l3lab/ntp-mathlib huggingface dataset:
theorem coeff_iterate_derivative {k} (p : R[X]) (m : ℕ) :
    (derivative^[k] p).coeff m = (m + k).descFactorial k • p.coeff (m + k) := by
  induction k generalizing m with
  | zero => simp
  | succ k ih =>
      calc
        (derivative^[k + 1] p).coeff m
        _ = Nat.descFactorial (Nat.succ (m + k)) k • p.coeff (m + k.succ) * (m + 1) := by
          rw [Function.iterate_succ_apply', coeff_derivative, ih m.succ, Nat.succ_add, Nat.add_succ]
        _ = ((m + 1) * Nat.descFactorial (Nat.succ (m + k)) k) • p.coeff (m + k.succ) := by
          rw [← Nat.cast_add_one, ← nsmul_eq_mul', smul_smul]
        _ = Nat.descFactorial (m.succ + k) k.succ • p.coeff (m + k.succ) := by
          rw [← Nat.succ_add, Nat.descFactorial_succ, add_tsub_cancel_right]
        _ = Nat.descFactorial (m + k.succ) k.succ • p.coeff (m + k.succ) := by
          

Proof state:
R : Type u
S : Type v
T : Type w
ι : Type y
A : Type z
a b : R
n : ℕ
inst✝ : Semiring R
p : R[X]
k : ℕ
ih : ∀ (m : ℕ), coeff ((⇑derivative)^[k] p) m = Nat.descFactorial (m + k) k • coeff p (m + k)
m : ℕ
⊢ Nat.descFactorial (Nat.succ m + k) (Nat.succ k) • coeff p (m + Nat.succ k) =
    Nat.descFactorial (m + Nat.succ k) (Nat.succ k) • coeff p (m + Nat.succ k)

Next tactic:
rw [Nat.succ_add_eq_add_succ]
theorem iterate_derivative_mul {n} (p q : R[X]) :
    derivative^[n] (p * q) =
      ∑ k in range n.succ, (n.choose k • (derivative^[n - k] p * derivative^[k] q)) := by
  

Proof state:
R : Type u
S : Type v
T : Type w
ι : Type y
A : Type z
a b : R
n✝ : ℕ
inst✝ : Semiring R
n : ℕ
p q : R[X]
⊢ (⇑derivative)^[n] (p * q) =
    ∑ k in range (Nat.succ n), Nat.choose n k • ((⇑derivative)^[n - k] p * (⇑derivative)^[k] q)

Next tactic:
induction' n with n IH
theorem iterate_derivative_mul {n} (p q : R[X]) :
    derivative^[n] (p * q) =
      ∑ k in range n.succ, (n.choose k • (derivative^[n - k] p * derivative^[k] q)) := by
  induction' n with n IH
  · 

Proof state:
case zero
R : Type u
S : Type v
T : Type w
ι : Type y
A : Type z
a b : R
n : ℕ
inst✝ : Semiring R
p q : R[X]
⊢ (⇑derivative)^[Nat.zero] (p * q) =
    ∑ k in range (Nat.succ Nat.zero), Nat.choose Nat.zero k • ((⇑derivative)^[Nat.zero - k] p * (⇑derivative)^[k] q)

Next tactic:
simp [Finset.range]
theorem iterate_derivative_mul {n} (p q : R[X]) :
    derivative^[n] (p * q) =
      ∑ k in range n.succ, (n.choose k • (derivative^[n - k] p * derivative^[k] q)) := by
  induction' n with n IH
  · simp [Finset.range]
  

Proof state:
case succ
R : Type u
S : Type v
T : Type w
ι : Type y
A : Type z
a b : R
n✝ : ℕ
inst✝ : Semiring R
p q : R[X]
n : ℕ
IH :
  (⇑derivative)^[n] (p * q) =
    ∑ k in range (Nat.succ n), Nat.choose n k • ((⇑derivative)^[n - k] p * (⇑derivative)^[k] q)
⊢ (⇑derivative)^[Nat.succ n] (p * q) =
    ∑ k in range (Nat.succ (Nat.succ n)),
      Nat.choose (Nat.succ n) k • ((⇑derivative)^[Nat.succ n - k] p * (⇑derivative)^[k] q)

