awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.NumberTheory.Bernoulli
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_r :
  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))
⊢ HMul.hMul (α := ℚ⟦X⟧) (PowerSeries.mk fun (n : ℕ) ↦ (algebraMap ℚ ℚ : (a : ℚ) → ℚ) (bernoulli n / (↑n ! : ℚ)))
      (PowerSeries.mk fun (p : ℕ) ↦ (coeff (p + 1) : (a : ℚ⟦X⟧) → ℚ) (exp ℚ ^ n)) =
    PowerSeries.mk fun (p : ℕ) ↦
      ∑ x ∈ range (p + 1),
        HPow.hPow (α := ℚ) (↑n : ℚ) (p + 1 - x) * (bernoulli x * (↑((p + 1).choose x) : ℚ)) / (↑(p + 1)! : ℚ)

Tactic:
Eq.trans (mul_eq_mul_right_iff.mpr ?_) (Eq.trans h_cauchy ?_)

Premises used: Iff.mpr, mul_eq_mul_right_iff, Eq.trans
-- Module: Mathlib.NumberTheory.Bernoulli
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_r :
  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))
refine_1 :
  Eq (α := ℚ⟦X⟧) (PowerSeries.mk fun (n : ℕ) ↦ (algebraMap ℚ ℚ : (a : ℚ) → ℚ) (bernoulli n / (↑n ! : ℚ)))
      (PowerSeries.mk fun (p : ℕ) ↦ bernoulli p / (↑p ! : ℚ)) ∨
    Eq (α := ℚ⟦X⟧) (PowerSeries.mk fun (p : ℕ) ↦ (coeff (p + 1) : (a : ℚ⟦X⟧) → ℚ) (exp ℚ ^ n)) 0
refine_2 :
  Eq (α := ℚ⟦X⟧)
    (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))
    (PowerSeries.mk fun (p : ℕ) ↦
      ∑ x ∈ range (p + 1),
        HPow.hPow (α := ℚ) (↑n : ℚ) (p + 1 - x) * (bernoulli x * (↑((p + 1).choose x) : ℚ)) / (↑(p + 1)! : ℚ))
⊢ HMul.hMul (α := ℚ⟦X⟧) (PowerSeries.mk fun (n : ℕ) ↦ (algebraMap ℚ ℚ : (a : ℚ) → ℚ) (bernoulli n / (↑n ! : ℚ)))
      (PowerSeries.mk fun (p : ℕ) ↦ (coeff (p + 1) : (a : ℚ⟦X⟧) → ℚ) (exp ℚ ^ n)) =
    PowerSeries.mk fun (p : ℕ) ↦
      ∑ x ∈ range (p + 1),
        HPow.hPow (α := ℚ) (↑n : ℚ) (p + 1 - x) * (bernoulli x * (↑((p + 1).choose x) : ℚ)) / (↑(p + 1)! : ℚ)

Tactic:
refine Eq.trans (mul_eq_mul_right_iff.mpr ?_) (Eq.trans h_cauchy ?_)

Premises used: Iff.mpr, mul_eq_mul_right_iff, Eq.trans
-- Module: Mathlib.NumberTheory.Bernoulli
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_r :
  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))
refine_1 :
  Eq (α := ℚ⟦X⟧) (PowerSeries.mk fun (n : ℕ) ↦ (algebraMap ℚ ℚ : (a : ℚ) → ℚ) (bernoulli n / (↑n ! : ℚ)))
    (PowerSeries.mk fun (p : ℕ) ↦ bernoulli p / (↑p ! : ℚ))
⊢ Eq (α := ℚ⟦X⟧) (PowerSeries.mk fun (n : ℕ) ↦ (algebraMap ℚ ℚ : (a : ℚ) → ℚ) (bernoulli n / (↑n ! : ℚ)))
      (PowerSeries.mk fun (p : ℕ) ↦ bernoulli p / (↑p ! : ℚ)) ∨
    Eq (α := ℚ⟦X⟧) (PowerSeries.mk fun (p : ℕ) ↦ (coeff (p + 1) : (a : ℚ⟦X⟧) → ℚ) (exp ℚ ^ n)) 0

