l3lab/ntp-mathlib huggingface dataset:
theorem bernoulli_spec' (n : ℕ) :
    (∑ k in antidiagonal n, ((k.1 + k.2).choose k.2 : ℚ) / (k.2 + 1) * bernoulli k.1) =
      if n = 0 then 1 else 0 := by
  cases' n with n n;
  · simp
  rw [if_neg (succ_ne_zero _)]
  -- algebra facts
  have h₁ : (1, n) ∈ antidiagonal n.succ := by simp [mem_antidiagonal, add_comm]
  have h₂ : (n : ℚ) + 1 ≠ 0 := by norm_cast; exact succ_ne_zero _
  have h₃ : (1 + n).choose n = n + 1 := by simp [add_comm]
  -- key equation: the corresponding fact for `bernoulli'`
  have H := bernoulli'_spec' n.succ
  -- massage it to match the structure of the goal, then convert piece by piece
  rw [sum_eq_add_sum_diff_singleton h₁] at H ⊢
  apply add_eq_of_eq_sub'
  convert eq_sub_of_add_eq' H using 1
  · 

Proof state:
case h.e'_2
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
h₁ : (1, n) ∈ antidiagonal (succ n)
h₂ : ↑n + 1 ≠ 0
h₃ : Nat.choose (1 + n) n = n + 1
H :
  ↑(Nat.choose ((1, n).1 + (1, n).2) (1, n).2) / (↑(1, n).2 + 1) * bernoulli' (1, n).1 +
      ∑ x in antidiagonal (succ n) \ {(1, n)}, ↑(Nat.choose (x.1 + x.2) x.2) / (↑x.2 + 1) * bernoulli' x.1 =
    1
⊢ ∑ x in antidiagonal (succ n) \ {(1, n)}, ↑(Nat.choose (x.1 + x.2) x.2) / (↑x.2 + 1) * bernoulli x.1 =
    ∑ x in antidiagonal (succ n) \ {(1, n)}, ↑(Nat.choose (x.1 + x.2) x.2) / (↑x.2 + 1) * bernoulli' x.1

Next tactic:
refine' sum_congr rfl fun p h => _
theorem bernoulli_spec' (n : ℕ) :
    (∑ k in antidiagonal n, ((k.1 + k.2).choose k.2 : ℚ) / (k.2 + 1) * bernoulli k.1) =
      if n = 0 then 1 else 0 := by
  cases' n with n n;
  · simp
  rw [if_neg (succ_ne_zero _)]
  -- algebra facts
  have h₁ : (1, n) ∈ antidiagonal n.succ := by simp [mem_antidiagonal, add_comm]
  have h₂ : (n : ℚ) + 1 ≠ 0 := by norm_cast; exact succ_ne_zero _
  have h₃ : (1 + n).choose n = n + 1 := by simp [add_comm]
  -- key equation: the corresponding fact for `bernoulli'`
  have H := bernoulli'_spec' n.succ
  -- massage it to match the structure of the goal, then convert piece by piece
  rw [sum_eq_add_sum_diff_singleton h₁] at H ⊢
  apply add_eq_of_eq_sub'
  convert eq_sub_of_add_eq' H using 1
  · refine' sum_congr rfl fun p h => _
    

Proof state:
case h.e'_2
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
h₁ : (1, n) ∈ antidiagonal (succ n)
h₂ : ↑n + 1 ≠ 0
h₃ : Nat.choose (1 + n) n = n + 1
H :
  ↑(Nat.choose ((1, n).1 + (1, n).2) (1, n).2) / (↑(1, n).2 + 1) * bernoulli' (1, n).1 +
      ∑ x in antidiagonal (succ n) \ {(1, n)}, ↑(Nat.choose (x.1 + x.2) x.2) / (↑x.2 + 1) * bernoulli' x.1 =
    1
p : ℕ × ℕ
h : p ∈ antidiagonal (succ n) \ {(1, n)}
⊢ ↑(Nat.choose (p.1 + p.2) p.2) / (↑p.2 + 1) * bernoulli p.1 =
    ↑(Nat.choose (p.1 + p.2) p.2) / (↑p.2 + 1) * bernoulli' p.1

