awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.Bernoulli (goal)
Goal:
A : Type u_1
inst : CommRing A
inst_1 : Algebra ℚ A
n : ℕ
h :
  Eq (α := A) ((coeff 0 : (a : A⟦X⟧) → A) (bernoulliPowerSeries A * (exp A - 1)) : A) ((coeff 0 : (a : A⟦X⟧) → A) X : A)
⊢ Eq (α := A) ((coeff n : (a : A⟦X⟧) → A) (bernoulliPowerSeries A * (exp A - 1)) : A) ((coeff n : (a : A⟦X⟧) → A) X : A)
have h₁ : ∀ (n : ℕ),
  Eq (α := A) ((coeff (n + 1) : (a : A⟦X⟧) → A) (bernoulliPowerSeries A * (exp A - 1)) : A)
    ((coeff (n + 1) : (a : A⟦X⟧) → A) X : A)
Tactic: cases n with | zero => simp | succ n =>
  simp only [bernoulliPowerSeries, coeff_mul, coeff_X, sum_antidiagonal_succ', one_div, coeff_mk,
    coeff_one, coeff_exp, map_sub, factorial, if_pos, cast_succ, cast_mul,
    sub_zero, add_eq_zero, if_false, one_ne_zero, and_false, ← map_mul, ← map_sum]
  cases n with | zero => simp | succ n =>
  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]
  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 _ _)]
  simp [field, mul_comm _ (bernoulli x.1), mul_assoc]
Cleared hypotheses: n
-- Lean 4 proof step from Mathlib.NumberTheory.Bernoulli (hypothesis)
Goal:
n p : ℕ
⊢ Eq (α := ℚ) (∑ k ∈ range n, HPow.hPow (α := ℚ) (↑k : ℚ) p)
    (∑ i ∈ range (p + 1),
      bernoulli i * (↑((p + 1).choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p + 1 - i) / HAdd.hAdd (α := ℚ) (↑p : ℚ) 1)
have hne : ∀ (m : ℕ), Ne (α := ℚ) (↑m ! : ℚ) 0
Tactic: have hne : ∀ m : ℕ, (m ! : ℚ) ≠ 0 := fun m => mod_cast factorial_ne_zero m
-- Lean 4 proof step from Mathlib.NumberTheory.Bernoulli (goal)
Goal:
n p : ℕ
hne : ∀ (m : ℕ), Ne (α := ℚ) (↑m ! : ℚ) 0
⊢ HMul.hMul (α := ℚ⟦X⟧) (PowerSeries.mk fun (p : ℕ) ↦ bernoulli p / (↑p ! : ℚ))
      (PowerSeries.mk fun (q : ℕ) ↦ (coeff (q + 1) : (a : ℚ⟦X⟧) → ℚ) (exp ℚ ^ n)) =
    PowerSeries.mk fun (p : ℕ) ↦
      ∑ i ∈ range (p + 1),
        bernoulli i * (↑((p + 1).choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p + 1 - i) / (↑(p + 1)! : ℚ)
have h : ∀ (q : ℕ),
  Eq (α := ℚ)
    ((coeff q : (a : ℚ⟦X⟧) → ℚ)
        (HMul.hMul (α := ℚ⟦X⟧) (PowerSeries.mk fun (p : ℕ) ↦ bernoulli p / (↑p ! : ℚ))
          (PowerSeries.mk fun (q : ℕ) ↦ (coeff (q + 1) : (a : ℚ⟦X⟧) → ℚ) (exp ℚ ^ n))) :
      ℚ)
    ((coeff q : (a : ℚ⟦X⟧) → ℚ)
        (PowerSeries.mk fun (p : ℕ) ↦
          ∑ i ∈ range (p + 1),
            bernoulli i * (↑((p + 1).choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p + 1 - i) / (↑(p + 1)! : ℚ)) :
      ℚ)
Tactic: ext q : 1
-- Lean 4 proof step from Mathlib.NumberTheory.Bernoulli (goal)
Goal:
n p : ℕ
hne : ∀ (m : ℕ), Ne (α := ℚ) (↑m ! : ℚ) 0
q : ℕ
f : (a b : ℕ) → ℚ :=
  fun (a b : ℕ) ↦ HMul.hMul (β := ℚ) (bernoulli a / (↑a ! : ℚ)) ((coeff (b + 1) : (a : ℚ⟦X⟧) → ℚ) (exp ℚ ^ n) : ℚ)
⊢ Eq (α := ℚ)
    ((coeff q : (a : ℚ⟦X⟧) → ℚ)
        (HMul.hMul (α := ℚ⟦X⟧) (PowerSeries.mk fun (p : ℕ) ↦ bernoulli p / (↑p ! : ℚ))
          (PowerSeries.mk fun (q : ℕ) ↦ (coeff (q + 1) : (a : ℚ⟦X⟧) → ℚ) (exp ℚ ^ n))) :
      ℚ)
    ((coeff q : (a : ℚ⟦X⟧) → ℚ)
        (PowerSeries.mk fun (p : ℕ) ↦
