awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Algebra.Order.CauSeq.BigOperators
Goal:
α : Type u_1
β : Type u_2
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
inst_3 : Ring β
abv : β → α
inst_4 : IsAbsoluteValue abv
f g : ℕ → β
ha : IsCauSeq.{u_1, u_1} (α := α) abs fun (m : ℕ) ↦ ∑ n ∈ range m, abv (f n)
hb : IsCauSeq abv fun (m : ℕ) ↦ ∑ n ∈ range m, g n
ε : α
ε0 : 0 < ε
P : α
hP : ∀ (i : ℕ), |∑ n ∈ range i, abv (f n)| < P
Q : α
hQ : ∀ (i : ℕ), abv (∑ n ∈ range i, g n) < Q
hP0 : 0 < P
hPε0 : 0 < ε / (2 * P)
N : ℕ
hN : ∀ j ≥ N, ∀ k ≥ N, abv (HSub.hSub (α := β) (∑ n ∈ range j, g n) (∑ n ∈ range k, g n)) < ε / (2 * P)
hQε0 : 0 < ε / (4 * Q)
M : ℕ
hM : ∀ j ≥ M, ∀ k ≥ M, |HSub.hSub (α := α) (∑ n ∈ range j, abv (f n)) (∑ n ∈ range k, abv (f n))| < ε / (4 * Q)
K : ℕ
hK : K ≥ 2 * (max N M + 1)
h₁ :
  Eq.{u_2 + 1} (α := β) (∑ m ∈ range K, ∑ k ∈ range (m + 1), f k * g (m - k))
    (∑ m ∈ range K, ∑ n ∈ range (K - m), f m * g n)
h₂ :
  Eq.{u_2 + 1} (α := (i : ℕ) → β) (fun (i : ℕ) ↦ ∑ k ∈ range (K - i), f i * g k) fun (i : ℕ) ↦
    f i * ∑ k ∈ range (K - i), g k
h₃ :
  ∑ i ∈ range K, f i * ∑ k ∈ range (K - i), g k =
    HAdd.hAdd (α := β) (∑ i ∈ range K, f i * HSub.hSub (α := β) (∑ k ∈ range (K - i), g k) (∑ k ∈ range K, g k))
      (∑ i ∈ range K, f i * ∑ k ∈ range K, g k)
two_mul_two : 4 = HMul.hMul (α := α) 2 2
hQ0 : Q ≠ 0
h2Q0 : 2 * Q ≠ 0
hε : ε / (2 * P) * P + ε / (4 * Q) * (2 * Q) = ε
hNMK : max N M + 1 < K
⊢ N < K