Next tactic:
obtain ⟨h', h''⟩ : p ∈ _ ∧ p ≠ _ := by rwa [mem_sdiff, mem_singleton] at h
theorem bernoulli_spec' (n : ℕ) :
    (∑ k in antidiagonal n, ((k.1 + k.2).choose k.2 : ℚ) / (k.2 + 1) * bernoulli k.1) =
      if n = 0 then 1 else 0 := by
  cases' n with n n;
  · simp
  rw [if_neg (succ_ne_zero _)]
  -- algebra facts
  have h₁ : (1, n) ∈ antidiagonal n.succ := by simp [mem_antidiagonal, add_comm]
  have h₂ : (n : ℚ) + 1 ≠ 0 := by norm_cast; exact succ_ne_zero _
  have h₃ : (1 + n).choose n = n + 1 := by simp [add_comm]
  -- key equation: the corresponding fact for `bernoulli'`
  have H := bernoulli'_spec' n.succ
  -- massage it to match the structure of the goal, then convert piece by piece
  rw [sum_eq_add_sum_diff_singleton h₁] at H ⊢
  apply add_eq_of_eq_sub'
  convert eq_sub_of_add_eq' H using 1
  · refine' sum_congr rfl fun p h => _
    obtain ⟨h', h''⟩ : p ∈ _ ∧ p ≠ _ := by 

Proof state:
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
h₁ : (1, n) ∈ antidiagonal (succ n)
h₂ : ↑n + 1 ≠ 0
h₃ : Nat.choose (1 + n) n = n + 1
H :
  ↑(Nat.choose ((1, n).1 + (1, n).2) (1, n).2) / (↑(1, n).2 + 1) * bernoulli' (1, n).1 +
      ∑ x in antidiagonal (succ n) \ {(1, n)}, ↑(Nat.choose (x.1 + x.2) x.2) / (↑x.2 + 1) * bernoulli' x.1 =
    1
p : ℕ × ℕ
h : p ∈ antidiagonal (succ n) \ {(1, n)}
⊢ p ∈ ?m.702535 ∧ p ≠ ?m.702586

Next tactic:
rwa [mem_sdiff, mem_singleton] at h
theorem bernoulli_spec' (n : ℕ) :
    (∑ k in antidiagonal n, ((k.1 + k.2).choose k.2 : ℚ) / (k.2 + 1) * bernoulli k.1) =
      if n = 0 then 1 else 0 := by
  cases' n with n n;
  · simp
  rw [if_neg (succ_ne_zero _)]
  -- algebra facts
  have h₁ : (1, n) ∈ antidiagonal n.succ := by simp [mem_antidiagonal, add_comm]
  have h₂ : (n : ℚ) + 1 ≠ 0 := by norm_cast; exact succ_ne_zero _
  have h₃ : (1 + n).choose n = n + 1 := by simp [add_comm]
  -- key equation: the corresponding fact for `bernoulli'`
  have H := bernoulli'_spec' n.succ
  -- massage it to match the structure of the goal, then convert piece by piece
  rw [sum_eq_add_sum_diff_singleton h₁] at H ⊢
  apply add_eq_of_eq_sub'
  convert eq_sub_of_add_eq' H using 1
  · refine' sum_congr rfl fun p h => _
    obtain ⟨h', h''⟩ : p ∈ _ ∧ p ≠ _ := by rwa [mem_sdiff, mem_singleton] at h
    

Proof state:
case h.e'_2.intro
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
h₁ : (1, n) ∈ antidiagonal (succ n)
h₂ : ↑n + 1 ≠ 0
h₃ : Nat.choose (1 + n) n = n + 1
H :
  ↑(Nat.choose ((1, n).1 + (1, n).2) (1, n).2) / (↑(1, n).2 + 1) * bernoulli' (1, n).1 +
      ∑ x in antidiagonal (succ n) \ {(1, n)}, ↑(Nat.choose (x.1 + x.2) x.2) / (↑x.2 + 1) * bernoulli' x.1 =
    1
p : ℕ × ℕ
h : p ∈ antidiagonal (succ n) \ {(1, n)}
h' : p ∈ antidiagonal (succ n)
h'' : p ≠ (1, n)
⊢ ↑(Nat.choose (p.1 + p.2) p.2) / (↑p.2 + 1) * bernoulli p.1 =
    ↑(Nat.choose (p.1 + p.2) p.2) / (↑p.2 + 1) * bernoulli' p.1

