l3lab/ntp-mathlib huggingface dataset:
/-- The sequence `log (stirlingSeq (m + 1)) - log (stirlingSeq (m + 2))` has the series expansion
   `∑ 1 / (2 * (k + 1) + 1) * (1 / 2 * (m + 1) + 1)^(2 * (k + 1))`
-/
theorem log_stirlingSeq_diff_hasSum (m : ℕ) :
    HasSum (fun k : ℕ => ↑1 / (↑2 * ↑(k + 1) + ↑1) * (((1:ℝ)/(↑2 * ↑(m + 1) + ↑1)) ^ 2) ^ ↑(k + 1))
      (log (stirlingSeq (m + 1)) - log (stirlingSeq (m + 2))) := by
  change HasSum
    ((fun b : ℕ => (1:ℝ) / ((2:ℝ) * b + (1:ℝ)) * (((1:ℝ) / (2 * (m + 1 :) + 1)) ^ 2) ^ b) ∘ succ) _
  refine' (hasSum_nat_add_iff (a := _ - _) -- Porting note: must give implicit arguments
    (f := (fun b : ℕ => ↑1 / (↑2 * b + ↑1) * (((1:ℝ) / (2 * ↑(m + 1) + 1)) ^ 2) ^ b)) 1).mpr _
  convert (hasSum_log_one_add_inv <|
    cast_pos.mpr (succ_pos m)).mul_left ((↑(m + 1) : ℝ) + 1 / 2) using 1
  · ext k
    rw [← pow_mul, pow_add]
    push_cast
    field_simp
    ring
  · have h : ∀ (x : ℝ) (_ : x ≠ 0), 1 + x⁻¹ = (x + 1) / x := by
      intro x hx; rw [_root_.add_div, div_self hx, inv_eq_one_div]
    

Proof state:
case h.e'_6
m : ℕ
h : ∀ (x : ℝ), x ≠ 0 → 1 + x⁻¹ = (x + 1) / x
⊢ Real.log (stirlingSeq (m + 1)) - Real.log (stirlingSeq (m + 2)) +
      ∑ i in range 1, 1 / (2 * ↑i + 1) * ((1 / (2 * ↑(m + 1) + 1)) ^ 2) ^ i =
    (↑(m + 1) + 1 / 2) * Real.log (1 + (↑(succ m))⁻¹)

Next tactic:
simp (disch := norm_cast <;> apply_rules [mul_ne_zero, succ_ne_zero, factorial_ne_zero,
      exp_ne_zero]) only [log_stirlingSeq_formula, log_div, log_mul, log_exp, factorial_succ,
      cast_mul, cast_succ, cast_zero, range_one, sum_singleton, h]
/-- The sequence `log (stirlingSeq (m + 1)) - log (stirlingSeq (m + 2))` has the series expansion
   `∑ 1 / (2 * (k + 1) + 1) * (1 / 2 * (m + 1) + 1)^(2 * (k + 1))`
-/
theorem log_stirlingSeq_diff_hasSum (m : ℕ) :
    HasSum (fun k : ℕ => ↑1 / (↑2 * ↑(k + 1) + ↑1) * (((1:ℝ)/(↑2 * ↑(m + 1) + ↑1)) ^ 2) ^ ↑(k + 1))
      (log (stirlingSeq (m + 1)) - log (stirlingSeq (m + 2))) := by
  change HasSum
    ((fun b : ℕ => (1:ℝ) / ((2:ℝ) * b + (1:ℝ)) * (((1:ℝ) / (2 * (m + 1 :) + 1)) ^ 2) ^ b) ∘ succ) _
  refine' (hasSum_nat_add_iff (a := _ - _) -- Porting note: must give implicit arguments
    (f := (fun b : ℕ => ↑1 / (↑2 * b + ↑1) * (((1:ℝ) / (2 * ↑(m + 1) + 1)) ^ 2) ^ b)) 1).mpr _
  convert (hasSum_log_one_add_inv <|
    cast_pos.mpr (succ_pos m)).mul_left ((↑(m + 1) : ℝ) + 1 / 2) using 1
  · ext k
    rw [← pow_mul, pow_add]
    push_cast
    field_simp
    ring
  · have h : ∀ (x : ℝ) (_ : x ≠ 0), 1 + x⁻¹ = (x + 1) / x := by
      intro x hx; rw [_root_.add_div, div_self hx, inv_eq_one_div]
    simp (disch := norm_cast <;> apply_rules [mul_ne_zero, succ_ne_zero, factorial_ne_zero,
      exp_ne_zero]) only [log_stirlingSeq_formula, log_div, log_mul, log_exp, factorial_succ,
      cast_mul, cast_succ, cast_zero, range_one, sum_singleton, h]
    