Tactic:
have hKN : N < K :=
    calc
      N ≤ max N M := le_max_left _ _
      _ < max N M + 1 := Nat.lt_succ_self _
      _ < K := hNMK
-- Module: Mathlib.Algebra.Order.CauSeq.BigOperators
Goal:
α : Type u_1
β : Type u_2
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
inst_3 : Ring β
abv : β → α
inst_4 : IsAbsoluteValue abv
f g : ℕ → β
ha : IsCauSeq.{u_1, u_1} (α := α) abs fun (m : ℕ) ↦ ∑ n ∈ range m, abv (f n)
hb : IsCauSeq abv fun (m : ℕ) ↦ ∑ n ∈ range m, g n
ε : α
ε0 : 0 < ε
P : α
hP : ∀ (i : ℕ), |∑ n ∈ range i, abv (f n)| < P
Q : α
hQ : ∀ (i : ℕ), abv (∑ n ∈ range i, g n) < Q
hP0 : 0 < P
hPε0 : 0 < ε / (2 * P)
N : ℕ
hN : ∀ j ≥ N, ∀ k ≥ N, abv (HSub.hSub (α := β) (∑ n ∈ range j, g n) (∑ n ∈ range k, g n)) < ε / (2 * P)
hQε0 : 0 < ε / (4 * Q)
M : ℕ
hM : ∀ j ≥ M, ∀ k ≥ M, |HSub.hSub (α := α) (∑ n ∈ range j, abv (f n)) (∑ n ∈ range k, abv (f n))| < ε / (4 * Q)
K : ℕ
hK : K ≥ 2 * (max N M + 1)
h₁ :
  Eq.{u_2 + 1} (α := β) (∑ m ∈ range K, ∑ k ∈ range (m + 1), f k * g (m - k))
    (∑ m ∈ range K, ∑ n ∈ range (K - m), f m * g n)
h₂ :
  Eq.{u_2 + 1} (α := (i : ℕ) → β) (fun (i : ℕ) ↦ ∑ k ∈ range (K - i), f i * g k) fun (i : ℕ) ↦
    f i * ∑ k ∈ range (K - i), g k
h₃ :
  ∑ i ∈ range K, f i * ∑ k ∈ range (K - i), g k =
    HAdd.hAdd (α := β) (∑ i ∈ range K, f i * HSub.hSub (α := β) (∑ k ∈ range (K - i), g k) (∑ k ∈ range K, g k))
      (∑ i ∈ range K, f i * ∑ k ∈ range K, g k)
two_mul_two : 4 = HMul.hMul (α := α) 2 2
hQ0 : Q ≠ 0
h2Q0 : 2 * Q ≠ 0
hε : ε / (2 * P) * P + ε / (4 * Q) * (2 * Q) = ε
hNMK : max N M + 1 < K
hKN : N < K
h :
  ∀ (m : ℕ) (hmJ : m ∈ range (max N M + 1)),
    abv (HSub.hSub (α := β) (∑ k ∈ range (K - m), g k) (∑ k ∈ range K, g k)) ≤ ε / (2 * P)
⊢ LE.le.{u_1} (α := α)
    (∑ i ∈ range (max N M + 1), abv (f i) * abv (HSub.hSub (α := β) (∑ k ∈ range (K - i), g k) (∑ k ∈ range K, g k)))
    (∑ i ∈ range (max N M + 1), abv (f i) * (ε / (2 * P)))

Tactic:
gcongr with m hmJ
-- Module: Mathlib.Algebra.Order.CauSeq.BigOperators
Goal:
α : Type u_1
β : Type u_2
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
inst_3 : Ring β
abv : β → α
inst_4 : IsAbsoluteValue abv
f g : ℕ → β
ha : IsCauSeq.{u_1, u_1} (α := α) abs fun (m : ℕ) ↦ ∑ n ∈ range m, abv (f n)
hb : IsCauSeq abv fun (m : ℕ) ↦ ∑ n ∈ range m, g n
ε : α
ε0 : 0 < ε
P : α
hP : ∀ (i : ℕ), |∑ n ∈ range i, abv (f n)| < P
Q : α
hQ : ∀ (i : ℕ), abv (∑ n ∈ range i, g n) < Q
hP0 : 0 < P
hPε0 : 0 < ε / (2 * P)
N : ℕ
hN : ∀ j ≥ N, ∀ k ≥ N, abv (HSub.hSub (α := β) (∑ n ∈ range j, g n) (∑ n ∈ range k, g n)) < ε / (2 * P)
hQε0 : 0 < ε / (4 * Q)
M : ℕ
hM : ∀ j ≥ M, ∀ k ≥ M, |HSub.hSub (α := α) (∑ n ∈ range j, abv (f n)) (∑ n ∈ range k, abv (f n))| < ε / (4 * Q)
K : ℕ
hK : K ≥ 2 * (max N M + 1)
h₁ :
  Eq.{u_2 + 1} (α := β) (∑ m ∈ range K, ∑ k ∈ range (m + 1), f k * g (m - k))
    (∑ m ∈ range K, ∑ n ∈ range (K - m), f m * g n)
h₂ :
  Eq.{u_2 + 1} (α := (i : ℕ) → β) (fun (i : ℕ) ↦ ∑ k ∈ range (K - i), f i * g k) fun (i : ℕ) ↦
    f i * ∑ k ∈ range (K - i), g k
h₃ :
  ∑ i ∈ range K, f i * ∑ k ∈ range (K - i), g k =
    HAdd.hAdd (α := β) (∑ i ∈ range K, f i * HSub.hSub (α := β) (∑ k ∈ range (K - i), g k) (∑ k ∈ range K, g k))
      (∑ i ∈ range K, f i * ∑ k ∈ range K, g k)
two_mul_two : 4 = HMul.hMul (α := α) 2 2
hQ0 : Q ≠ 0
h2Q0 : 2 * Q ≠ 0
hε : ε / (2 * P) * P + ε / (4 * Q) * (2 * Q) = ε
hNMK : max N M + 1 < K
hKN : N < K
m : ℕ
hmJ : m ∈ range (max N M + 1)
⊢ abv (HSub.hSub (α := β) (∑ k ∈ range (K - m), g k) (∑ k ∈ range K, g k)) ≤ ε / (2 * P)