Tactic:
left
-- Module: Mathlib.NumberTheory.Bernoulli
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_r :
  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))
⊢ Eq (α := ℚ⟦X⟧) (PowerSeries.mk fun (n : ℕ) ↦ (algebraMap ℚ ℚ : (a : ℚ) → ℚ) (bernoulli n / (↑n ! : ℚ)))
    (PowerSeries.mk fun (p : ℕ) ↦ bernoulli p / (↑p ! : ℚ))

Tactic:
congr
-- Module: Mathlib.NumberTheory.Bernoulli
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_r :
  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))
⊢ Eq (α := ℚ⟦X⟧)
    (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))
    (PowerSeries.mk fun (p : ℕ) ↦
      ∑ x ∈ range (p + 1),
        HPow.hPow (α := ℚ) (↑n : ℚ) (p + 1 - x) * (bernoulli x * (↑((p + 1).choose x) : ℚ)) / (↑(p + 1)! : ℚ))

Tactic:
simp only [mul_comm, factorial]

Premises used: Nat.factorial, mul_comm
-- Module: Mathlib.NumberTheory.Bernoulli
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)

Tactic:
have hps :
    (∑ k ∈ range n, (k : ℚ) ^ p) =
      (∑ i ∈ range (p + 1),
          bernoulli i * (p + 1).choose i * (n : ℚ) ^ (p + 1 - i) / (p + 1)!) * p ! := by
    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
    -- the power series `exp ℚ - 1` is non-zero, a fact we need in order to use `mul_right_inj'`
    have hexp : exp ℚ - 1 ≠ 0 := by
      simp only [exp, PowerSeries.ext_iff, Ne, not_forall]
      use 1
      simp
    have h_r : exp ℚ ^ n - 1 = X * mk fun p => coeff (p + 1) (exp ℚ ^ n) := by
      have h_const : C (constantCoeff (exp ℚ ^ n)) = 1 := by simp
      rw [← h_const, sub_const_eq_X_mul_shift]
    -- key step: a chain of equalities of power series
    rw [← mul_right_inj' hexp, mul_comm]
    rw [← exp_pow_sum, geom_sum_mul, h_r, ← bernoulliPowerSeries_mul_exp_sub_one,
      bernoulliPowerSeries, mul_right_comm]
    simp only [mul_comm, mul_eq_mul_left_iff, hexp, or_false]
    refine Eq.trans (mul_eq_mul_right_iff.mpr ?_) (Eq.trans h_cauchy ?_)
    · left
      congr
    · simp only [mul_comm, factorial]
-- Module: Mathlib.NumberTheory.Bernoulli
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)
hps :
  ∑ 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)
rw :
  Eq (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))
    (∑ 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)
⊢ 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)

Tactic:
rw [hps, sum_mul]

Premises used: Finset.sum_mul
-- Module: Mathlib.NumberTheory.Bernoulli
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)
hps :
  ∑ 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)
⊢ Eq (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))
    (∑ 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)

Tactic:
sum_congr rfl fun x _ => ?_

Premises used: Finset.sum_congr, rfl
-- Module: Mathlib.NumberTheory.Bernoulli
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)
hps :
  ∑ 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)
refine :
  ∀ x ∈ range (p + 1),
    bernoulli x * (↑((p + 1).choose x) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p + 1 - x) / (↑(p + 1)! : ℚ) * (↑p ! : ℚ) =
      bernoulli x * (↑((p + 1).choose x) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p + 1 - x) / HAdd.hAdd (α := ℚ) (↑p : ℚ) 1
⊢ Eq (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))
    (∑ 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)