Next tactic:
simp [bernoulli_eq_bernoulli'_of_ne_one ((not_congr (antidiagonal_congr h' h₁)).mp h'')]
theorem bernoulli_spec' (n : ℕ) :
    (∑ k in antidiagonal n, ((k.1 + k.2).choose k.2 : ℚ) / (k.2 + 1) * bernoulli k.1) =
      if n = 0 then 1 else 0 := by
  cases' n with n n;
  · simp
  rw [if_neg (succ_ne_zero _)]
  -- algebra facts
  have h₁ : (1, n) ∈ antidiagonal n.succ := by simp [mem_antidiagonal, add_comm]
  have h₂ : (n : ℚ) + 1 ≠ 0 := by norm_cast; exact succ_ne_zero _
  have h₃ : (1 + n).choose n = n + 1 := by simp [add_comm]
  -- key equation: the corresponding fact for `bernoulli'`
  have H := bernoulli'_spec' n.succ
  -- massage it to match the structure of the goal, then convert piece by piece
  rw [sum_eq_add_sum_diff_singleton h₁] at H ⊢
  apply add_eq_of_eq_sub'
  convert eq_sub_of_add_eq' H using 1
  · refine' sum_congr rfl fun p h => _
    obtain ⟨h', h''⟩ : p ∈ _ ∧ p ≠ _ := by rwa [mem_sdiff, mem_singleton] at h
    simp [bernoulli_eq_bernoulli'_of_ne_one ((not_congr (antidiagonal_congr h' h₁)).mp h'')]
  · 

Proof state:
case h.e'_3
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
h₁ : (1, n) ∈ antidiagonal (succ n)
h₂ : ↑n + 1 ≠ 0
h₃ : Nat.choose (1 + n) n = n + 1
H :
  ↑(Nat.choose ((1, n).1 + (1, n).2) (1, n).2) / (↑(1, n).2 + 1) * bernoulli' (1, n).1 +
      ∑ x in antidiagonal (succ n) \ {(1, n)}, ↑(Nat.choose (x.1 + x.2) x.2) / (↑x.2 + 1) * bernoulli' x.1 =
    1
⊢ 0 - ↑(Nat.choose ((1, n).1 + (1, n).2) (1, n).2) / (↑(1, n).2 + 1) * bernoulli (1, n).1 =
    1 - ↑(Nat.choose ((1, n).1 + (1, n).2) (1, n).2) / (↑(1, n).2 + 1) * bernoulli' (1, n).1

Next tactic:
field_simp [h₃]
theorem bernoulli_spec' (n : ℕ) :
    (∑ k in antidiagonal n, ((k.1 + k.2).choose k.2 : ℚ) / (k.2 + 1) * bernoulli k.1) =
      if n = 0 then 1 else 0 := by
  cases' n with n n;
  · simp
  rw [if_neg (succ_ne_zero _)]
  -- algebra facts
  have h₁ : (1, n) ∈ antidiagonal n.succ := by simp [mem_antidiagonal, add_comm]
  have h₂ : (n : ℚ) + 1 ≠ 0 := by norm_cast; exact succ_ne_zero _
  have h₃ : (1 + n).choose n = n + 1 := by simp [add_comm]
  -- key equation: the corresponding fact for `bernoulli'`
  have H := bernoulli'_spec' n.succ
  -- massage it to match the structure of the goal, then convert piece by piece
  rw [sum_eq_add_sum_diff_singleton h₁] at H ⊢
  apply add_eq_of_eq_sub'
  convert eq_sub_of_add_eq' H using 1
  · refine' sum_congr rfl fun p h => _
    obtain ⟨h', h''⟩ : p ∈ _ ∧ p ≠ _ := by rwa [mem_sdiff, mem_singleton] at h
    simp [bernoulli_eq_bernoulli'_of_ne_one ((not_congr (antidiagonal_congr h' h₁)).mp h'')]
  · field_simp [h₃]
    

Proof state:
case h.e'_3
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
h₁ : (1, n) ∈ antidiagonal (succ n)
h₂ : ↑n + 1 ≠ 0
h₃ : Nat.choose (1 + n) n = n + 1
H :
  ↑(Nat.choose ((1, n).1 + (1, n).2) (1, n).2) / (↑(1, n).2 + 1) * bernoulli' (1, n).1 +
      ∑ x in antidiagonal (succ n) \ {(1, n)}, ↑(Nat.choose (x.1 + x.2) x.2) / (↑x.2 + 1) * bernoulli' x.1 =
    1
⊢ 1 = 2 - 1