Tactic:
le_of_lt <| hN (K - m) (le_tsub_of_add_le_left <| hK.trans' ?_) K hKN.le

Premises used: le_tsub_of_add_le_left, HSub.hSub, le_of_lt, LT.lt.le, LE.le.trans'
-- Module: Mathlib.Algebra.Order.CauSeq.BigOperators
Goal:
α : Type u_1
β : Type u_2
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
inst_3 : Ring β
abv : β → α
inst_4 : IsAbsoluteValue abv
f g : ℕ → β
ha : IsCauSeq.{u_1, u_1} (α := α) abs fun (m : ℕ) ↦ ∑ n ∈ range m, abv (f n)
hb : IsCauSeq abv fun (m : ℕ) ↦ ∑ n ∈ range m, g n
ε : α
ε0 : 0 < ε
P : α
hP : ∀ (i : ℕ), |∑ n ∈ range i, abv (f n)| < P
Q : α
hQ : ∀ (i : ℕ), abv (∑ n ∈ range i, g n) < Q
hP0 : 0 < P
hPε0 : 0 < ε / (2 * P)
N : ℕ
hN : ∀ j ≥ N, ∀ k ≥ N, abv (HSub.hSub (α := β) (∑ n ∈ range j, g n) (∑ n ∈ range k, g n)) < ε / (2 * P)
hQε0 : 0 < ε / (4 * Q)
M : ℕ
hM : ∀ j ≥ M, ∀ k ≥ M, |HSub.hSub (α := α) (∑ n ∈ range j, abv (f n)) (∑ n ∈ range k, abv (f n))| < ε / (4 * Q)
K : ℕ
hK : K ≥ 2 * (max N M + 1)
h₁ :
  Eq.{u_2 + 1} (α := β) (∑ m ∈ range K, ∑ k ∈ range (m + 1), f k * g (m - k))
    (∑ m ∈ range K, ∑ n ∈ range (K - m), f m * g n)
h₂ :
  Eq.{u_2 + 1} (α := (i : ℕ) → β) (fun (i : ℕ) ↦ ∑ k ∈ range (K - i), f i * g k) fun (i : ℕ) ↦
    f i * ∑ k ∈ range (K - i), g k
h₃ :
  ∑ i ∈ range K, f i * ∑ k ∈ range (K - i), g k =
    HAdd.hAdd (α := β) (∑ i ∈ range K, f i * HSub.hSub (α := β) (∑ k ∈ range (K - i), g k) (∑ k ∈ range K, g k))
      (∑ i ∈ range K, f i * ∑ k ∈ range K, g k)
two_mul_two : 4 = HMul.hMul (α := α) 2 2
hQ0 : Q ≠ 0
h2Q0 : 2 * Q ≠ 0
hε : ε / (2 * P) * P + ε / (4 * Q) * (2 * Q) = ε
hNMK : max N M + 1 < K
hKN : N < K
m : ℕ
hmJ : m ∈ range (max N M + 1)
h : m + N ≤ 2 * (max N M + 1)
⊢ abv (HSub.hSub (α := β) (∑ k ∈ range (K - m), g k) (∑ k ∈ range K, g k)) ≤ ε / (2 * P)