Next tactic:
calc
    derivative^[n + 1] (p * q) =
        derivative (∑ k : ℕ in range n.succ,
            n.choose k • (derivative^[n - k] p * derivative^[k] q)) :=
      by rw [Function.iterate_succ_apply', IH]
    _ = (∑ k : ℕ in range n.succ,
          n.choose k • (derivative^[n - k + 1] p * derivative^[k] q)) +
        ∑ k : ℕ in range n.succ,
          n.choose k • (derivative^[n - k] p * derivative^[k + 1] q) := by
      simp_rw [derivative_sum, derivative_smul, derivative_mul, Function.iterate_succ_apply',
        smul_add, sum_add_distrib]
    _ = (∑ k : ℕ in range n.succ,
              n.choose k.succ • (derivative^[n - k] p * derivative^[k + 1] q)) +
            1 • (derivative^[n + 1] p * derivative^[0] q) +
          ∑ k : ℕ in range n.succ, n.choose k • (derivative^[n - k] p * derivative^[k + 1] q) :=
      ?_
    _ = ((∑ k : ℕ in range n.succ, n.choose k • (derivative^[n - k] p * derivative^[k + 1] q)) +
            ∑ k : ℕ in range n.succ,
              n.choose k.succ • (derivative^[n - k] p * derivative^[k + 1] q)) +
          1 • (derivative^[n + 1] p * derivative^[0] q) :=
      by rw [add_comm, add_assoc]
    _ = (∑ i : ℕ in range n.succ,
            (n + 1).choose (i + 1) • (derivative^[n + 1 - (i + 1)] p * derivative^[i + 1] q)) +
          1 • (derivative^[n + 1] p * derivative^[0] q) :=
      by simp_rw [Nat.choose_succ_succ, Nat.succ_sub_succ, add_smul, sum_add_distrib]
    _ = ∑ k : ℕ in range n.succ.succ,
          n.succ.choose k • (derivative^[n.succ - k] p * derivative^[k] q) :=
      by rw [sum_range_succ' _ n.succ, Nat.choose_zero_right, tsub_zero]
theorem iterate_derivative_mul {n} (p q : R[X]) :
    derivative^[n] (p * q) =
      ∑ k in range n.succ, (n.choose k • (derivative^[n - k] p * derivative^[k] q)) := by
  induction' n with n IH
  · simp [Finset.range]
  calc
    derivative^[n + 1] (p * q) =
        derivative (∑ k : ℕ in range n.succ,
            n.choose k • (derivative^[n - k] p * derivative^[k] q)) :=
      by 

Proof state:
R : Type u
S : Type v
T : Type w
ι : Type y
A : Type z
a b : R
n✝ : ℕ
inst✝ : Semiring R
p q : R[X]
n : ℕ
IH :
  (⇑derivative)^[n] (p * q) =
    ∑ k in range (Nat.succ n), Nat.choose n k • ((⇑derivative)^[n - k] p * (⇑derivative)^[k] q)
⊢ (⇑derivative)^[n + 1] (p * q) =
    derivative (∑ k in range (Nat.succ n), Nat.choose n k • ((⇑derivative)^[n - k] p * (⇑derivative)^[k] q))

Next tactic:
rw [Function.iterate_succ_apply', IH]
theorem iterate_derivative_mul {n} (p q : R[X]) :
    derivative^[n] (p * q) =
      ∑ k in range n.succ, (n.choose k • (derivative^[n - k] p * derivative^[k] q)) := by
  induction' n with n IH
  · simp [Finset.range]
  calc
    derivative^[n + 1] (p * q) =
        derivative (∑ k : ℕ in range n.succ,
            n.choose k • (derivative^[n - k] p * derivative^[k] q)) :=
      by rw [Function.iterate_succ_apply', IH]
    _ = (∑ k : ℕ in range n.succ,
          n.choose k • (derivative^[n - k + 1] p * derivative^[k] q)) +
        ∑ k : ℕ in range n.succ,
          n.choose k • (derivative^[n - k] p * derivative^[k + 1] q) := by
      