          ∑ i ∈ range (p + 1),
            bernoulli i * (↑((p + 1).choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p + 1 - i) / (↑(p + 1)! : ℚ)) :
      ℚ)
have h : Eq (α := ℚ)
  (∑ x ∈ range q.succ,
    HMul.hMul (β := ℚ) (bernoulli x / (↑x ! : ℚ)) ((coeff (q - x + 1) : (a : ℚ⟦X⟧) → ℚ) (exp ℚ ^ n) : ℚ))
  (∑ i ∈ range (q + 1),
    bernoulli i * (↑((q + 1).choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (q + 1 - i) / (↑(q + 1)! : ℚ))
Tactic: simp only [f, coeff_mul, coeff_mk, sum_antidiagonal_eq_sum_range_succ f]
-- Lean 4 proof step from Mathlib.NumberTheory.Bernoulli (goal)
Goal:
n p : ℕ
hne : ∀ (m : ℕ), Ne (α := ℚ) (↑m ! : ℚ) 0
q : ℕ
f : (a b : ℕ) → ℚ :=
  fun (a b : ℕ) ↦ HMul.hMul (β := ℚ) (bernoulli a / (↑a ! : ℚ)) ((coeff (b + 1) : (a : ℚ⟦X⟧) → ℚ) (exp ℚ ^ n) : ℚ)
⊢ Eq (α := ℚ)
    (∑ x ∈ range q.succ,
      HMul.hMul (β := ℚ) (bernoulli x / (↑x ! : ℚ)) ((coeff (q - x + 1) : (a : ℚ⟦X⟧) → ℚ) (exp ℚ ^ n) : ℚ))
    (∑ i ∈ range (q + 1),
      bernoulli i * (↑((q + 1).choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (q + 1 - i) / (↑(q + 1)! : ℚ))
have h : ∀ x ∈ range q.succ,
  HMul.hMul (β := ℚ) (bernoulli x / (↑x ! : ℚ)) ((coeff (q - x + 1) : (a : ℚ⟦X⟧) → ℚ) (exp ℚ ^ n) : ℚ) =
    bernoulli x * (↑((q + 1).choose x) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (q + 1 - x) / (↑(q + 1)! : ℚ)
Tactic: apply sum_congr rfl
-- Lean 4 proof step from Mathlib.NumberTheory.Bernoulli (goal)
Goal:
n p : ℕ
hne : ∀ (m : ℕ), Ne (α := ℚ) (↑m ! : ℚ) 0
q : ℕ
f : (a b : ℕ) → ℚ :=
  fun (a b : ℕ) ↦ HMul.hMul (β := ℚ) (bernoulli a / (↑a ! : ℚ)) ((coeff (b + 1) : (a : ℚ⟦X⟧) → ℚ) (exp ℚ ^ n) : ℚ)
m : ℕ
h : m ∈ range q.succ
⊢ HMul.hMul (β := ℚ) (bernoulli m / (↑m ! : ℚ)) ((coeff (q - m + 1) : (a : ℚ⟦X⟧) → ℚ) (exp ℚ ^ n) : ℚ) =
    bernoulli m * (↑((q + 1).choose m) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (q + 1 - m) / (↑(q + 1)! : ℚ)
have h₁ : HMul.hMul (β := ℚ) (bernoulli m / (↑m ! : ℚ))
    ((coeff (q - m + 1) : (a : ℚ⟦X⟧) → ℚ)
        (({
                toFun := fun (f : ℚ⟦X⟧) ↦
                  PowerSeries.mk fun (n_1 : ℕ) ↦
                    HMul.hMul (β := ℚ) (HPow.hPow (α := ℚ) (↑n : ℚ) n_1) ((coeff n_1 : (a : ℚ⟦X⟧) → ℚ) f : ℚ),
                map_one' := sorry, map_mul' := sorry : ℚ⟦X⟧ →* ℚ⟦X⟧ } :
              (a : ℚ⟦X⟧) → ℚ⟦X⟧)
            (exp ℚ) :
          ℚ⟦X⟧) :
      ℚ) =
  bernoulli m * (↑((q + 1).choose m) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (q + 1 - m) / (↑(q + 1)! : ℚ)
Tactic: simp only [exp_pow_eq_rescale_exp, rescale, RingHom.coe_mk]
-- Lean 4 proof step from Mathlib.NumberTheory.Bernoulli (hypothesis)
Goal:
n p : ℕ
hne : ∀ (m : ℕ), Ne (α := ℚ) (↑m ! : ℚ) 0
q : ℕ
f : (a b : ℕ) → ℚ :=
  fun (a b : ℕ) ↦ HMul.hMul (β := ℚ) (bernoulli a / (↑a ! : ℚ)) ((coeff (b + 1) : (a : ℚ⟦X⟧) → ℚ) (exp ℚ ^ n) : ℚ)
m : ℕ
h : m ∈ range q.succ
⊢ HMul.hMul (β := ℚ) (bernoulli m / (↑m ! : ℚ))
      ((coeff (q - m + 1) : (a : ℚ⟦X⟧) → ℚ)
          (({
                  toFun := fun (f : ℚ⟦X⟧) ↦
                    PowerSeries.mk fun (n_1 : ℕ) ↦
