l3lab/ntp-mathlib huggingface dataset:
@[simp]
theorem LinearMap.det_zero'' {R M : Type*} [CommRing R] [AddCommGroup M] [Module R M]
    [Module.Free R M] [Module.Finite R M] [Nontrivial M] : LinearMap.det (0 : M →ₗ[R] M) = 0 := by
  letI : Nonempty (Module.Free.ChooseBasisIndex R M) := (Module.Free.chooseBasis R M).index_nonempty
  nontriviality R
  

Proof state:
R : Type u_1
M : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Module.Free R M
inst✝¹ : Module.Finite R M
inst✝ : Nontrivial M
this : Nonempty (Module.Free.ChooseBasisIndex R M) := Basis.index_nonempty (Module.Free.chooseBasis R M)
✝ : Nontrivial R
⊢ LinearMap.det 0 = 0

Next tactic:
exact LinearMap.det_zero' (Module.Free.chooseBasis R M)
theorem isThin_iff : Quiver.IsThin C ↔ ∀ c : C, Subsingleton (c ⟶ c) := by
  

Proof state:
C : Type u_1
inst✝ : Groupoid C
⊢ Quiver.IsThin C ↔ ∀ (c : C), Subsingleton (c ⟶ c)

Next tactic:
refine' ⟨fun h c => h c c, fun h c d => Subsingleton.intro fun f g => _⟩
theorem isThin_iff : Quiver.IsThin C ↔ ∀ c : C, Subsingleton (c ⟶ c) := by
  refine' ⟨fun h c => h c c, fun h c d => Subsingleton.intro fun f g => _⟩
  

Proof state:
C : Type u_1
inst✝ : Groupoid C
h : ∀ (c : C), Subsingleton (c ⟶ c)
c d : C
f g : c ⟶ d
⊢ f = g

Next tactic:
haveI := h d
theorem isThin_iff : Quiver.IsThin C ↔ ∀ c : C, Subsingleton (c ⟶ c) := by
  refine' ⟨fun h c => h c c, fun h c d => Subsingleton.intro fun f g => _⟩
  haveI := h d
  

Proof state:
C : Type u_1
inst✝ : Groupoid C
h : ∀ (c : C), Subsingleton (c ⟶ c)
c d : C
f g : c ⟶ d
this : Subsingleton (d ⟶ d)
⊢ f = g

Next tactic:
calc
    f = f ≫ inv g ≫ g := by simp only [inv_eq_inv, IsIso.inv_hom_id, Category.comp_id]
    _ = f ≫ inv f ≫ g := by congr 1
                            simp only [inv_eq_inv, IsIso.inv_hom_id, eq_iff_true_of_subsingleton]
    _ = g := by simp only [inv_eq_inv, IsIso.hom_inv_id_assoc]
theorem isThin_iff : Quiver.IsThin C ↔ ∀ c : C, Subsingleton (c ⟶ c) := by
  refine' ⟨fun h c => h c c, fun h c d => Subsingleton.intro fun f g => _⟩
  haveI := h d
  calc
    f = f ≫ inv g ≫ g := by 

Proof state:
C : Type u_1
inst✝ : Groupoid C
h : ∀ (c : C), Subsingleton (c ⟶ c)
c d : C
f g : c ⟶ d
this : Subsingleton (d ⟶ d)
⊢ f = f ≫ inv g ≫ g

Next tactic:
simp only [inv_eq_inv, IsIso.inv_hom_id, Category.comp_id]
theorem isThin_iff : Quiver.IsThin C ↔ ∀ c : C, Subsingleton (c ⟶ c) := by
  refine' ⟨fun h c => h c c, fun h c d => Subsingleton.intro fun f g => _⟩
  haveI := h d
  calc
    f = f ≫ inv g ≫ g := by simp only [inv_eq_inv, IsIso.inv_hom_id, Category.comp_id]
    _ = f ≫ inv f ≫ g := by 

Proof state:
C : Type u_1
inst✝ : Groupoid C
h : ∀ (c : C), Subsingleton (c ⟶ c)
c d : C
f g : c ⟶ d
this : Subsingleton (d ⟶ d)
⊢ f ≫ inv g ≫ g = f ≫ inv f ≫ g