Proof state:
R : Type u
S : Type v
T : Type w
ι : Type y
A : Type z
a b : R
n✝ : ℕ
inst✝ : Semiring R
p q : R[X]
n : ℕ
IH :
  (⇑derivative)^[n] (p * q) =
    ∑ k in range (Nat.succ n), Nat.choose n k • ((⇑derivative)^[n - k] p * (⇑derivative)^[k] q)
⊢ derivative (∑ k in range (Nat.succ n), Nat.choose n k • ((⇑derivative)^[n - k] p * (⇑derivative)^[k] q)) =
    ∑ k in range (Nat.succ n), Nat.choose n k • ((⇑derivative)^[n - k + 1] p * (⇑derivative)^[k] q) +
      ∑ k in range (Nat.succ n), Nat.choose n k • ((⇑derivative)^[n - k] p * (⇑derivative)^[k + 1] q)

Next tactic:
simp_rw [derivative_sum, derivative_smul, derivative_mul, Function.iterate_succ_apply',
        smul_add, sum_add_distrib]
theorem iterate_derivative_mul {n} (p q : R[X]) :
    derivative^[n] (p * q) =
      ∑ k in range n.succ, (n.choose k • (derivative^[n - k] p * derivative^[k] q)) := by
  induction' n with n IH
  · simp [Finset.range]
  calc
    derivative^[n + 1] (p * q) =
        derivative (∑ k : ℕ in range n.succ,
            n.choose k • (derivative^[n - k] p * derivative^[k] q)) :=
      by rw [Function.iterate_succ_apply', IH]
    _ = (∑ k : ℕ in range n.succ,
          n.choose k • (derivative^[n - k + 1] p * derivative^[k] q)) +
        ∑ k : ℕ in range n.succ,
          n.choose k • (derivative^[n - k] p * derivative^[k + 1] q) := by
      simp_rw [derivative_sum, derivative_smul, derivative_mul, Function.iterate_succ_apply',
        smul_add, sum_add_distrib]
    _ = (∑ k : ℕ in range n.succ,
              n.choose k.succ • (derivative^[n - k] p * derivative^[k + 1] q)) +
            1 • (derivative^[n + 1] p * derivative^[0] q) +
          ∑ k : ℕ in range n.succ, n.choose k • (derivative^[n - k] p * derivative^[k + 1] q) :=
      ?_
    _ = ((∑ k : ℕ in range n.succ, n.choose k • (derivative^[n - k] p * derivative^[k + 1] q)) +
            ∑ k : ℕ in range n.succ,
              n.choose k.succ • (derivative^[n - k] p * derivative^[k + 1] q)) +
          1 • (derivative^[n + 1] p * derivative^[0] q) :=
      by 

Proof state:
R : Type u
S : Type v
T : Type w
ι : Type y
A : Type z
a b : R
n✝ : ℕ
inst✝ : Semiring R
p q : R[X]
n : ℕ
IH :
  (⇑derivative)^[n] (p * q) =
    ∑ k in range (Nat.succ n), Nat.choose n k • ((⇑derivative)^[n - k] p * (⇑derivative)^[k] q)
⊢ ∑ k in range (Nat.succ n), Nat.choose n (Nat.succ k) • ((⇑derivative)^[n - k] p * (⇑derivative)^[k + 1] q) +
        1 • ((⇑derivative)^[n + 1] p * (⇑derivative)^[0] q) +
      ∑ k in range (Nat.succ n), Nat.choose n k • ((⇑derivative)^[n - k] p * (⇑derivative)^[k + 1] q) =
    ∑ k in range (Nat.succ n), Nat.choose n k • ((⇑derivative)^[n - k] p * (⇑derivative)^[k + 1] q) +
        ∑ k in range (Nat.succ n), Nat.choose n (Nat.succ k) • ((⇑derivative)^[n - k] p * (⇑derivative)^[k + 1] q) +
      1 • ((⇑derivative)^[n + 1] p * (⇑derivative)^[0] q)