Tactic:
refine le_of_lt <| hN (K - m) (le_tsub_of_add_le_left <| hK.trans' ?_) K hKN.le

Premises used: le_tsub_of_add_le_left, HSub.hSub, le_of_lt, LT.lt.le, LE.le.trans'
-- Module: Mathlib.Algebra.Order.CauSeq.BigOperators
Goal:
α : Type u_1
β : Type u_2
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
inst_3 : Ring β
abv : β → α
inst_4 : IsAbsoluteValue abv
f g : ℕ → β
ha : IsCauSeq.{u_1, u_1} (α := α) abs fun (m : ℕ) ↦ ∑ n ∈ range m, abv (f n)
hb : IsCauSeq abv fun (m : ℕ) ↦ ∑ n ∈ range m, g n
ε : α
ε0 : 0 < ε
P : α
hP : ∀ (i : ℕ), |∑ n ∈ range i, abv (f n)| < P
Q : α
hQ : ∀ (i : ℕ), abv (∑ n ∈ range i, g n) < Q
hP0 : 0 < P
hPε0 : 0 < ε / (2 * P)
N : ℕ
hN : ∀ j ≥ N, ∀ k ≥ N, abv (HSub.hSub (α := β) (∑ n ∈ range j, g n) (∑ n ∈ range k, g n)) < ε / (2 * P)
hQε0 : 0 < ε / (4 * Q)
M : ℕ
hM : ∀ j ≥ M, ∀ k ≥ M, |HSub.hSub (α := α) (∑ n ∈ range j, abv (f n)) (∑ n ∈ range k, abv (f n))| < ε / (4 * Q)
K : ℕ
hK : K ≥ 2 * (max N M + 1)
h₁ :
  Eq.{u_2 + 1} (α := β) (∑ m ∈ range K, ∑ k ∈ range (m + 1), f k * g (m - k))
    (∑ m ∈ range K, ∑ n ∈ range (K - m), f m * g n)
h₂ :
  Eq.{u_2 + 1} (α := (i : ℕ) → β) (fun (i : ℕ) ↦ ∑ k ∈ range (K - i), f i * g k) fun (i : ℕ) ↦
    f i * ∑ k ∈ range (K - i), g k
h₃ :
  ∑ i ∈ range K, f i * ∑ k ∈ range (K - i), g k =
    HAdd.hAdd (α := β) (∑ i ∈ range K, f i * HSub.hSub (α := β) (∑ k ∈ range (K - i), g k) (∑ k ∈ range K, g k))
      (∑ i ∈ range K, f i * ∑ k ∈ range K, g k)
two_mul_two : 4 = HMul.hMul (α := α) 2 2
hQ0 : Q ≠ 0
h2Q0 : 2 * Q ≠ 0
hε : ε / (2 * P) * P + ε / (4 * Q) * (2 * Q) = ε
hNMK : max N M + 1 < K
hKN : N < K
m : ℕ
hmJ : m ∈ range (max N M + 1)
h : m + N ≤ max N M + 1 + (max N M + 1)
⊢ m + N ≤ 2 * (max N M + 1)

Tactic:
rw [two_mul]