Proof state:
case h.e'_6
m : ℕ
h : ∀ (x : ℝ), x ≠ 0 → 1 + x⁻¹ = (x + 1) / x
⊢ Real.log (↑m + 1) + Real.log ↑m ! - 1 / 2 * (Real.log 2 + Real.log (↑m + 1)) - (↑m + 1) * (Real.log (↑m + 1) - 1) -
        (Real.log (↑m + 1 + 1) + (Real.log (↑m + 1) + Real.log ↑m !) - 1 / 2 * (Real.log 2 + Real.log (↑m + 1 + 1)) -
          (↑m + 1 + 1) * (Real.log (↑m + 1 + 1) - 1)) +
      1 / (2 * 0 + 1) * ((1 / (2 * (↑m + 1) + 1)) ^ 2) ^ 0 =
    (↑m + 1 + 1 / 2) * (Real.log (↑m + 1 + 1) - Real.log (↑m + 1))

Next tactic:
ring
/-- The sequence `log ∘ stirlingSeq ∘ succ` is monotone decreasing -/
theorem log_stirlingSeq'_antitone : Antitone (Real.log ∘ stirlingSeq ∘ succ) :=
  antitone_nat_of_succ_le fun n =>
    sub_nonneg.mp <| (log_stirlingSeq_diff_hasSum n).nonneg fun m => by 

Proof state:
n m : ℕ
⊢ 0 ≤ 1 / (2 * ↑(m + 1) + 1) * ((1 / (2 * ↑(n + 1) + 1)) ^ 2) ^ (m + 1)

Next tactic:
positivity
/-- We have a bound for successive elements in the sequence `log (stirlingSeq k)`.
-/
theorem log_stirlingSeq_diff_le_geo_sum (n : ℕ) :
    log (stirlingSeq (n + 1)) - log (stirlingSeq (n + 2)) ≤
      ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2 / (↑1 - ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2) := by
  

Proof state:
n : ℕ
⊢ Real.log (stirlingSeq (n + 1)) - Real.log (stirlingSeq (n + 2)) ≤
    (1 / (2 * ↑(n + 1) + 1)) ^ 2 / (1 - (1 / (2 * ↑(n + 1) + 1)) ^ 2)

Next tactic:
have h_nonneg : (0 : ℝ) ≤ ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2 := sq_nonneg _
/-- We have a bound for successive elements in the sequence `log (stirlingSeq k)`.
-/
theorem log_stirlingSeq_diff_le_geo_sum (n : ℕ) :
    log (stirlingSeq (n + 1)) - log (stirlingSeq (n + 2)) ≤
      ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2 / (↑1 - ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2) := by
  have h_nonneg : (0 : ℝ) ≤ ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2 := sq_nonneg _
  

Proof state:
n : ℕ
h_nonneg : 0 ≤ (1 / (2 * ↑(n + 1) + 1)) ^ 2
⊢ Real.log (stirlingSeq (n + 1)) - Real.log (stirlingSeq (n + 2)) ≤
    (1 / (2 * ↑(n + 1) + 1)) ^ 2 / (1 - (1 / (2 * ↑(n + 1) + 1)) ^ 2)

Next tactic:
have g : HasSum (fun k : ℕ => (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2) ^ ↑(k + 1))
      (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2 / (↑1 - ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2)) := by
    have := (hasSum_geometric_of_lt_1 h_nonneg ?_).mul_left (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2)
    · simp_rw [← _root_.pow_succ] at this
      exact this
    rw [one_div, inv_pow]
    exact inv_lt_one (one_lt_pow ((lt_add_iff_pos_left 1).mpr <| by positivity) two_ne_zero)
/-- We have a bound for successive elements in the sequence `log (stirlingSeq k)`.
-/
theorem log_stirlingSeq_diff_le_geo_sum (n : ℕ) :
    log (stirlingSeq (n + 1)) - log (stirlingSeq (n + 2)) ≤
      ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2 / (↑1 - ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2) := by
  have h_nonneg : (0 : ℝ) ≤ ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2 := sq_nonneg _
  have g : HasSum (fun k : ℕ => (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2) ^ ↑(k + 1))
      (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2 / (↑1 - ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2)) := by
    