Next tactic:
rw [add_comm, add_assoc]
theorem iterate_derivative_mul {n} (p q : R[X]) :
    derivative^[n] (p * q) =
      ∑ k in range n.succ, (n.choose k • (derivative^[n - k] p * derivative^[k] q)) := by
  induction' n with n IH
  · simp [Finset.range]
  calc
    derivative^[n + 1] (p * q) =
        derivative (∑ k : ℕ in range n.succ,
            n.choose k • (derivative^[n - k] p * derivative^[k] q)) :=
      by rw [Function.iterate_succ_apply', IH]
    _ = (∑ k : ℕ in range n.succ,
          n.choose k • (derivative^[n - k + 1] p * derivative^[k] q)) +
        ∑ k : ℕ in range n.succ,
          n.choose k • (derivative^[n - k] p * derivative^[k + 1] q) := by
      simp_rw [derivative_sum, derivative_smul, derivative_mul, Function.iterate_succ_apply',
        smul_add, sum_add_distrib]
    _ = (∑ k : ℕ in range n.succ,
              n.choose k.succ • (derivative^[n - k] p * derivative^[k + 1] q)) +
            1 • (derivative^[n + 1] p * derivative^[0] q) +
          ∑ k : ℕ in range n.succ, n.choose k • (derivative^[n - k] p * derivative^[k + 1] q) :=
      ?_
    _ = ((∑ k : ℕ in range n.succ, n.choose k • (derivative^[n - k] p * derivative^[k + 1] q)) +
            ∑ k : ℕ in range n.succ,
              n.choose k.succ • (derivative^[n - k] p * derivative^[k + 1] q)) +
          1 • (derivative^[n + 1] p * derivative^[0] q) :=
      by rw [add_comm, add_assoc]
    _ = (∑ i : ℕ in range n.succ,
            (n + 1).choose (i + 1) • (derivative^[n + 1 - (i + 1)] p * derivative^[i + 1] q)) +
          1 • (derivative^[n + 1] p * derivative^[0] q) :=
      by 

Proof state:
R : Type u
S : Type v
T : Type w
ι : Type y
A : Type z
a b : R
n✝ : ℕ
inst✝ : Semiring R
p q : R[X]
n : ℕ
IH :
  (⇑derivative)^[n] (p * q) =
    ∑ k in range (Nat.succ n), Nat.choose n k • ((⇑derivative)^[n - k] p * (⇑derivative)^[k] q)
⊢ ∑ k in range (Nat.succ n), Nat.choose n k • ((⇑derivative)^[n - k] p * (⇑derivative)^[k + 1] q) +
        ∑ k in range (Nat.succ n), Nat.choose n (Nat.succ k) • ((⇑derivative)^[n - k] p * (⇑derivative)^[k + 1] q) +
      1 • ((⇑derivative)^[n + 1] p * (⇑derivative)^[0] q) =
    ∑ i in range (Nat.succ n),
        Nat.choose (n + 1) (i + 1) • ((⇑derivative)^[n + 1 - (i + 1)] p * (⇑derivative)^[i + 1] q) +
      1 • ((⇑derivative)^[n + 1] p * (⇑derivative)^[0] q)