Tactic:
refine sum_congr rfl fun x _ => ?_

Premises used: Finset.sum_congr, rfl
-- Module: Mathlib.NumberTheory.Bernoulli
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)
hps :
  ∑ 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)
x : ℕ
h : x ∈ range (p + 1)
⊢ bernoulli x * (↑((p + 1).choose x) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p + 1 - x) / (↑(p + 1)! : ℚ) * (↑p ! : ℚ) =
    bernoulli x * (↑((p + 1).choose x) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p + 1 - x) / HAdd.hAdd (α := ℚ) (↑p : ℚ) 1

Tactic:
simp [field, factorial]

Premises used: Nat.factorial
-- Module: Mathlib.NumberTheory.Bernoulli
Goal:
n p : ℕ
rw :
  Eq (α := ℚ) (∑ k ∈ Ico 1 (n + 1), 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) / (↑p.succ : ℚ))
⊢ Eq (α := ℚ) (∑ k ∈ Ico 1 (n + 1), 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)

Tactic:
rw [← Nat.cast_succ]

Premises used: Nat.cast_succ
-- Module: Mathlib.NumberTheory.Bernoulli
Goal:
n : ℕ
⊢ Eq (α := ℚ) (∑ k ∈ Ico 1 (n + 1), HPow.hPow (α := ℚ) (β := ℕ) (↑k : ℚ) 0)
    (∑ i ∈ range (HAdd.hAdd (α := ℕ) 0 1),
      bernoulli' i * (↑((HAdd.hAdd (α := ℕ) 0 1).choose i) : ℚ) *
          HPow.hPow (α := ℚ) (↑n : ℚ) (HAdd.hAdd (α := ℕ) 0 1 - i) /
        (↑(succ 0) : ℚ))

Tactic:
simp
-- Module: Mathlib.NumberTheory.Bernoulli
Goal:
n p : ℕ
f : (i : ℕ) → ℚ :=
  fun (i : ℕ) ↦
    bernoulli i * (↑(p.succ.succ.choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p.succ.succ - i) / (↑p.succ.succ : ℚ)
f' : (i : ℕ) → ℚ :=
  fun (i : ℕ) ↦
    bernoulli' i * (↑(p.succ.succ.choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p.succ.succ - i) / (↑p.succ.succ : ℚ)
this : Eq (α := ℚ) (∑ k ∈ Ico 1 n.succ, HPow.hPow (α := ℚ) (↑k : ℚ) p.succ) (∑ i ∈ range p.succ.succ, f' i)
⊢ Eq (α := ℚ) (∑ k ∈ Ico 1 n.succ, HPow.hPow (α := ℚ) (↑k : ℚ) p.succ) (∑ i ∈ range p.succ.succ, f' i)

Tactic:
this
-- Module: Mathlib.NumberTheory.Bernoulli
Goal:
n p : ℕ
f : (i : ℕ) → ℚ :=
  fun (i : ℕ) ↦
    bernoulli i * (↑(p.succ.succ.choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p.succ.succ - i) / (↑p.succ.succ : ℚ)
f' : (i : ℕ) → ℚ :=
  fun (i : ℕ) ↦
    bernoulli' i * (↑(p.succ.succ.choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p.succ.succ - i) / (↑p.succ.succ : ℚ)
this : Eq (α := ℚ) (∑ k ∈ Ico 1 n.succ, HPow.hPow (α := ℚ) (↑k : ℚ) p.succ) (∑ i ∈ range p.succ.succ, f' i)
⊢ Eq (α := ℚ) (∑ k ∈ Ico 1 (n + 1), HPow.hPow (α := ℚ) (↑k : ℚ) (p + 1))
    (∑ i ∈ range (p + 1 + 1),
      bernoulli' i * (↑((p + 1 + 1).choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p + 1 + 1 - i) / (↑(p + 1).succ : ℚ))