                      HMul.hMul (β := ℚ) (HPow.hPow (α := ℚ) (↑n : ℚ) n_1) ((coeff n_1 : (a : ℚ⟦X⟧) → ℚ) f : ℚ),
                  map_one' := ⋯, map_mul' := ⋯ : ℚ⟦X⟧ →* ℚ⟦X⟧ } :
                (a : ℚ⟦X⟧) → ℚ⟦X⟧)
              (exp ℚ) :
            ℚ⟦X⟧) :
        ℚ) =
    bernoulli m * (↑((q + 1).choose m) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (q + 1 - m) / (↑(q + 1)! : ℚ)
have h₁ : m < q + 1
Tactic: have h : m < q + 1 := by simpa using h
-- Lean 4 proof step from Mathlib.NumberTheory.Bernoulli (goal)
Goal:
n p : ℕ
hne : ∀ (m : ℕ), Ne (α := ℚ) (↑m ! : ℚ) 0
q : ℕ
f : (a b : ℕ) → ℚ :=
  fun (a b : ℕ) ↦ HMul.hMul (β := ℚ) (bernoulli a / (↑a ! : ℚ)) ((coeff (b + 1) : (a : ℚ⟦X⟧) → ℚ) (exp ℚ ^ n) : ℚ)
m : ℕ
h_1 : m ∈ range q.succ
h : m < q + 1
⊢ HMul.hMul (β := ℚ) (bernoulli m / (↑m ! : ℚ))
      ((coeff (q - m + 1) : (a : ℚ⟦X⟧) → ℚ)
          (({
                  toFun := fun (f : ℚ⟦X⟧) ↦
                    PowerSeries.mk fun (n_1 : ℕ) ↦
                      HMul.hMul (β := ℚ) (HPow.hPow (α := ℚ) (↑n : ℚ) n_1) ((coeff n_1 : (a : ℚ⟦X⟧) → ℚ) f : ℚ),
                  map_one' := ⋯, map_mul' := ⋯ : ℚ⟦X⟧ →* ℚ⟦X⟧ } :
                (a : ℚ⟦X⟧) → ℚ⟦X⟧)
              (exp ℚ) :
            ℚ⟦X⟧) :
        ℚ) =
    bernoulli m * (↑((q + 1).choose m) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (q + 1 - m) / (↑(q + 1)! : ℚ)
have h₁ : bernoulli m * (↑((q + 1)! / (m ! * (q + 1 - m)!)) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (q + 1 - m) =
  HMul.hMul (β := ℚ) (bernoulli m * (↑q.succ ! : ℚ) / (↑m ! : ℚ))
    ((coeff (q - m + 1) : (a : ℚ⟦X⟧) → ℚ)
        (({
                toFun := fun (f : ℚ⟦X⟧) ↦
                  PowerSeries.mk fun (n_1 : ℕ) ↦
                    HMul.hMul (β := ℚ) (HPow.hPow (α := ℚ) (↑n : ℚ) n_1) ((coeff n_1 : (a : ℚ⟦X⟧) → ℚ) f : ℚ),
                map_one' := sorry, map_mul' := sorry } :
              (a : ℚ⟦X⟧) → ℚ⟦X⟧)
            (exp ℚ) :
          ℚ⟦X⟧) :
      ℚ)
Tactic: rw [choose_eq_factorial_div_factorial h.le, eq_comm, div_eq_iff (hne q.succ), succ_eq_add_one,
      mul_assoc _ _ (q.succ ! : ℚ), mul_comm _ (q.succ ! : ℚ), ← mul_assoc, div_mul_eq_mul_div]
-- Lean 4 proof step from Mathlib.NumberTheory.Bernoulli (goal)
Goal:
n p : ℕ
hne : ∀ (m : ℕ), Ne (α := ℚ) (↑m ! : ℚ) 0
q : ℕ
f : (a b : ℕ) → ℚ :=
  fun (a b : ℕ) ↦ HMul.hMul (β := ℚ) (bernoulli a / (↑a ! : ℚ)) ((coeff (b + 1) : (a : ℚ⟦X⟧) → ℚ) (exp ℚ ^ n) : ℚ)
m : ℕ
h_1 : m ∈ range q.succ
h : m < q + 1
⊢ bernoulli m * (↑((q + 1)! / (m ! * (q + 1 - m)!)) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (q + 1 - m) =
    HMul.hMul (β := ℚ) (bernoulli m * (↑q.succ ! : ℚ) / (↑m ! : ℚ))
      ((coeff (q - m + 1) : (a : ℚ⟦X⟧) → ℚ)
          (({
                  toFun := fun (f : ℚ⟦X⟧) ↦
                    PowerSeries.mk fun (n_1 : ℕ) ↦
                      HMul.hMul (β := ℚ) (HPow.hPow (α := ℚ) (↑n : ℚ) n_1) ((coeff n_1 : (a : ℚ⟦X⟧) → ℚ) f : ℚ),
                  map_one' := ⋯, map_mul' := ⋯ } :
                (a : ℚ⟦X⟧) → ℚ⟦X⟧)
              (exp ℚ) :
            ℚ⟦X⟧) :