Next tactic:
simp_rw [Nat.choose_succ_succ, Nat.succ_sub_succ, add_smul, sum_add_distrib]
theorem iterate_derivative_mul {n} (p q : R[X]) :
    derivative^[n] (p * q) =
      ∑ k in range n.succ, (n.choose k • (derivative^[n - k] p * derivative^[k] q)) := by
  induction' n with n IH
  · simp [Finset.range]
  calc
    derivative^[n + 1] (p * q) =
        derivative (∑ k : ℕ in range n.succ,
            n.choose k • (derivative^[n - k] p * derivative^[k] q)) :=
      by rw [Function.iterate_succ_apply', IH]
    _ = (∑ k : ℕ in range n.succ,
          n.choose k • (derivative^[n - k + 1] p * derivative^[k] q)) +
        ∑ k : ℕ in range n.succ,
          n.choose k • (derivative^[n - k] p * derivative^[k + 1] q) := by
      simp_rw [derivative_sum, derivative_smul, derivative_mul, Function.iterate_succ_apply',
        smul_add, sum_add_distrib]
    _ = (∑ k : ℕ in range n.succ,
              n.choose k.succ • (derivative^[n - k] p * derivative^[k + 1] q)) +
            1 • (derivative^[n + 1] p * derivative^[0] q) +
          ∑ k : ℕ in range n.succ, n.choose k • (derivative^[n - k] p * derivative^[k + 1] q) :=
      ?_
    _ = ((∑ k : ℕ in range n.succ, n.choose k • (derivative^[n - k] p * derivative^[k + 1] q)) +
            ∑ k : ℕ in range n.succ,
              n.choose k.succ • (derivative^[n - k] p * derivative^[k + 1] q)) +
          1 • (derivative^[n + 1] p * derivative^[0] q) :=
      by rw [add_comm, add_assoc]
    _ = (∑ i : ℕ in range n.succ,
            (n + 1).choose (i + 1) • (derivative^[n + 1 - (i + 1)] p * derivative^[i + 1] q)) +
          1 • (derivative^[n + 1] p * derivative^[0] q) :=
      by simp_rw [Nat.choose_succ_succ, Nat.succ_sub_succ, add_smul, sum_add_distrib]
    _ = ∑ k : ℕ in range n.succ.succ,
          n.succ.choose k • (derivative^[n.succ - k] p * derivative^[k] q) :=
      by 

Proof state:
R : Type u
S : Type v
T : Type w
ι : Type y
A : Type z
a b : R
n✝ : ℕ
inst✝ : Semiring R
p q : R[X]
n : ℕ
IH :
  (⇑derivative)^[n] (p * q) =
    ∑ k in range (Nat.succ n), Nat.choose n k • ((⇑derivative)^[n - k] p * (⇑derivative)^[k] q)
⊢ ∑ i in range (Nat.succ n),
        Nat.choose (n + 1) (i + 1) • ((⇑derivative)^[n + 1 - (i + 1)] p * (⇑derivative)^[i + 1] q) +
      1 • ((⇑derivative)^[n + 1] p * (⇑derivative)^[0] q) =
    ∑ k in range (Nat.succ (Nat.succ n)),
      Nat.choose (Nat.succ n) k • ((⇑derivative)^[Nat.succ n - k] p * (⇑derivative)^[k] q)

Next tactic:
rw [sum_range_succ' _ n.succ, Nat.choose_zero_right, tsub_zero]
theorem iterate_derivative_mul {n} (p q : R[X]) :
    derivative^[n] (p * q) =
      ∑ k in range n.succ, (n.choose k • (derivative^[n - k] p * derivative^[k] q)) := by
  induction' n with n IH
  · simp [Finset.range]
  calc
    derivative^[n + 1] (p * q) =
        derivative (∑ k : ℕ in range n.succ,
            n.choose k • (derivative^[n - k] p * derivative^[k] q)) :=
      by rw [Function.iterate_succ_apply', IH]
    _ = (∑ k : ℕ in range n.succ,
          n.choose k • (derivative^[n - k + 1] p * derivative^[k] q)) +
        ∑ k : ℕ in range n.succ,
          n.choose k • (derivative^[n - k] p * derivative^[k + 1] q) := by
      simp_rw [derivative_sum, derivative_smul, derivative_mul, Function.iterate_succ_apply',
        smul_add, sum_add_distrib]
    _ = (∑ k : ℕ in range n.succ,
              n.choose k.succ • (derivative^[n - k] p * derivative^[k + 1] q)) +
            1 • (derivative^[n + 1] p * derivative^[0] q) +
          ∑ k : ℕ in range n.succ, n.choose k • (derivative^[n - k] p * derivative^[k + 1] q) :=
      ?_
    _ = ((∑ k : ℕ in range n.succ, n.choose k • (derivative^[n - k] p * derivative^[k + 1] q)) +
            ∑ k : ℕ in range n.succ,
              n.choose k.succ • (derivative^[n - k] p * derivative^[k + 1] q)) +
          1 • (derivative^[n + 1] p * derivative^[0] q) :=
      by rw [add_comm, add_assoc]
    _ = (∑ i : ℕ in range n.succ,
            (n + 1).choose (i + 1) • (derivative^[n + 1 - (i + 1)] p * derivative^[i + 1] q)) +
          1 • (derivative^[n + 1] p * derivative^[0] q) :=
      by simp_rw [Nat.choose_succ_succ, Nat.succ_sub_succ, add_smul, sum_add_distrib]
    _ = ∑ k : ℕ in range n.succ.succ,
          n.succ.choose k • (derivative^[n.succ - k] p * derivative^[k] q) :=
      by rw [sum_range_succ' _ n.succ, Nat.choose_zero_right, tsub_zero]
  

