Theorem: ZetaAsymptotics.term_of_lt
File: Mathlib/NumberTheory/Harmonic/ZetaAsymp.lean

tactic: have hv : ∀ x ∈ uIcc (n : ℝ) (n + 1), 0 < x := by
  intro x hx
  rw [uIcc_of_le (by simp only [le_add_iff_nonneg_right, zero_le_one])] at hx
  exact (Nat.cast_pos.mpr hn).trans_le hx.1
state_before: n : ℕ
hn : 0 < n
s : ℝ
hs : 1 < s
⊢ term n s = 1 / (s - 1) * (1 / ↑n ^ (s - 1) - 1 / (↑n + 1) ^ (s - 1)) - ↑n / s * (1 / ↑n ^ s - 1 / (↑n + 1) ^ s)
state_after: n : ℕ
hn : 0 < n
s : ℝ
hs : 1 < s
hv : ∀ x ∈ uIcc (↑n) (↑n + 1), 0 < x
⊢ term n s = 1 / (s - 1) * (1 / ↑n ^ (s - 1) - 1 / (↑n + 1) ^ (s - 1)) - ↑n / s * (1 / ↑n ^ s - 1 / (↑n + 1) ^ s)

tactic: intro x hx
state_before: n : ℕ
hn : 0 < n
s : ℝ
hs : 1 < s
⊢ ∀ x ∈ uIcc (↑n) (↑n + 1), 0 < x
state_after: n : ℕ
hn : 0 < n
s : ℝ
hs : 1 < s
x : ℝ
hx : x ∈ uIcc (↑n) (↑n + 1)
⊢ 0 < x

tactic: rw [uIcc_of_le (by simp only [le_add_iff_nonneg_right, zero_le_one])] at hx
state_before: n : ℕ
hn : 0 < n
s : ℝ
hs : 1 < s
x : ℝ
hx : x ∈ uIcc (↑n) (↑n + 1)
⊢ 0 < x
state_after: n : ℕ
hn : 0 < n
s : ℝ
hs : 1 < s
x : ℝ
hx : x ∈ Icc (↑n) (↑n + 1)
⊢ 0 < x

tactic: exact (Nat.cast_pos.mpr hn).trans_le hx.1
state_before: n : ℕ
hn : 0 < n
s : ℝ
hs : 1 < s
x : ℝ
hx : x ∈ Icc (↑n) (↑n + 1)
⊢ 0 < x
state_after: no goals

tactic: simp only [le_add_iff_nonneg_right, zero_le_one]
state_before: n : ℕ
hn : 0 < n
s : ℝ
hs : 1 < s
x : ℝ
hx : x ∈ uIcc (↑n) (↑n + 1)
⊢ ↑n ≤ ↑n + 1
state_after: no goals

tactic: rfl
state_before: n : ℕ
hn : 0 < n
s : ℝ
hs : 1 < s
hv : ∀ x ∈ uIcc (↑n) (↑n + 1), 0 < x
⊢ term n s = ∫ (x : ℝ) in ↑n..↑n + 1, (x - ↑n) / x ^ (s + 1)
state_after: no goals

tactic: refine intervalIntegral.integral_congr (fun x hx ↦ ?_)
state_before: n : ℕ
hn : 0 < n
s : ℝ
hs : 1 < s
hv : ∀ x ∈ uIcc (↑n) (↑n + 1), 0 < x
⊢ ∫ (x : ℝ) in ↑n..↑n + 1, (x - ↑n) / x ^ (s + 1) = ∫ (x : ℝ) in ↑n..↑n + 1, x ^ (-s) - ↑n * x ^ (-(s + 1))
state_after: n : ℕ
hn : 0 < n
s : ℝ
hs : 1 < s
hv : ∀ x ∈ uIcc (↑n) (↑n + 1), 0 < x
x : ℝ
hx : x ∈ uIcc (↑n) (↑n + 1)
⊢ (x - ↑n) / x ^ (s + 1) = x ^ (-s) - ↑n * x ^ (-(s + 1))