Next tactic:
congr 1
theorem isThin_iff : Quiver.IsThin C ↔ ∀ c : C, Subsingleton (c ⟶ c) := by
  refine' ⟨fun h c => h c c, fun h c d => Subsingleton.intro fun f g => _⟩
  haveI := h d
  calc
    f = f ≫ inv g ≫ g := by simp only [inv_eq_inv, IsIso.inv_hom_id, Category.comp_id]
    _ = f ≫ inv f ≫ g := by congr 1
                            

Proof state:
case e_a
C : Type u_1
inst✝ : Groupoid C
h : ∀ (c : C), Subsingleton (c ⟶ c)
c d : C
f g : c ⟶ d
this : Subsingleton (d ⟶ d)
⊢ inv g ≫ g = inv f ≫ g

Next tactic:
simp only [inv_eq_inv, IsIso.inv_hom_id, eq_iff_true_of_subsingleton]
theorem isThin_iff : Quiver.IsThin C ↔ ∀ c : C, Subsingleton (c ⟶ c) := by
  refine' ⟨fun h c => h c c, fun h c d => Subsingleton.intro fun f g => _⟩
  haveI := h d
  calc
    f = f ≫ inv g ≫ g := by simp only [inv_eq_inv, IsIso.inv_hom_id, Category.comp_id]
    _ = f ≫ inv f ≫ g := by congr 1
                            simp only [inv_eq_inv, IsIso.inv_hom_id, eq_iff_true_of_subsingleton]
    _ = g := by 

Proof state:
C : Type u_1
inst✝ : Groupoid C
h : ∀ (c : C), Subsingleton (c ⟶ c)
c d : C
f g : c ⟶ d
this : Subsingleton (d ⟶ d)
⊢ f ≫ inv f ≫ g = g

Next tactic:
simp only [inv_eq_inv, IsIso.hom_inv_id_assoc]
theorem bernoulli'_def (n : ℕ) :
    bernoulli' n = 1 - ∑ k in range n, n.choose k / (n - k + 1) * bernoulli' k := by
  

Proof state:
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
⊢ bernoulli' n = 1 - ∑ k in range n, ↑(Nat.choose n k) / (↑n - ↑k + 1) * bernoulli' k

Next tactic:
rw [bernoulli'_def', ← Fin.sum_univ_eq_sum_range]
theorem bernoulli'_spec (n : ℕ) :
    (∑ k in range n.succ, (n.choose (n - k) : ℚ) / (n - k + 1) * bernoulli' k) = 1 := by
  

Proof state:
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
⊢ ∑ k in range (succ n), ↑(Nat.choose n (n - k)) / (↑n - ↑k + 1) * bernoulli' k = 1

Next tactic:
rw [sum_range_succ_comm, bernoulli'_def n, tsub_self, choose_zero_right, sub_self, zero_add,
    div_one, cast_one, one_mul, sub_add, ← sum_sub_distrib, ← sub_eq_zero, sub_sub_cancel_left,
    neg_eq_zero]
theorem bernoulli'_spec (n : ℕ) :
    (∑ k in range n.succ, (n.choose (n - k) : ℚ) / (n - k + 1) * bernoulli' k) = 1 := by
  rw [sum_range_succ_comm, bernoulli'_def n, tsub_self, choose_zero_right, sub_self, zero_add,
    div_one, cast_one, one_mul, sub_add, ← sum_sub_distrib, ← sub_eq_zero, sub_sub_cancel_left,
    neg_eq_zero]
  

Proof state:
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
⊢ ∑ x in range n,
      (↑(Nat.choose n x) / (↑n - ↑x + 1) * bernoulli' x - ↑(Nat.choose n (n - x)) / (↑n - ↑x + 1) * bernoulli' x) =
    0