        ℚ)
have h₁ : bernoulli m * (↑((q + 1)! / (m ! * (q + 1 - m)!)) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (q + 1 - m) =
  bernoulli m * (↑q.succ ! : ℚ) / (↑m ! : ℚ) * (HPow.hPow (α := ℚ) (↑n : ℚ) (q - m + 1) * (↑(q - m + 1)! : ℚ)⁻¹)
Tactic: simp only [MonoidHom.coe_mk, OneHom.coe_mk, coeff_exp, Algebra.algebraMap_self, one_div,
      map_inv₀, map_natCast, coeff_mk]
-- Lean 4 proof step from Mathlib.NumberTheory.Bernoulli (goal)
Goal:
n p : ℕ
hne : ∀ (m : ℕ), Ne (α := ℚ) (↑m ! : ℚ) 0
q : ℕ
f : (a b : ℕ) → ℚ :=
  fun (a b : ℕ) ↦ HMul.hMul (β := ℚ) (bernoulli a / (↑a ! : ℚ)) ((coeff (b + 1) : (a : ℚ⟦X⟧) → ℚ) (exp ℚ ^ n) : ℚ)
m : ℕ
h_1 : m ∈ range q.succ
h : m < q + 1
⊢ bernoulli m * (↑((q + 1)! / (m ! * (q + 1 - m)!)) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (q + 1 - m) =
    bernoulli m * (↑q.succ ! : ℚ) / (↑m ! : ℚ) * (HPow.hPow (α := ℚ) (↑n : ℚ) (q - m + 1) * (↑(q - m + 1)! : ℚ)⁻¹)
have h₁ : bernoulli m * ((↑(q + 1)! : ℚ) / HMul.hMul (α := ℚ) (↑m ! : ℚ) (↑(q + 1 - m)! : ℚ)) *
    HPow.hPow (α := ℚ) (↑n : ℚ) (q + 1 - m) =
  bernoulli m * (↑q.succ ! : ℚ) / HMul.hMul (α := ℚ) (↑m ! : ℚ) (↑(q + 1 - m)! : ℚ) *
    HPow.hPow (α := ℚ) (↑n : ℚ) (q + 1 - m)
Tactic: rw [mul_comm ((n : ℚ) ^ (q - m + 1)), ← mul_assoc _ _ ((n : ℚ) ^ (q - m + 1)), ← one_div,
      mul_one_div, div_div, tsub_add_eq_add_tsub (le_of_lt_succ h), cast_div, cast_mul]
-- Lean 4 proof step from Mathlib.NumberTheory.Bernoulli (goal)
Goal:
n p : ℕ
hne : ∀ (m : ℕ), Ne (α := ℚ) (↑m ! : ℚ) 0
q : ℕ
f : (a b : ℕ) → ℚ :=
  fun (a b : ℕ) ↦ HMul.hMul (β := ℚ) (bernoulli a / (↑a ! : ℚ)) ((coeff (b + 1) : (a : ℚ⟦X⟧) → ℚ) (exp ℚ ^ n) : ℚ)
m : ℕ
h_1 : m ∈ range q.succ
h : m < q + 1
h₁ :
  bernoulli m * ((↑(q + 1)! : ℚ) / HMul.hMul (α := ℚ) (↑m ! : ℚ) (↑(q + 1 - m)! : ℚ)) *
      HPow.hPow (α := ℚ) (↑n : ℚ) (q + 1 - m) =
    bernoulli m * (↑q.succ ! : ℚ) / HMul.hMul (α := ℚ) (↑m ! : ℚ) (↑(q + 1 - m)! : ℚ) *
      HPow.hPow (α := ℚ) (↑n : ℚ) (q + 1 - m)
⊢ bernoulli m * (↑((q + 1)! / (m ! * (q + 1 - m)!)) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (q + 1 - m) =
    bernoulli m * (↑q.succ ! : ℚ) / (↑m ! : ℚ) * (HPow.hPow (α := ℚ) (↑n : ℚ) (q - m + 1) * (↑(q - m + 1)! : ℚ)⁻¹)
have h₂ : m ! * (q + 1 - m)! ∣ (q + 1)!
Tactic: rw [mul_comm ((n : ℚ) ^ (q - m + 1)), ← mul_assoc _ _ ((n : ℚ) ^ (q - m + 1)), ← one_div,
      mul_one_div, div_div, tsub_add_eq_add_tsub (le_of_lt_succ h), cast_div, cast_mul]
-- Lean 4 proof step from Mathlib.NumberTheory.Bernoulli (goal)
Goal:
n p : ℕ
hne : ∀ (m : ℕ), Ne (α := ℚ) (↑m ! : ℚ) 0
q : ℕ
f : (a b : ℕ) → ℚ :=
  fun (a b : ℕ) ↦ HMul.hMul (β := ℚ) (bernoulli a / (↑a ! : ℚ)) ((coeff (b + 1) : (a : ℚ⟦X⟧) → ℚ) (exp ℚ ^ n) : ℚ)
m : ℕ
h_1 : m ∈ range q.succ
h : m < q + 1
h₁ :
  bernoulli m * ((↑(q + 1)! : ℚ) / HMul.hMul (α := ℚ) (↑m ! : ℚ) (↑(q + 1 - m)! : ℚ)) *
      HPow.hPow (α := ℚ) (↑n : ℚ) (q + 1 - m) =
    bernoulli m * (↑q.succ ! : ℚ) / HMul.hMul (α := ℚ) (↑m ! : ℚ) (↑(q + 1 - m)! : ℚ) *
      HPow.hPow (α := ℚ) (↑n : ℚ) (q + 1 - m)
h₂ : m ! * (q + 1 - m)! ∣ (q + 1)!
