l3lab/ntp-mathlib huggingface dataset:
theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
    (h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
    IsCauSeq abv fun m => ∑ n in range m, f n := by
  have har1 : |r| < 1 := by rwa [abs_of_nonneg hr0]
  refine'
    isCauSeq_series_of_abv_le_of_isCauSeq n.succ _
      (isCauSeq_geo_series_const (abv (f n.succ) * r⁻¹ ^ n.succ) har1)
  intro m hmn
  cases' Classical.em (r = 0) with r_zero r_ne_zero
  · have m_pos := lt_of_lt_of_le (Nat.succ_pos n) hmn
    have := h m.pred (Nat.le_of_succ_le_succ (by rwa [Nat.succ_pred_eq_of_pos m_pos]))
    simpa [r_zero, Nat.succ_pred_eq_of_pos m_pos, pow_succ] using this
  generalize hk : m - n.succ = k
  have r_pos : 0 < r := lt_of_le_of_ne hr0 (Ne.symm r_ne_zero)
  replace hk : m = k + n.succ := (tsub_eq_iff_eq_add_of_le hmn).1 hk
  induction' k with k ih generalizing m n
  · rw [hk, Nat.zero_add, mul_right_comm, inv_pow _ _, ← div_eq_mul_inv, mul_div_cancel]
    exact (ne_of_lt (pow_pos r_pos _)).symm
  · have kn : k + n.succ ≥ n.succ := by
      rw [← zero_add n.succ]; exact add_le_add (Nat.zero_le _) (by simp)
    erw [hk, Nat.succ_add, pow_succ' r, ← mul_assoc]
    refine
      le_trans (by rw [mul_comm] <;> 

Proof state:
α : Type u_1
inst✝³ : LinearOrderedField α
inst✝² : Archimedean α
β : Type u_2
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
r : α
hr0 : 0 ≤ r
hr1 : r < 1
har1 : abs' r < 1
r_ne_zero : ¬r = 0
r_pos : 0 < r
k : ℕ
ih :
  ∀ (n : ℕ),
    (∀ (m : ℕ), n ≤ m → abv (f (Nat.succ m)) ≤ r * abv (f m)) →
      ∀ (m : ℕ), Nat.succ n ≤ m → m = k + Nat.succ n → abv (f m) ≤ abv (f (Nat.succ n)) * r⁻¹ ^ Nat.succ n * r ^ m
n : ℕ
h : ∀ (m : ℕ), n ≤ m → abv (f (Nat.succ m)) ≤ r * abv (f m)
m : ℕ
hmn : Nat.succ n ≤ m
hk : m = Nat.succ k + Nat.succ n
kn : k + Nat.succ n ≥ Nat.succ n
⊢ abv (f (Nat.succ (k + Nat.succ n))) ≤ r * ?a

Next tactic:
exact h _ (Nat.le_of_succ_le kn)
theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
    (h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
    IsCauSeq abv fun m => ∑ n in range m, f n := by
  have har1 : |r| < 1 := by rwa [abs_of_nonneg hr0]
  refine'
    isCauSeq_series_of_abv_le_of_isCauSeq n.succ _
      (isCauSeq_geo_series_const (abv (f n.succ) * r⁻¹ ^ n.succ) har1)
  intro m hmn
  cases' Classical.em (r = 0) with r_zero r_ne_zero
  · have m_pos := lt_of_lt_of_le (Nat.succ_pos n) hmn
    have := h m.pred (Nat.le_of_succ_le_succ (by rwa [Nat.succ_pred_eq_of_pos m_pos]))
    simpa [r_zero, Nat.succ_pred_eq_of_pos m_pos, pow_succ] using this
  generalize hk : m - n.succ = k
  have r_pos : 0 < r := lt_of_le_of_ne hr0 (Ne.symm r_ne_zero)
  replace hk : m = k + n.succ := (tsub_eq_iff_eq_add_of_le hmn).1 hk
  induction' k with k ih generalizing m n
  · rw [hk, Nat.zero_add, mul_right_comm, inv_pow _ _, ← div_eq_mul_inv, mul_div_cancel]
    exact (ne_of_lt (pow_pos r_pos _)).symm
  · have kn : k + n.succ ≥ n.succ := by
      rw [← zero_add n.succ]; exact add_le_add (Nat.zero_le _) (by simp)
    erw [hk, Nat.succ_add, pow_succ' r, ← mul_assoc]
    refine
      le_trans (by rw [mul_comm] <;> exact h _ (Nat.le_of_succ_le kn))
        (mul_le_mul_of_nonneg_right ?_ hr0)
    