Proof state:
n : ℕ
h_nonneg : 0 ≤ (1 / (2 * ↑(n + 1) + 1)) ^ 2
⊢ HasSum (fun k => ((1 / (2 * ↑(n + 1) + 1)) ^ 2) ^ (k + 1))
    ((1 / (2 * ↑(n + 1) + 1)) ^ 2 / (1 - (1 / (2 * ↑(n + 1) + 1)) ^ 2))

Next tactic:
have := (hasSum_geometric_of_lt_1 h_nonneg ?_).mul_left (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2)
/-- We have a bound for successive elements in the sequence `log (stirlingSeq k)`.
-/
theorem log_stirlingSeq_diff_le_geo_sum (n : ℕ) :
    log (stirlingSeq (n + 1)) - log (stirlingSeq (n + 2)) ≤
      ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2 / (↑1 - ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2) := by
  have h_nonneg : (0 : ℝ) ≤ ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2 := sq_nonneg _
  have g : HasSum (fun k : ℕ => (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2) ^ ↑(k + 1))
      (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2 / (↑1 - ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2)) := by
    have := (hasSum_geometric_of_lt_1 h_nonneg ?_).mul_left (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2)
    · 

Proof state:
case refine_2
n : ℕ
h_nonneg : 0 ≤ (1 / (2 * ↑(n + 1) + 1)) ^ 2
this :
  HasSum (fun i => (1 / (2 * ↑(n + 1) + 1)) ^ 2 * ((1 / (2 * ↑(n + 1) + 1)) ^ 2) ^ i)
    ((1 / (2 * ↑(n + 1) + 1)) ^ 2 * (1 - (1 / (2 * ↑(n + 1) + 1)) ^ 2)⁻¹)
⊢ HasSum (fun k => ((1 / (2 * ↑(n + 1) + 1)) ^ 2) ^ (k + 1))
    ((1 / (2 * ↑(n + 1) + 1)) ^ 2 / (1 - (1 / (2 * ↑(n + 1) + 1)) ^ 2))

Next tactic:
simp_rw [← _root_.pow_succ] at this
/-- We have a bound for successive elements in the sequence `log (stirlingSeq k)`.
-/
theorem log_stirlingSeq_diff_le_geo_sum (n : ℕ) :
    log (stirlingSeq (n + 1)) - log (stirlingSeq (n + 2)) ≤
      ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2 / (↑1 - ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2) := by
  have h_nonneg : (0 : ℝ) ≤ ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2 := sq_nonneg _
  have g : HasSum (fun k : ℕ => (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2) ^ ↑(k + 1))
      (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2 / (↑1 - ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2)) := by
    have := (hasSum_geometric_of_lt_1 h_nonneg ?_).mul_left (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2)
    · simp_rw [← _root_.pow_succ] at this
      

Proof state:
case refine_2
n : ℕ
h_nonneg : 0 ≤ (1 / (2 * ↑(n + 1) + 1)) ^ 2
this :
  HasSum (fun i => ((1 / (2 * ↑(n + 1) + 1)) ^ 2) ^ (i + 1))
    ((1 / (2 * ↑(n + 1) + 1)) ^ 2 * (1 - (1 / (2 * ↑(n + 1) + 1)) ^ 2)⁻¹)
⊢ HasSum (fun k => ((1 / (2 * ↑(n + 1) + 1)) ^ 2) ^ (k + 1))
    ((1 / (2 * ↑(n + 1) + 1)) ^ 2 / (1 - (1 / (2 * ↑(n + 1) + 1)) ^ 2))

Next tactic:
exact this
/-- We have a bound for successive elements in the sequence `log (stirlingSeq k)`.
-/
theorem log_stirlingSeq_diff_le_geo_sum (n : ℕ) :
    log (stirlingSeq (n + 1)) - log (stirlingSeq (n + 2)) ≤
      ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2 / (↑1 - ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2) := by
  have h_nonneg : (0 : ℝ) ≤ ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2 := sq_nonneg _
  have g : HasSum (fun k : ℕ => (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2) ^ ↑(k + 1))
      (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2 / (↑1 - ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2)) := by
    have := (hasSum_geometric_of_lt_1 h_nonneg ?_).mul_left (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2)
    · simp_rw [← _root_.pow_succ] at this
      exact this
    

Proof state:
case refine_1
n : ℕ
h_nonneg : 0 ≤ (1 / (2 * ↑(n + 1) + 1)) ^ 2
⊢ (1 / (2 * ↑(n + 1) + 1)) ^ 2 < 1