Next tactic:
norm_num
theorem bernoulliPowerSeries_mul_exp_sub_one : bernoulliPowerSeries A * (exp A - 1) = X := by
  

Proof state:
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
⊢ bernoulliPowerSeries A * (exp A - 1) = X

Next tactic:
ext n
theorem bernoulliPowerSeries_mul_exp_sub_one : bernoulliPowerSeries A * (exp A - 1) = X := by
  ext n
  -- constant coefficient is a special case
  

Proof state:
case h
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
⊢ (coeff A n) (bernoulliPowerSeries A * (exp A - 1)) = (coeff A n) X

Next tactic:
cases' n with n n
theorem bernoulliPowerSeries_mul_exp_sub_one : bernoulliPowerSeries A * (exp A - 1) = X := by
  ext n
  -- constant coefficient is a special case
  cases' n with n n;
  · 

Proof state:
case h.zero
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
⊢ (coeff A zero) (bernoulliPowerSeries A * (exp A - 1)) = (coeff A zero) X

Next tactic:
simp
theorem bernoulliPowerSeries_mul_exp_sub_one : bernoulliPowerSeries A * (exp A - 1) = X := by
  ext n
  -- constant coefficient is a special case
  cases' n with n n;
  · simp
  

Proof state:
case h.succ
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
⊢ (coeff A (succ n)) (bernoulliPowerSeries A * (exp A - 1)) = (coeff A (succ n)) X

Next tactic:
simp only [bernoulliPowerSeries, coeff_mul, coeff_X, sum_antidiagonal_succ', one_div, coeff_mk,
    coeff_one, coeff_exp, LinearMap.map_sub, factorial, if_pos, cast_succ, cast_one, cast_mul,
    sub_zero, RingHom.map_one, add_eq_zero_iff, if_false, _root_.inv_one, zero_add, one_ne_zero,
    mul_zero, and_false_iff, sub_self, ← RingHom.map_mul, ← map_sum]
theorem bernoulliPowerSeries_mul_exp_sub_one : bernoulliPowerSeries A * (exp A - 1) = X := by
  ext n
  -- constant coefficient is a special case
  cases' n with n n;
  · simp
  simp only [bernoulliPowerSeries, coeff_mul, coeff_X, sum_antidiagonal_succ', one_div, coeff_mk,
    coeff_one, coeff_exp, LinearMap.map_sub, factorial, if_pos, cast_succ, cast_one, cast_mul,
    sub_zero, RingHom.map_one, add_eq_zero_iff, if_false, _root_.inv_one, zero_add, one_ne_zero,
    mul_zero, and_false_iff, sub_self, ← RingHom.map_mul, ← map_sum]
  

Proof state:
case h.succ
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
⊢ (algebraMap ℚ A) (bernoulli (n + 1) / ((↑n + 1) * ↑(Nat.add n 0)!)) * ((algebraMap ℚ A) (↑0 + 1)⁻¹ - 1) +
      (algebraMap ℚ A) (∑ x in antidiagonal n, bernoulli x.1 / ↑x.1! * ((↑x.2 + 1) * ↑(Nat.add x.2 0)!)⁻¹) =
    if succ n = 1 then 1 else 0

Next tactic:
cases' n with n
theorem bernoulliPowerSeries_mul_exp_sub_one : bernoulliPowerSeries A * (exp A - 1) = X := by
  ext n
  -- constant coefficient is a special case
  cases' n with n n;
  · simp
  simp only [bernoulliPowerSeries, coeff_mul, coeff_X, sum_antidiagonal_succ', one_div, coeff_mk,
    coeff_one, coeff_exp, LinearMap.map_sub, factorial, if_pos, cast_succ, cast_one, cast_mul,
    sub_zero, RingHom.map_one, add_eq_zero_iff, if_false, _root_.inv_one, zero_add, one_ne_zero,
    mul_zero, and_false_iff, sub_self, ← RingHom.map_mul, ← map_sum]
  cases' n with n
  · 

