l3lab/ntp-mathlib huggingface dataset:
theorem natDegree_derivative_lt {p : R[X]} (hp : p.natDegree ≠ 0) :
    p.derivative.natDegree < p.natDegree := by
  rcases eq_or_ne (derivative p) 0 with hp' | hp'
  · rw [hp', Polynomial.natDegree_zero]
    

Proof state:
case inl
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]
hp : natDegree p ≠ 0
hp' : derivative p = 0
⊢ 0 < natDegree p

Next tactic:
exact hp.bot_lt
theorem natDegree_derivative_lt {p : R[X]} (hp : p.natDegree ≠ 0) :
    p.derivative.natDegree < p.natDegree := by
  rcases eq_or_ne (derivative p) 0 with hp' | hp'
  · rw [hp', Polynomial.natDegree_zero]
    exact hp.bot_lt
  · 

Proof state:
case inr
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]
hp : natDegree p ≠ 0
hp' : derivative p ≠ 0
⊢ natDegree (derivative p) < natDegree p

Next tactic:
rw [natDegree_lt_natDegree_iff hp']
theorem natDegree_derivative_lt {p : R[X]} (hp : p.natDegree ≠ 0) :
    p.derivative.natDegree < p.natDegree := by
  rcases eq_or_ne (derivative p) 0 with hp' | hp'
  · rw [hp', Polynomial.natDegree_zero]
    exact hp.bot_lt
  · rw [natDegree_lt_natDegree_iff hp']
    

Proof state:
case inr
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]
hp : natDegree p ≠ 0
hp' : derivative p ≠ 0
⊢ degree (derivative p) < degree p

Next tactic:
exact degree_derivative_lt fun h => hp (h.symm ▸ natDegree_zero)
theorem natDegree_derivative_le (p : R[X]) : p.derivative.natDegree ≤ p.natDegree - 1 := 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]
⊢ natDegree (derivative p) ≤ natDegree p - 1

Next tactic:
by_cases p0 : p.natDegree = 0
theorem natDegree_derivative_le (p : R[X]) : p.derivative.natDegree ≤ p.natDegree - 1 := by
  by_cases p0 : p.natDegree = 0
  · 

Proof state:
case pos
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]
p0 : natDegree p = 0
⊢ natDegree (derivative p) ≤ natDegree p - 1

Next tactic:
simp [p0, derivative_of_natDegree_zero]
theorem natDegree_derivative_le (p : R[X]) : p.derivative.natDegree ≤ p.natDegree - 1 := by
  by_cases p0 : p.natDegree = 0
  · simp [p0, derivative_of_natDegree_zero]
  · 

Proof state:
case neg
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]
p0 : ¬natDegree p = 0
⊢ natDegree (derivative p) ≤ natDegree p - 1

Next tactic:
exact Nat.le_sub_one_of_lt (natDegree_derivative_lt p0)
theorem natDegree_iterate_derivative (p : R[X]) (k : ℕ) :
    (derivative^[k] p).natDegree ≤ p.natDegree - k := 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
⊢ natDegree ((⇑derivative)^[k] p) ≤ natDegree p - k

Next tactic:
induction k with
  | zero => rw [Function.iterate_zero_apply, Nat.sub_zero]
  | succ d hd =>
      rw [Function.iterate_succ_apply', Nat.sub_succ']
      exact (natDegree_derivative_le _).trans <| Nat.sub_le_sub_right hd 1
theorem natDegree_iterate_derivative (p : R[X]) (k : ℕ) :
    (derivative^[k] p).natDegree ≤ p.natDegree - k := by
  induction k with
  

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 : R[X]
⊢ natDegree ((⇑derivative)^[Nat.zero] p) ≤ natDegree p - Nat.zero

Next tactic:
| zero => rw [Function.iterate_zero_apply, Nat.sub_zero]
theorem natDegree_iterate_derivative (p : R[X]) (k : ℕ) :
    (derivative^[k] p).natDegree ≤ p.natDegree - k := by
  induction k with
  | zero => 