Premises used: two_mul
-- Module: Mathlib.Algebra.Order.CauSeq.BigOperators
Goal:
α : Type u_1
β : Type u_2
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
inst_3 : Ring β
abv : β → α
inst_4 : IsAbsoluteValue abv
f g : ℕ → β
ha : IsCauSeq.{u_1, u_1} (α := α) abs fun (m : ℕ) ↦ ∑ n ∈ range m, abv (f n)
hb : IsCauSeq abv fun (m : ℕ) ↦ ∑ n ∈ range m, g n
ε : α
ε0 : 0 < ε
P : α
hP : ∀ (i : ℕ), |∑ n ∈ range i, abv (f n)| < P
Q : α
hQ : ∀ (i : ℕ), abv (∑ n ∈ range i, g n) < Q
hP0 : 0 < P
hPε0 : 0 < ε / (2 * P)
N : ℕ
hN : ∀ j ≥ N, ∀ k ≥ N, abv (HSub.hSub (α := β) (∑ n ∈ range j, g n) (∑ n ∈ range k, g n)) < ε / (2 * P)
hQε0 : 0 < ε / (4 * Q)
M : ℕ
hM : ∀ j ≥ M, ∀ k ≥ M, |HSub.hSub (α := α) (∑ n ∈ range j, abv (f n)) (∑ n ∈ range k, abv (f n))| < ε / (4 * Q)
K : ℕ
hK : K ≥ 2 * (max N M + 1)
h₁ :
  Eq.{u_2 + 1} (α := β) (∑ m ∈ range K, ∑ k ∈ range (m + 1), f k * g (m - k))
    (∑ m ∈ range K, ∑ n ∈ range (K - m), f m * g n)
h₂ :
  Eq.{u_2 + 1} (α := (i : ℕ) → β) (fun (i : ℕ) ↦ ∑ k ∈ range (K - i), f i * g k) fun (i : ℕ) ↦
    f i * ∑ k ∈ range (K - i), g k
h₃ :
  ∑ i ∈ range K, f i * ∑ k ∈ range (K - i), g k =
    HAdd.hAdd (α := β) (∑ i ∈ range K, f i * HSub.hSub (α := β) (∑ k ∈ range (K - i), g k) (∑ k ∈ range K, g k))
      (∑ i ∈ range K, f i * ∑ k ∈ range K, g k)
two_mul_two : 4 = HMul.hMul (α := α) 2 2
hQ0 : Q ≠ 0
h2Q0 : 2 * Q ≠ 0
hε : ε / (2 * P) * P + ε / (4 * Q) * (2 * Q) = ε
hNMK : max N M + 1 < K
hKN : N < K
m : ℕ
hmJ : m ∈ range (max N M + 1)
h : m ≤ max N M + 1
h₄ : N ≤ max N M + 1
⊢ m + N ≤ max N M + 1 + (max N M + 1)

Tactic:
gcongr
-- Module: Mathlib.Algebra.Order.CauSeq.BigOperators
Goal:
α : Type u_1
β : Type u_2
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
inst_3 : Ring β
abv : β → α
inst_4 : IsAbsoluteValue abv
f g : ℕ → β
ha : IsCauSeq.{u_1, u_1} (α := α) abs fun (m : ℕ) ↦ ∑ n ∈ range m, abv (f n)
hb : IsCauSeq abv fun (m : ℕ) ↦ ∑ n ∈ range m, g n
ε : α
ε0 : 0 < ε
P : α
hP : ∀ (i : ℕ), |∑ n ∈ range i, abv (f n)| < P
Q : α
hQ : ∀ (i : ℕ), abv (∑ n ∈ range i, g n) < Q
hP0 : 0 < P
hPε0 : 0 < ε / (2 * P)
N : ℕ
hN : ∀ j ≥ N, ∀ k ≥ N, abv (HSub.hSub (α := β) (∑ n ∈ range j, g n) (∑ n ∈ range k, g n)) < ε / (2 * P)
hQε0 : 0 < ε / (4 * Q)
M : ℕ
hM : ∀ j ≥ M, ∀ k ≥ M, |HSub.hSub (α := α) (∑ n ∈ range j, abv (f n)) (∑ n ∈ range k, abv (f n))| < ε / (4 * Q)
K : ℕ
hK : K ≥ 2 * (max N M + 1)
h₁ :
  Eq.{u_2 + 1} (α := β) (∑ m ∈ range K, ∑ k ∈ range (m + 1), f k * g (m - k))
    (∑ m ∈ range K, ∑ n ∈ range (K - m), f m * g n)
h₂ :
  Eq.{u_2 + 1} (α := (i : ℕ) → β) (fun (i : ℕ) ↦ ∑ k ∈ range (K - i), f i * g k) fun (i : ℕ) ↦
    f i * ∑ k ∈ range (K - i), g k
h₃ :
  ∑ i ∈ range K, f i * ∑ k ∈ range (K - i), g k =
    HAdd.hAdd (α := β) (∑ i ∈ range K, f i * HSub.hSub (α := β) (∑ k ∈ range (K - i), g k) (∑ k ∈ range K, g k))
      (∑ i ∈ range K, f i * ∑ k ∈ range K, g k)
two_mul_two : 4 = HMul.hMul (α := α) 2 2
hQ0 : Q ≠ 0
h2Q0 : 2 * Q ≠ 0
hε : ε / (2 * P) * P + ε / (4 * Q) * (2 * Q) = ε
hNMK : max N M + 1 < K
hKN : N < K
m : ℕ
hmJ : m ∈ range (max N M + 1)
⊢ m ≤ max N M + 1