Proof state:
case inr.succ
α : Type u_1
inst✝³ : LinearOrderedField α
inst✝² : Archimedean α
β : Type u_2
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
r : α
hr0 : 0 ≤ r
hr1 : r < 1
har1 : abs' r < 1
r_ne_zero : ¬r = 0
r_pos : 0 < r
k : ℕ
ih :
  ∀ (n : ℕ),
    (∀ (m : ℕ), n ≤ m → abv (f (Nat.succ m)) ≤ r * abv (f m)) →
      ∀ (m : ℕ), Nat.succ n ≤ m → m = k + Nat.succ n → abv (f m) ≤ abv (f (Nat.succ n)) * r⁻¹ ^ Nat.succ n * r ^ m
n : ℕ
h : ∀ (m : ℕ), n ≤ m → abv (f (Nat.succ m)) ≤ r * abv (f m)
m : ℕ
hmn : Nat.succ n ≤ m
hk : m = Nat.succ k + Nat.succ n
kn : k + Nat.succ n ≥ Nat.succ n
⊢ abv (f (k + Nat.succ n)) ≤ abv (f (Nat.succ n)) * r⁻¹ ^ Nat.succ n * r ^ (k + Nat.succ n)

Next tactic:
exact ih _ h _ (by simp) rfl
theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
    (h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
    IsCauSeq abv fun m => ∑ n in range m, f n := by
  have har1 : |r| < 1 := by rwa [abs_of_nonneg hr0]
  refine'
    isCauSeq_series_of_abv_le_of_isCauSeq n.succ _
      (isCauSeq_geo_series_const (abv (f n.succ) * r⁻¹ ^ n.succ) har1)
  intro m hmn
  cases' Classical.em (r = 0) with r_zero r_ne_zero
  · have m_pos := lt_of_lt_of_le (Nat.succ_pos n) hmn
    have := h m.pred (Nat.le_of_succ_le_succ (by rwa [Nat.succ_pred_eq_of_pos m_pos]))
    simpa [r_zero, Nat.succ_pred_eq_of_pos m_pos, pow_succ] using this
  generalize hk : m - n.succ = k
  have r_pos : 0 < r := lt_of_le_of_ne hr0 (Ne.symm r_ne_zero)
  replace hk : m = k + n.succ := (tsub_eq_iff_eq_add_of_le hmn).1 hk
  induction' k with k ih generalizing m n
  · rw [hk, Nat.zero_add, mul_right_comm, inv_pow _ _, ← div_eq_mul_inv, mul_div_cancel]
    exact (ne_of_lt (pow_pos r_pos _)).symm
  · have kn : k + n.succ ≥ n.succ := by
      rw [← zero_add n.succ]; exact add_le_add (Nat.zero_le _) (by simp)
    erw [hk, Nat.succ_add, pow_succ' r, ← mul_assoc]
    refine
      le_trans (by rw [mul_comm] <;> exact h _ (Nat.le_of_succ_le kn))
        (mul_le_mul_of_nonneg_right ?_ hr0)
    exact ih _ h _ (by 