Proof state:
case h.succ.zero
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
⊢ (algebraMap ℚ A) (bernoulli (zero + 1) / ((↑zero + 1) * ↑(Nat.add zero 0)!)) * ((algebraMap ℚ A) (↑0 + 1)⁻¹ - 1) +
      (algebraMap ℚ A) (∑ x in antidiagonal zero, bernoulli x.1 / ↑x.1! * ((↑x.2 + 1) * ↑(Nat.add x.2 0)!)⁻¹) =
    if succ zero = 1 then 1 else 0

Next tactic:
simp
theorem bernoulliPowerSeries_mul_exp_sub_one : bernoulliPowerSeries A * (exp A - 1) = X := by
  ext n
  -- constant coefficient is a special case
  cases' n with n n;
  · simp
  simp only [bernoulliPowerSeries, coeff_mul, coeff_X, sum_antidiagonal_succ', one_div, coeff_mk,
    coeff_one, coeff_exp, LinearMap.map_sub, factorial, if_pos, cast_succ, cast_one, cast_mul,
    sub_zero, RingHom.map_one, add_eq_zero_iff, if_false, _root_.inv_one, zero_add, one_ne_zero,
    mul_zero, and_false_iff, sub_self, ← RingHom.map_mul, ← map_sum]
  cases' n with n
  · simp
  

Proof state:
case h.succ.succ
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
⊢ (algebraMap ℚ A) (bernoulli (succ n + 1) / ((↑(succ n) + 1) * ↑(Nat.add (succ n) 0)!)) *
        ((algebraMap ℚ A) (↑0 + 1)⁻¹ - 1) +
      (algebraMap ℚ A) (∑ x in antidiagonal (succ n), bernoulli x.1 / ↑x.1! * ((↑x.2 + 1) * ↑(Nat.add x.2 0)!)⁻¹) =
    if succ (succ n) = 1 then 1 else 0

Next tactic:
rw [if_neg n.succ_succ_ne_one]
theorem bernoulliPowerSeries_mul_exp_sub_one : bernoulliPowerSeries A * (exp A - 1) = X := by
  ext n
  -- constant coefficient is a special case
  cases' n with n n;
  · simp
  simp only [bernoulliPowerSeries, coeff_mul, coeff_X, sum_antidiagonal_succ', one_div, coeff_mk,
    coeff_one, coeff_exp, LinearMap.map_sub, factorial, if_pos, cast_succ, cast_one, cast_mul,
    sub_zero, RingHom.map_one, add_eq_zero_iff, if_false, _root_.inv_one, zero_add, one_ne_zero,
    mul_zero, and_false_iff, sub_self, ← RingHom.map_mul, ← map_sum]
  cases' n with n
  · simp
  rw [if_neg n.succ_succ_ne_one]
  

Proof state:
case h.succ.succ
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
⊢ (algebraMap ℚ A) (bernoulli (succ n + 1) / ((↑(succ n) + 1) * ↑(Nat.add (succ n) 0)!)) *
        ((algebraMap ℚ A) (↑0 + 1)⁻¹ - 1) +
      (algebraMap ℚ A) (∑ x in antidiagonal (succ n), bernoulli x.1 / ↑x.1! * ((↑x.2 + 1) * ↑(Nat.add x.2 0)!)⁻¹) =
    0

Next tactic:
have hfact : ∀ m, (m ! : ℚ) ≠ 0 := fun m => mod_cast factorial_ne_zero m
theorem bernoulliPowerSeries_mul_exp_sub_one : bernoulliPowerSeries A * (exp A - 1) = X := by
  ext n
  -- constant coefficient is a special case
  cases' n with n n;
  · simp
  simp only [bernoulliPowerSeries, coeff_mul, coeff_X, sum_antidiagonal_succ', one_div, coeff_mk,
    coeff_one, coeff_exp, LinearMap.map_sub, factorial, if_pos, cast_succ, cast_one, cast_mul,
    sub_zero, RingHom.map_one, add_eq_zero_iff, if_false, _root_.inv_one, zero_add, one_ne_zero,
    mul_zero, and_false_iff, sub_self, ← RingHom.map_mul, ← map_sum]
  cases' n with n
  · simp
  rw [if_neg n.succ_succ_ne_one]
  have hfact : ∀ m, (m ! : ℚ) ≠ 0 := fun m => mod_cast factorial_ne_zero m
  