⊢ bernoulli m * (↑((q + 1)! / (m ! * (q + 1 - m)!)) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (q + 1 - m) =
    bernoulli m * (↑q.succ ! : ℚ) / (↑m ! : ℚ) * (HPow.hPow (α := ℚ) (↑n : ℚ) (q - m + 1) * (↑(q - m + 1)! : ℚ)⁻¹)
have h₃ : Ne (α := ℚ) (↑(m ! * (q + 1 - m)!) : ℚ) 0
Tactic: rw [mul_comm ((n : ℚ) ^ (q - m + 1)), ← mul_assoc _ _ ((n : ℚ) ^ (q - m + 1)), ← one_div,
      mul_one_div, div_div, tsub_add_eq_add_tsub (le_of_lt_succ h), cast_div, cast_mul]
-- Lean 4 proof step from Mathlib.NumberTheory.Bernoulli (hypothesis)
Goal:
n p : ℕ
hne : ∀ (m : ℕ), Ne (α := ℚ) (↑m ! : ℚ) 0
⊢ Eq (α := ℚ) (∑ k ∈ range n, HPow.hPow (α := ℚ) (↑k : ℚ) p)
    (∑ i ∈ range (p + 1),
      bernoulli i * (↑((p + 1).choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p + 1 - i) / HAdd.hAdd (α := ℚ) (↑p : ℚ) 1)
have h_cauchy : HMul.hMul (α := ℚ⟦X⟧) (PowerSeries.mk fun (p : ℕ) ↦ bernoulli p / (↑p ! : ℚ))
    (PowerSeries.mk fun (q : ℕ) ↦ (coeff (q + 1) : (a : ℚ⟦X⟧) → ℚ) (exp ℚ ^ n)) =
  PowerSeries.mk fun (p : ℕ) ↦
    ∑ i ∈ range (p + 1),
      bernoulli i * (↑((p + 1).choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p + 1 - i) / (↑(p + 1)! : ℚ)
Tactic: have h_cauchy :
    ((mk fun p => bernoulli p / p !) * mk fun q => coeff (q + 1) (exp ℚ ^ n)) =
      mk fun p => ∑ i ∈ range (p + 1),
          bernoulli i * (p + 1).choose i * (n : ℚ) ^ (p + 1 - i) / (p + 1)! := by
    ext q : 1
    let f a b := bernoulli a / a ! * coeff (b + 1) (exp ℚ ^ n)
    -- key step: use `PowerSeries.coeff_mul` and then rewrite sums
    simp only [f, coeff_mul, coeff_mk, sum_antidiagonal_eq_sum_range_succ f]
    apply sum_congr rfl
    intro m h
    simp only [exp_pow_eq_rescale_exp, rescale, RingHom.coe_mk]
    -- manipulate factorials and binomial coefficients
    have h : m < q + 1 := by simpa using h
    rw [choose_eq_factorial_div_factorial h.le, eq_comm, div_eq_iff (hne q.succ), succ_eq_add_one,
      mul_assoc _ _ (q.succ ! : ℚ), mul_comm _ (q.succ ! : ℚ), ← mul_assoc, div_mul_eq_mul_div]
    simp only [MonoidHom.coe_mk, OneHom.coe_mk, coeff_exp, Algebra.algebraMap_self, one_div,
      map_inv₀, map_natCast, coeff_mk]
    rw [mul_comm ((n : ℚ) ^ (q - m + 1)), ← mul_assoc _ _ ((n : ℚ) ^ (q - m + 1)), ← one_div,
      mul_one_div, div_div, tsub_add_eq_add_tsub (le_of_lt_succ h), cast_div, cast_mul]
    · ring
    · exact factorial_mul_factorial_dvd_factorial h.le
    · simp [factorial_ne_zero]
-- Lean 4 proof step from Mathlib.NumberTheory.Bernoulli (goal)
Goal:
n p : ℕ
hne : ∀ (m : ℕ), Ne (α := ℚ) (↑m ! : ℚ) 0
h_cauchy :
  HMul.hMul (α := ℚ⟦X⟧) (PowerSeries.mk fun (p : ℕ) ↦ bernoulli p / (↑p ! : ℚ))
      (PowerSeries.mk fun (q : ℕ) ↦ (coeff (q + 1) : (a : ℚ⟦X⟧) → ℚ) (exp ℚ ^ n)) =
    PowerSeries.mk fun (p : ℕ) ↦
      ∑ i ∈ range (p + 1),
        bernoulli i * (↑((p + 1).choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p + 1 - i) / (↑(p + 1)! : ℚ)