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 : R[X]
⊢ natDegree ((⇑derivative)^[Nat.zero] p) ≤ natDegree p - Nat.zero

Next tactic:
rw [Function.iterate_zero_apply, Nat.sub_zero]
theorem natDegree_iterate_derivative (p : R[X]) (k : ℕ) :
    (derivative^[k] p).natDegree ≤ p.natDegree - k := by
  induction k with
  | zero => rw [Function.iterate_zero_apply, Nat.sub_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 : R[X]
d : ℕ
hd : natDegree ((⇑derivative)^[d] p) ≤ natDegree p - d
⊢ natDegree ((⇑derivative)^[Nat.succ d] p) ≤ natDegree p - Nat.succ d

Next tactic:
| succ d hd =>
      rw [Function.iterate_succ_apply', Nat.sub_succ']
      exact (natDegree_derivative_le _).trans <| Nat.sub_le_sub_right hd 1
theorem natDegree_iterate_derivative (p : R[X]) (k : ℕ) :
    (derivative^[k] p).natDegree ≤ p.natDegree - k := by
  induction k with
  | zero => rw [Function.iterate_zero_apply, Nat.sub_zero]
  | succ d hd =>
      

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 : R[X]
d : ℕ
hd : natDegree ((⇑derivative)^[d] p) ≤ natDegree p - d
⊢ natDegree ((⇑derivative)^[Nat.succ d] p) ≤ natDegree p - Nat.succ d

Next tactic:
rw [Function.iterate_succ_apply', Nat.sub_succ']
theorem natDegree_iterate_derivative (p : R[X]) (k : ℕ) :
    (derivative^[k] p).natDegree ≤ p.natDegree - k := by
  induction k with
  | zero => rw [Function.iterate_zero_apply, Nat.sub_zero]
  | succ d hd =>
      rw [Function.iterate_succ_apply', Nat.sub_succ']
      

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 : R[X]
d : ℕ
hd : natDegree ((⇑derivative)^[d] p) ≤ natDegree p - d
⊢ natDegree (derivative ((⇑derivative)^[d] p)) ≤ natDegree p - d - 1

Next tactic:
exact (natDegree_derivative_le _).trans <| Nat.sub_le_sub_right hd 1
@[simp]
theorem derivative_nat_cast {n : ℕ} : derivative (n : R[X]) = 0 := 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
⊢ derivative ↑n = 0

Next tactic:
rw [← map_natCast C n]
@[simp]
theorem derivative_nat_cast {n : ℕ} : derivative (n : R[X]) = 0 := by
  rw [← map_natCast C n]
  

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
⊢ derivative (C ↑n) = 0

Next tactic:
exact derivative_C
theorem iterate_derivative_eq_zero {p : R[X]} {x : ℕ} (hx : p.natDegree < x) :
    Polynomial.derivative^[x] p = 0 := 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]
x : ℕ
hx : natDegree p < x
⊢ (⇑derivative)^[x] p = 0

Next tactic:
induction' h : p.natDegree using Nat.strong_induction_on with _ ih generalizing p x
theorem iterate_derivative_eq_zero {p : R[X]} {x : ℕ} (hx : p.natDegree < x) :
    Polynomial.derivative^[x] p = 0 := by
  induction' h : p.natDegree using Nat.strong_induction_on with _ ih generalizing p x
  

Proof state:
case h
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
ih : ∀ m < n✝, ∀ {p : R[X]} {x : ℕ}, natDegree p < x → natDegree p = m → (⇑derivative)^[x] p = 0
p : R[X]
x : ℕ
hx : natDegree p < x
h : natDegree p = n✝
⊢ (⇑derivative)^[x] p = 0

Next tactic:
subst h
theorem iterate_derivative_eq_zero {p : R[X]} {x : ℕ} (hx : p.natDegree < x) :
    Polynomial.derivative^[x] p = 0 := by
  induction' h : p.natDegree using Nat.strong_induction_on with _ ih generalizing p x
  subst h
  