Proof state:
α : Type u_1
inst✝³ : LinearOrderedField α
inst✝² : Archimedean α
β : Type u_2
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
r : α
hr0 : 0 ≤ r
hr1 : r < 1
har1 : abs' r < 1
r_ne_zero : ¬r = 0
r_pos : 0 < r
k : ℕ
ih :
  ∀ (n : ℕ),
    (∀ (m : ℕ), n ≤ m → abv (f (Nat.succ m)) ≤ r * abv (f m)) →
      ∀ (m : ℕ), Nat.succ n ≤ m → m = k + Nat.succ n → abv (f m) ≤ abv (f (Nat.succ n)) * r⁻¹ ^ Nat.succ n * r ^ m
n : ℕ
h : ∀ (m : ℕ), n ≤ m → abv (f (Nat.succ m)) ≤ r * abv (f m)
m : ℕ
hmn : Nat.succ n ≤ m
hk : m = Nat.succ k + Nat.succ n
kn : k + Nat.succ n ≥ Nat.succ n
⊢ Nat.succ n ≤ k + Nat.succ n

Next tactic:
simp
theorem sum_range_diag_flip {α : Type*} [AddCommMonoid α] (n : ℕ) (f : ℕ → ℕ → α) :
    (∑ m in range n, ∑ k in range (m + 1), f k (m - k)) =
      ∑ m in range n, ∑ k in range (n - m), f m k := by
  

Proof state:
α✝ : Type u_1
inst✝⁴ : LinearOrderedField α✝
inst✝³ : Archimedean α✝
β : Type u_2
inst✝² : Ring β
abv : β → α✝
inst✝¹ : IsAbsoluteValue abv
α : Type u_3
inst✝ : AddCommMonoid α
n : ℕ
f : ℕ → ℕ → α
⊢ ∑ m in range n, ∑ k in range (m + 1), f k (m - k) = ∑ m in range n, ∑ k in range (n - m), f m k

Next tactic:
rw [sum_sigma', sum_sigma']
theorem sum_range_diag_flip {α : Type*} [AddCommMonoid α] (n : ℕ) (f : ℕ → ℕ → α) :
    (∑ m in range n, ∑ k in range (m + 1), f k (m - k)) =
      ∑ m in range n, ∑ k in range (n - m), f m k := by
  rw [sum_sigma', sum_sigma']
  

Proof state:
α✝ : Type u_1
inst✝⁴ : LinearOrderedField α✝
inst✝³ : Archimedean α✝
β : Type u_2
inst✝² : Ring β
abv : β → α✝
inst✝¹ : IsAbsoluteValue abv
α : Type u_3
inst✝ : AddCommMonoid α
n : ℕ
f : ℕ → ℕ → α
⊢ ∑ x in Finset.sigma (range n) fun m => range (m + 1), f x.snd (x.fst - x.snd) =
    ∑ x in Finset.sigma (range n) fun m => range (n - m), f x.fst x.snd