Next tactic:
exact Finset.sum_eq_zero (fun x hx => by rw [choose_symm (le_of_lt (mem_range.1 hx)), sub_self])
theorem bernoulli'_spec (n : ℕ) :
    (∑ k in range n.succ, (n.choose (n - k) : ℚ) / (n - k + 1) * bernoulli' k) = 1 := by
  rw [sum_range_succ_comm, bernoulli'_def n, tsub_self, choose_zero_right, sub_self, zero_add,
    div_one, cast_one, one_mul, sub_add, ← sum_sub_distrib, ← sub_eq_zero, sub_sub_cancel_left,
    neg_eq_zero]
  exact Finset.sum_eq_zero (fun x hx => by 

Proof state:
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n x : ℕ
hx : x ∈ range n
⊢ ↑(Nat.choose n x) / (↑n - ↑x + 1) * bernoulli' x - ↑(Nat.choose n (n - x)) / (↑n - ↑x + 1) * bernoulli' x = 0

Next tactic:
rw [choose_symm (le_of_lt (mem_range.1 hx)), sub_self]
theorem bernoulli'_spec' (n : ℕ) :
    (∑ k in antidiagonal n, ((k.1 + k.2).choose k.2 : ℚ) / (k.2 + 1) * bernoulli' k.1) = 1 := by
  

Proof state:
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
⊢ ∑ k in antidiagonal n, ↑(Nat.choose (k.1 + k.2) k.2) / (↑k.2 + 1) * bernoulli' k.1 = 1

Next tactic:
refine' ((sum_antidiagonal_eq_sum_range_succ_mk _ n).trans _).trans (bernoulli'_spec n)
theorem bernoulli'_spec' (n : ℕ) :
    (∑ k in antidiagonal n, ((k.1 + k.2).choose k.2 : ℚ) / (k.2 + 1) * bernoulli' k.1) = 1 := by
  refine' ((sum_antidiagonal_eq_sum_range_succ_mk _ n).trans _).trans (bernoulli'_spec n)
  

Proof state:
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
⊢ ∑ k in range (succ n),
      ↑(Nat.choose ((k, n - k).1 + (k, n - k).2) (k, n - k).2) / (↑(k, n - k).2 + 1) * bernoulli' (k, n - k).1 =
    ∑ k in range (succ n), ↑(Nat.choose n (n - k)) / (↑n - ↑k + 1) * bernoulli' k

Next tactic:
refine' sum_congr rfl fun x hx => _
theorem bernoulli'_spec' (n : ℕ) :
    (∑ k in antidiagonal n, ((k.1 + k.2).choose k.2 : ℚ) / (k.2 + 1) * bernoulli' k.1) = 1 := by
  refine' ((sum_antidiagonal_eq_sum_range_succ_mk _ n).trans _).trans (bernoulli'_spec n)
  refine' sum_congr rfl fun x hx => _
  

Proof state:
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n x : ℕ
hx : x ∈ range (succ n)
⊢ ↑(Nat.choose ((x, n - x).1 + (x, n - x).2) (x, n - x).2) / (↑(x, n - x).2 + 1) * bernoulli' (x, n - x).1 =
    ↑(Nat.choose n (n - x)) / (↑n - ↑x + 1) * bernoulli' x

Next tactic:
simp only [add_tsub_cancel_of_le, mem_range_succ_iff.mp hx, cast_sub]
@[simp]
theorem bernoulli'_zero : bernoulli' 0 = 1 := by
  

Proof state:
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
⊢ bernoulli' 0 = 1

Next tactic:
rw [bernoulli'_def]
@[simp]
theorem bernoulli'_zero : bernoulli' 0 = 1 := by
  rw [bernoulli'_def]
  

Proof state:
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
⊢ 1 - ∑ k in range 0, ↑(Nat.choose 0 k) / (↑0 - ↑k + 1) * bernoulli' k = 1

Next tactic:
norm_num
@[simp]
theorem bernoulli'_one : bernoulli' 1 = 1 / 2 := by
  

Proof state:
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
⊢ bernoulli' 1 = 1 / 2

Next tactic:
rw [bernoulli'_def]
@[simp]
theorem bernoulli'_one : bernoulli' 1 = 1 / 2 := by
  rw [bernoulli'_def]
  