Proof state:
case h
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]
x : ℕ
hx : natDegree p < x
ih : ∀ m < natDegree p, ∀ {p : R[X]} {x : ℕ}, natDegree p < x → natDegree p = m → (⇑derivative)^[x] p = 0
⊢ (⇑derivative)^[x] p = 0

Next tactic:
obtain ⟨t, rfl⟩ := Nat.exists_eq_succ_of_ne_zero (pos_of_gt hx).ne'
theorem iterate_derivative_eq_zero {p : R[X]} {x : ℕ} (hx : p.natDegree < x) :
    Polynomial.derivative^[x] p = 0 := by
  induction' h : p.natDegree using Nat.strong_induction_on with _ ih generalizing p x
  subst h
  obtain ⟨t, rfl⟩ := Nat.exists_eq_succ_of_ne_zero (pos_of_gt hx).ne'
  

Proof state:
case h.intro
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]
ih : ∀ m < natDegree p, ∀ {p : R[X]} {x : ℕ}, natDegree p < x → natDegree p = m → (⇑derivative)^[x] p = 0
t : ℕ
hx : natDegree p < Nat.succ t
⊢ (⇑derivative)^[Nat.succ t] p = 0

Next tactic:
rw [Function.iterate_succ_apply]
theorem iterate_derivative_eq_zero {p : R[X]} {x : ℕ} (hx : p.natDegree < x) :
    Polynomial.derivative^[x] p = 0 := by
  induction' h : p.natDegree using Nat.strong_induction_on with _ ih generalizing p x
  subst h
  obtain ⟨t, rfl⟩ := Nat.exists_eq_succ_of_ne_zero (pos_of_gt hx).ne'
  rw [Function.iterate_succ_apply]
  

Proof state:
case h.intro
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]
ih : ∀ m < natDegree p, ∀ {p : R[X]} {x : ℕ}, natDegree p < x → natDegree p = m → (⇑derivative)^[x] p = 0
t : ℕ
hx : natDegree p < Nat.succ t
⊢ (⇑derivative)^[t] (derivative p) = 0

Next tactic:
by_cases hp : p.natDegree = 0
theorem iterate_derivative_eq_zero {p : R[X]} {x : ℕ} (hx : p.natDegree < x) :
    Polynomial.derivative^[x] p = 0 := by
  induction' h : p.natDegree using Nat.strong_induction_on with _ ih generalizing p x
  subst h
  obtain ⟨t, rfl⟩ := Nat.exists_eq_succ_of_ne_zero (pos_of_gt hx).ne'
  rw [Function.iterate_succ_apply]
  by_cases hp : p.natDegree = 0
  · 

Proof state:
case pos
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]
ih : ∀ m < natDegree p, ∀ {p : R[X]} {x : ℕ}, natDegree p < x → natDegree p = m → (⇑derivative)^[x] p = 0
t : ℕ
hx : natDegree p < Nat.succ t
hp : natDegree p = 0
⊢ (⇑derivative)^[t] (derivative p) = 0

Next tactic:
rw [derivative_of_natDegree_zero hp, iterate_derivative_zero]
theorem iterate_derivative_eq_zero {p : R[X]} {x : ℕ} (hx : p.natDegree < x) :
    Polynomial.derivative^[x] p = 0 := by
  induction' h : p.natDegree using Nat.strong_induction_on with _ ih generalizing p x
  subst h
  obtain ⟨t, rfl⟩ := Nat.exists_eq_succ_of_ne_zero (pos_of_gt hx).ne'
  rw [Function.iterate_succ_apply]
  by_cases hp : p.natDegree = 0
  · rw [derivative_of_natDegree_zero hp, iterate_derivative_zero]
  

