Theorem: sum_Ico_pow
File: Mathlib/NumberTheory/Bernoulli.lean

Proof steps:
state_before: A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n p : ℕ
⊢ ∑ k ∈ Ico 1 (n + 1), ↑k ^ p = ∑ i ∈ range (p + 1), bernoulli' i * ↑((p + 1).choose i) * ↑n ^ (p + 1 - i) / (↑p + 1)
state_after: A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n p : ℕ
⊢ ∑ k ∈ Ico 1 (n + 1), ↑k ^ p = ∑ i ∈ range (p + 1), bernoulli' i * ↑((p + 1).choose i) * ↑n ^ (p + 1 - i) / ↑p.succ
tactic: rw [← Nat.cast_succ]
state_before: A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n p : ℕ
⊢ ∑ k ∈ Ico 1 (n + 1), ↑k ^ p = ∑ i ∈ range (p + 1), bernoulli' i * ↑((p + 1).choose i) * ↑n ^ (p + 1 - i) / ↑p.succ
state_after: case zero
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
⊢ ∑ k ∈ Ico 1 (n + 1), ↑k ^ 0 = ∑ i ∈ range (0 + 1), bernoulli' i * ↑((0 + 1).choose i) * ↑n ^ (0 + 1 - i) / ↑(succ 0)

case succ
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n p : ℕ
⊢ ∑ k ∈ Ico 1 (n + 1), ↑k ^ (p + 1) =
    ∑ i ∈ range (p + 1 + 1), bernoulli' i * ↑((p + 1 + 1).choose i) * ↑n ^ (p + 1 + 1 - i) / ↑(p + 1).succ
tactic: cases' p with p
state_before: case succ
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n p : ℕ
⊢ ∑ k ∈ Ico 1 (n + 1), ↑k ^ (p + 1) =
    ∑ i ∈ range (p + 1 + 1), bernoulli' i * ↑((p + 1 + 1).choose i) * ↑n ^ (p + 1 + 1 - i) / ↑(p + 1).succ
state_after: case succ
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n p : ℕ
f : ℕ → ℚ := fun i => bernoulli i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ
⊢ ∑ k ∈ Ico 1 (n + 1), ↑k ^ (p + 1) =
    ∑ i ∈ range (p + 1 + 1), bernoulli' i * ↑((p + 1 + 1).choose i) * ↑n ^ (p + 1 + 1 - i) / ↑(p + 1).succ
tactic: let f i := bernoulli i * p.succ.succ.choose i * (n : ℚ) ^ (p.succ.succ - i) / p.succ.succ
state_before: case succ
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n p : ℕ
f : ℕ → ℚ := fun i => bernoulli i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ
⊢ ∑ k ∈ Ico 1 (n + 1), ↑k ^ (p + 1) =
    ∑ i ∈ range (p + 1 + 1), bernoulli' i * ↑((p + 1 + 1).choose i) * ↑n ^ (p + 1 + 1 - i) / ↑(p + 1).succ
state_after: case succ
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n p : ℕ
f : ℕ → ℚ := fun i => bernoulli i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ
f' : ℕ → ℚ := fun i => bernoulli' i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ
⊢ ∑ k ∈ Ico 1 (n + 1), ↑k ^ (p + 1) =
    ∑ i ∈ range (p + 1 + 1), bernoulli' i * ↑((p + 1 + 1).choose i) * ↑n ^ (p + 1 + 1 - i) / ↑(p + 1).succ
tactic: let f' i := bernoulli' i * p.succ.succ.choose i * (n : ℚ) ^ (p.succ.succ - i) / p.succ.succ
state_before: case succ
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n p : ℕ
f : ℕ → ℚ := fun i => bernoulli i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ
f' : ℕ → ℚ := fun i => bernoulli' i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ
⊢ ∑ k ∈ Ico 1 (n + 1), ↑k ^ (p + 1) =
    ∑ i ∈ range (p + 1 + 1), bernoulli' i * ↑((p + 1 + 1).choose i) * ↑n ^ (p + 1 + 1 - i) / ↑(p + 1).succ
state_after: case succ
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n p : ℕ
f : ℕ → ℚ := fun i => bernoulli i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ
f' : ℕ → ℚ := fun i => bernoulli' i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ
⊢ ∑ k ∈ Ico 1 n.succ, ↑k ^ p.succ = ∑ i ∈ range p.succ.succ, f' i
tactic: suffices (∑ k ∈ Ico 1 n.succ, (k : ℚ) ^ p.succ) = ∑ i ∈ range p.succ.succ, f' i by convert this
state_before: case succ
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n p : ℕ
f : ℕ → ℚ := fun i => bernoulli i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ
f' : ℕ → ℚ := fun i => bernoulli' i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ
⊢ ∑ k ∈ Ico 1 n.succ, ↑k ^ p.succ = ∑ i ∈ range p.succ.succ, f' i
state_after: case succ
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n p : ℕ
f : ℕ → ℚ := fun i => bernoulli i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ
f' : ℕ → ℚ := fun i => bernoulli' i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ
hle : 1 ≤ n + 1
⊢ ∑ k ∈ Ico 1 n.succ, ↑k ^ p.succ = ∑ i ∈ range p.succ.succ, f' i
tactic: have hle := Nat.le_add_left 1 n
state_before: case succ
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n p : ℕ
f : ℕ → ℚ := fun i => bernoulli i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ
f' : ℕ → ℚ := fun i => bernoulli' i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ
hle : 1 ≤ n + 1
⊢ ∑ k ∈ Ico 1 n.succ, ↑k ^ p.succ = ∑ i ∈ range p.succ.succ, f' i
state_after: case succ
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n p : ℕ
f : ℕ → ℚ := fun i => bernoulli i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ
f' : ℕ → ℚ := fun i => bernoulli' i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ
hle : 1 ≤ n + 1
hne : ↑p + 1 + 1 ≠ 0
⊢ ∑ k ∈ Ico 1 n.succ, ↑k ^ p.succ = ∑ i ∈ range p.succ.succ, f' i
tactic: have hne : (p + 1 + 1 : ℚ) ≠ 0 := by norm_cast
state_before: case succ
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n p : ℕ
f : ℕ → ℚ := fun i => bernoulli i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ
f' : ℕ → ℚ := fun i => bernoulli' i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ
hle : 1 ≤ n + 1
hne : ↑p + 1 + 1 ≠ 0
⊢ ∑ k ∈ Ico 1 n.succ, ↑k ^ p.succ = ∑ i ∈ range p.succ.succ, f' i
state_after: case succ
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n p : ℕ
f : ℕ → ℚ := fun i => bernoulli i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ
f' : ℕ → ℚ := fun i => bernoulli' i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ
hle : 1 ≤ n + 1
hne : ↑p + 1 + 1 ≠ 0
h1 : ∀ (r : ℚ), r * (↑p + 1 + 1) * ↑n ^ p.succ / (↑p + 1 + 1) = r * ↑n ^ p.succ
⊢ ∑ k ∈ Ico 1 n.succ, ↑k ^ p.succ = ∑ i ∈ range p.succ.succ, f' i
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]
state_before: case succ
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n p : ℕ
f : ℕ → ℚ := fun i => bernoulli i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ
f' : ℕ → ℚ := fun i => bernoulli' i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ
hle : 1 ≤ n + 1
hne : ↑p + 1 + 1 ≠ 0
h1 : ∀ (r : ℚ), r * (↑p + 1 + 1) * ↑n ^ p.succ / (↑p + 1 + 1) = r * ↑n ^ p.succ
⊢ ∑ k ∈ Ico 1 n.succ, ↑k ^ p.succ = ∑ i ∈ range p.succ.succ, f' i
state_after: case succ
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n p : ℕ
f : ℕ → ℚ := fun i => bernoulli i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ
f' : ℕ → ℚ := fun i => bernoulli' i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ
hle : 1 ≤ n + 1
hne : ↑p + 1 + 1 ≠ 0