Tactic:
(mem_range.1 hmJ).le

Premises used: LT.lt.le, Finset.mem_range, Iff.mp
-- Module: Mathlib.Algebra.Order.CauSeq.BigOperators
Goal:
α : Type u_1
β : Type u_2
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
inst_3 : Ring β
abv : β → α
inst_4 : IsAbsoluteValue abv
f g : ℕ → β
ha : IsCauSeq.{u_1, u_1} (α := α) abs fun (m : ℕ) ↦ ∑ n ∈ range m, abv (f n)
hb : IsCauSeq abv fun (m : ℕ) ↦ ∑ n ∈ range m, g n
ε : α
ε0 : 0 < ε
P : α
hP : ∀ (i : ℕ), |∑ n ∈ range i, abv (f n)| < P
Q : α
hQ : ∀ (i : ℕ), abv (∑ n ∈ range i, g n) < Q
hP0 : 0 < P
hPε0 : 0 < ε / (2 * P)
N : ℕ
hN : ∀ j ≥ N, ∀ k ≥ N, abv (HSub.hSub (α := β) (∑ n ∈ range j, g n) (∑ n ∈ range k, g n)) < ε / (2 * P)
hQε0 : 0 < ε / (4 * Q)
M : ℕ
hM : ∀ j ≥ M, ∀ k ≥ M, |HSub.hSub (α := α) (∑ n ∈ range j, abv (f n)) (∑ n ∈ range k, abv (f n))| < ε / (4 * Q)
K : ℕ
hK : K ≥ 2 * (max N M + 1)
h₁ :
  Eq.{u_2 + 1} (α := β) (∑ m ∈ range K, ∑ k ∈ range (m + 1), f k * g (m - k))
    (∑ m ∈ range K, ∑ n ∈ range (K - m), f m * g n)
h₂ :
  Eq.{u_2 + 1} (α := (i : ℕ) → β) (fun (i : ℕ) ↦ ∑ k ∈ range (K - i), f i * g k) fun (i : ℕ) ↦
    f i * ∑ k ∈ range (K - i), g k
h₃ :
  ∑ i ∈ range K, f i * ∑ k ∈ range (K - i), g k =
    HAdd.hAdd (α := β) (∑ i ∈ range K, f i * HSub.hSub (α := β) (∑ k ∈ range (K - i), g k) (∑ k ∈ range K, g k))
      (∑ i ∈ range K, f i * ∑ k ∈ range K, g k)
two_mul_two : 4 = HMul.hMul (α := α) 2 2
hQ0 : Q ≠ 0
h2Q0 : 2 * Q ≠ 0
hε : ε / (2 * P) * P + ε / (4 * Q) * (2 * Q) = ε
hNMK : max N M + 1 < K
hKN : N < K
m : ℕ
hmJ : m ∈ range (max N M + 1)
⊢ m ≤ max N M + 1