Proof state:
case neg
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]
ih : ∀ m < natDegree p, ∀ {p : R[X]} {x : ℕ}, natDegree p < x → natDegree p = m → (⇑derivative)^[x] p = 0
t : ℕ
hx : natDegree p < Nat.succ t
hp : ¬natDegree p = 0
⊢ (⇑derivative)^[t] (derivative p) = 0

Next tactic:
have := natDegree_derivative_lt hp
theorem iterate_derivative_eq_zero {p : R[X]} {x : ℕ} (hx : p.natDegree < x) :
    Polynomial.derivative^[x] p = 0 := by
  induction' h : p.natDegree using Nat.strong_induction_on with _ ih generalizing p x
  subst h
  obtain ⟨t, rfl⟩ := Nat.exists_eq_succ_of_ne_zero (pos_of_gt hx).ne'
  rw [Function.iterate_succ_apply]
  by_cases hp : p.natDegree = 0
  · rw [derivative_of_natDegree_zero hp, iterate_derivative_zero]
  have := natDegree_derivative_lt hp
  

Proof state:
case neg
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]
ih : ∀ m < natDegree p, ∀ {p : R[X]} {x : ℕ}, natDegree p < x → natDegree p = m → (⇑derivative)^[x] p = 0
t : ℕ
hx : natDegree p < Nat.succ t
hp : ¬natDegree p = 0
this : natDegree (derivative p) < natDegree p
⊢ (⇑derivative)^[t] (derivative p) = 0

Next tactic:
exact ih _ this (this.trans_le <| Nat.le_of_lt_succ hx) rfl
theorem natDegree_eq_zero_of_derivative_eq_zero [NoZeroSMulDivisors ℕ R] {f : R[X]}
    (h : derivative f = 0) : f.natDegree = 0 := by
  

Proof state:
R : Type u
S : Type v
T : Type w
ι : Type y
A : Type z
a b : R
n : ℕ
inst✝¹ : Semiring R
inst✝ : NoZeroSMulDivisors ℕ R
f : R[X]
h : derivative f = 0
⊢ natDegree f = 0

Next tactic:
rcases eq_or_ne f 0 with (rfl | hf)
theorem natDegree_eq_zero_of_derivative_eq_zero [NoZeroSMulDivisors ℕ R] {f : R[X]}
    (h : derivative f = 0) : f.natDegree = 0 := by
  rcases eq_or_ne f 0 with (rfl | hf)
  · 

Proof state:
case inl
R : Type u
S : Type v
T : Type w
ι : Type y
A : Type z
a b : R
n : ℕ
inst✝¹ : Semiring R
inst✝ : NoZeroSMulDivisors ℕ R
h : derivative 0 = 0
⊢ natDegree 0 = 0

Next tactic:
exact natDegree_zero
theorem natDegree_eq_zero_of_derivative_eq_zero [NoZeroSMulDivisors ℕ R] {f : R[X]}
    (h : derivative f = 0) : f.natDegree = 0 := by
  rcases eq_or_ne f 0 with (rfl | hf)
  · exact natDegree_zero
  

Proof state:
case inr
R : Type u
S : Type v
T : Type w
ι : Type y
A : Type z
a b : R
n : ℕ
inst✝¹ : Semiring R
inst✝ : NoZeroSMulDivisors ℕ R
f : R[X]
h : derivative f = 0
hf : f ≠ 0
⊢ natDegree f = 0

Next tactic:
rw [natDegree_eq_zero_iff_degree_le_zero]
theorem natDegree_eq_zero_of_derivative_eq_zero [NoZeroSMulDivisors ℕ R] {f : R[X]}
    (h : derivative f = 0) : f.natDegree = 0 := by
  rcases eq_or_ne f 0 with (rfl | hf)
  · exact natDegree_zero
  rw [natDegree_eq_zero_iff_degree_le_zero]
  

Proof state:
case inr
R : Type u
S : Type v
T : Type w
ι : Type y
A : Type z
a b : R
n : ℕ
inst✝¹ : Semiring R
inst✝ : NoZeroSMulDivisors ℕ R
f : R[X]
h : derivative f = 0
hf : f ≠ 0
⊢ degree f ≤ 0