Proof state:
case h.succ.succ
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
hfact : ∀ (m : ℕ), ↑m ! ≠ 0
⊢ (algebraMap ℚ A) (bernoulli (succ n + 1) / ((↑(succ n) + 1) * ↑(Nat.add (succ n) 0)!)) *
        ((algebraMap ℚ A) (↑0 + 1)⁻¹ - 1) +
      (algebraMap ℚ A) (∑ x in antidiagonal (succ n), bernoulli x.1 / ↑x.1! * ((↑x.2 + 1) * ↑(Nat.add x.2 0)!)⁻¹) =
    0

Next tactic:
have hite2 : ite (n.succ = 0) 1 0 = (0 : ℚ) := if_neg n.succ_ne_zero
theorem bernoulliPowerSeries_mul_exp_sub_one : bernoulliPowerSeries A * (exp A - 1) = X := by
  ext n
  -- constant coefficient is a special case
  cases' n with n n;
  · simp
  simp only [bernoulliPowerSeries, coeff_mul, coeff_X, sum_antidiagonal_succ', one_div, coeff_mk,
    coeff_one, coeff_exp, LinearMap.map_sub, factorial, if_pos, cast_succ, cast_one, cast_mul,
    sub_zero, RingHom.map_one, add_eq_zero_iff, if_false, _root_.inv_one, zero_add, one_ne_zero,
    mul_zero, and_false_iff, sub_self, ← RingHom.map_mul, ← map_sum]
  cases' n with n
  · simp
  rw [if_neg n.succ_succ_ne_one]
  have hfact : ∀ m, (m ! : ℚ) ≠ 0 := fun m => mod_cast factorial_ne_zero m
  have hite2 : ite (n.succ = 0) 1 0 = (0 : ℚ) := if_neg n.succ_ne_zero
  

Proof state:
case h.succ.succ
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
hfact : ∀ (m : ℕ), ↑m ! ≠ 0
hite2 : (if succ n = 0 then 1 else 0) = 0
⊢ (algebraMap ℚ A) (bernoulli (succ n + 1) / ((↑(succ n) + 1) * ↑(Nat.add (succ n) 0)!)) *
        ((algebraMap ℚ A) (↑0 + 1)⁻¹ - 1) +
      (algebraMap ℚ A) (∑ x in antidiagonal (succ n), bernoulli x.1 / ↑x.1! * ((↑x.2 + 1) * ↑(Nat.add x.2 0)!)⁻¹) =
    0

Next tactic:
simp only [CharP.cast_eq_zero, zero_add, inv_one, map_one, sub_self, mul_zero, add_eq]
theorem bernoulliPowerSeries_mul_exp_sub_one : bernoulliPowerSeries A * (exp A - 1) = X := by
  ext n
  -- constant coefficient is a special case
  cases' n with n n;
  · simp
  simp only [bernoulliPowerSeries, coeff_mul, coeff_X, sum_antidiagonal_succ', one_div, coeff_mk,
    coeff_one, coeff_exp, LinearMap.map_sub, factorial, if_pos, cast_succ, cast_one, cast_mul,
    sub_zero, RingHom.map_one, add_eq_zero_iff, if_false, _root_.inv_one, zero_add, one_ne_zero,
    mul_zero, and_false_iff, sub_self, ← RingHom.map_mul, ← map_sum]
  cases' n with n
  · simp
  rw [if_neg n.succ_succ_ne_one]
  have hfact : ∀ m, (m ! : ℚ) ≠ 0 := fun m => mod_cast factorial_ne_zero m
  have hite2 : ite (n.succ = 0) 1 0 = (0 : ℚ) := if_neg n.succ_ne_zero
  simp only [CharP.cast_eq_zero, zero_add, inv_one, map_one, sub_self, mul_zero, add_eq]
  

Proof state:
case h.succ.succ
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
hfact : ∀ (m : ℕ), ↑m ! ≠ 0
hite2 : (if succ n = 0 then 1 else 0) = 0
⊢ (algebraMap ℚ A) (∑ x in antidiagonal (succ n), bernoulli x.1 / ↑x.1! * ((↑x.2 + 1) * ↑(x.2 + 0)!)⁻¹) = 0