Tactic:
convert!
    this
-- Module: Mathlib.NumberTheory.Bernoulli
Goal:
n p : ℕ
f : (i : ℕ) → ℚ :=
  fun (i : ℕ) ↦
    bernoulli i * (↑(p.succ.succ.choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p.succ.succ - i) / (↑p.succ.succ : ℚ)
f' : (i : ℕ) → ℚ :=
  fun (i : ℕ) ↦
    bernoulli' i * (↑(p.succ.succ.choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p.succ.succ - i) / (↑p.succ.succ : ℚ)
succ : Eq (α := ℚ) (∑ k ∈ Ico 1 n.succ, HPow.hPow (α := ℚ) (↑k : ℚ) p.succ) (∑ i ∈ range p.succ.succ, f' i)
⊢ Eq (α := ℚ) (∑ k ∈ Ico 1 (n + 1), HPow.hPow (α := ℚ) (↑k : ℚ) (p + 1))
    (∑ i ∈ range (p + 1 + 1),
      bernoulli' i * (↑((p + 1 + 1).choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p + 1 + 1 - i) / (↑(p + 1).succ : ℚ))

Tactic:
suffices (∑ k ∈ Ico 1 n.succ, (k : ℚ) ^ p.succ) = ∑ i ∈ range p.succ.succ, f' i by convert!
    this
-- Module: Mathlib.NumberTheory.Bernoulli
Goal:
n p : ℕ
f : (i : ℕ) → ℚ :=
  fun (i : ℕ) ↦
    bernoulli i * (↑(p.succ.succ.choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p.succ.succ - i) / (↑p.succ.succ : ℚ)
f' : (i : ℕ) → ℚ :=
  fun (i : ℕ) ↦
    bernoulli' i * (↑(p.succ.succ.choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p.succ.succ - i) / (↑p.succ.succ : ℚ)
⊢ 1 ≤ n + 1

Tactic:
have hle := Nat.le_add_left 1 n
-- Module: Mathlib.NumberTheory.Bernoulli
Goal:
n p : ℕ
f : (i : ℕ) → ℚ :=
  fun (i : ℕ) ↦
    bernoulli i * (↑(p.succ.succ.choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p.succ.succ - i) / (↑p.succ.succ : ℚ)
f' : (i : ℕ) → ℚ :=
  fun (i : ℕ) ↦
    bernoulli' i * (↑(p.succ.succ.choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p.succ.succ - i) / (↑p.succ.succ : ℚ)
hle : 1 ≤ n + 1
norm_cast norm_cast₁ norm_cast₂ norm_cast₃ : ¬False
⊢ HAdd.hAdd (α := ℚ) (↑p : ℚ) 1 + 1 ≠ 0

Tactic:
norm_cast
-- Module: Mathlib.NumberTheory.Bernoulli
Goal:
n p : ℕ
f : (i : ℕ) → ℚ :=
  fun (i : ℕ) ↦
    bernoulli i * (↑(p.succ.succ.choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p.succ.succ - i) / (↑p.succ.succ : ℚ)
f' : (i : ℕ) → ℚ :=
  fun (i : ℕ) ↦
    bernoulli' i * (↑(p.succ.succ.choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p.succ.succ - i) / (↑p.succ.succ : ℚ)
hle : 1 ≤ n + 1
⊢ HAdd.hAdd (α := ℚ) (↑p : ℚ) 1 + 1 ≠ 0