Tactic:
exact (mem_range.1 hmJ).le

Premises used: LT.lt.le, Finset.mem_range, Iff.mp
-- Module: Mathlib.Algebra.Order.CauSeq.BigOperators
Goal:
α : Type u_1
β : Type u_2
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
inst_3 : Ring β
abv : β → α
inst_4 : IsAbsoluteValue abv
f g : ℕ → β
ha : IsCauSeq.{u_1, u_1} (α := α) abs fun (m : ℕ) ↦ ∑ n ∈ range m, abv (f n)
hb : IsCauSeq abv fun (m : ℕ) ↦ ∑ n ∈ range m, g n
ε : α
ε0 : 0 < ε
P : α
hP : ∀ (i : ℕ), |∑ n ∈ range i, abv (f n)| < P
Q : α
hQ : ∀ (i : ℕ), abv (∑ n ∈ range i, g n) < Q
hP0 : 0 < P
hPε0 : 0 < ε / (2 * P)
N : ℕ
hN : ∀ j ≥ N, ∀ k ≥ N, abv (HSub.hSub (α := β) (∑ n ∈ range j, g n) (∑ n ∈ range k, g n)) < ε / (2 * P)
hQε0 : 0 < ε / (4 * Q)
M : ℕ
hM : ∀ j ≥ M, ∀ k ≥ M, |HSub.hSub (α := α) (∑ n ∈ range j, abv (f n)) (∑ n ∈ range k, abv (f n))| < ε / (4 * Q)
K : ℕ
hK : K ≥ 2 * (max N M + 1)
h₁ :
  Eq.{u_2 + 1} (α := β) (∑ m ∈ range K, ∑ k ∈ range (m + 1), f k * g (m - k))
    (∑ m ∈ range K, ∑ n ∈ range (K - m), f m * g n)
h₂ :
  Eq.{u_2 + 1} (α := (i : ℕ) → β) (fun (i : ℕ) ↦ ∑ k ∈ range (K - i), f i * g k) fun (i : ℕ) ↦
    f i * ∑ k ∈ range (K - i), g k
h₃ :
  ∑ i ∈ range K, f i * ∑ k ∈ range (K - i), g k =
    HAdd.hAdd (α := β) (∑ i ∈ range K, f i * HSub.hSub (α := β) (∑ k ∈ range (K - i), g k) (∑ k ∈ range K, g k))
      (∑ i ∈ range K, f i * ∑ k ∈ range K, g k)
two_mul_two : 4 = HMul.hMul (α := α) 2 2
hQ0 : Q ≠ 0
h2Q0 : 2 * Q ≠ 0
hε : ε / (2 * P) * P + ε / (4 * Q) * (2 * Q) = ε
hNMK : max N M + 1 < K
hKN : N < K
m : ℕ
hmJ : m ∈ range (max N M + 1)
⊢ N ≤ max N M + 1

Tactic:
Nat.le_succ_of_le (le_max_left _ _)