Next tactic:
rw [one_div, inv_pow]
/-- We have a bound for successive elements in the sequence `log (stirlingSeq k)`.
-/
theorem log_stirlingSeq_diff_le_geo_sum (n : ℕ) :
    log (stirlingSeq (n + 1)) - log (stirlingSeq (n + 2)) ≤
      ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2 / (↑1 - ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2) := by
  have h_nonneg : (0 : ℝ) ≤ ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2 := sq_nonneg _
  have g : HasSum (fun k : ℕ => (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2) ^ ↑(k + 1))
      (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2 / (↑1 - ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2)) := by
    have := (hasSum_geometric_of_lt_1 h_nonneg ?_).mul_left (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2)
    · simp_rw [← _root_.pow_succ] at this
      exact this
    rw [one_div, inv_pow]
    

Proof state:
case refine_1
n : ℕ
h_nonneg : 0 ≤ (1 / (2 * ↑(n + 1) + 1)) ^ 2
⊢ ((2 * ↑(n + 1) + 1) ^ 2)⁻¹ < 1

Next tactic:
exact inv_lt_one (one_lt_pow ((lt_add_iff_pos_left 1).mpr <| by positivity) two_ne_zero)
/-- We have a bound for successive elements in the sequence `log (stirlingSeq k)`.
-/
theorem log_stirlingSeq_diff_le_geo_sum (n : ℕ) :
    log (stirlingSeq (n + 1)) - log (stirlingSeq (n + 2)) ≤
      ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2 / (↑1 - ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2) := by
  have h_nonneg : (0 : ℝ) ≤ ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2 := sq_nonneg _
  have g : HasSum (fun k : ℕ => (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2) ^ ↑(k + 1))
      (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2 / (↑1 - ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2)) := by
    have := (hasSum_geometric_of_lt_1 h_nonneg ?_).mul_left (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2)
    · simp_rw [← _root_.pow_succ] at this
      exact this
    rw [one_div, inv_pow]
    exact inv_lt_one (one_lt_pow ((lt_add_iff_pos_left 1).mpr <| by 

Proof state:
n : ℕ
h_nonneg : 0 ≤ (1 / (2 * ↑(n + 1) + 1)) ^ 2
⊢ 0 < 2 * ↑(n + 1)

Next tactic:
positivity
/-- We have a bound for successive elements in the sequence `log (stirlingSeq k)`.
-/
theorem log_stirlingSeq_diff_le_geo_sum (n : ℕ) :
    log (stirlingSeq (n + 1)) - log (stirlingSeq (n + 2)) ≤
      ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2 / (↑1 - ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2) := by
  have h_nonneg : (0 : ℝ) ≤ ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2 := sq_nonneg _
  have g : HasSum (fun k : ℕ => (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2) ^ ↑(k + 1))
      (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2 / (↑1 - ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2)) := by
    have := (hasSum_geometric_of_lt_1 h_nonneg ?_).mul_left (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2)
    · simp_rw [← _root_.pow_succ] at this
      exact this
    rw [one_div, inv_pow]
    exact inv_lt_one (one_lt_pow ((lt_add_iff_pos_left 1).mpr <| by positivity) two_ne_zero)
  

Proof state:
n : ℕ
h_nonneg : 0 ≤ (1 / (2 * ↑(n + 1) + 1)) ^ 2
g :
  HasSum (fun k => ((1 / (2 * ↑(n + 1) + 1)) ^ 2) ^ (k + 1))
    ((1 / (2 * ↑(n + 1) + 1)) ^ 2 / (1 - (1 / (2 * ↑(n + 1) + 1)) ^ 2))
⊢ Real.log (stirlingSeq (n + 1)) - Real.log (stirlingSeq (n + 2)) ≤
    (1 / (2 * ↑(n + 1) + 1)) ^ 2 / (1 - (1 / (2 * ↑(n + 1) + 1)) ^ 2)

Next tactic:
have hab : ∀ k : ℕ, (1:ℝ) / (↑2 * ↑(k + 1) + ↑1) * (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2) ^ ↑(k + 1) ≤
      (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2) ^ ↑(k + 1) := by
    refine' fun k => mul_le_of_le_one_left (pow_nonneg h_nonneg ↑(k + 1)) _
    rw [one_div]
    exact inv_le_one (le_add_of_nonneg_left <| by positivity)
/-- We have a bound for successive elements in the sequence `log (stirlingSeq k)`.
-/
theorem log_stirlingSeq_diff_le_geo_sum (n : ℕ) :
    log (stirlingSeq (n + 1)) - log (stirlingSeq (n + 2)) ≤
      ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2 / (↑1 - ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2) := by
  have h_nonneg : (0 : ℝ) ≤ ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2 := sq_nonneg _
  have g : HasSum (fun k : ℕ => (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2) ^ ↑(k + 1))
      (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2 / (↑1 - ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2)) := by
    have := (hasSum_geometric_of_lt_1 h_nonneg ?_).mul_left (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2)
    · simp_rw [← _root_.pow_succ] at this
      exact this
    rw [one_div, inv_pow]
    exact inv_lt_one (one_lt_pow ((lt_add_iff_pos_left 1).mpr <| by positivity) two_ne_zero)
  have hab : ∀ k : ℕ, (1:ℝ) / (↑2 * ↑(k + 1) + ↑1) * (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2) ^ ↑(k + 1) ≤
      (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2) ^ ↑(k + 1) := by
    