Next tactic:
exact
    sum_bij (fun a _ => ⟨a.2, a.1 - a.2⟩)
      (fun a ha =>
        have h₁ : a.1 < n := mem_range.1 (mem_sigma.1 ha).1
        have h₂ : a.2 < Nat.succ a.1 := mem_range.1 (mem_sigma.1 ha).2
        mem_sigma.2
          ⟨mem_range.2 (lt_of_lt_of_le h₂ h₁),
            mem_range.2 ((tsub_lt_tsub_iff_right (Nat.le_of_lt_succ h₂)).2 h₁)⟩)
      (fun _ _ => rfl)
      (fun ⟨a₁, a₂⟩ ⟨b₁, b₂⟩ ha hb h =>
        have ha : a₁ < n ∧ a₂ ≤ a₁ :=
          ⟨mem_range.1 (mem_sigma.1 ha).1, Nat.le_of_lt_succ (mem_range.1 (mem_sigma.1 ha).2)⟩
        have hb : b₁ < n ∧ b₂ ≤ b₁ :=
          ⟨mem_range.1 (mem_sigma.1 hb).1, Nat.le_of_lt_succ (mem_range.1 (mem_sigma.1 hb).2)⟩
        have h : a₂ = b₂ ∧ _ := by simpa using h
        have h' : a₁ = b₁ - b₂ + a₂ := (tsub_eq_iff_eq_add_of_le ha.2).1 (eq_of_heq h.2)
        Sigma.mk.inj_iff.2 ⟨tsub_add_cancel_of_le hb.2 ▸ h'.symm ▸ h.1 ▸ rfl, heq_of_eq h.1⟩)
      fun ⟨a₁, a₂⟩ ha =>
      have ha : a₁ < n ∧ a₂ < n - a₁ :=
        ⟨mem_range.1 (mem_sigma.1 ha).1, mem_range.1 (mem_sigma.1 ha).2⟩
      ⟨⟨a₂ + a₁, a₁⟩,
        ⟨mem_sigma.2
            ⟨mem_range.2 (lt_tsub_iff_right.1 ha.2),
              mem_range.2 (Nat.lt_succ_of_le (Nat.le_add_left _ _))⟩,
          Sigma.mk.inj_iff.2 ⟨rfl, heq_of_eq (add_tsub_cancel_right _ _).symm⟩⟩⟩
theorem sum_range_diag_flip {α : Type*} [AddCommMonoid α] (n : ℕ) (f : ℕ → ℕ → α) :
    (∑ m in range n, ∑ k in range (m + 1), f k (m - k)) =
      ∑ m in range n, ∑ k in range (n - m), f m k := by
  rw [sum_sigma', sum_sigma']
  exact
    sum_bij (fun a _ => ⟨a.2, a.1 - a.2⟩)
      (fun a ha =>
        have h₁ : a.1 < n := mem_range.1 (mem_sigma.1 ha).1
        have h₂ : a.2 < Nat.succ a.1 := mem_range.1 (mem_sigma.1 ha).2
        mem_sigma.2
          ⟨mem_range.2 (lt_of_lt_of_le h₂ h₁),
            mem_range.2 ((tsub_lt_tsub_iff_right (Nat.le_of_lt_succ h₂)).2 h₁)⟩)
      (fun _ _ => rfl)
      (fun ⟨a₁, a₂⟩ ⟨b₁, b₂⟩ ha hb h =>
        have ha : a₁ < n ∧ a₂ ≤ a₁ :=
          ⟨mem_range.1 (mem_sigma.1 ha).1, Nat.le_of_lt_succ (mem_range.1 (mem_sigma.1 ha).2)⟩
        have hb : b₁ < n ∧ b₂ ≤ b₁ :=
          ⟨mem_range.1 (mem_sigma.1 hb).1, Nat.le_of_lt_succ (mem_range.1 (mem_sigma.1 hb).2)⟩
        have h : a₂ = b₂ ∧ _ := by 

Proof state:
α✝ : Type u_1
inst✝⁴ : LinearOrderedField α✝
inst✝³ : Archimedean α✝
β : Type u_2
inst✝² : Ring β
abv : β → α✝
inst✝¹ : IsAbsoluteValue abv
α : Type u_3
inst✝ : AddCommMonoid α
n : ℕ
f : ℕ → ℕ → α
x✝¹ x✝ : (_ : ℕ) × ℕ
a₁ a₂ : ℕ
ha✝ : { fst := a₁, snd := a₂ } ∈ Finset.sigma (range n) fun m => range (m + 1)
b₁ b₂ : ℕ
hb✝ : { fst := b₁, snd := b₂ } ∈ Finset.sigma (range n) fun m => range (m + 1)
h :
  (fun a x => { fst := a.snd, snd := a.fst - a.snd }) { fst := a₁, snd := a₂ } ha✝ =
    (fun a x => { fst := a.snd, snd := a.fst - a.snd }) { fst := b₁, snd := b₂ } hb✝
ha : a₁ < n ∧ a₂ ≤ a₁
hb : b₁ < n ∧ b₂ ≤ b₁
⊢ a₂ = b₂ ∧ HEq (a₁ - a₂) (b₁ - b₂)

Next tactic:
simpa using h
theorem abv_sum_le_sum_abv {γ : Type*} (f : γ → β) (s : Finset γ) :
    abv (∑ k in s, f k) ≤ ∑ k in s, abv (f k) :=
  haveI := Classical.decEq γ
  Finset.induction_on s (by 