Next tactic:
rw [← map_zero (algebraMap ℚ A), ← zero_div (n.succ ! : ℚ), ← hite2, ← bernoulli_spec', sum_div]
theorem bernoulliPowerSeries_mul_exp_sub_one : bernoulliPowerSeries A * (exp A - 1) = X := by
  ext n
  -- constant coefficient is a special case
  cases' n with n n;
  · simp
  simp only [bernoulliPowerSeries, coeff_mul, coeff_X, sum_antidiagonal_succ', one_div, coeff_mk,
    coeff_one, coeff_exp, LinearMap.map_sub, factorial, if_pos, cast_succ, cast_one, cast_mul,
    sub_zero, RingHom.map_one, add_eq_zero_iff, if_false, _root_.inv_one, zero_add, one_ne_zero,
    mul_zero, and_false_iff, sub_self, ← RingHom.map_mul, ← map_sum]
  cases' n with n
  · simp
  rw [if_neg n.succ_succ_ne_one]
  have hfact : ∀ m, (m ! : ℚ) ≠ 0 := fun m => mod_cast factorial_ne_zero m
  have hite2 : ite (n.succ = 0) 1 0 = (0 : ℚ) := if_neg n.succ_ne_zero
  simp only [CharP.cast_eq_zero, zero_add, inv_one, map_one, sub_self, mul_zero, add_eq]
  rw [← map_zero (algebraMap ℚ A), ← zero_div (n.succ ! : ℚ), ← hite2, ← bernoulli_spec', sum_div]
  

Proof state:
case h.succ.succ
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
hfact : ∀ (m : ℕ), ↑m ! ≠ 0
hite2 : (if succ n = 0 then 1 else 0) = 0
⊢ (algebraMap ℚ A) (∑ x in antidiagonal (succ n), bernoulli x.1 / ↑x.1! * ((↑x.2 + 1) * ↑(x.2 + 0)!)⁻¹) =
    (algebraMap ℚ A)
      (∑ x in antidiagonal (succ n), ↑(Nat.choose (x.1 + x.2) x.2) / (↑x.2 + 1) * bernoulli x.1 / ↑(succ n)!)

Next tactic:
refine' congr_arg (algebraMap ℚ A) (sum_congr rfl fun x h => eq_div_of_mul_eq (hfact n.succ) _)
theorem bernoulliPowerSeries_mul_exp_sub_one : bernoulliPowerSeries A * (exp A - 1) = X := by
  ext n
  -- constant coefficient is a special case
  cases' n with n n;
  · simp
  simp only [bernoulliPowerSeries, coeff_mul, coeff_X, sum_antidiagonal_succ', one_div, coeff_mk,
    coeff_one, coeff_exp, LinearMap.map_sub, factorial, if_pos, cast_succ, cast_one, cast_mul,
    sub_zero, RingHom.map_one, add_eq_zero_iff, if_false, _root_.inv_one, zero_add, one_ne_zero,
    mul_zero, and_false_iff, sub_self, ← RingHom.map_mul, ← map_sum]
  cases' n with n
  · simp
  rw [if_neg n.succ_succ_ne_one]
  have hfact : ∀ m, (m ! : ℚ) ≠ 0 := fun m => mod_cast factorial_ne_zero m
  have hite2 : ite (n.succ = 0) 1 0 = (0 : ℚ) := if_neg n.succ_ne_zero
  simp only [CharP.cast_eq_zero, zero_add, inv_one, map_one, sub_self, mul_zero, add_eq]
  rw [← map_zero (algebraMap ℚ A), ← zero_div (n.succ ! : ℚ), ← hite2, ← bernoulli_spec', sum_div]
  refine' congr_arg (algebraMap ℚ A) (sum_congr rfl fun x h => eq_div_of_mul_eq (hfact n.succ) _)
  

Proof state:
case h.succ.succ
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
hfact : ∀ (m : ℕ), ↑m ! ≠ 0
hite2 : (if succ n = 0 then 1 else 0) = 0
x : ℕ × ℕ
h : x ∈ antidiagonal (succ n)
⊢ bernoulli x.1 / ↑x.1! * ((↑x.2 + 1) * ↑(x.2 + 0)!)⁻¹ * ↑(succ n)! =
    ↑(Nat.choose (x.1 + x.2) x.2) / (↑x.2 + 1) * bernoulli x.1