Proof state:
n : ℕ
h_nonneg : 0 ≤ (1 / (2 * ↑(n + 1) + 1)) ^ 2
g :
  HasSum (fun k => ((1 / (2 * ↑(n + 1) + 1)) ^ 2) ^ (k + 1))
    ((1 / (2 * ↑(n + 1) + 1)) ^ 2 / (1 - (1 / (2 * ↑(n + 1) + 1)) ^ 2))
⊢ ∀ (k : ℕ),
    1 / (2 * ↑(k + 1) + 1) * ((1 / (2 * ↑(n + 1) + 1)) ^ 2) ^ (k + 1) ≤ ((1 / (2 * ↑(n + 1) + 1)) ^ 2) ^ (k + 1)

Next tactic:
refine' fun k => mul_le_of_le_one_left (pow_nonneg h_nonneg ↑(k + 1)) _
/-- We have a bound for successive elements in the sequence `log (stirlingSeq k)`.
-/
theorem log_stirlingSeq_diff_le_geo_sum (n : ℕ) :
    log (stirlingSeq (n + 1)) - log (stirlingSeq (n + 2)) ≤
      ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2 / (↑1 - ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2) := by
  have h_nonneg : (0 : ℝ) ≤ ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2 := sq_nonneg _
  have g : HasSum (fun k : ℕ => (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2) ^ ↑(k + 1))
      (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2 / (↑1 - ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2)) := by
    have := (hasSum_geometric_of_lt_1 h_nonneg ?_).mul_left (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2)
    · simp_rw [← _root_.pow_succ] at this
      exact this
    rw [one_div, inv_pow]
    exact inv_lt_one (one_lt_pow ((lt_add_iff_pos_left 1).mpr <| by positivity) two_ne_zero)
  have hab : ∀ k : ℕ, (1:ℝ) / (↑2 * ↑(k + 1) + ↑1) * (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2) ^ ↑(k + 1) ≤
      (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2) ^ ↑(k + 1) := by
    refine' fun k => mul_le_of_le_one_left (pow_nonneg h_nonneg ↑(k + 1)) _
    

Proof state:
n : ℕ
h_nonneg : 0 ≤ (1 / (2 * ↑(n + 1) + 1)) ^ 2
g :
  HasSum (fun k => ((1 / (2 * ↑(n + 1) + 1)) ^ 2) ^ (k + 1))
    ((1 / (2 * ↑(n + 1) + 1)) ^ 2 / (1 - (1 / (2 * ↑(n + 1) + 1)) ^ 2))
k : ℕ
⊢ 1 / (2 * ↑(k + 1) + 1) ≤ 1

Next tactic:
rw [one_div]
/-- We have a bound for successive elements in the sequence `log (stirlingSeq k)`.
-/
theorem log_stirlingSeq_diff_le_geo_sum (n : ℕ) :
    log (stirlingSeq (n + 1)) - log (stirlingSeq (n + 2)) ≤
      ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2 / (↑1 - ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2) := by
  have h_nonneg : (0 : ℝ) ≤ ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2 := sq_nonneg _
  have g : HasSum (fun k : ℕ => (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2) ^ ↑(k + 1))
      (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2 / (↑1 - ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2)) := by
    have := (hasSum_geometric_of_lt_1 h_nonneg ?_).mul_left (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2)
    · simp_rw [← _root_.pow_succ] at this
      exact this
    rw [one_div, inv_pow]
    exact inv_lt_one (one_lt_pow ((lt_add_iff_pos_left 1).mpr <| by positivity) two_ne_zero)
  have hab : ∀ k : ℕ, (1:ℝ) / (↑2 * ↑(k + 1) + ↑1) * (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2) ^ ↑(k + 1) ≤
      (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2) ^ ↑(k + 1) := by
    refine' fun k => mul_le_of_le_one_left (pow_nonneg h_nonneg ↑(k + 1)) _
    rw [one_div]
    