Proof state:
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Semiring β
inst✝ : IsAbsoluteValue abv
γ : Type u_3
f : γ → β
s : Finset γ
this : DecidableEq γ
⊢ abv (∑ k in ∅, f k) ≤ ∑ k in ∅, abv (f k)

Next tactic:
simp [abv_zero abv]
theorem abv_sum_le_sum_abv {γ : Type*} (f : γ → β) (s : Finset γ) :
    abv (∑ k in s, f k) ≤ ∑ k in s, abv (f k) :=
  haveI := Classical.decEq γ
  Finset.induction_on s (by simp [abv_zero abv]) fun a s has ih => by
    

Proof state:
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Semiring β
inst✝ : IsAbsoluteValue abv
γ : Type u_3
f : γ → β
s✝ : Finset γ
this : DecidableEq γ
a : γ
s : Finset γ
has : a ∉ s
ih : abv (∑ k in s, f k) ≤ ∑ k in s, abv (f k)
⊢ abv (∑ k in insert a s, f k) ≤ ∑ k in insert a s, abv (f k)

Next tactic:
rw [sum_insert has, sum_insert has]
theorem abv_sum_le_sum_abv {γ : Type*} (f : γ → β) (s : Finset γ) :
    abv (∑ k in s, f k) ≤ ∑ k in s, abv (f k) :=
  haveI := Classical.decEq γ
  Finset.induction_on s (by simp [abv_zero abv]) fun a s has ih => by
    rw [sum_insert has, sum_insert has]; 

Proof state:
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Semiring β
inst✝ : IsAbsoluteValue abv
γ : Type u_3
f : γ → β
s✝ : Finset γ
this : DecidableEq γ
a : γ
s : Finset γ
has : a ∉ s
ih : abv (∑ k in s, f k) ≤ ∑ k in s, abv (f k)
⊢ abv (f a + ∑ x in s, f x) ≤ abv (f a) + ∑ x in s, abv (f x)

Next tactic:
exact le_trans (abv_add abv _ _) (add_le_add_left ih _)
theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
    (hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
    ∃ i : ℕ,
      ∀ j ≥ i,
        abv
            (((∑ k in range j, a k) * ∑ k in range j, b k) -
              ∑ n in range j, ∑ m in range (n + 1), a m * b (n - m)) <
          ε :=
  let ⟨Q, hQ⟩ := CauSeq.bounded ⟨_, hb⟩
  let ⟨P, hP⟩ := CauSeq.bounded ⟨_, ha⟩
  have hP0 : 0 < P := lt_of_le_of_lt (abs_nonneg _) (hP 0)
  have hPε0 : 0 < ε / (2 * P) := div_pos ε0 (mul_pos (show (2 : α) > 0 by 

Proof state:
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, property := hb } i) < Q
P : α
hP : ∀ (i : ℕ), abs' (↑{ val := fun m => ∑ n in range m, abv (a n), property := ha } i) < P
hP0 : 0 < P
⊢ 2 > 0

Next tactic:
norm_num
theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
    (hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
    ∃ i : ℕ,
      ∀ j ≥ i,
        abv
            (((∑ k in range j, a k) * ∑ k in range j, b k) -
              ∑ n in range j, ∑ m in range (n + 1), a m * b (n - m)) <
          ε :=
  let ⟨Q, hQ⟩ := CauSeq.bounded ⟨_, hb⟩
  let ⟨P, hP⟩ := CauSeq.bounded ⟨_, ha⟩
  have hP0 : 0 < P := lt_of_le_of_lt (abs_nonneg _) (hP 0)
  have hPε0 : 0 < ε / (2 * P) := div_pos ε0 (mul_pos (show (2 : α) > 0 by norm_num) hP0)
  let ⟨N, hN⟩ := CauSeq.cauchy₂ ⟨_, hb⟩ hPε0
  have hQε0 : 0 < ε / (4 * Q) :=
    div_pos ε0 (mul_pos (show (0 : α) < 4 by 