Proof state:
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
⊢ 1 - ∑ k in range 1, ↑(Nat.choose 1 k) / (↑1 - ↑k + 1) * bernoulli' k = 1 / 2

Next tactic:
norm_num
@[simp]
theorem bernoulli'_two : bernoulli' 2 = 1 / 6 := by
  

Proof state:
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
⊢ bernoulli' 2 = 1 / 6

Next tactic:
rw [bernoulli'_def]
@[simp]
theorem bernoulli'_two : bernoulli' 2 = 1 / 6 := by
  rw [bernoulli'_def]
  

Proof state:
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
⊢ 1 - ∑ k in range 2, ↑(Nat.choose 2 k) / (↑2 - ↑k + 1) * bernoulli' k = 1 / 6

Next tactic:
norm_num [sum_range_succ, sum_range_succ, sum_range_zero]
@[simp]
theorem bernoulli'_three : bernoulli' 3 = 0 := by
  

Proof state:
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
⊢ bernoulli' 3 = 0

Next tactic:
rw [bernoulli'_def]
@[simp]
theorem bernoulli'_three : bernoulli' 3 = 0 := by
  rw [bernoulli'_def]
  

Proof state:
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
⊢ 1 - ∑ k in range 3, ↑(Nat.choose 3 k) / (↑3 - ↑k + 1) * bernoulli' k = 0

Next tactic:
norm_num [sum_range_succ, sum_range_succ, sum_range_zero]
@[simp]
theorem bernoulli'_four : bernoulli' 4 = -1 / 30 := by
  

Proof state:
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
⊢ bernoulli' 4 = -1 / 30

Next tactic:
have : Nat.choose 4 2 = 6 := by decide
@[simp]
theorem bernoulli'_four : bernoulli' 4 = -1 / 30 := by
  have : Nat.choose 4 2 = 6 := by 

Proof state:
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
⊢ Nat.choose 4 2 = 6

Next tactic:
decide
@[simp]
theorem bernoulli'_four : bernoulli' 4 = -1 / 30 := by
  have : Nat.choose 4 2 = 6 := by decide -- shrug
  

Proof state:
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
this : Nat.choose 4 2 = 6
⊢ bernoulli' 4 = -1 / 30

Next tactic:
rw [bernoulli'_def]
@[simp]
theorem bernoulli'_four : bernoulli' 4 = -1 / 30 := by
  have : Nat.choose 4 2 = 6 := by decide -- shrug
  rw [bernoulli'_def]
  

Proof state:
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
this : Nat.choose 4 2 = 6
⊢ 1 - ∑ k in range 4, ↑(Nat.choose 4 k) / (↑4 - ↑k + 1) * bernoulli' k = -1 / 30

Next tactic:
norm_num [sum_range_succ, sum_range_succ, sum_range_zero, this]
@[simp]
theorem sum_bernoulli' (n : ℕ) : (∑ k in range n, (n.choose k : ℚ) * bernoulli' k) = n := by
  

Proof state:
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
⊢ ∑ k in range n, ↑(Nat.choose n k) * bernoulli' k = ↑n

Next tactic:
cases' n with n
@[simp]
theorem sum_bernoulli' (n : ℕ) : (∑ k in range n, (n.choose k : ℚ) * bernoulli' k) = n := by
  cases' n with n
  · 

Proof state:
case zero
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
⊢ ∑ k in range zero, ↑(Nat.choose zero k) * bernoulli' k = ↑zero

Next tactic:
simp
@[simp]
theorem sum_bernoulli' (n : ℕ) : (∑ k in range n, (n.choose k : ℚ) * bernoulli' k) = n := by
  cases' n with n
  · simp
  

Proof state:
case succ
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
⊢ ∑ k in range (succ n), ↑(Nat.choose (succ n) k) * bernoulli' k = ↑(succ n)

