Theorem: Polynomial.coeff_hermite_explicit
File: Mathlib/RingTheory/Polynomial/Hermite/Basic.lean

Proof steps:
state_before: x✝ : ℕ
⊢ (hermite (2 * 0 + x✝)).coeff x✝ = (-1) ^ 0 * ↑(2 * 0 - 1)‼ * ↑((2 * 0 + x✝).choose x✝)
state_after: no goals
tactic: simp
state_before: n : ℕ
⊢ (hermite (2 * (n + 1) + 0)).coeff 0 = (-1) ^ (n + 1) * ↑(2 * (n + 1) - 1)‼ * ↑((2 * (n + 1) + 0).choose 0)
state_after: case h.e'_3
n : ℕ
⊢ (-1) ^ (n + 1) * ↑(2 * (n + 1) - 1)‼ * ↑((2 * (n + 1) + 0).choose 0) = -(hermite (2 * n + 1)).coeff 1
tactic: convert coeff_hermite_succ_zero (2 * n + 1) using 1
state_before: case h.e'_3
n : ℕ
⊢ (-1) ^ (n + 1) * ↑(2 * (n + 1) - 1)‼ * ↑((2 * (n + 1) + 0).choose 0) = -(hermite (2 * n + 1)).coeff 1
state_after: case h.e'_3
n : ℕ
⊢ (-1) ^ n * -1 * ↑((2 * n + 1) * (2 * n - 1)‼) * ↑1 = -((-1) ^ n * ↑(2 * n - 1)‼ * ↑(2 * n + 1))
tactic: rw [coeff_hermite_explicit n 1, (by rw [Nat.left_distrib, mul_one, Nat.add_one_sub_one] :
      2 * (n + 1) - 1 = 2 * n + 1), Nat.doubleFactorial_add_one, Nat.choose_zero_right,
      Nat.choose_one_right, pow_succ]
state_before: case h.e'_3
n : ℕ
⊢ (-1) ^ n * -1 * ↑((2 * n + 1) * (2 * n - 1)‼) * ↑1 = -((-1) ^ n * ↑(2 * n - 1)‼ * ↑(2 * n + 1))
state_after: case h.e'_3
n : ℕ
⊢ (-1) ^ n * -1 * ((2 * ↑n + 1) * ↑(2 * n - 1)‼) * 1 = -((-1) ^ n * ↑(2 * n - 1)‼ * (2 * ↑n + 1))
tactic: push_cast
state_before: case h.e'_3
n : ℕ
⊢ (-1) ^ n * -1 * ((2 * ↑n + 1) * ↑(2 * n - 1)‼) * 1 = -((-1) ^ n * ↑(2 * n - 1)‼ * (2 * ↑n + 1))
state_after: no goals
tactic: ring
state_before: n k : ℕ
⊢ (hermite (2 * (n + 1) + (k + 1))).coeff (k + 1) =
    (-1) ^ (n + 1) * ↑(2 * (n + 1) - 1)‼ * ↑((2 * (n + 1) + (k + 1)).choose (k + 1))
state_after: n k : ℕ
hermite_explicit : ℕ → ℕ → ℤ := fun n k => (-1) ^ n * ↑(2 * n - 1)‼ * ↑((2 * n + k).choose k)
⊢ (hermite (2 * (n + 1) + (k + 1))).coeff (k + 1) =
    (-1) ^ (n + 1) * ↑(2 * (n + 1) - 1)‼ * ↑((2 * (n + 1) + (k + 1)).choose (k + 1))
tactic: let hermite_explicit : ℕ → ℕ → ℤ := fun n k =>
      (-1) ^ n * (2 * n - 1)‼ * Nat.choose (2 * n + k) k
state_before: n k : ℕ
hermite_explicit : ℕ → ℕ → ℤ := fun n k => (-1) ^ n * ↑(2 * n - 1)‼ * ↑((2 * n + k).choose k)
⊢ (hermite (2 * (n + 1) + (k + 1))).coeff (k + 1) =
    (-1) ^ (n + 1) * ↑(2 * (n + 1) - 1)‼ * ↑((2 * (n + 1) + (k + 1)).choose (k + 1))
state_after: n k : ℕ
hermite_explicit : ℕ → ℕ → ℤ := fun n k => (-1) ^ n * ↑(2 * n - 1)‼ * ↑((2 * n + k).choose k)
hermite_explicit_recur :
  ∀ (n k : ℕ), hermite_explicit (n + 1) (k + 1) = hermite_explicit (n + 1) k - (↑k + 2) * hermite_explicit n (k + 2)
⊢ (hermite (2 * (n + 1) + (k + 1))).coeff (k + 1) =
    (-1) ^ (n + 1) * ↑(2 * (n + 1) - 1)‼ * ↑((2 * (n + 1) + (k + 1)).choose (k + 1))
tactic: have hermite_explicit_recur :
      ∀ n k : ℕ,
        hermite_explicit (n + 1) (k + 1) =
          hermite_explicit (n + 1) k - (k + 2) * hermite_explicit n (k + 2) := by
      intro n k
      simp only [hermite_explicit]
      -- Factor out (-1)'s.
      rw [mul_comm (↑k + _ : ℤ), sub_eq_add_neg]
      nth_rw 3 [neg_eq_neg_one_mul]
      simp only [mul_assoc, ← mul_add, pow_succ']
      congr 2
      -- Factor out double factorials.