Tactic:
have hne : (p + 1 + 1 : ℚ) ≠ 0 := by norm_cast
-- Module: Mathlib.NumberTheory.Bernoulli
Goal:
n p : ℕ
f : (i : ℕ) → ℚ :=
  fun (i : ℕ) ↦
    bernoulli i * (↑(p.succ.succ.choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p.succ.succ - i) / (↑p.succ.succ : ℚ)
f' : (i : ℕ) → ℚ :=
  fun (i : ℕ) ↦
    bernoulli' i * (↑(p.succ.succ.choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p.succ.succ - i) / (↑p.succ.succ : ℚ)
hle : 1 ≤ n + 1
hne : HAdd.hAdd (α := ℚ) (↑p : ℚ) 1 + 1 ≠ 0
r : ℚ
rw :
  r * (HAdd.hAdd (α := ℚ) (↑p : ℚ) 1 + 1) / (HAdd.hAdd (α := ℚ) (↑p : ℚ) 1 + 1) * HPow.hPow (α := ℚ) (↑n : ℚ) p.succ =
    r * HPow.hPow (α := ℚ) (↑n : ℚ) p.succ
rw₁ rw₂ : r * HPow.hPow (α := ℚ) (↑n : ℚ) p.succ = r * HPow.hPow (α := ℚ) (↑n : ℚ) p.succ
⊢ r * (HAdd.hAdd (α := ℚ) (↑p : ℚ) 1 + 1) * HPow.hPow (α := ℚ) (↑n : ℚ) p.succ / (HAdd.hAdd (α := ℚ) (↑p : ℚ) 1 + 1) =
    r * HPow.hPow (α := ℚ) (↑n : ℚ) p.succ

Tactic:
rw [mul_div_right_comm, mul_div_cancel_right₀ _ hne]

Premises used: mul_div_right_comm, mul_div_cancel_right₀
-- Module: Mathlib.NumberTheory.Bernoulli
Goal:
n p : ℕ
f : (i : ℕ) → ℚ :=
  fun (i : ℕ) ↦
    bernoulli i * (↑(p.succ.succ.choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p.succ.succ - i) / (↑p.succ.succ : ℚ)
f' : (i : ℕ) → ℚ :=
  fun (i : ℕ) ↦
    bernoulli' i * (↑(p.succ.succ.choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p.succ.succ - i) / (↑p.succ.succ : ℚ)
hle : 1 ≤ n + 1
hne : HAdd.hAdd (α := ℚ) (↑p : ℚ) 1 + 1 ≠ 0
⊢ ∀ (r : ℚ),
    r * (HAdd.hAdd (α := ℚ) (↑p : ℚ) 1 + 1) * HPow.hPow (α := ℚ) (↑n : ℚ) p.succ / (HAdd.hAdd (α := ℚ) (↑p : ℚ) 1 + 1) =
      r * HPow.hPow (α := ℚ) (↑n : ℚ) p.succ

Tactic:
have h1 : ∀ r : ℚ, r * (p + 1 + 1) * (n : ℚ) ^ p.succ / (p + 1 + 1 : ℚ) = r * (n : ℚ) ^ p.succ :=
      fun r => by rw [mul_div_right_comm, mul_div_cancel_right₀ _ hne]
-- Module: Mathlib.NumberTheory.Bernoulli
Goal:
n p : ℕ
f : (i : ℕ) → ℚ :=
  fun (i : ℕ) ↦
    bernoulli i * (↑(p.succ.succ.choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p.succ.succ - i) / (↑p.succ.succ : ℚ)
f' : (i : ℕ) → ℚ :=
  fun (i : ℕ) ↦
    bernoulli' i * (↑(p.succ.succ.choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p.succ.succ - i) / (↑p.succ.succ : ℚ)
hle : 1 ≤ n + 1
hne : HAdd.hAdd (α := ℚ) (↑p : ℚ) 1 + 1 ≠ 0
h1 :
  ∀ (r : ℚ),
    r * (HAdd.hAdd (α := ℚ) (↑p : ℚ) 1 + 1) * HPow.hPow (α := ℚ) (↑n : ℚ) p.succ / (HAdd.hAdd (α := ℚ) (↑p : ℚ) 1 + 1) =
      r * HPow.hPow (α := ℚ) (↑n : ℚ) p.succ
simp_rw :
  -1 / 2 * HPow.hPow (α := ℚ) (↑n : ℚ) p.succ + HPow.hPow (α := ℚ) (↑n : ℚ) p.succ =
    1 / 2 * HPow.hPow (α := ℚ) (↑n : ℚ) p.succ
⊢ f 1 + HPow.hPow (α := ℚ) (↑n : ℚ) p.succ = 1 / 2 * HPow.hPow (α := ℚ) (↑n : ℚ) p.succ