this :
  Eq (α := ℚ⟦X⟧)
    (PowerSeries.mk fun (p : ℕ) ↦
      ∑ k ∈ range n,
        HMul.hMul (β := ℚ) (HPow.hPow (α := ℚ) (↑k : ℚ) p) ((algebraMap ℚ ℚ : (a : ℚ) → ℚ) (↑p ! : ℚ)⁻¹ : ℚ))
    (PowerSeries.mk fun (p : ℕ) ↦
      ∑ i ∈ range (p + 1),
        bernoulli i * (↑((p + 1).choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p + 1 - i) / (↑(p + 1)! : ℚ))
⊢ ∑ k ∈ range n, HPow.hPow (α := ℚ) (↑k : ℚ) p =
    HMul.hMul (α := ℚ)
      (∑ i ∈ range (p + 1),
        bernoulli i * (↑((p + 1).choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p + 1 - i) / (↑(p + 1)! : ℚ))
      (↑p ! : ℚ)
have rw : Eq (α := ℚ) (∑ i ∈ range n, HPow.hPow (α := ℚ) (↑i : ℚ) p * (↑p ! : ℚ)⁻¹)
  (∑ i ∈ range (p + 1),
    bernoulli i * (↑((p + 1).choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p + 1 - i) / (↑(p + 1)! : ℚ))
Tactic: rw [← div_eq_iff (hne p), div_eq_mul_inv, sum_mul]
-- Lean 4 proof step from Mathlib.NumberTheory.Bernoulli (hypothesis)
Goal:
n p : ℕ
hne : ∀ (m : ℕ), Ne (α := ℚ) (↑m ! : ℚ) 0
h_cauchy :
  HMul.hMul (α := ℚ⟦X⟧) (PowerSeries.mk fun (p : ℕ) ↦ bernoulli p / (↑p ! : ℚ))
      (PowerSeries.mk fun (q : ℕ) ↦ (coeff (q + 1) : (a : ℚ⟦X⟧) → ℚ) (exp ℚ ^ n)) =
    PowerSeries.mk fun (p : ℕ) ↦
      ∑ i ∈ range (p + 1),
        bernoulli i * (↑((p + 1).choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p + 1 - i) / (↑(p + 1)! : ℚ)
this :
  Eq (α := ℚ⟦X⟧)
    (PowerSeries.mk fun (p : ℕ) ↦
      ∑ k ∈ range n,
        HMul.hMul (β := ℚ) (HPow.hPow (α := ℚ) (↑k : ℚ) p) ((algebraMap ℚ ℚ : (a : ℚ) → ℚ) (↑p ! : ℚ)⁻¹ : ℚ))
    (PowerSeries.mk fun (p : ℕ) ↦
      ∑ i ∈ range (p + 1),
        bernoulli i * (↑((p + 1).choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p + 1 - i) / (↑(p + 1)! : ℚ))
⊢ Eq (α := ℚ) (∑ i ∈ range n, HPow.hPow (α := ℚ) (↑i : ℚ) p * (↑p ! : ℚ)⁻¹)
    (∑ i ∈ range (p + 1),
      bernoulli i * (↑((p + 1).choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p + 1 - i) / (↑(p + 1)! : ℚ))
have this₁ : ∀ (n_1 : ℕ),
  Eq (α := ℚ)
    ((coeff n_1 : (a : ℚ⟦X⟧) → ℚ)
        (PowerSeries.mk fun (p : ℕ) ↦
          ∑ k ∈ range n,
            HMul.hMul (β := ℚ) (HPow.hPow (α := ℚ) (↑k : ℚ) p) ((algebraMap ℚ ℚ : (a : ℚ) → ℚ) (↑p ! : ℚ)⁻¹ : ℚ)) :
      ℚ)
    ((coeff n_1 : (a : ℚ⟦X⟧) → ℚ)
        (PowerSeries.mk fun (p : ℕ) ↦
          ∑ i ∈ range (p + 1),
            bernoulli i * (↑((p + 1).choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p + 1 - i) / (↑(p + 1)! : ℚ)) :
      ℚ)
Tactic: rw [PowerSeries.ext_iff] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.NumberTheory.Bernoulli (goal)
Goal:
n p : ℕ
hne : ∀ (m : ℕ), Ne (α := ℚ) (↑m ! : ℚ) 0
h_cauchy :
  HMul.hMul (α := ℚ⟦X⟧) (PowerSeries.mk fun (p : ℕ) ↦ bernoulli p / (↑p ! : ℚ))
      (PowerSeries.mk fun (q : ℕ) ↦ (coeff (q + 1) : (a : ℚ⟦X⟧) → ℚ) (exp ℚ ^ n)) =
    PowerSeries.mk fun (p : ℕ) ↦