Next tactic:
by_contra! f_nat_degree_pos
theorem natDegree_eq_zero_of_derivative_eq_zero [NoZeroSMulDivisors ℕ R] {f : R[X]}
    (h : derivative f = 0) : f.natDegree = 0 := by
  rcases eq_or_ne f 0 with (rfl | hf)
  · exact natDegree_zero
  rw [natDegree_eq_zero_iff_degree_le_zero]
  by_contra! f_nat_degree_pos
  

Proof state:
case inr
R : Type u
S : Type v
T : Type w
ι : Type y
A : Type z
a b : R
n : ℕ
inst✝¹ : Semiring R
inst✝ : NoZeroSMulDivisors ℕ R
f : R[X]
h : derivative f = 0
hf : f ≠ 0
f_nat_degree_pos : 0 < degree f
⊢ False

Next tactic:
rw [← natDegree_pos_iff_degree_pos] at f_nat_degree_pos
theorem natDegree_eq_zero_of_derivative_eq_zero [NoZeroSMulDivisors ℕ R] {f : R[X]}
    (h : derivative f = 0) : f.natDegree = 0 := by
  rcases eq_or_ne f 0 with (rfl | hf)
  · exact natDegree_zero
  rw [natDegree_eq_zero_iff_degree_le_zero]
  by_contra! f_nat_degree_pos
  rw [← natDegree_pos_iff_degree_pos] at f_nat_degree_pos
  

Proof state:
case inr
R : Type u
S : Type v
T : Type w
ι : Type y
A : Type z
a b : R
n : ℕ
inst✝¹ : Semiring R
inst✝ : NoZeroSMulDivisors ℕ R
f : R[X]
h : derivative f = 0
hf : f ≠ 0
f_nat_degree_pos : 0 < natDegree f
⊢ False

Next tactic:
let m := f.natDegree - 1
theorem natDegree_eq_zero_of_derivative_eq_zero [NoZeroSMulDivisors ℕ R] {f : R[X]}
    (h : derivative f = 0) : f.natDegree = 0 := by
  rcases eq_or_ne f 0 with (rfl | hf)
  · exact natDegree_zero
  rw [natDegree_eq_zero_iff_degree_le_zero]
  by_contra! f_nat_degree_pos
  rw [← natDegree_pos_iff_degree_pos] at f_nat_degree_pos
  let m := f.natDegree - 1
  

Proof state:
case inr
R : Type u
S : Type v
T : Type w
ι : Type y
A : Type z
a b : R
n : ℕ
inst✝¹ : Semiring R
inst✝ : NoZeroSMulDivisors ℕ R
f : R[X]
h : derivative f = 0
hf : f ≠ 0
f_nat_degree_pos : 0 < natDegree f
m : ℕ := natDegree f - 1
⊢ False

Next tactic:
have hm : m + 1 = f.natDegree := tsub_add_cancel_of_le f_nat_degree_pos
theorem natDegree_eq_zero_of_derivative_eq_zero [NoZeroSMulDivisors ℕ R] {f : R[X]}
    (h : derivative f = 0) : f.natDegree = 0 := by
  rcases eq_or_ne f 0 with (rfl | hf)
  · exact natDegree_zero
  rw [natDegree_eq_zero_iff_degree_le_zero]
  by_contra! f_nat_degree_pos
  rw [← natDegree_pos_iff_degree_pos] at f_nat_degree_pos
  let m := f.natDegree - 1
  have hm : m + 1 = f.natDegree := tsub_add_cancel_of_le f_nat_degree_pos
  

Proof state:
case inr
R : Type u
S : Type v
T : Type w
ι : Type y
A : Type z
a b : R
n : ℕ
inst✝¹ : Semiring R
inst✝ : NoZeroSMulDivisors ℕ R
f : R[X]
h : derivative f = 0
hf : f ≠ 0
f_nat_degree_pos : 0 < natDegree f
m : ℕ := natDegree f - 1
hm : m + 1 = natDegree f
⊢ False