Tactic:
simp_rw [f, bernoulli_one, choose_one_right, succ_sub_succ_eq_sub, cast_succ, tsub_zero, h1]

Premises used: Nat.cast_succ, Nat.choose_one_right, tsub_zero, bernoulli_one, Nat.succ_sub_succ_eq_sub
-- Module: Mathlib.NumberTheory.Bernoulli
Goal:
n p : ℕ
f : (i : ℕ) → ℚ :=
  fun (i : ℕ) ↦
    bernoulli i * (↑(p.succ.succ.choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p.succ.succ - i) / (↑p.succ.succ : ℚ)
f' : (i : ℕ) → ℚ :=
  fun (i : ℕ) ↦
    bernoulli' i * (↑(p.succ.succ.choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p.succ.succ - i) / (↑p.succ.succ : ℚ)
hle : 1 ≤ n + 1
hne : HAdd.hAdd (α := ℚ) (↑p : ℚ) 1 + 1 ≠ 0
h1 :
  ∀ (r : ℚ),
    r * (HAdd.hAdd (α := ℚ) (↑p : ℚ) 1 + 1) * HPow.hPow (α := ℚ) (↑n : ℚ) p.succ / (HAdd.hAdd (α := ℚ) (↑p : ℚ) 1 + 1) =
      r * HPow.hPow (α := ℚ) (↑n : ℚ) p.succ
⊢ -1 / 2 * HPow.hPow (α := ℚ) (↑n : ℚ) p.succ + HPow.hPow (α := ℚ) (↑n : ℚ) p.succ =
    1 / 2 * HPow.hPow (α := ℚ) (↑n : ℚ) p.succ

Tactic:
ring
-- Module: Mathlib.NumberTheory.Bernoulli
Goal:
n p : ℕ
f : (i : ℕ) → ℚ :=
  fun (i : ℕ) ↦
    bernoulli i * (↑(p.succ.succ.choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p.succ.succ - i) / (↑p.succ.succ : ℚ)
f' : (i : ℕ) → ℚ :=
  fun (i : ℕ) ↦
    bernoulli' i * (↑(p.succ.succ.choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p.succ.succ - i) / (↑p.succ.succ : ℚ)
hle : 1 ≤ n + 1
hne : HAdd.hAdd (α := ℚ) (↑p : ℚ) 1 + 1 ≠ 0
h1 :
  ∀ (r : ℚ),
    r * (HAdd.hAdd (α := ℚ) (↑p : ℚ) 1 + 1) * HPow.hPow (α := ℚ) (↑n : ℚ) p.succ / (HAdd.hAdd (α := ℚ) (↑p : ℚ) 1 + 1) =
      r * HPow.hPow (α := ℚ) (↑n : ℚ) p.succ
⊢ f 1 + HPow.hPow (α := ℚ) (↑n : ℚ) p.succ = 1 / 2 * HPow.hPow (α := ℚ) (↑n : ℚ) p.succ