Proof state:
case succ
R : Type u
S : Type v
T : Type w
ι : Type y
A : Type z
a b : R
n✝ : ℕ
inst✝ : Semiring R
p q : R[X]
n : ℕ
IH :
  (⇑derivative)^[n] (p * q) =
    ∑ k in range (Nat.succ n), Nat.choose n k • ((⇑derivative)^[n - k] p * (⇑derivative)^[k] q)
⊢ ∑ k in range (Nat.succ n), Nat.choose n k • ((⇑derivative)^[n - k + 1] p * (⇑derivative)^[k] q) +
      ∑ k in range (Nat.succ n), Nat.choose n k • ((⇑derivative)^[n - k] p * (⇑derivative)^[k + 1] q) =
    ∑ k in range (Nat.succ n), Nat.choose n (Nat.succ k) • ((⇑derivative)^[n - k] p * (⇑derivative)^[k + 1] q) +
        1 • ((⇑derivative)^[n + 1] p * (⇑derivative)^[0] q) +
      ∑ k in range (Nat.succ n), Nat.choose n k • ((⇑derivative)^[n - k] p * (⇑derivative)^[k + 1] q)

Next tactic:
congr
theorem iterate_derivative_mul {n} (p q : R[X]) :
    derivative^[n] (p * q) =
      ∑ k in range n.succ, (n.choose k • (derivative^[n - k] p * derivative^[k] q)) := by
  induction' n with n IH
  · simp [Finset.range]
  calc
    derivative^[n + 1] (p * q) =
        derivative (∑ k : ℕ in range n.succ,
            n.choose k • (derivative^[n - k] p * derivative^[k] q)) :=
      by rw [Function.iterate_succ_apply', IH]
    _ = (∑ k : ℕ in range n.succ,
          n.choose k • (derivative^[n - k + 1] p * derivative^[k] q)) +
        ∑ k : ℕ in range n.succ,
          n.choose k • (derivative^[n - k] p * derivative^[k + 1] q) := by
      simp_rw [derivative_sum, derivative_smul, derivative_mul, Function.iterate_succ_apply',
        smul_add, sum_add_distrib]
    _ = (∑ k : ℕ in range n.succ,
              n.choose k.succ • (derivative^[n - k] p * derivative^[k + 1] q)) +
            1 • (derivative^[n + 1] p * derivative^[0] q) +
          ∑ k : ℕ in range n.succ, n.choose k • (derivative^[n - k] p * derivative^[k + 1] q) :=
      ?_
    _ = ((∑ k : ℕ in range n.succ, n.choose k • (derivative^[n - k] p * derivative^[k + 1] q)) +
            ∑ k : ℕ in range n.succ,
              n.choose k.succ • (derivative^[n - k] p * derivative^[k + 1] q)) +
          1 • (derivative^[n + 1] p * derivative^[0] q) :=
      by rw [add_comm, add_assoc]
    _ = (∑ i : ℕ in range n.succ,
            (n + 1).choose (i + 1) • (derivative^[n + 1 - (i + 1)] p * derivative^[i + 1] q)) +
          1 • (derivative^[n + 1] p * derivative^[0] q) :=
      by simp_rw [Nat.choose_succ_succ, Nat.succ_sub_succ, add_smul, sum_add_distrib]
    _ = ∑ k : ℕ in range n.succ.succ,
          n.succ.choose k • (derivative^[n.succ - k] p * derivative^[k] q) :=
      by rw [sum_range_succ' _ n.succ, Nat.choose_zero_right, tsub_zero]
  congr
  