tactic: rw [sub_div, rpow_add_one (hv x hx).ne', mul_comm, ← div_div, div_self (hv x hx).ne',
  rpow_neg (hv x hx).le, rpow_neg (hv x hx).le, one_div, rpow_add_one (hv x hx).ne', mul_comm,
  div_eq_mul_inv]
state_before: n : ℕ
hn : 0 < n
s : ℝ
hs : 1 < s
hv : ∀ x ∈ uIcc (↑n) (↑n + 1), 0 < x
x : ℝ
hx : x ∈ uIcc (↑n) (↑n + 1)
⊢ (x - ↑n) / x ^ (s + 1) = x ^ (-s) - ↑n * x ^ (-(s + 1))
state_after: no goals

tactic: refine intervalIntegral.intervalIntegrable_rpow (Or.inr <| not_mem_uIcc_of_lt ?_ ?_)
state_before: case hg.hf
n : ℕ
hn : 0 < n
s : ℝ
hs : 1 < s
hv : ∀ x ∈ uIcc (↑n) (↑n + 1), 0 < x
⊢ IntervalIntegrable (fun x => x ^ (-(s + 1))) volume (↑n) (↑n + 1)
state_after: case hg.hf.refine_1
n : ℕ
hn : 0 < n
s : ℝ
hs : 1 < s
hv : ∀ x ∈ uIcc (↑n) (↑n + 1), 0 < x
⊢ 0 < ↑n

case hg.hf.refine_2
n : ℕ
hn : 0 < n
s : ℝ
hs : 1 < s
hv : ∀ x ∈ uIcc (↑n) (↑n + 1), 0 < x
⊢ 0 < ↑n + 1

tactic: exact_mod_cast hn
state_before: case hg.hf.refine_1
n : ℕ
hn : 0 < n
s : ℝ
hs : 1 < s
hv : ∀ x ∈ uIcc (↑n) (↑n + 1), 0 < x
⊢ 0 < ↑n
state_after: no goals

tactic: linarith
state_before: case hg.hf.refine_2
n : ℕ
hn : 0 < n
s : ℝ
hs : 1 < s
hv : ∀ x ∈ uIcc (↑n) (↑n + 1), 0 < x
⊢ 0 < ↑n + 1
state_after: no goals

tactic: rw [integral_rpow (Or.inr ⟨by linarith, this⟩), integral_rpow (Or.inr ⟨by linarith, this⟩)]
state_before: n : ℕ
hn : 0 < n
s : ℝ
hs : 1 < s
hv : ∀ x ∈ uIcc (↑n) (↑n + 1), 0 < x
this : 0 ∉ uIcc (↑n) (↑n + 1)
⊢ (∫ (x : ℝ) in ↑n..↑n + 1, x ^ (-s)) - ↑n * ∫ (x : ℝ) in ↑n..↑n + 1, x ^ (-(s + 1)) =
    1 / (s - 1) * (1 / ↑n ^ (s - 1) - 1 / (↑n + 1) ^ (s - 1)) - ↑n / s * (1 / ↑n ^ s - 1 / (↑n + 1) ^ s)
state_after: n : ℕ
hn : 0 < n
s : ℝ
hs : 1 < s
hv : ∀ x ∈ uIcc (↑n) (↑n + 1), 0 < x
this : 0 ∉ uIcc (↑n) (↑n + 1)
⊢ ((↑n + 1) ^ (-s + 1) - ↑n ^ (-s + 1)) / (-s + 1) -
      ↑n * (((↑n + 1) ^ (-(s + 1) + 1) - ↑n ^ (-(s + 1) + 1)) / (-(s + 1) + 1)) =
    1 / (s - 1) * (1 / ↑n ^ (s - 1) - 1 / (↑n + 1) ^ (s - 1)) - ↑n / s * (1 / ↑n ^ s - 1 / (↑n + 1) ^ s)

tactic: congr 1
state_before: n : ℕ
hn : 0 < n
s : ℝ
hs : 1 < s
hv : ∀ x ∈ uIcc (↑n) (↑n + 1), 0 < x
this : 0 ∉ uIcc (↑n) (↑n + 1)
⊢ ((↑n + 1) ^ (-s + 1) - ↑n ^ (-s + 1)) / (-s + 1) -
      ↑n * (((↑n + 1) ^ (-(s + 1) + 1) - ↑n ^ (-(s + 1) + 1)) / (-(s + 1) + 1)) =
    1 / (s - 1) * (1 / ↑n ^ (s - 1) - 1 / (↑n + 1) ^ (s - 1)) - ↑n / s * (1 / ↑n ^ s - 1 / (↑n + 1) ^ s)
state_after: case e_a
n : ℕ
hn : 0 < n
s : ℝ
hs : 1 < s
hv : ∀ x ∈ uIcc (↑n) (↑n + 1), 0 < x
this : 0 ∉ uIcc (↑n) (↑n + 1)
⊢ ((↑n + 1) ^ (-s + 1) - ↑n ^ (-s + 1)) / (-s + 1) = 1 / (s - 1) * (1 / ↑n ^ (s - 1) - 1 / (↑n + 1) ^ (s - 1))