Proof state:
n : ℕ
h_nonneg : 0 ≤ (1 / (2 * ↑(n + 1) + 1)) ^ 2
g :
  HasSum (fun k => ((1 / (2 * ↑(n + 1) + 1)) ^ 2) ^ (k + 1))
    ((1 / (2 * ↑(n + 1) + 1)) ^ 2 / (1 - (1 / (2 * ↑(n + 1) + 1)) ^ 2))
k : ℕ
⊢ (2 * ↑(k + 1) + 1)⁻¹ ≤ 1

Next tactic:
exact inv_le_one (le_add_of_nonneg_left <| by positivity)
/-- We have a bound for successive elements in the sequence `log (stirlingSeq k)`.
-/
theorem log_stirlingSeq_diff_le_geo_sum (n : ℕ) :
    log (stirlingSeq (n + 1)) - log (stirlingSeq (n + 2)) ≤
      ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2 / (↑1 - ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2) := by
  have h_nonneg : (0 : ℝ) ≤ ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2 := sq_nonneg _
  have g : HasSum (fun k : ℕ => (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2) ^ ↑(k + 1))
      (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2 / (↑1 - ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2)) := by
    have := (hasSum_geometric_of_lt_1 h_nonneg ?_).mul_left (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2)
    · simp_rw [← _root_.pow_succ] at this
      exact this
    rw [one_div, inv_pow]
    exact inv_lt_one (one_lt_pow ((lt_add_iff_pos_left 1).mpr <| by positivity) two_ne_zero)
  have hab : ∀ k : ℕ, (1:ℝ) / (↑2 * ↑(k + 1) + ↑1) * (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2) ^ ↑(k + 1) ≤
      (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2) ^ ↑(k + 1) := by
    refine' fun k => mul_le_of_le_one_left (pow_nonneg h_nonneg ↑(k + 1)) _
    rw [one_div]
    exact inv_le_one (le_add_of_nonneg_left <| by 

Proof state:
n : ℕ
h_nonneg : 0 ≤ (1 / (2 * ↑(n + 1) + 1)) ^ 2
g :
  HasSum (fun k => ((1 / (2 * ↑(n + 1) + 1)) ^ 2) ^ (k + 1))
    ((1 / (2 * ↑(n + 1) + 1)) ^ 2 / (1 - (1 / (2 * ↑(n + 1) + 1)) ^ 2))
k : ℕ
⊢ 0 ≤ 2 * ↑(k + 1)

Next tactic:
positivity
/-- We have a bound for successive elements in the sequence `log (stirlingSeq k)`.
-/
theorem log_stirlingSeq_diff_le_geo_sum (n : ℕ) :
    log (stirlingSeq (n + 1)) - log (stirlingSeq (n + 2)) ≤
      ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2 / (↑1 - ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2) := by
  have h_nonneg : (0 : ℝ) ≤ ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2 := sq_nonneg _
  have g : HasSum (fun k : ℕ => (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2) ^ ↑(k + 1))
      (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2 / (↑1 - ((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2)) := by
    have := (hasSum_geometric_of_lt_1 h_nonneg ?_).mul_left (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2)
    · simp_rw [← _root_.pow_succ] at this
      exact this
    rw [one_div, inv_pow]
    exact inv_lt_one (one_lt_pow ((lt_add_iff_pos_left 1).mpr <| by positivity) two_ne_zero)
  have hab : ∀ k : ℕ, (1:ℝ) / (↑2 * ↑(k + 1) + ↑1) * (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2) ^ ↑(k + 1) ≤
      (((1:ℝ) / (2 * ↑(n + 1) + 1)) ^ 2) ^ ↑(k + 1) := by
    refine' fun k => mul_le_of_le_one_left (pow_nonneg h_nonneg ↑(k + 1)) _
    rw [one_div]
    exact inv_le_one (le_add_of_nonneg_left <| by positivity)
  

Proof state:
n : ℕ
h_nonneg : 0 ≤ (1 / (2 * ↑(n + 1) + 1)) ^ 2
g :
  HasSum (fun k => ((1 / (2 * ↑(n + 1) + 1)) ^ 2) ^ (k + 1))
    ((1 / (2 * ↑(n + 1) + 1)) ^ 2 / (1 - (1 / (2 * ↑(n + 1) + 1)) ^ 2))
hab :
  ∀ (k : ℕ),
    1 / (2 * ↑(k + 1) + 1) * ((1 / (2 * ↑(n + 1) + 1)) ^ 2) ^ (k + 1) ≤ ((1 / (2 * ↑(n + 1) + 1)) ^ 2) ^ (k + 1)
⊢ Real.log (stirlingSeq (n + 1)) - Real.log (stirlingSeq (n + 2)) ≤
    (1 / (2 * ↑(n + 1) + 1)) ^ 2 / (1 - (1 / (2 * ↑(n + 1) + 1)) ^ 2)