Proof state:
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, property := hb } i) < Q
P : α
hP : ∀ (i : ℕ), abs' (↑{ val := fun m => ∑ n in range m, abv (a n), property := ha } i) < P
hP0 : 0 < P
hPε0 : 0 < ε / (2 * P)
N : ℕ
hN :
  ∀ j ≥ N,
    ∀ k ≥ N,
      abv
          (↑{ val := fun m => ∑ n in range m, b n, property := hb } j -
            ↑{ val := fun m => ∑ n in range m, b n, property := hb } k) <
        ε / (2 * P)
⊢ 0 < 4

Next tactic:
norm_num
theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
    (hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
    ∃ i : ℕ,
      ∀ j ≥ i,
        abv
            (((∑ k in range j, a k) * ∑ k in range j, b k) -
              ∑ n in range j, ∑ m in range (n + 1), a m * b (n - m)) <
          ε :=
  let ⟨Q, hQ⟩ := CauSeq.bounded ⟨_, hb⟩
  let ⟨P, hP⟩ := CauSeq.bounded ⟨_, ha⟩
  have hP0 : 0 < P := lt_of_le_of_lt (abs_nonneg _) (hP 0)
  have hPε0 : 0 < ε / (2 * P) := div_pos ε0 (mul_pos (show (2 : α) > 0 by norm_num) hP0)
  let ⟨N, hN⟩ := CauSeq.cauchy₂ ⟨_, hb⟩ hPε0
  have hQε0 : 0 < ε / (4 * Q) :=
    div_pos ε0 (mul_pos (show (0 : α) < 4 by norm_num) (lt_of_le_of_lt (abv_nonneg _ _) (hQ 0)))
  let ⟨M, hM⟩ := CauSeq.cauchy₂ ⟨_, ha⟩ hQε0
  ⟨2 * (max N M + 1), fun K hK => by
    

Proof state:
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, property := hb } i) < Q
P : α
hP : ∀ (i : ℕ), abs' (↑{ val := fun m => ∑ n in range m, abv (a n), property := ha } i) < P
hP0 : 0 < P
hPε0 : 0 < ε / (2 * P)
N : ℕ
hN :
  ∀ j ≥ N,
    ∀ k ≥ N,
      abv
          (↑{ val := fun m => ∑ n in range m, b n, property := hb } j -
            ↑{ val := fun m => ∑ n in range m, b n, property := hb } k) <
        ε / (2 * P)
hQε0 : 0 < ε / (4 * Q)
M : ℕ
hM :
  ∀ j ≥ M,
    ∀ k ≥ M,
      abs'
          (↑{ val := fun m => ∑ n in range m, abv (a n), property := ha } j -
            ↑{ val := fun m => ∑ n in range m, abv (a n), property := ha } k) <
        ε / (4 * Q)
K : ℕ
hK : K ≥ 2 * (max N M + 1)
⊢ abv ((∑ k in range K, a k) * ∑ k in range K, b k - ∑ n in range K, ∑ m in range (n + 1), a m * b (n - m)) < ε

Next tactic:
have h₁ :
      (∑ m in range K, ∑ k in range (m + 1), a k * b (m - k)) =
        ∑ m in range K, ∑ n in range (K - m), a m * b n :=
      by simpa using sum_range_diag_flip K fun m n => a m * b n
theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
    (hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
    ∃ i : ℕ,
      ∀ j ≥ i,
        abv
            (((∑ k in range j, a k) * ∑ k in range j, b k) -
              ∑ n in range j, ∑ m in range (n + 1), a m * b (n - m)) <
          ε :=
  let ⟨Q, hQ⟩ := CauSeq.bounded ⟨_, hb⟩
  let ⟨P, hP⟩ := CauSeq.bounded ⟨_, ha⟩
  have hP0 : 0 < P := lt_of_le_of_lt (abs_nonneg _) (hP 0)
  have hPε0 : 0 < ε / (2 * P) := div_pos ε0 (mul_pos (show (2 : α) > 0 by norm_num) hP0)
  let ⟨N, hN⟩ := CauSeq.cauchy₂ ⟨_, hb⟩ hPε0
  have hQε0 : 0 < ε / (4 * Q) :=
    div_pos ε0 (mul_pos (show (0 : α) < 4 by norm_num) (lt_of_le_of_lt (abv_nonneg _ _) (hQ 0)))
  let ⟨M, hM⟩ := CauSeq.cauchy₂ ⟨_, ha⟩ hQε0
  ⟨2 * (max N M + 1), fun K hK => by
    have h₁ :
      (∑ m in range K, ∑ k in range (m + 1), a k * b (m - k)) =
        ∑ m in range K, ∑ n in range (K - m), a m * b n :=
      by 