Next tactic:
have h2 := coeff_derivative f m
theorem natDegree_eq_zero_of_derivative_eq_zero [NoZeroSMulDivisors ℕ R] {f : R[X]}
    (h : derivative f = 0) : f.natDegree = 0 := by
  rcases eq_or_ne f 0 with (rfl | hf)
  · exact natDegree_zero
  rw [natDegree_eq_zero_iff_degree_le_zero]
  by_contra! f_nat_degree_pos
  rw [← natDegree_pos_iff_degree_pos] at f_nat_degree_pos
  let m := f.natDegree - 1
  have hm : m + 1 = f.natDegree := tsub_add_cancel_of_le f_nat_degree_pos
  have h2 := coeff_derivative f m
  

Proof state:
case inr
R : Type u
S : Type v
T : Type w
ι : Type y
A : Type z
a b : R
n : ℕ
inst✝¹ : Semiring R
inst✝ : NoZeroSMulDivisors ℕ R
f : R[X]
h : derivative f = 0
hf : f ≠ 0
f_nat_degree_pos : 0 < natDegree f
m : ℕ := natDegree f - 1
hm : m + 1 = natDegree f
h2 : coeff (derivative f) m = coeff f (m + 1) * (↑m + 1)
⊢ False

Next tactic:
rw [Polynomial.ext_iff] at h
theorem natDegree_eq_zero_of_derivative_eq_zero [NoZeroSMulDivisors ℕ R] {f : R[X]}
    (h : derivative f = 0) : f.natDegree = 0 := by
  rcases eq_or_ne f 0 with (rfl | hf)
  · exact natDegree_zero
  rw [natDegree_eq_zero_iff_degree_le_zero]
  by_contra! f_nat_degree_pos
  rw [← natDegree_pos_iff_degree_pos] at f_nat_degree_pos
  let m := f.natDegree - 1
  have hm : m + 1 = f.natDegree := tsub_add_cancel_of_le f_nat_degree_pos
  have h2 := coeff_derivative f m
  rw [Polynomial.ext_iff] at h
  

Proof state:
case inr
R : Type u
S : Type v
T : Type w
ι : Type y
A : Type z
a b : R
n : ℕ
inst✝¹ : Semiring R
inst✝ : NoZeroSMulDivisors ℕ R
f : R[X]
h : ∀ (n : ℕ), coeff (derivative f) n = coeff 0 n
hf : f ≠ 0
f_nat_degree_pos : 0 < natDegree f
m : ℕ := natDegree f - 1
hm : m + 1 = natDegree f
h2 : coeff (derivative f) m = coeff f (m + 1) * (↑m + 1)
⊢ False

Next tactic:
rw [h m, coeff_zero, ← Nat.cast_add_one, ← nsmul_eq_mul', eq_comm, smul_eq_zero] at h2
theorem natDegree_eq_zero_of_derivative_eq_zero [NoZeroSMulDivisors ℕ R] {f : R[X]}
    (h : derivative f = 0) : f.natDegree = 0 := by
  rcases eq_or_ne f 0 with (rfl | hf)
  · exact natDegree_zero
  rw [natDegree_eq_zero_iff_degree_le_zero]
  by_contra! f_nat_degree_pos
  rw [← natDegree_pos_iff_degree_pos] at f_nat_degree_pos
  let m := f.natDegree - 1
  have hm : m + 1 = f.natDegree := tsub_add_cancel_of_le f_nat_degree_pos
  have h2 := coeff_derivative f m
  rw [Polynomial.ext_iff] at h
  rw [h m, coeff_zero, ← Nat.cast_add_one, ← nsmul_eq_mul', eq_comm, smul_eq_zero] at h2
  