Tactic:
have h2 : f 1 + (n : ℚ) ^ p.succ = 1 / 2 * (n : ℚ) ^ p.succ := by
    simp_rw [f, bernoulli_one, choose_one_right, succ_sub_succ_eq_sub, cast_succ, tsub_zero, h1]
    ring
-- Module: Mathlib.NumberTheory.Bernoulli
Goal:
n p : ℕ
f : (i : ℕ) → ℚ :=
  fun (i : ℕ) ↦
    bernoulli i * (↑(p.succ.succ.choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p.succ.succ - i) / (↑p.succ.succ : ℚ)
f' : (i : ℕ) → ℚ :=
  fun (i : ℕ) ↦
    bernoulli' i * (↑(p.succ.succ.choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p.succ.succ - i) / (↑p.succ.succ : ℚ)
hle : 1 ≤ n + 1
hne : HAdd.hAdd (α := ℚ) (↑p : ℚ) 1 + 1 ≠ 0
h1 :
  ∀ (r : ℚ),
    r * (HAdd.hAdd (α := ℚ) (↑p : ℚ) 1 + 1) * HPow.hPow (α := ℚ) (↑n : ℚ) p.succ / (HAdd.hAdd (α := ℚ) (↑p : ℚ) 1 + 1) =
      r * HPow.hPow (α := ℚ) (↑n : ℚ) p.succ
h2 : f 1 + HPow.hPow (α := ℚ) (↑n : ℚ) p.succ = 1 / 2 * HPow.hPow (α := ℚ) (↑n : ℚ) p.succ
i : ℕ
h : i ∈ range p
rw rw₁ :
  bernoulli' i.succ.succ * (↑((p + 2).choose (i + 2)) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p - i) / (↑(p + 2) : ℚ) =
    bernoulli' (i + 2) * (↑((p + 2).choose (i + 2)) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p - i) / (↑(p + 2) : ℚ)
⊢ bernoulli (i + 2) * (↑((p + 2).choose (i + 2)) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p - i) / (↑(p + 2) : ℚ) =
    bernoulli' (i + 2) * (↑((p + 2).choose (i + 2)) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p - i) / (↑(p + 2) : ℚ)

Tactic:
rw [bernoulli_eq_bernoulli'_of_ne_one (succ_succ_ne_one i)]

Premises used: Nat.succ_succ_ne_one, bernoulli_eq_bernoulli'_of_ne_one
-- Module: Mathlib.NumberTheory.Bernoulli
Goal:
n p : ℕ
f : (i : ℕ) → ℚ :=
  fun (i : ℕ) ↦
    bernoulli i * (↑(p.succ.succ.choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p.succ.succ - i) / (↑p.succ.succ : ℚ)
f' : (i : ℕ) → ℚ :=
  fun (i : ℕ) ↦
    bernoulli' i * (↑(p.succ.succ.choose i) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p.succ.succ - i) / (↑p.succ.succ : ℚ)
hle : 1 ≤ n + 1
hne : HAdd.hAdd (α := ℚ) (↑p : ℚ) 1 + 1 ≠ 0
h1 :
  ∀ (r : ℚ),
    r * (HAdd.hAdd (α := ℚ) (↑p : ℚ) 1 + 1) * HPow.hPow (α := ℚ) (↑n : ℚ) p.succ / (HAdd.hAdd (α := ℚ) (↑p : ℚ) 1 + 1) =
      r * HPow.hPow (α := ℚ) (↑n : ℚ) p.succ
h2 : f 1 + HPow.hPow (α := ℚ) (↑n : ℚ) p.succ = 1 / 2 * HPow.hPow (α := ℚ) (↑n : ℚ) p.succ
⊢ Eq (α := ℚ)
    (∑ i ∈ range p,
      bernoulli (i + 2) * (↑((p + 2).choose (i + 2)) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p - i) / (↑(p + 2) : ℚ))
    (∑ i ∈ range p,
      bernoulli' (i + 2) * (↑((p + 2).choose (i + 2)) : ℚ) * HPow.hPow (α := ℚ) (↑n : ℚ) (p - i) / (↑(p + 2) : ℚ))

Tactic:
have :
    (∑ i ∈ range p, bernoulli (i + 2) * (p + 2).choose (i + 2) * (n : ℚ) ^ (p - i) / ↑(p + 2)) =
      ∑ i ∈ range p, bernoulli' (i + 2) * (p + 2).choose (i + 2) * (n : ℚ) ^ (p - i) / ↑(p + 2) :=
    sum_congr rfl fun i _ => by rw [bernoulli_eq_bernoulli'_of_ne_one (succ_succ_ne_one i)]