Theorem: centralBinom_factorization_small
File: Mathlib/NumberTheory/Bertrand.lean

Proof steps:
state_before: n : ℕ
n_large : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
⊢ n.centralBinom = ∏ p ∈ Finset.range (2 * n / 3 + 1), p ^ n.centralBinom.factorization p
state_after: n : ℕ
n_large : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
⊢ ∏ p ∈ Finset.range (2 * n / 3 + 1), p ^ n.centralBinom.factorization p =
    ∏ p ∈ Finset.range (2 * n + 1), p ^ n.centralBinom.factorization p
tactic: refine (Eq.trans ?_ n.prod_pow_factorization_centralBinom).symm
state_before: n : ℕ
n_large : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
⊢ ∏ p ∈ Finset.range (2 * n / 3 + 1), p ^ n.centralBinom.factorization p =
    ∏ p ∈ Finset.range (2 * n + 1), p ^ n.centralBinom.factorization p
state_after: case h
n : ℕ
n_large : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
⊢ Finset.range (2 * n / 3 + 1) ⊆ Finset.range (2 * n + 1)

case hf
n : ℕ
n_large : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
⊢ ∀ x ∈ Finset.range (2 * n + 1), x ∉ Finset.range (2 * n / 3 + 1) → x ^ n.centralBinom.factorization x = 1
tactic: apply Finset.prod_subset
state_before: case hf
n : ℕ
n_large : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
⊢ ∀ x ∈ Finset.range (2 * n + 1), x ∉ Finset.range (2 * n / 3 + 1) → x ^ n.centralBinom.factorization x = 1
state_after: case hf
n : ℕ
n_large : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
x : ℕ
hx : x ∈ Finset.range (2 * n + 1)
h2x : x ∉ Finset.range (2 * n / 3 + 1)
⊢ x ^ n.centralBinom.factorization x = 1
tactic: intro x hx h2x
state_before: case hf
n : ℕ
n_large : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
x : ℕ
hx : x ∈ Finset.range (2 * n + 1)
h2x : x ∉ Finset.range (2 * n / 3 + 1)
⊢ x ^ n.centralBinom.factorization x = 1
state_after: case hf
n : ℕ
n_large : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
x : ℕ
hx : x ≤ 2 * n
h2x : ¬x ≤ 2 * n / 3
⊢ x ^ n.centralBinom.factorization x = 1
tactic: rw [Finset.mem_range, Nat.lt_succ_iff] at hx h2x
state_before: case hf
n : ℕ
n_large : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
x : ℕ
hx : x ≤ 2 * n
h2x : ¬x ≤ 2 * n / 3
⊢ x ^ n.centralBinom.factorization x = 1
state_after: case hf
n : ℕ
n_large : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
x : ℕ
hx : x ≤ 2 * n
h2x : 2 * n < 3 * x
⊢ x ^ n.centralBinom.factorization x = 1
tactic: rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x
state_before: case hf
n : ℕ
n_large : 2 < n
no_prime : ¬∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n
x : ℕ
hx : x ≤ 2 * n
h2x : 2 * n < 3 * x
⊢ x ^ n.centralBinom.factorization x = 1
state_after: case hf
n : ℕ
n_large : 2 < n
x : ℕ
hx : x ≤ 2 * n
h2x : 2 * n < 3 * x
no_prime : ¬(Nat.Prime x ∧ n < x ∧ x ≤ 2 * n)
⊢ x ^ n.centralBinom.factorization x = 1
tactic: replace no_prime := not_exists.mp no_prime x
state_before: case hf
n : ℕ
n_large : 2 < n
x : ℕ
hx : x ≤ 2 * n
h2x : 2 * n < 3 * x
no_prime : ¬(Nat.Prime x ∧ n < x ∧ x ≤ 2 * n)
⊢ x ^ n.centralBinom.factorization x = 1
state_after: case hf
n : ℕ
n_large : 2 < n
x : ℕ
hx : x ≤ 2 * n
h2x : 2 * n < 3 * x
no_prime : x ≤ 2 * n → ¬Nat.Prime x ∨ x ≤ n
⊢ x ^ n.centralBinom.factorization x = 1
tactic: rw [← and_assoc, not_and', not_and_or, not_lt] at no_prime
state_before: case hf
n : ℕ
n_large : 2 < n
x : ℕ
hx : x ≤ 2 * n
h2x : 2 * n < 3 * x
no_prime : x ≤ 2 * n → ¬Nat.Prime x ∨ x ≤ n
⊢ x ^ n.centralBinom.factorization x = 1
state_after: case hf.inl
n : ℕ
n_large : 2 < n
x : ℕ
hx : x ≤ 2 * n
h2x : 2 * n < 3 * x
no_prime : x ≤ 2 * n → ¬Nat.Prime x ∨ x ≤ n
h : ¬Nat.Prime x
⊢ x ^ n.centralBinom.factorization x = 1

case hf.inr
n : ℕ
n_large : 2 < n
x : ℕ
hx : x ≤ 2 * n
h2x : 2 * n < 3 * x
no_prime : x ≤ 2 * n → ¬Nat.Prime x ∨ x ≤ n
h : x ≤ n
⊢ x ^ n.centralBinom.factorization x = 1
tactic: cases' no_prime hx with h h