x => x ^ n.centralBinom.factorization x
this : ∏ x ∈ S, f x = ∏ x ∈ Finset.range (2 * n / 3 + 1), f x
⊢ (Finset.Icc 1 (2 * n).sqrt).card = (2 * n).sqrt

Tactic:
rw [card_Icc, Nat.add_sub_cancel]

Result:
no goals

Proof state:
case h₁.refine_2
n : ℕ
n_large : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
n_pos : 0 < n
n2_pos : 1 ≤ 2 * n
S : Finset ℕ := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : ℕ → ℕ := fun x => x ^ n.centralBinom.factorization x
this✝ : ∏ x ∈ S, f x = ∏ x ∈ Finset.range (2 * n / 3 + 1), f x
this : (Finset.Icc 1 (2 * n).sqrt).card = (2 * n).sqrt
⊢ (2 * n) ^ (Finset.filter (fun x => x ≤ (2 * n).sqrt) S).card ≤ (2 * n) ^ (2 * n).sqrt

Tactic:
refine pow_right_mono₀ n2_pos ((Finset.card_le_card fun x hx => ?_).trans this.le)

Result:
case h₁.refine_2
n : ℕ
n_large : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
n_pos : 0 < n
n2_pos : 1 ≤ 2 * n
S : Finset ℕ := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : ℕ → ℕ := fun x => x ^ n.centralBinom.factorization x
this✝ : ∏ x ∈ S, f x = ∏ x ∈ Finset.range (2 * n / 3 + 1), f x
this : (Finset.Icc 1 (2 * n).sqrt).card = (2 * n).sqrt
x : ℕ
hx : x ∈ Finset.filter (fun x => x ≤ (2 * n).sqrt) S
⊢ x ∈ Finset.Icc 1 (2 * n).sqrt

Proof state:
case h₁.refine_2
n : ℕ
n_large : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
n_pos : 0 < n
n2_pos : 1 ≤ 2 * n
S : Finset ℕ := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : ℕ → ℕ := fun x => x ^ n.centralBinom.factorization x
this : ∏ x ∈ S, f x = ∏ x ∈ Finset.range (2 * n / 3 + 1), f x
⊢ ∏ i ∈ Finset.filter (fun x => x ≤ (2 * n).sqrt) S, 2 * n ≤ (2 * n) ^ (2 * n).sqrt

Tactic:
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]

Result:
case h₁.refine_2
n : ℕ
n_large : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
n_pos : 0 < n
n2_pos : 1 ≤ 2 * n
S : Finset ℕ := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : ℕ → ℕ := fun x => x ^ n.centralBinom.factorization x
this✝ : ∏ x ∈ S, f x = ∏ x ∈ Finset.range (2 * n / 3 + 1), f x
this : (Finset.Icc 1 (2 * n).sqrt).card = (2 * n).sqrt
⊢ ∏ i ∈ Finset.filter (fun x => x ≤ (2 * n).sqrt) S, 2 * n ≤ (2 * n) ^ (2 * n).sqrt

Proof state:
case h₁.refine_2
n : ℕ
n_large : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
n_pos : 0 < n
n2_pos : 1 ≤ 2 * n
S : Finset ℕ := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : ℕ → ℕ := fun x => x ^ n.centralBinom.factorization x
this✝ : ∏ x ∈ S, f x = ∏ x ∈ Finset.range (2 * n / 3 + 1), f x
this : (Finset.Icc 1 (2 * n).sqrt).card = (2 * n).sqrt
x : ℕ
hx : x ∈ Finset.filter (fun x => x ≤ (2 * n).sqrt) S
⊢ x ∈ Finset.Icc 1 (2 * n).sqrt

Tactic:
obtain ⟨h1, h2⟩ := Finset.mem_filter.1 hx

Result:
case h₁.refine_2.intro
n : ℕ
n_large : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
n_pos : 0 < n
n2_pos : 1 ≤ 2 * n
S : Finset ℕ := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : ℕ → ℕ := fun x => x ^ n.centralBinom.factorization x
this✝ : ∏ x ∈ S, f x = ∏ x ∈ Finset.range (2 * n / 3 + 1), f x
this : (Finset.Icc 1 (2 * n).sqrt).card = (2 * n).sqrt
x : ℕ
hx : x ∈ Finset.filter (fun x => x ≤ (2 * n).sqrt) S
h1 : x ∈ S
h2 : x ≤ (2 * n).sqrt
⊢ x ∈ Finset.Icc 1 (2 * n).sqrt

Proof state:
n : ℕ
n_large : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
n_pos : 0 < n
n2_pos : 1 ≤ 2 * n
S : Finset ℕ := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : ℕ → ℕ := fun x => x ^ n.centralBinom.factorization x
⊢ n.centralBinom ≤ (2 * n) ^ (2 * n).sqrt * 4 ^ (2 * n / 3)

Tactic:
have : ∏ x ∈ S, f x = ∏ x ∈ Finset.range (2 * n / 3 + 1), f x := by
  refine Finset.prod_filter_of_ne fun p _ h => ?_
  contrapose! h; dsimp only [f]
  rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]

Result:
n : ℕ
n_large : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
n_pos : 0 < n
n2_pos : 1 ≤ 2 * n
S : Finset ℕ := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : ℕ → ℕ := fun x => x ^ n.centralBinom.factorization x
this : ∏ x ∈ S, f x = ∏ x ∈ Finset.range (2 * n / 3 + 1), f x
⊢ n.centralBinom ≤ (2 * n) ^ (2 * n).sqrt * 4 ^ (2 * n / 3)

Proof state:
n : ℕ
n_large : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
n_pos : 0 < n
n2_pos : 1 ≤ 2 * n
S : Finset ℕ := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : ℕ → ℕ := fun x => x ^ n.centralBinom.factorization x
⊢ ∏ x ∈ S, f x = ∏ x ∈ Finset.range (2 * n / 3 + 1), f x

Tactic:
refine Finset.prod_filter_of_ne fun p _ h => ?_

Result:
n : ℕ
n_large : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
n_pos : 0 < n
n2_pos : 1 ≤ 2 * n
S : Finset ℕ := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : ℕ → ℕ := fun x => x ^ n.centralBinom.factorization x
p : ℕ
x✝ : p ∈ Finset.range (2 * n / 3 + 1)
h : f p ≠ 1
⊢ Nat.Prime p