      ∑ i ∈ range (p + 1),
        bernoulli i * (↑((p + 1).choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p + 1 - i) / (↑(p + 1)! : ℚ)
⊢ ∑ k ∈ range n, HPow.hPow (α := ℚ) (↑k : ℚ) p =
    HMul.hMul (α := ℚ)
      (∑ i ∈ range (p + 1),
        bernoulli i * (↑((p + 1).choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p + 1 - i) / (↑(p + 1)! : ℚ))
      (↑p ! : ℚ)
have fun : Eq (α := ℚ⟦X⟧)
  (PowerSeries.mk fun (p : ℕ) ↦
    ∑ k ∈ range n, HMul.hMul (β := ℚ) (HPow.hPow (α := ℚ) (↑k : ℚ) p) ((algebraMap ℚ ℚ : (a : ℚ) → ℚ) (↑p ! : ℚ)⁻¹ : ℚ))
  (PowerSeries.mk fun (p : ℕ) ↦
    ∑ i ∈ range (p + 1),
      bernoulli i * (↑((p + 1).choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p + 1 - i) / (↑(p + 1)! : ℚ))
Tactic: suffices
      (mk fun p => ∑ k ∈ range n, (k : ℚ) ^ p * algebraMap ℚ ℚ p !⁻¹) =
        mk fun p =>
          ∑ i ∈ range (p + 1), bernoulli i * (p + 1).choose i * (n : ℚ) ^ (p + 1 - i) / (p + 1)! by
      rw [← div_eq_iff (hne p), div_eq_mul_inv, sum_mul]
      rw [PowerSeries.ext_iff] at this
      simpa using this p
-- Lean 4 proof step from Mathlib.NumberTheory.Bernoulli (goal)
Goal:
n p : ℕ
hne : ∀ (m : ℕ), Ne (α := ℚ) (↑m ! : ℚ) 0
h_cauchy :
  HMul.hMul (α := ℚ⟦X⟧) (PowerSeries.mk fun (p : ℕ) ↦ bernoulli p / (↑p ! : ℚ))
      (PowerSeries.mk fun (q : ℕ) ↦ (coeff (q + 1) : (a : ℚ⟦X⟧) → ℚ) (exp ℚ ^ n)) =
    PowerSeries.mk fun (p : ℕ) ↦
      ∑ i ∈ range (p + 1),
        bernoulli i * (↑((p + 1).choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p + 1 - i) / (↑(p + 1)! : ℚ)
⊢ exp ℚ - 1 ≠ 0
have simp : ∃ (x : ℕ),
  ¬Eq (α := ℚ)
      ((coeff x : (a : ℚ⟦X⟧) → ℚ)
          (HSub.hSub (α := ℚ⟦X⟧)
            (PowerSeries.mk fun (n : ℕ) ↦ (algebraMap ℚ ℚ : (a : ℚ) → ℚ) (HDiv.hDiv (α := ℚ) 1 (↑n ! : ℚ))) 1) :
        ℚ)
      ((coeff x : (a : ℚ⟦X⟧) → ℚ) 0 : ℚ)
Tactic: simp only [exp, PowerSeries.ext_iff, Ne, not_forall]
-- Lean 4 proof step from Mathlib.NumberTheory.Bernoulli (goal)
Goal:
n p : ℕ
hne : ∀ (m : ℕ), Ne (α := ℚ) (↑m ! : ℚ) 0
h_cauchy :
  HMul.hMul (α := ℚ⟦X⟧) (PowerSeries.mk fun (p : ℕ) ↦ bernoulli p / (↑p ! : ℚ))
      (PowerSeries.mk fun (q : ℕ) ↦ (coeff (q + 1) : (a : ℚ⟦X⟧) → ℚ) (exp ℚ ^ n)) =
    PowerSeries.mk fun (p : ℕ) ↦
      ∑ i ∈ range (p + 1),
        bernoulli i * (↑((p + 1).choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p + 1 - i) / (↑(p + 1)! : ℚ)
⊢ ∃ (x : ℕ),
    ¬Eq (α := ℚ)
        ((coeff x : (a : ℚ⟦X⟧) → ℚ)
            (HSub.hSub (α := ℚ⟦X⟧)
              (PowerSeries.mk fun (n : ℕ) ↦ (algebraMap ℚ ℚ : (a : ℚ) → ℚ) (HDiv.hDiv (α := ℚ) 1 (↑n ! : ℚ))) 1) :
          ℚ)
        ((coeff x : (a : ℚ⟦X⟧) → ℚ) 0 : ℚ)
have h : ¬Eq (α := ℚ)
    ((coeff 1 : (a : ℚ⟦X⟧) → ℚ)
        (HSub.hSub (α := ℚ⟦X⟧)
          (PowerSeries.mk fun (n : ℕ) ↦ (algebraMap ℚ ℚ : (a : ℚ) → ℚ) (HDiv.hDiv (α := ℚ) 1 (↑n ! : ℚ))) 1) :
      ℚ)