Next tactic:
suffices
    ((n + 1 : ℚ) * ∑ k in range n, ↑(n.choose k) / (n - k + 1) * bernoulli' k) =
      ∑ x in range n, ↑(n.succ.choose x) * bernoulli' x by
    rw_mod_cast [sum_range_succ, bernoulli'_def, ← this, choose_succ_self_right]
    ring
@[simp]
theorem sum_bernoulli' (n : ℕ) : (∑ k in range n, (n.choose k : ℚ) * bernoulli' k) = n := by
  cases' n with n
  · simp
  suffices
    ((n + 1 : ℚ) * ∑ k in range n, ↑(n.choose k) / (n - k + 1) * bernoulli' k) =
      ∑ x in range n, ↑(n.succ.choose x) * bernoulli' x by
    

Proof state:
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
this :
  (↑n + 1) * ∑ k in range n, ↑(Nat.choose n k) / (↑n - ↑k + 1) * bernoulli' k =
    ∑ x in range n, ↑(Nat.choose (succ n) x) * bernoulli' x
⊢ ∑ k in range (succ n), ↑(Nat.choose (succ n) k) * bernoulli' k = ↑(succ n)

Next tactic:
rw_mod_cast [sum_range_succ, bernoulli'_def, ← this, choose_succ_self_right]
@[simp]
theorem sum_bernoulli' (n : ℕ) : (∑ k in range n, (n.choose k : ℚ) * bernoulli' k) = n := by
  cases' n with n
  · simp
  suffices
    ((n + 1 : ℚ) * ∑ k in range n, ↑(n.choose k) / (n - k + 1) * bernoulli' k) =
      ∑ x in range n, ↑(n.succ.choose x) * bernoulli' x by
    rw_mod_cast [sum_range_succ, bernoulli'_def, ← this, choose_succ_self_right]
    

Proof state:
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
this :
  ↑(n + 1) * ∑ x in range n, Rat.divInt (↑(Nat.choose n x)) (Int.subNatNat n x + 1) * bernoulli' x =
    ∑ x in range n, ↑(Nat.choose (succ n) x) * bernoulli' x
⊢ ↑(n + 1) * ∑ x in range n, Rat.divInt (↑(Nat.choose n x)) (Int.subNatNat n x + 1) * bernoulli' x +
      ↑(n + 1) * (1 - ∑ x in range n, Rat.divInt (↑(Nat.choose n x)) (Int.subNatNat n x + 1) * bernoulli' x) =
    ↑(succ n)

Next tactic:
ring
@[simp]
theorem sum_bernoulli' (n : ℕ) : (∑ k in range n, (n.choose k : ℚ) * bernoulli' k) = n := by
  cases' n with n
  · simp
  suffices
    ((n + 1 : ℚ) * ∑ k in range n, ↑(n.choose k) / (n - k + 1) * bernoulli' k) =
      ∑ x in range n, ↑(n.succ.choose x) * bernoulli' x by
    rw_mod_cast [sum_range_succ, bernoulli'_def, ← this, choose_succ_self_right]
    ring
  

Proof state:
case succ
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
⊢ (↑n + 1) * ∑ k in range n, ↑(Nat.choose n k) / (↑n - ↑k + 1) * bernoulli' k =
    ∑ x in range n, ↑(Nat.choose (succ n) x) * bernoulli' x

Next tactic:
simp_rw [mul_sum, ← mul_assoc]
@[simp]
theorem sum_bernoulli' (n : ℕ) : (∑ k in range n, (n.choose k : ℚ) * bernoulli' k) = n := by
  cases' n with n
  · simp
  suffices
    ((n + 1 : ℚ) * ∑ k in range n, ↑(n.choose k) / (n - k + 1) * bernoulli' k) =
      ∑ x in range n, ↑(n.succ.choose x) * bernoulli' x by
    rw_mod_cast [sum_range_succ, bernoulli'_def, ← this, choose_succ_self_right]
    ring
  simp_rw [mul_sum, ← mul_assoc]
  

Proof state:
case succ
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
⊢ ∑ x in range n, (↑n + 1) * (↑(Nat.choose n x) / (↑n - ↑x + 1)) * bernoulli' x =
    ∑ x in range n, ↑(Nat.choose (succ n) x) * bernoulli' x