Proof state:
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, property := hb } i) < Q
P : α
hP : ∀ (i : ℕ), abs' (↑{ val := fun m => ∑ n in range m, abv (a n), property := ha } i) < P
hP0 : 0 < P
hPε0 : 0 < ε / (2 * P)
N : ℕ
hN :
  ∀ j ≥ N,
    ∀ k ≥ N,
      abv
          (↑{ val := fun m => ∑ n in range m, b n, property := hb } j -
            ↑{ val := fun m => ∑ n in range m, b n, property := hb } k) <
        ε / (2 * P)
hQε0 : 0 < ε / (4 * Q)
M : ℕ
hM :
  ∀ j ≥ M,
    ∀ k ≥ M,
      abs'
          (↑{ val := fun m => ∑ n in range m, abv (a n), property := ha } j -
            ↑{ val := fun m => ∑ n in range m, abv (a n), property := ha } k) <
        ε / (4 * Q)
K : ℕ
hK : K ≥ 2 * (max N M + 1)
⊢ ∑ m in range K, ∑ k in range (m + 1), a k * b (m - k) = ∑ m in range K, ∑ n in range (K - m), a m * b n

Next tactic:
simpa using sum_range_diag_flip K fun m n => a m * b n
theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
    (hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
    ∃ i : ℕ,
      ∀ j ≥ i,
        abv
            (((∑ k in range j, a k) * ∑ k in range j, b k) -
              ∑ n in range j, ∑ m in range (n + 1), a m * b (n - m)) <
          ε :=
  let ⟨Q, hQ⟩ := CauSeq.bounded ⟨_, hb⟩
  let ⟨P, hP⟩ := CauSeq.bounded ⟨_, ha⟩
  have hP0 : 0 < P := lt_of_le_of_lt (abs_nonneg _) (hP 0)
  have hPε0 : 0 < ε / (2 * P) := div_pos ε0 (mul_pos (show (2 : α) > 0 by norm_num) hP0)
  let ⟨N, hN⟩ := CauSeq.cauchy₂ ⟨_, hb⟩ hPε0
  have hQε0 : 0 < ε / (4 * Q) :=
    div_pos ε0 (mul_pos (show (0 : α) < 4 by norm_num) (lt_of_le_of_lt (abv_nonneg _ _) (hQ 0)))
  let ⟨M, hM⟩ := CauSeq.cauchy₂ ⟨_, ha⟩ hQε0
  ⟨2 * (max N M + 1), fun K hK => by
    have h₁ :
      (∑ m in range K, ∑ k in range (m + 1), a k * b (m - k)) =
        ∑ m in range K, ∑ n in range (K - m), a m * b n :=
      by simpa using sum_range_diag_flip K fun m n => a m * b n
    