    ((coeff 1 : (a : ℚ⟦X⟧) → ℚ) 0 : ℚ)
Tactic: use 1
-- Lean 4 proof step from Mathlib.NumberTheory.Bernoulli (hypothesis)
Goal:
n p : ℕ
hne : ∀ (m : ℕ), Ne (α := ℚ) (↑m ! : ℚ) 0
h_cauchy :
  HMul.hMul (α := ℚ⟦X⟧) (PowerSeries.mk fun (p : ℕ) ↦ bernoulli p / (↑p ! : ℚ))
      (PowerSeries.mk fun (q : ℕ) ↦ (coeff (q + 1) : (a : ℚ⟦X⟧) → ℚ) (exp ℚ ^ n)) =
    PowerSeries.mk fun (p : ℕ) ↦
      ∑ i ∈ range (p + 1),
        bernoulli i * (↑((p + 1).choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p + 1 - i) / (↑(p + 1)! : ℚ)
⊢ Eq (α := ℚ⟦X⟧)
    (PowerSeries.mk fun (p : ℕ) ↦
      ∑ k ∈ range n,
        HMul.hMul (β := ℚ) (HPow.hPow (α := ℚ) (↑k : ℚ) p) ((algebraMap ℚ ℚ : (a : ℚ) → ℚ) (↑p ! : ℚ)⁻¹ : ℚ))
    (PowerSeries.mk fun (p : ℕ) ↦
      ∑ i ∈ range (p + 1),
        bernoulli i * (↑((p + 1).choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p + 1 - i) / (↑(p + 1)! : ℚ))
have hexp : exp ℚ - 1 ≠ 0
Tactic: have hexp : exp ℚ - 1 ≠ 0 := by
      simp only [exp, PowerSeries.ext_iff, Ne, not_forall]
      use 1
      simp
-- Lean 4 proof step from Mathlib.NumberTheory.Bernoulli (hypothesis)
Goal:
n p : ℕ
hne : ∀ (m : ℕ), Ne (α := ℚ) (↑m ! : ℚ) 0
h_cauchy :
  HMul.hMul (α := ℚ⟦X⟧) (PowerSeries.mk fun (p : ℕ) ↦ bernoulli p / (↑p ! : ℚ))
      (PowerSeries.mk fun (q : ℕ) ↦ (coeff (q + 1) : (a : ℚ⟦X⟧) → ℚ) (exp ℚ ^ n)) =
    PowerSeries.mk fun (p : ℕ) ↦
      ∑ i ∈ range (p + 1),
        bernoulli i * (↑((p + 1).choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p + 1 - i) / (↑(p + 1)! : ℚ)
hexp : exp ℚ - 1 ≠ 0
⊢ exp ℚ ^ n - 1 = HMul.hMul (α := ℚ⟦X⟧) X (PowerSeries.mk fun (p : ℕ) ↦ (coeff (p + 1) : (a : ℚ⟦X⟧) → ℚ) (exp ℚ ^ n))
have h_const : Eq (α := ℚ⟦X⟧) ((C : (a : ℚ) → ℚ⟦X⟧) ((constantCoeff : (a : ℚ⟦X⟧) → ℚ) (exp ℚ ^ n) : ℚ) : ℚ⟦X⟧) (1 : ℚ⟦X⟧)
Tactic: have h_const : C (constantCoeff (exp ℚ ^ n)) = 1 := by simp
-- Lean 4 proof step from Mathlib.NumberTheory.Bernoulli (goal)
Goal:
n p : ℕ
hne : ∀ (m : ℕ), Ne (α := ℚ) (↑m ! : ℚ) 0
h_cauchy :
  HMul.hMul (α := ℚ⟦X⟧) (PowerSeries.mk fun (p : ℕ) ↦ bernoulli p / (↑p ! : ℚ))
      (PowerSeries.mk fun (q : ℕ) ↦ (coeff (q + 1) : (a : ℚ⟦X⟧) → ℚ) (exp ℚ ^ n)) =
    PowerSeries.mk fun (p : ℕ) ↦
      ∑ i ∈ range (p + 1),
        bernoulli i * (↑((p + 1).choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p + 1 - i) / (↑(p + 1)! : ℚ)
hexp : exp ℚ - 1 ≠ 0
h_const : Eq (α := ℚ⟦X⟧) ((C : (a : ℚ) → ℚ⟦X⟧) ((constantCoeff : (a : ℚ⟦X⟧) → ℚ) (exp ℚ ^ n) : ℚ) : ℚ⟦X⟧) (1 : ℚ⟦X⟧)
⊢ exp ℚ ^ n - 1 = HMul.hMul (α := ℚ⟦X⟧) X (PowerSeries.mk fun (p : ℕ) ↦ (coeff (p + 1) : (a : ℚ⟦X⟧) → ℚ) (exp ℚ ^ n))
have rw : HSub.hSub (β := ℚ⟦X⟧) (exp ℚ ^ n) ((C : (a : ℚ) → ℚ⟦X⟧) ((constantCoeff : (a : ℚ⟦X⟧) → ℚ) (exp ℚ ^ n) : ℚ) : ℚ⟦X⟧) =
  HMul.hMul (α := ℚ⟦X⟧) X (PowerSeries.mk fun (p : ℕ) ↦ (coeff (p + 1) : (a : ℚ⟦X⟧) → ℚ) (exp ℚ ^ n))
Tactic: rw [← h_const, sub_const_eq_X_mul_shift]