Next tactic:
refine' sum_congr rfl fun k hk => _
@[simp]
theorem sum_bernoulli' (n : ℕ) : (∑ k in range n, (n.choose k : ℚ) * bernoulli' k) = n := by
  cases' n with n
  · simp
  suffices
    ((n + 1 : ℚ) * ∑ k in range n, ↑(n.choose k) / (n - k + 1) * bernoulli' k) =
      ∑ x in range n, ↑(n.succ.choose x) * bernoulli' x by
    rw_mod_cast [sum_range_succ, bernoulli'_def, ← this, choose_succ_self_right]
    ring
  simp_rw [mul_sum, ← mul_assoc]
  refine' sum_congr rfl fun k hk => _
  

Proof state:
case succ
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n k : ℕ
hk : k ∈ range n
⊢ (↑n + 1) * (↑(Nat.choose n k) / (↑n - ↑k + 1)) * bernoulli' k = ↑(Nat.choose (succ n) k) * bernoulli' k

Next tactic:
congr
@[simp]
theorem sum_bernoulli' (n : ℕ) : (∑ k in range n, (n.choose k : ℚ) * bernoulli' k) = n := by
  cases' n with n
  · simp
  suffices
    ((n + 1 : ℚ) * ∑ k in range n, ↑(n.choose k) / (n - k + 1) * bernoulli' k) =
      ∑ x in range n, ↑(n.succ.choose x) * bernoulli' x by
    rw_mod_cast [sum_range_succ, bernoulli'_def, ← this, choose_succ_self_right]
    ring
  simp_rw [mul_sum, ← mul_assoc]
  refine' sum_congr rfl fun k hk => _
  congr
  

Proof state:
case succ.e_a
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n k : ℕ
hk : k ∈ range n
⊢ (↑n + 1) * (↑(Nat.choose n k) / (↑n - ↑k + 1)) = ↑(Nat.choose (succ n) k)

Next tactic:
have : ((n - k : ℕ) : ℚ) + 1 ≠ 0 := by norm_cast; exact succ_ne_zero _
@[simp]
theorem sum_bernoulli' (n : ℕ) : (∑ k in range n, (n.choose k : ℚ) * bernoulli' k) = n := by
  cases' n with n
  · simp
  suffices
    ((n + 1 : ℚ) * ∑ k in range n, ↑(n.choose k) / (n - k + 1) * bernoulli' k) =
      ∑ x in range n, ↑(n.succ.choose x) * bernoulli' x by
    rw_mod_cast [sum_range_succ, bernoulli'_def, ← this, choose_succ_self_right]
    ring
  simp_rw [mul_sum, ← mul_assoc]
  refine' sum_congr rfl fun k hk => _
  congr
  have : ((n - k : ℕ) : ℚ) + 1 ≠ 0 := by 

Proof state:
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n k : ℕ
hk : k ∈ range n
⊢ ↑(n - k) + 1 ≠ 0

Next tactic:
norm_cast
@[simp]
theorem sum_bernoulli' (n : ℕ) : (∑ k in range n, (n.choose k : ℚ) * bernoulli' k) = n := by
  cases' n with n
  · simp
  suffices
    ((n + 1 : ℚ) * ∑ k in range n, ↑(n.choose k) / (n - k + 1) * bernoulli' k) =
      ∑ x in range n, ↑(n.succ.choose x) * bernoulli' x by
    rw_mod_cast [sum_range_succ, bernoulli'_def, ← this, choose_succ_self_right]
    ring
  simp_rw [mul_sum, ← mul_assoc]
  refine' sum_congr rfl fun k hk => _
  congr
  have : ((n - k : ℕ) : ℚ) + 1 ≠ 0 := by norm_cast; 

Proof state:
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n k : ℕ
hk : k ∈ range n
⊢ ¬n - k + 1 = 0