Next tactic:
exact hasSum_le hab (log_stirlingSeq_diff_hasSum n) g
/-- We have the bound `log (stirlingSeq n) - log (stirlingSeq (n+1))` ≤ 1/(4 n^2)
-/
theorem log_stirlingSeq_sub_log_stirlingSeq_succ (n : ℕ) :
    log (stirlingSeq (n + 1)) - log (stirlingSeq (n + 2)) ≤ 1 / (4 * (↑(n + 1):ℝ) ^ 2) := by
  

Proof state:
n : ℕ
⊢ Real.log (stirlingSeq (n + 1)) - Real.log (stirlingSeq (n + 2)) ≤ 1 / (4 * ↑(n + 1) ^ 2)

Next tactic:
have h₁ : (0 : ℝ) < ↑4 * ((n:ℝ) + 1) ^ 2 := by positivity
/-- We have the bound `log (stirlingSeq n) - log (stirlingSeq (n+1))` ≤ 1/(4 n^2)
-/
theorem log_stirlingSeq_sub_log_stirlingSeq_succ (n : ℕ) :
    log (stirlingSeq (n + 1)) - log (stirlingSeq (n + 2)) ≤ 1 / (4 * (↑(n + 1):ℝ) ^ 2) := by
  have h₁ : (0 : ℝ) < ↑4 * ((n:ℝ) + 1) ^ 2 := by 

Proof state:
n : ℕ
⊢ 0 < 4 * (↑n + 1) ^ 2

Next tactic:
positivity
/-- We have the bound `log (stirlingSeq n) - log (stirlingSeq (n+1))` ≤ 1/(4 n^2)
-/
theorem log_stirlingSeq_sub_log_stirlingSeq_succ (n : ℕ) :
    log (stirlingSeq (n + 1)) - log (stirlingSeq (n + 2)) ≤ 1 / (4 * (↑(n + 1):ℝ) ^ 2) := by
  have h₁ : (0 : ℝ) < ↑4 * ((n:ℝ) + 1) ^ 2 := by positivity
  

Proof state:
n : ℕ
h₁ : 0 < 4 * (↑n + 1) ^ 2
⊢ Real.log (stirlingSeq (n + 1)) - Real.log (stirlingSeq (n + 2)) ≤ 1 / (4 * ↑(n + 1) ^ 2)

Next tactic:
have h₃ : (0 : ℝ) < (2 * ((n:ℝ) + 1) + 1) ^ 2 := by positivity
/-- We have the bound `log (stirlingSeq n) - log (stirlingSeq (n+1))` ≤ 1/(4 n^2)
-/
theorem log_stirlingSeq_sub_log_stirlingSeq_succ (n : ℕ) :
    log (stirlingSeq (n + 1)) - log (stirlingSeq (n + 2)) ≤ 1 / (4 * (↑(n + 1):ℝ) ^ 2) := by
  have h₁ : (0 : ℝ) < ↑4 * ((n:ℝ) + 1) ^ 2 := by positivity
  have h₃ : (0 : ℝ) < (2 * ((n:ℝ) + 1) + 1) ^ 2 := by 

Proof state:
n : ℕ
h₁ : 0 < 4 * (↑n + 1) ^ 2
⊢ 0 < (2 * (↑n + 1) + 1) ^ 2

Next tactic:
positivity
/-- We have the bound `log (stirlingSeq n) - log (stirlingSeq (n+1))` ≤ 1/(4 n^2)
-/
theorem log_stirlingSeq_sub_log_stirlingSeq_succ (n : ℕ) :
    log (stirlingSeq (n + 1)) - log (stirlingSeq (n + 2)) ≤ 1 / (4 * (↑(n + 1):ℝ) ^ 2) := by
  have h₁ : (0 : ℝ) < ↑4 * ((n:ℝ) + 1) ^ 2 := by positivity
  have h₃ : (0 : ℝ) < (2 * ((n:ℝ) + 1) + 1) ^ 2 := by positivity
  