h1 : ∀ (r : ℚ), r * (↑p + 1 + 1) * ↑n ^ p.succ / (↑p + 1 + 1) = r * ↑n ^ p.succ
h2 : f 1 + ↑n ^ p.succ = 1 / 2 * ↑n ^ p.succ
⊢ ∑ k ∈ Ico 1 n.succ, ↑k ^ p.succ = ∑ i ∈ range p.succ.succ, f' i
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
state_before: case succ
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n p : ℕ
f : ℕ → ℚ := fun i => bernoulli i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ
f' : ℕ → ℚ := fun i => bernoulli' i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ
hle : 1 ≤ n + 1
hne : ↑p + 1 + 1 ≠ 0
h1 : ∀ (r : ℚ), r * (↑p + 1 + 1) * ↑n ^ p.succ / (↑p + 1 + 1) = r * ↑n ^ p.succ
h2 : f 1 + ↑n ^ p.succ = 1 / 2 * ↑n ^ p.succ
⊢ ∑ k ∈ Ico 1 n.succ, ↑k ^ p.succ = ∑ i ∈ range p.succ.succ, f' i
state_after: case succ
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n p : ℕ
f : ℕ → ℚ := fun i => bernoulli i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ
f' : ℕ → ℚ := fun i => bernoulli' i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ
hle : 1 ≤ n + 1
hne : ↑p + 1 + 1 ≠ 0
h1 : ∀ (r : ℚ), r * (↑p + 1 + 1) * ↑n ^ p.succ / (↑p + 1 + 1) = r * ↑n ^ p.succ
h2 : f 1 + ↑n ^ p.succ = 1 / 2 * ↑n ^ p.succ
this :
  ∑ i ∈ range p, bernoulli (i + 2) * ↑((p + 2).choose (i + 2)) * ↑n ^ (p - i) / ↑(p + 2) =
    ∑ i ∈ range p, bernoulli' (i + 2) * ↑((p + 2).choose (i + 2)) * ↑n ^ (p - i) / ↑(p + 2)
⊢ ∑ k ∈ Ico 1 n.succ, ↑k ^ p.succ = ∑ i ∈ range p.succ.succ, f' i
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)]
state_before: case succ
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n p : ℕ
f : ℕ → ℚ := fun i => bernoulli i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ
f' : ℕ → ℚ := fun i => bernoulli' i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ
hle : 1 ≤ n + 1
hne : ↑p + 1 + 1 ≠ 0
h1 : ∀ (r : ℚ), r * (↑p + 1 + 1) * ↑n ^ p.succ / (↑p + 1 + 1) = r * ↑n ^ p.succ
h2 : f 1 + ↑n ^ p.succ = 1 / 2 * ↑n ^ p.succ
this :
  ∑ i ∈ range p, bernoulli (i + 2) * ↑((p + 2).choose (i + 2)) * ↑n ^ (p - i) / ↑(p + 2) =
    ∑ i ∈ range p, bernoulli' (i + 2) * ↑((p + 2).choose (i + 2)) * ↑n ^ (p - i) / ↑(p + 2)
⊢ ∑ k ∈ Ico 1 n.succ, ↑k ^ p.succ = ∑ i ∈ range p.succ.succ, f' i
state_after: no goals
tactic: calc
    (-- replace sum over `Ico` with sum over `range` and simplify
        ∑ k ∈ Ico 1 n.succ, (k : ℚ) ^ p.succ)
    _ = ∑ k ∈ range n.succ, (k : ℚ) ^ p.succ := by simp [sum_Ico_eq_sub _ hle, succ_ne_zero]
    -- extract the last term of the sum
    _ = (∑ k ∈ range n, (k : ℚ) ^ p.succ) + (n : ℚ) ^ p.succ := by rw [sum_range_succ]
    -- apply the key lemma, `sum_range_pow`
    _ = (∑ i ∈ range p.succ.succ, f i) + (n : ℚ) ^ p.succ := by simp [f, sum_range_pow]
    -- extract the first two terms of the sum
    _ = (∑ i ∈ range p, f i.succ.succ) + f 1 + f 0 + (n : ℚ) ^ p.succ := by
      simp_rw [sum_range_succ']
    _ = (∑ i ∈ range p, f i.succ.succ) + (f 1 + (n : ℚ) ^ p.succ) + f 0 := by ring
    _ = (∑ i ∈ range p, f i.succ.succ) + 1 / 2 * (n : ℚ) ^ p.succ + f 0 := by rw [h2]
    -- convert from `bernoulli` to `bernoulli'`
    _ = (∑ i ∈ range p, f' i.succ.succ) + f' 1 + f' 0 := by
      simpa [f, f', h1, fun i => show i + 2 = i + 1 + 1 from rfl]
    -- rejoin the first two terms of the sum
    _ = ∑ i ∈ range p.succ.succ, f' i := by simp_rw [sum_range_succ']