Proof state:
case inr
R : Type u
S : Type v
T : Type w
ι : Type y
A : Type z
a b : R
n : ℕ
inst✝¹ : Semiring R
inst✝ : NoZeroSMulDivisors ℕ R
f : R[X]
h : ∀ (n : ℕ), coeff (derivative f) n = coeff 0 n
hf : f ≠ 0
f_nat_degree_pos : 0 < natDegree f
m : ℕ := natDegree f - 1
hm : m + 1 = natDegree f
h2 : m + 1 = 0 ∨ coeff f (m + 1) = 0
⊢ False

Next tactic:
replace h2 := h2.resolve_left m.succ_ne_zero
theorem natDegree_eq_zero_of_derivative_eq_zero [NoZeroSMulDivisors ℕ R] {f : R[X]}
    (h : derivative f = 0) : f.natDegree = 0 := by
  rcases eq_or_ne f 0 with (rfl | hf)
  · exact natDegree_zero
  rw [natDegree_eq_zero_iff_degree_le_zero]
  by_contra! f_nat_degree_pos
  rw [← natDegree_pos_iff_degree_pos] at f_nat_degree_pos
  let m := f.natDegree - 1
  have hm : m + 1 = f.natDegree := tsub_add_cancel_of_le f_nat_degree_pos
  have h2 := coeff_derivative f m
  rw [Polynomial.ext_iff] at h
  rw [h m, coeff_zero, ← Nat.cast_add_one, ← nsmul_eq_mul', eq_comm, smul_eq_zero] at h2
  replace h2 := h2.resolve_left m.succ_ne_zero
  

Proof state:
case inr
R : Type u
S : Type v
T : Type w
ι : Type y
A : Type z
a b : R
n : ℕ
inst✝¹ : Semiring R
inst✝ : NoZeroSMulDivisors ℕ R
f : R[X]
h : ∀ (n : ℕ), coeff (derivative f) n = coeff 0 n
hf : f ≠ 0
f_nat_degree_pos : 0 < natDegree f
m : ℕ := natDegree f - 1
hm : m + 1 = natDegree f
h2 : coeff f (m + 1) = 0
⊢ False

Next tactic:
rw [hm, ← leadingCoeff, leadingCoeff_eq_zero] at h2
theorem natDegree_eq_zero_of_derivative_eq_zero [NoZeroSMulDivisors ℕ R] {f : R[X]}
    (h : derivative f = 0) : f.natDegree = 0 := by
  rcases eq_or_ne f 0 with (rfl | hf)
  · exact natDegree_zero
  rw [natDegree_eq_zero_iff_degree_le_zero]
  by_contra! f_nat_degree_pos
  rw [← natDegree_pos_iff_degree_pos] at f_nat_degree_pos
  let m := f.natDegree - 1
  have hm : m + 1 = f.natDegree := tsub_add_cancel_of_le f_nat_degree_pos
  have h2 := coeff_derivative f m
  rw [Polynomial.ext_iff] at h
  rw [h m, coeff_zero, ← Nat.cast_add_one, ← nsmul_eq_mul', eq_comm, smul_eq_zero] at h2
  replace h2 := h2.resolve_left m.succ_ne_zero
  rw [hm, ← leadingCoeff, leadingCoeff_eq_zero] at h2
  

Proof state:
case inr
R : Type u
S : Type v
T : Type w
ι : Type y
A : Type z
a b : R
n : ℕ
inst✝¹ : Semiring R
inst✝ : NoZeroSMulDivisors ℕ R
f : R[X]
h : ∀ (n : ℕ), coeff (derivative f) n = coeff 0 n
hf : f ≠ 0
f_nat_degree_pos : 0 < natDegree f
m : ℕ := natDegree f - 1
hm : m + 1 = natDegree f
h2 : f = 0
⊢ False