Next tactic:
rw [mem_antidiagonal] at h
theorem bernoulliPowerSeries_mul_exp_sub_one : bernoulliPowerSeries A * (exp A - 1) = X := by
  ext n
  -- constant coefficient is a special case
  cases' n with n n;
  · simp
  simp only [bernoulliPowerSeries, coeff_mul, coeff_X, sum_antidiagonal_succ', one_div, coeff_mk,
    coeff_one, coeff_exp, LinearMap.map_sub, factorial, if_pos, cast_succ, cast_one, cast_mul,
    sub_zero, RingHom.map_one, add_eq_zero_iff, if_false, _root_.inv_one, zero_add, one_ne_zero,
    mul_zero, and_false_iff, sub_self, ← RingHom.map_mul, ← map_sum]
  cases' n with n
  · simp
  rw [if_neg n.succ_succ_ne_one]
  have hfact : ∀ m, (m ! : ℚ) ≠ 0 := fun m => mod_cast factorial_ne_zero m
  have hite2 : ite (n.succ = 0) 1 0 = (0 : ℚ) := if_neg n.succ_ne_zero
  simp only [CharP.cast_eq_zero, zero_add, inv_one, map_one, sub_self, mul_zero, add_eq]
  rw [← map_zero (algebraMap ℚ A), ← zero_div (n.succ ! : ℚ), ← hite2, ← bernoulli_spec', sum_div]
  refine' congr_arg (algebraMap ℚ A) (sum_congr rfl fun x h => eq_div_of_mul_eq (hfact n.succ) _)
  rw [mem_antidiagonal] at h
  

Proof state:
case h.succ.succ
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
hfact : ∀ (m : ℕ), ↑m ! ≠ 0
hite2 : (if succ n = 0 then 1 else 0) = 0
x : ℕ × ℕ
h : x.1 + x.2 = succ n
⊢ bernoulli x.1 / ↑x.1! * ((↑x.2 + 1) * ↑(x.2 + 0)!)⁻¹ * ↑(succ n)! =
    ↑(Nat.choose (x.1 + x.2) x.2) / (↑x.2 + 1) * bernoulli x.1

Next tactic:
rw [← h, add_choose, cast_div_charZero (factorial_mul_factorial_dvd_factorial_add _ _)]
theorem bernoulliPowerSeries_mul_exp_sub_one : bernoulliPowerSeries A * (exp A - 1) = X := by
  ext n
  -- constant coefficient is a special case
  cases' n with n n;
  · simp
  simp only [bernoulliPowerSeries, coeff_mul, coeff_X, sum_antidiagonal_succ', one_div, coeff_mk,
    coeff_one, coeff_exp, LinearMap.map_sub, factorial, if_pos, cast_succ, cast_one, cast_mul,
    sub_zero, RingHom.map_one, add_eq_zero_iff, if_false, _root_.inv_one, zero_add, one_ne_zero,
    mul_zero, and_false_iff, sub_self, ← RingHom.map_mul, ← map_sum]
  cases' n with n
  · simp
  rw [if_neg n.succ_succ_ne_one]
  have hfact : ∀ m, (m ! : ℚ) ≠ 0 := fun m => mod_cast factorial_ne_zero m
  have hite2 : ite (n.succ = 0) 1 0 = (0 : ℚ) := if_neg n.succ_ne_zero
  simp only [CharP.cast_eq_zero, zero_add, inv_one, map_one, sub_self, mul_zero, add_eq]
  rw [← map_zero (algebraMap ℚ A), ← zero_div (n.succ ! : ℚ), ← hite2, ← bernoulli_spec', sum_div]
  refine' congr_arg (algebraMap ℚ A) (sum_congr rfl fun x h => eq_div_of_mul_eq (hfact n.succ) _)
  rw [mem_antidiagonal] at h
  rw [← h, add_choose, cast_div_charZero (factorial_mul_factorial_dvd_factorial_add _ _)]
  

Proof state:
case h.succ.succ
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
hfact : ∀ (m : ℕ), ↑m ! ≠ 0
hite2 : (if succ n = 0 then 1 else 0) = 0
x : ℕ × ℕ
h : x.1 + x.2 = succ n
⊢ bernoulli x.1 / ↑x.1! * ((↑x.2 + 1) * ↑(x.2 + 0)!)⁻¹ * ↑(x.1 + x.2)! =
    ↑(x.1 + x.2)! / ↑(x.1! * x.2!) / (↑x.2 + 1) * bernoulli x.1

Next tactic:
field_simp [hfact x.1, mul_comm _ (bernoulli x.1), mul_assoc]