Proof state:
case succ.e_a
R : Type u
S : Type v
T : Type w
ι : Type y
A : Type z
a b : R
n✝ : ℕ
inst✝ : Semiring R
p q : R[X]
n : ℕ
IH :
  (⇑derivative)^[n] (p * q) =
    ∑ k in range (Nat.succ n), Nat.choose n k • ((⇑derivative)^[n - k] p * (⇑derivative)^[k] q)
⊢ ∑ k in range (Nat.succ n), Nat.choose n k • ((⇑derivative)^[n - k + 1] p * (⇑derivative)^[k] q) =
    ∑ k in range (Nat.succ n), Nat.choose n (Nat.succ k) • ((⇑derivative)^[n - k] p * (⇑derivative)^[k + 1] q) +
      1 • ((⇑derivative)^[n + 1] p * (⇑derivative)^[0] q)

Next tactic:
refine' (sum_range_succ' _ _).trans (congr_arg₂ (· + ·) _ _)
theorem iterate_derivative_mul {n} (p q : R[X]) :
    derivative^[n] (p * q) =
      ∑ k in range n.succ, (n.choose k • (derivative^[n - k] p * derivative^[k] q)) := by
  induction' n with n IH
  · simp [Finset.range]
  calc
    derivative^[n + 1] (p * q) =
        derivative (∑ k : ℕ in range n.succ,
            n.choose k • (derivative^[n - k] p * derivative^[k] q)) :=
      by rw [Function.iterate_succ_apply', IH]
    _ = (∑ k : ℕ in range n.succ,
          n.choose k • (derivative^[n - k + 1] p * derivative^[k] q)) +
        ∑ k : ℕ in range n.succ,
          n.choose k • (derivative^[n - k] p * derivative^[k + 1] q) := by
      simp_rw [derivative_sum, derivative_smul, derivative_mul, Function.iterate_succ_apply',
        smul_add, sum_add_distrib]
    _ = (∑ k : ℕ in range n.succ,
              n.choose k.succ • (derivative^[n - k] p * derivative^[k + 1] q)) +
            1 • (derivative^[n + 1] p * derivative^[0] q) +
          ∑ k : ℕ in range n.succ, n.choose k • (derivative^[n - k] p * derivative^[k + 1] q) :=
      ?_
    _ = ((∑ k : ℕ in range n.succ, n.choose k • (derivative^[n - k] p * derivative^[k + 1] q)) +
            ∑ k : ℕ in range n.succ,
              n.choose k.succ • (derivative^[n - k] p * derivative^[k + 1] q)) +
          1 • (derivative^[n + 1] p * derivative^[0] q) :=
      by rw [add_comm, add_assoc]
    _ = (∑ i : ℕ in range n.succ,
            (n + 1).choose (i + 1) • (derivative^[n + 1 - (i + 1)] p * derivative^[i + 1] q)) +
          1 • (derivative^[n + 1] p * derivative^[0] q) :=
      by simp_rw [Nat.choose_succ_succ, Nat.succ_sub_succ, add_smul, sum_add_distrib]
    _ = ∑ k : ℕ in range n.succ.succ,
          n.succ.choose k • (derivative^[n.succ - k] p * derivative^[k] q) :=
      by rw [sum_range_succ' _ n.succ, Nat.choose_zero_right, tsub_zero]
  congr
  refine' (sum_range_succ' _ _).trans (congr_arg₂ (· + ·) _ _)
  · 