Proof state:
n : ℕ
h₁ : 0 < 4 * (↑n + 1) ^ 2
h₃ : 0 < (2 * (↑n + 1) + 1) ^ 2
⊢ Real.log (stirlingSeq (n + 1)) - Real.log (stirlingSeq (n + 2)) ≤ 1 / (4 * ↑(n + 1) ^ 2)

Next tactic:
have h₂ : (0 : ℝ) < ↑1 - (1 / (2 * ((n:ℝ) + 1) + 1)) ^ 2 := by
    rw [← mul_lt_mul_right h₃]
    have H : ↑0 < (2 * ((n:ℝ) + 1) + 1) ^ 2 - 1 := by nlinarith [@cast_nonneg ℝ _ n]
    convert H using 1 <;> field_simp [h₃.ne']
/-- We have the bound `log (stirlingSeq n) - log (stirlingSeq (n+1))` ≤ 1/(4 n^2)
-/
theorem log_stirlingSeq_sub_log_stirlingSeq_succ (n : ℕ) :
    log (stirlingSeq (n + 1)) - log (stirlingSeq (n + 2)) ≤ 1 / (4 * (↑(n + 1):ℝ) ^ 2) := by
  have h₁ : (0 : ℝ) < ↑4 * ((n:ℝ) + 1) ^ 2 := by positivity
  have h₃ : (0 : ℝ) < (2 * ((n:ℝ) + 1) + 1) ^ 2 := by positivity
  have h₂ : (0 : ℝ) < ↑1 - (1 / (2 * ((n:ℝ) + 1) + 1)) ^ 2 := by
    

Proof state:
n : ℕ
h₁ : 0 < 4 * (↑n + 1) ^ 2
h₃ : 0 < (2 * (↑n + 1) + 1) ^ 2
⊢ 0 < 1 - (1 / (2 * (↑n + 1) + 1)) ^ 2

Next tactic:
rw [← mul_lt_mul_right h₃]
/-- We have the bound `log (stirlingSeq n) - log (stirlingSeq (n+1))` ≤ 1/(4 n^2)
-/
theorem log_stirlingSeq_sub_log_stirlingSeq_succ (n : ℕ) :
    log (stirlingSeq (n + 1)) - log (stirlingSeq (n + 2)) ≤ 1 / (4 * (↑(n + 1):ℝ) ^ 2) := by
  have h₁ : (0 : ℝ) < ↑4 * ((n:ℝ) + 1) ^ 2 := by positivity
  have h₃ : (0 : ℝ) < (2 * ((n:ℝ) + 1) + 1) ^ 2 := by positivity
  have h₂ : (0 : ℝ) < ↑1 - (1 / (2 * ((n:ℝ) + 1) + 1)) ^ 2 := by
    rw [← mul_lt_mul_right h₃]
    

Proof state:
n : ℕ
h₁ : 0 < 4 * (↑n + 1) ^ 2
h₃ : 0 < (2 * (↑n + 1) + 1) ^ 2
⊢ 0 * (2 * (↑n + 1) + 1) ^ 2 < (1 - (1 / (2 * (↑n + 1) + 1)) ^ 2) * (2 * (↑n + 1) + 1) ^ 2

Next tactic:
have H : ↑0 < (2 * ((n:ℝ) + 1) + 1) ^ 2 - 1 := by nlinarith [@cast_nonneg ℝ _ n]
/-- We have the bound `log (stirlingSeq n) - log (stirlingSeq (n+1))` ≤ 1/(4 n^2)
-/
theorem log_stirlingSeq_sub_log_stirlingSeq_succ (n : ℕ) :
    log (stirlingSeq (n + 1)) - log (stirlingSeq (n + 2)) ≤ 1 / (4 * (↑(n + 1):ℝ) ^ 2) := by
  have h₁ : (0 : ℝ) < ↑4 * ((n:ℝ) + 1) ^ 2 := by positivity
  have h₃ : (0 : ℝ) < (2 * ((n:ℝ) + 1) + 1) ^ 2 := by positivity
  have h₂ : (0 : ℝ) < ↑1 - (1 / (2 * ((n:ℝ) + 1) + 1)) ^ 2 := by
    rw [← mul_lt_mul_right h₃]
    have H : ↑0 < (2 * ((n:ℝ) + 1) + 1) ^ 2 - 1 := by 

Proof state:
n : ℕ
h₁ : 0 < 4 * (↑n + 1) ^ 2
h₃ : 0 < (2 * (↑n + 1) + 1) ^ 2
⊢ 0 < (2 * (↑n + 1) + 1) ^ 2 - 1

Next tactic:
nlinarith [@cast_nonneg ℝ _ n]