case e_a
n : ℕ
hn : 0 < n
s : ℝ
hs : 1 < s
hv : ∀ x ∈ uIcc (↑n) (↑n + 1), 0 < x
this : 0 ∉ uIcc (↑n) (↑n + 1)
⊢ ↑n * (((↑n + 1) ^ (-(s + 1) + 1) - ↑n ^ (-(s + 1) + 1)) / (-(s + 1) + 1)) = ↑n / s * (1 / ↑n ^ s - 1 / (↑n + 1) ^ s)

tactic: linarith
state_before: n : ℕ
hn : 0 < n
s : ℝ
hs : 1 < s
hv : ∀ x ∈ uIcc (↑n) (↑n + 1), 0 < x
this : 0 ∉ uIcc (↑n) (↑n + 1)
⊢ -s ≠ -1
state_after: no goals

tactic: linarith
state_before: n : ℕ
hn : 0 < n
s : ℝ
hs : 1 < s
hv : ∀ x ∈ uIcc (↑n) (↑n + 1), 0 < x
this : 0 ∉ uIcc (↑n) (↑n + 1)
⊢ -(s + 1) ≠ -1
state_after: no goals

tactic: rw [show -s + 1 = -(s - 1) by ring, div_neg, ← neg_div, mul_comm, mul_one_div, neg_sub,
  rpow_neg (Nat.cast_nonneg _), one_div, rpow_neg (by linarith), one_div]
state_before: case e_a
n : ℕ
hn : 0 < n
s : ℝ
hs : 1 < s
hv : ∀ x ∈ uIcc (↑n) (↑n + 1), 0 < x
this : 0 ∉ uIcc (↑n) (↑n + 1)
⊢ ((↑n + 1) ^ (-s + 1) - ↑n ^ (-s + 1)) / (-s + 1) = 1 / (s - 1) * (1 / ↑n ^ (s - 1) - 1 / (↑n + 1) ^ (s - 1))
state_after: no goals

tactic: ring
state_before: n : ℕ
hn : 0 < n
s : ℝ
hs : 1 < s
hv : ∀ x ∈ uIcc (↑n) (↑n + 1), 0 < x
this : 0 ∉ uIcc (↑n) (↑n + 1)
⊢ -s + 1 = -(s - 1)
state_after: no goals

tactic: linarith
state_before: n : ℕ
hn : 0 < n
s : ℝ
hs : 1 < s
hv : ∀ x ∈ uIcc (↑n) (↑n + 1), 0 < x
this : 0 ∉ uIcc (↑n) (↑n + 1)
⊢ 0 ≤ ↑n + 1
state_after: no goals

tactic: rw [show -(s + 1) + 1 = -s by ring, div_neg, ← neg_div, neg_sub, div_mul_eq_mul_div,
  mul_div_assoc, rpow_neg (Nat.cast_nonneg _), one_div, rpow_neg (by linarith), one_div]
state_before: case e_a
n : ℕ
hn : 0 < n
s : ℝ
hs : 1 < s
hv : ∀ x ∈ uIcc (↑n) (↑n + 1), 0 < x
this : 0 ∉ uIcc (↑n) (↑n + 1)
⊢ ↑n * (((↑n + 1) ^ (-(s + 1) + 1) - ↑n ^ (-(s + 1) + 1)) / (-(s + 1) + 1)) = ↑n / s * (1 / ↑n ^ s - 1 / (↑n + 1) ^ s)
state_after: no goals

tactic: ring
state_before: n : ℕ
hn : 0 < n
s : ℝ
hs : 1 < s
hv : ∀ x ∈ uIcc (↑n) (↑n + 1), 0 < x
this : 0 ∉ uIcc (↑n) (↑n + 1)
⊢ -(s + 1) + 1 = -s
state_after: no goals