Proof state:
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, property := hb } i) < Q
P : α
hP : ∀ (i : ℕ), abs' (↑{ val := fun m => ∑ n in range m, abv (a n), property := ha } i) < P
hP0 : 0 < P
hPε0 : 0 < ε / (2 * P)
N : ℕ
hN :
  ∀ j ≥ N,
    ∀ k ≥ N,
      abv
          (↑{ val := fun m => ∑ n in range m, b n, property := hb } j -
            ↑{ val := fun m => ∑ n in range m, b n, property := hb } k) <
        ε / (2 * P)
hQε0 : 0 < ε / (4 * Q)
M : ℕ
hM :
  ∀ j ≥ M,
    ∀ k ≥ M,
      abs'
          (↑{ val := fun m => ∑ n in range m, abv (a n), property := ha } j -
            ↑{ val := fun m => ∑ n in range m, abv (a n), property := ha } k) <
        ε / (4 * Q)
K : ℕ
hK : K ≥ 2 * (max N M + 1)
h₁ : ∑ m in range K, ∑ k in range (m + 1), a k * b (m - k) = ∑ m in range K, ∑ n in range (K - m), a m * b n
⊢ abv ((∑ k in range K, a k) * ∑ k in range K, b k - ∑ n in range K, ∑ m in range (n + 1), a m * b (n - m)) < ε

Next tactic:
have h₂ :
      (fun i => ∑ k in range (K - i), a i * b k) = fun i => a i * ∑ k in range (K - i), b k := by
      simp [Finset.mul_sum]
theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
    (hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
    ∃ i : ℕ,
      ∀ j ≥ i,
        abv
            (((∑ k in range j, a k) * ∑ k in range j, b k) -
              ∑ n in range j, ∑ m in range (n + 1), a m * b (n - m)) <
          ε :=
  let ⟨Q, hQ⟩ := CauSeq.bounded ⟨_, hb⟩
  let ⟨P, hP⟩ := CauSeq.bounded ⟨_, ha⟩
  have hP0 : 0 < P := lt_of_le_of_lt (abs_nonneg _) (hP 0)
  have hPε0 : 0 < ε / (2 * P) := div_pos ε0 (mul_pos (show (2 : α) > 0 by norm_num) hP0)
  let ⟨N, hN⟩ := CauSeq.cauchy₂ ⟨_, hb⟩ hPε0
  have hQε0 : 0 < ε / (4 * Q) :=
    div_pos ε0 (mul_pos (show (0 : α) < 4 by norm_num) (lt_of_le_of_lt (abv_nonneg _ _) (hQ 0)))
  let ⟨M, hM⟩ := CauSeq.cauchy₂ ⟨_, ha⟩ hQε0
  ⟨2 * (max N M + 1), fun K hK => by
    have h₁ :
      (∑ m in range K, ∑ k in range (m + 1), a k * b (m - k)) =
        ∑ m in range K, ∑ n in range (K - m), a m * b n :=
      by simpa using sum_range_diag_flip K fun m n => a m * b n
    have h₂ :
      (fun i => ∑ k in range (K - i), a i * b k) = fun i => a i * ∑ k in range (K - i), b k := by
      

Proof state:
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, property := hb } i) < Q
P : α
hP : ∀ (i : ℕ), abs' (↑{ val := fun m => ∑ n in range m, abv (a n), property := ha } i) < P
hP0 : 0 < P
hPε0 : 0 < ε / (2 * P)
N : ℕ
hN :
  ∀ j ≥ N,
    ∀ k ≥ N,
      abv
          (↑{ val := fun m => ∑ n in range m, b n, property := hb } j -
            ↑{ val := fun m => ∑ n in range m, b n, property := hb } k) <
        ε / (2 * P)
hQε0 : 0 < ε / (4 * Q)
M : ℕ
hM :
  ∀ j ≥ M,
    ∀ k ≥ M,
      abs'
          (↑{ val := fun m => ∑ n in range m, abv (a n), property := ha } j -
            ↑{ val := fun m => ∑ n in range m, abv (a n), property := ha } k) <
        ε / (4 * Q)
K : ℕ
hK : K ≥ 2 * (max N M + 1)
h₁ : ∑ m in range K, ∑ k in range (m + 1), a k * b (m - k) = ∑ m in range K, ∑ n in range (K - m), a m * b n
⊢ (fun i => ∑ k in range (K - i), a i * b k) = fun i => a i * ∑ k in range (K - i), b k

Next tactic:
simp [Finset.mul_sum]