      norm_cast
      -- Porting note: ring_nf did not solve the goal on line 186
      rw [(by rw [Nat.left_distrib, mul_one, Nat.add_one_sub_one] : 2 * (n + 1) - 1 = 2 * n + 1),
        Nat.doubleFactorial_add_one, mul_comm (2 * n + 1)]
      simp only [mul_assoc, ← mul_add]
      congr 1
      -- Match up binomial coefficients using `Nat.choose_succ_right_eq`.
      rw [(by ring : 2 * (n + 1) + (k + 1) = 2 * n + 1 + (k + 1) + 1),
        (by ring : 2 * (n + 1) + k = 2 * n + 1 + (k + 1)),
        (by ring : 2 * n + (k + 2) = 2 * n + 1 + (k + 1))]
      rw [Nat.choose, Nat.choose_succ_right_eq (2 * n + 1 + (k + 1)) (k + 1), Nat.add_sub_cancel]
      ring
state_before: n k : ℕ
hermite_explicit : ℕ → ℕ → ℤ := fun n k => (-1) ^ n * ↑(2 * n - 1)‼ * ↑((2 * n + k).choose k)
hermite_explicit_recur :
  ∀ (n k : ℕ), hermite_explicit (n + 1) (k + 1) = hermite_explicit (n + 1) k - (↑k + 2) * hermite_explicit n (k + 2)
⊢ (hermite (2 * (n + 1) + (k + 1))).coeff (k + 1) =
    (-1) ^ (n + 1) * ↑(2 * (n + 1) - 1)‼ * ↑((2 * (n + 1) + (k + 1)).choose (k + 1))
state_after: n k : ℕ
hermite_explicit : ℕ → ℕ → ℤ := fun n k => (-1) ^ n * ↑(2 * n - 1)‼ * ↑((2 * n + k).choose k)
hermite_explicit_recur :
  ∀ (n k : ℕ), hermite_explicit (n + 1) (k + 1) = hermite_explicit (n + 1) k - (↑k + 2) * hermite_explicit n (k + 2)
⊢ (hermite (2 * (n + 1) + (k + 1))).coeff (k + 1) = hermite_explicit (n + 1) (k + 1)
tactic: change _ = hermite_explicit _ _
state_before: n k : ℕ
hermite_explicit : ℕ → ℕ → ℤ := fun n k => (-1) ^ n * ↑(2 * n - 1)‼ * ↑((2 * n + k).choose k)
hermite_explicit_recur :
  ∀ (n k : ℕ), hermite_explicit (n + 1) (k + 1) = hermite_explicit (n + 1) k - (↑k + 2) * hermite_explicit n (k + 2)
⊢ (hermite (2 * (n + 1) + (k + 1))).coeff (k + 1) = hermite_explicit (n + 1) (k + 1)
state_after: n k : ℕ
hermite_explicit : ℕ → ℕ → ℤ := fun n k => (-1) ^ n * ↑(2 * n - 1)‼ * ↑((2 * n + k).choose k)
hermite_explicit_recur :
  ∀ (n k : ℕ), hermite_explicit (n + 1) (k + 1) = hermite_explicit (n + 1) k - (↑k + 2) * hermite_explicit n (k + 2)
⊢ (hermite (2 * (n + 1) + k)).coeff k - (↑k + 2) * (hermite (2 * (n + 1) + k)).coeff (k + 2) =
    hermite_explicit (n + 1) k - (↑k + 2) * hermite_explicit n (k + 2)
tactic: rw [← add_assoc, coeff_hermite_succ_succ, hermite_explicit_recur]
state_before: n k : ℕ
hermite_explicit : ℕ → ℕ → ℤ := fun n k => (-1) ^ n * ↑(2 * n - 1)‼ * ↑((2 * n + k).choose k)
hermite_explicit_recur :
  ∀ (n k : ℕ), hermite_explicit (n + 1) (k + 1) = hermite_explicit (n + 1) k - (↑k + 2) * hermite_explicit n (k + 2)
⊢ (hermite (2 * (n + 1) + k)).coeff k - (↑k + 2) * (hermite (2 * (n + 1) + k)).coeff (k + 2) =
    hermite_explicit (n + 1) k - (↑k + 2) * hermite_explicit n (k + 2)
state_after: case e_a
n k : ℕ
hermite_explicit : ℕ → ℕ → ℤ := fun n k => (-1) ^ n * ↑(2 * n - 1)‼ * ↑((2 * n + k).choose k)
hermite_explicit_recur :
  ∀ (n k : ℕ), hermite_explicit (n + 1) (k + 1) = hermite_explicit (n + 1) k - (↑k + 2) * hermite_explicit n (k + 2)
⊢ (hermite (2 * (n + 1) + k)).coeff k = hermite_explicit (n + 1) k

case e_a.e_a
n k : ℕ
hermite_explicit : ℕ → ℕ → ℤ := fun n k => (-1) ^ n * ↑(2 * n - 1)‼ * ↑((2 * n + k).choose k)
hermite_explicit_recur :
  ∀ (n k : ℕ), hermite_explicit (n + 1) (k + 1) = hermite_explicit (n + 1) k - (↑k + 2) * hermite_explicit n (k + 2)
⊢ (hermite (2 * (n + 1) + k)).coeff (k + 2) = hermite_explicit n (k + 2)
tactic: congr