Proof state:
case succ.e_a.refine'_1
R : Type u
S : Type v
T : Type w
ι : Type y
A : Type z
a b : R
n✝ : ℕ
inst✝ : Semiring R
p q : R[X]
n : ℕ
IH :
  (⇑derivative)^[n] (p * q) =
    ∑ k in range (Nat.succ n), Nat.choose n k • ((⇑derivative)^[n - k] p * (⇑derivative)^[k] q)
⊢ ∑ k in range n, Nat.choose n (k + 1) • ((⇑derivative)^[n - (k + 1) + 1] p * (⇑derivative)^[k + 1] q) =
    ∑ k in range (Nat.succ n), Nat.choose n (Nat.succ k) • ((⇑derivative)^[n - k] p * (⇑derivative)^[k + 1] q)

Next tactic:
rw [sum_range_succ, Nat.choose_succ_self, zero_smul, add_zero]
theorem iterate_derivative_mul {n} (p q : R[X]) :
    derivative^[n] (p * q) =
      ∑ k in range n.succ, (n.choose k • (derivative^[n - k] p * derivative^[k] q)) := by
  induction' n with n IH
  · simp [Finset.range]
  calc
    derivative^[n + 1] (p * q) =
        derivative (∑ k : ℕ in range n.succ,
            n.choose k • (derivative^[n - k] p * derivative^[k] q)) :=
      by rw [Function.iterate_succ_apply', IH]
    _ = (∑ k : ℕ in range n.succ,
          n.choose k • (derivative^[n - k + 1] p * derivative^[k] q)) +
        ∑ k : ℕ in range n.succ,
          n.choose k • (derivative^[n - k] p * derivative^[k + 1] q) := by
      simp_rw [derivative_sum, derivative_smul, derivative_mul, Function.iterate_succ_apply',
        smul_add, sum_add_distrib]
    _ = (∑ k : ℕ in range n.succ,
              n.choose k.succ • (derivative^[n - k] p * derivative^[k + 1] q)) +
            1 • (derivative^[n + 1] p * derivative^[0] q) +
          ∑ k : ℕ in range n.succ, n.choose k • (derivative^[n - k] p * derivative^[k + 1] q) :=
      ?_
    _ = ((∑ k : ℕ in range n.succ, n.choose k • (derivative^[n - k] p * derivative^[k + 1] q)) +
            ∑ k : ℕ in range n.succ,
              n.choose k.succ • (derivative^[n - k] p * derivative^[k + 1] q)) +
          1 • (derivative^[n + 1] p * derivative^[0] q) :=
      by rw [add_comm, add_assoc]
    _ = (∑ i : ℕ in range n.succ,
            (n + 1).choose (i + 1) • (derivative^[n + 1 - (i + 1)] p * derivative^[i + 1] q)) +
          1 • (derivative^[n + 1] p * derivative^[0] q) :=
      by simp_rw [Nat.choose_succ_succ, Nat.succ_sub_succ, add_smul, sum_add_distrib]
    _ = ∑ k : ℕ in range n.succ.succ,
          n.succ.choose k • (derivative^[n.succ - k] p * derivative^[k] q) :=
      by rw [sum_range_succ' _ n.succ, Nat.choose_zero_right, tsub_zero]
  congr
  refine' (sum_range_succ' _ _).trans (congr_arg₂ (· + ·) _ _)
  · rw [sum_range_succ, Nat.choose_succ_self, zero_smul, add_zero]
    

Proof state:
case succ.e_a.refine'_1
R : Type u
S : Type v
T : Type w
ι : Type y
A : Type z
a b : R
n✝ : ℕ
inst✝ : Semiring R
p q : R[X]
n : ℕ
IH :
  (⇑derivative)^[n] (p * q) =
    ∑ k in range (Nat.succ n), Nat.choose n k • ((⇑derivative)^[n - k] p * (⇑derivative)^[k] q)
⊢ ∑ k in range n, Nat.choose n (k + 1) • ((⇑derivative)^[n - (k + 1) + 1] p * (⇑derivative)^[k + 1] q) =
    ∑ x in range n, Nat.choose n (Nat.succ x) • ((⇑derivative)^[n - x] p * (⇑derivative)^[x + 1] q)

Next tactic:
refine' sum_congr rfl fun k hk => _