Next tactic:
exact succ_ne_zero _
@[simp]
theorem sum_bernoulli' (n : ℕ) : (∑ k in range n, (n.choose k : ℚ) * bernoulli' k) = n := by
  cases' n with n
  · simp
  suffices
    ((n + 1 : ℚ) * ∑ k in range n, ↑(n.choose k) / (n - k + 1) * bernoulli' k) =
      ∑ x in range n, ↑(n.succ.choose x) * bernoulli' x by
    rw_mod_cast [sum_range_succ, bernoulli'_def, ← this, choose_succ_self_right]
    ring
  simp_rw [mul_sum, ← mul_assoc]
  refine' sum_congr rfl fun k hk => _
  congr
  have : ((n - k : ℕ) : ℚ) + 1 ≠ 0 := by norm_cast; exact succ_ne_zero _
  

Proof state:
case succ.e_a
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n k : ℕ
hk : k ∈ range n
this : ↑(n - k) + 1 ≠ 0
⊢ (↑n + 1) * (↑(Nat.choose n k) / (↑n - ↑k + 1)) = ↑(Nat.choose (succ n) k)

Next tactic:
field_simp [← cast_sub (mem_range.1 hk).le, mul_comm]
@[simp]
theorem sum_bernoulli' (n : ℕ) : (∑ k in range n, (n.choose k : ℚ) * bernoulli' k) = n := by
  cases' n with n
  · simp
  suffices
    ((n + 1 : ℚ) * ∑ k in range n, ↑(n.choose k) / (n - k + 1) * bernoulli' k) =
      ∑ x in range n, ↑(n.succ.choose x) * bernoulli' x by
    rw_mod_cast [sum_range_succ, bernoulli'_def, ← this, choose_succ_self_right]
    ring
  simp_rw [mul_sum, ← mul_assoc]
  refine' sum_congr rfl fun k hk => _
  congr
  have : ((n - k : ℕ) : ℚ) + 1 ≠ 0 := by norm_cast; exact succ_ne_zero _
  field_simp [← cast_sub (mem_range.1 hk).le, mul_comm]
  

Proof state:
case succ.e_a
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n k : ℕ
hk : k ∈ range n
this : ↑(n - k) + 1 ≠ 0
⊢ ↑(Nat.choose n k) * (↑n + 1) = ↑(Nat.choose (succ n) k) * (↑(n - k) + 1)

Next tactic:
rw_mod_cast [tsub_add_eq_add_tsub (mem_range.1 hk).le, choose_mul_succ_eq]
theorem bernoulli'PowerSeries_mul_exp_sub_one :
    bernoulli'PowerSeries A * (exp A - 1) = X * exp A := by
  

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

Next tactic:
ext n
theorem bernoulli'PowerSeries_mul_exp_sub_one :
    bernoulli'PowerSeries A * (exp A - 1) = X * exp A := 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) (bernoulli'PowerSeries A * (exp A - 1)) = (coeff A n) (X * exp A)

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

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

Next tactic:
simp
theorem bernoulli'PowerSeries_mul_exp_sub_one :
    bernoulli'PowerSeries A * (exp A - 1) = X * exp A := by
  ext n
  -- constant coefficient is a special case
  cases' n with 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)) (bernoulli'PowerSeries A * (exp A - 1)) = (coeff A (succ n)) (X * exp A)

Next tactic:
rw [bernoulli'PowerSeries, coeff_mul, mul_comm X, sum_antidiagonal_succ']
theorem bernoulli'PowerSeries_mul_exp_sub_one :
    bernoulli'PowerSeries A * (exp A - 1) = X * exp A := by
  ext n
  -- constant coefficient is a special case
  cases' n with n
  · simp
  rw [bernoulli'PowerSeries, coeff_mul, mul_comm X, sum_antidiagonal_succ']
  

Proof state:
case h.succ
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
⊢ (coeff A (n + 1, 0).1) (PowerSeries.mk fun n => (algebraMap ℚ A) (bernoulli' n / ↑n !)) *
        (coeff A (n + 1, 0).2) (exp A - 1) +
      ∑ p in antidiagonal n,
        (coeff A (p.1, p.2 + 1).1) (PowerSeries.mk fun n => (algebraMap ℚ A) (bernoulli' n / ↑n !)) *
          (coeff A (p.1, p.2 + 1).2) (exp A - 1) =
    (coeff A (succ n)) (exp A * X)