Next tactic:
exact hf h2
@[simp]
theorem derivative_mul {f g : R[X]} : derivative (f * g) = derivative f * g + f * derivative g :=
  calc
    derivative (f * g) =
        f.sum fun n a => g.sum fun m b => (n + m) • (C (a * b) * X ^ (n + m - 1)) := 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
f g : R[X]
⊢ derivative (f * g) = sum f fun n a => sum g fun m b => (n + m) • (C (a * b) * X ^ (n + m - 1))

Next tactic:
rw [mul_eq_sum_sum, derivative_sum]
@[simp]
theorem derivative_mul {f g : R[X]} : derivative (f * g) = derivative f * g + f * derivative g :=
  calc
    derivative (f * g) =
        f.sum fun n a => g.sum fun m b => (n + m) • (C (a * b) * X ^ (n + m - 1)) := by
      rw [mul_eq_sum_sum, derivative_sum]
      

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
f g : R[X]
⊢ ∑ b in support f, derivative (sum g fun j a => (monomial (b + j)) (coeff f b * a)) =
    sum f fun n a => sum g fun m b => (n + m) • (C (a * b) * X ^ (n + m - 1))

Next tactic:
trans
@[simp]
theorem derivative_mul {f g : R[X]} : derivative (f * g) = derivative f * g + f * derivative g :=
  calc
    derivative (f * g) =
        f.sum fun n a => g.sum fun m b => (n + m) • (C (a * b) * X ^ (n + m - 1)) := by
      rw [mul_eq_sum_sum, derivative_sum]
      trans
      · 

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
f g : R[X]
⊢ ∑ b in support f, derivative (sum g fun j a => (monomial (b + j)) (coeff f b * a)) = ?m.409787

Next tactic:
apply Finset.sum_congr rfl
@[simp]
theorem derivative_mul {f g : R[X]} : derivative (f * g) = derivative f * g + f * derivative g :=
  calc
    derivative (f * g) =
        f.sum fun n a => g.sum fun m b => (n + m) • (C (a * b) * X ^ (n + m - 1)) := by
      rw [mul_eq_sum_sum, derivative_sum]
      trans
      · apply Finset.sum_congr rfl
        

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
f g : R[X]
⊢ ∀ x ∈ support f, derivative (sum g fun j a => (monomial (x + j)) (coeff f x * a)) = ?m.409837 x
R : Type u S : Type v T : Type w ι : Type y A : Type z a b : R n : ℕ inst✝ : Semiring R f g : R[X] ⊢ ℕ → R[X]

Next tactic:
intro x _
@[simp]
theorem derivative_mul {f g : R[X]} : derivative (f * g) = derivative f * g + f * derivative g :=
  calc
    derivative (f * g) =
        f.sum fun n a => g.sum fun m b => (n + m) • (C (a * b) * X ^ (n + m - 1)) := by
      rw [mul_eq_sum_sum, derivative_sum]
      trans
      · apply Finset.sum_congr rfl
        intro x _
        

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
f g : R[X]
x : ℕ
a✝ : x ∈ support f
⊢ derivative (sum g fun j a => (monomial (x + j)) (coeff f x * a)) = ?m.409837 x
R : Type u S : Type v T : Type w ι : Type y A : Type z a b : R n : ℕ inst✝ : Semiring R f g : R[X] ⊢ ℕ → R[X]

Next tactic:
exact derivative_sum
@[simp]
theorem derivative_mul {f g : R[X]} : derivative (f * g) = derivative f * g + f * derivative g :=
  calc
    derivative (f * g) =
        f.sum fun n a => g.sum fun m b => (n + m) • (C (a * b) * X ^ (n + m - 1)) := by
      rw [mul_eq_sum_sum, derivative_sum]
      trans
      · apply Finset.sum_congr rfl
        intro x _
        exact derivative_sum
      

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
f g : R[X]
⊢ ∑ x in support f, ∑ b in support g, derivative ((fun j a => (monomial (x + j)) (coeff f x * a)) b (coeff g b)) =
    sum f fun n a => sum g fun m b => (n + m) • (C (a * b) * X ^ (n + m - 1))

Next tactic:
apply Finset.sum_congr rfl