Next tactic:
suffices (∑ p in antidiagonal n,
      bernoulli' p.1 / p.1! * ((p.2 + 1) * p.2! : ℚ)⁻¹) = (n ! : ℚ)⁻¹ by
    simpa [map_sum, Nat.factorial] using congr_arg (algebraMap ℚ A) this
theorem bernoulli'PowerSeries_mul_exp_sub_one :
    bernoulli'PowerSeries A * (exp A - 1) = X * exp A := by
  ext n
  -- constant coefficient is a special case
  cases' n with n
  · simp
  rw [bernoulli'PowerSeries, coeff_mul, mul_comm X, sum_antidiagonal_succ']
  suffices (∑ p in antidiagonal n,
      bernoulli' p.1 / p.1! * ((p.2 + 1) * p.2! : ℚ)⁻¹) = (n ! : ℚ)⁻¹ by
    

Proof state:
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
this : ∑ p in antidiagonal n, bernoulli' p.1 / ↑p.1! * ((↑p.2 + 1) * ↑p.2!)⁻¹ = (↑n !)⁻¹
⊢ (coeff A (n + 1, 0).1) (PowerSeries.mk fun n => (algebraMap ℚ A) (bernoulli' n / ↑n !)) *
        (coeff A (n + 1, 0).2) (exp A - 1) +
      ∑ p in antidiagonal n,
        (coeff A (p.1, p.2 + 1).1) (PowerSeries.mk fun n => (algebraMap ℚ A) (bernoulli' n / ↑n !)) *
          (coeff A (p.1, p.2 + 1).2) (exp A - 1) =
    (coeff A (succ n)) (exp A * X)

Next tactic:
simpa [map_sum, Nat.factorial] using congr_arg (algebraMap ℚ A) this
theorem bernoulli'PowerSeries_mul_exp_sub_one :
    bernoulli'PowerSeries A * (exp A - 1) = X * exp A := by
  ext n
  -- constant coefficient is a special case
  cases' n with n
  · simp
  rw [bernoulli'PowerSeries, coeff_mul, mul_comm X, sum_antidiagonal_succ']
  suffices (∑ p in antidiagonal n,
      bernoulli' p.1 / p.1! * ((p.2 + 1) * p.2! : ℚ)⁻¹) = (n ! : ℚ)⁻¹ by
    simpa [map_sum, Nat.factorial] using congr_arg (algebraMap ℚ A) this
  

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

Next tactic:
apply eq_inv_of_mul_eq_one_left
theorem bernoulli'PowerSeries_mul_exp_sub_one :
    bernoulli'PowerSeries A * (exp A - 1) = X * exp A := by
  ext n
  -- constant coefficient is a special case
  cases' n with n
  · simp
  rw [bernoulli'PowerSeries, coeff_mul, mul_comm X, sum_antidiagonal_succ']
  suffices (∑ p in antidiagonal n,
      bernoulli' p.1 / p.1! * ((p.2 + 1) * p.2! : ℚ)⁻¹) = (n ! : ℚ)⁻¹ by
    simpa [map_sum, Nat.factorial] using congr_arg (algebraMap ℚ A) this
  apply eq_inv_of_mul_eq_one_left
  

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

Next tactic:
rw [sum_mul]
theorem bernoulli'PowerSeries_mul_exp_sub_one :
    bernoulli'PowerSeries A * (exp A - 1) = X * exp A := by
  ext n
  -- constant coefficient is a special case
  cases' n with n
  · simp
  rw [bernoulli'PowerSeries, coeff_mul, mul_comm X, sum_antidiagonal_succ']
  suffices (∑ p in antidiagonal n,
      bernoulli' p.1 / p.1! * ((p.2 + 1) * p.2! : ℚ)⁻¹) = (n ! : ℚ)⁻¹ by
    simpa [map_sum, Nat.factorial] using congr_arg (algebraMap ℚ A) this
  apply eq_inv_of_mul_eq_one_left
  rw [sum_mul]
  

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

Next tactic:
convert bernoulli'_spec' n using 1