Premises used: Nat.le_succ_of_le, le_max_left
-- Module: Mathlib.Algebra.Order.CauSeq.BigOperators
Goal:
α : Type u_1
β : Type u_2
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
inst_3 : Ring β
abv : β → α
inst_4 : IsAbsoluteValue abv
f g : ℕ → β
ha : IsCauSeq.{u_1, u_1} (α := α) abs fun (m : ℕ) ↦ ∑ n ∈ range m, abv (f n)
hb : IsCauSeq abv fun (m : ℕ) ↦ ∑ n ∈ range m, g n
ε : α
ε0 : 0 < ε
P : α
hP : ∀ (i : ℕ), |∑ n ∈ range i, abv (f n)| < P
Q : α
hQ : ∀ (i : ℕ), abv (∑ n ∈ range i, g n) < Q
hP0 : 0 < P
hPε0 : 0 < ε / (2 * P)
N : ℕ
hN : ∀ j ≥ N, ∀ k ≥ N, abv (HSub.hSub (α := β) (∑ n ∈ range j, g n) (∑ n ∈ range k, g n)) < ε / (2 * P)
hQε0 : 0 < ε / (4 * Q)
M : ℕ
hM : ∀ j ≥ M, ∀ k ≥ M, |HSub.hSub (α := α) (∑ n ∈ range j, abv (f n)) (∑ n ∈ range k, abv (f n))| < ε / (4 * Q)
K : ℕ
hK : K ≥ 2 * (max N M + 1)
h₁ :
  Eq.{u_2 + 1} (α := β) (∑ m ∈ range K, ∑ k ∈ range (m + 1), f k * g (m - k))
    (∑ m ∈ range K, ∑ n ∈ range (K - m), f m * g n)
h₂ :
  Eq.{u_2 + 1} (α := (i : ℕ) → β) (fun (i : ℕ) ↦ ∑ k ∈ range (K - i), f i * g k) fun (i : ℕ) ↦
    f i * ∑ k ∈ range (K - i), g k
h₃ :
  ∑ i ∈ range K, f i * ∑ k ∈ range (K - i), g k =
    HAdd.hAdd (α := β) (∑ i ∈ range K, f i * HSub.hSub (α := β) (∑ k ∈ range (K - i), g k) (∑ k ∈ range K, g k))
      (∑ i ∈ range K, f i * ∑ k ∈ range K, g k)
two_mul_two : 4 = HMul.hMul (α := α) 2 2
hQ0 : Q ≠ 0
h2Q0 : 2 * Q ≠ 0
hε : ε / (2 * P) * P + ε / (4 * Q) * (2 * Q) = ε
hNMK : max N M + 1 < K
hKN : N < K
m : ℕ
hmJ : m ∈ range (max N M + 1)
⊢ N ≤ max N M + 1

Tactic:
exact Nat.le_succ_of_le (le_max_left _ _)

Premises used: Nat.le_succ_of_le, le_max_left
-- Module: Mathlib.Algebra.Order.CauSeq.BigOperators
Goal:
α : Type u_1
β : Type u_2
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
inst_3 : Ring β
abv : β → α
inst_4 : IsAbsoluteValue abv
f g : ℕ → β
ha : IsCauSeq.{u_1, u_1} (α := α) abs fun (m : ℕ) ↦ ∑ n ∈ range m, abv (f n)
hb : IsCauSeq abv fun (m : ℕ) ↦ ∑ n ∈ range m, g n
ε : α
ε0 : 0 < ε
P : α
hP : ∀ (i : ℕ), |∑ n ∈ range i, abv (f n)| < P
Q : α
hQ : ∀ (i : ℕ), abv (∑ n ∈ range i, g n) < Q
hP0 : 0 < P
hPε0 : 0 < ε / (2 * P)
N : ℕ
hN : ∀ j ≥ N, ∀ k ≥ N, abv (HSub.hSub (α := β) (∑ n ∈ range j, g n) (∑ n ∈ range k, g n)) < ε / (2 * P)
hQε0 : 0 < ε / (4 * Q)
M : ℕ
hM : ∀ j ≥ M, ∀ k ≥ M, |HSub.hSub (α := α) (∑ n ∈ range j, abv (f n)) (∑ n ∈ range k, abv (f n))| < ε / (4 * Q)
K : ℕ
hK : K ≥ 2 * (max N M + 1)
h₁ :
  Eq.{u_2 + 1} (α := β) (∑ m ∈ range K, ∑ k ∈ range (m + 1), f k * g (m - k))
    (∑ m ∈ range K, ∑ n ∈ range (K - m), f m * g n)
h₂ :
  Eq.{u_2 + 1} (α := (i : ℕ) → β) (fun (i : ℕ) ↦ ∑ k ∈ range (K - i), f i * g k) fun (i : ℕ) ↦
    f i * ∑ k ∈ range (K - i), g k
h₃ :
  ∑ i ∈ range K, f i * ∑ k ∈ range (K - i), g k =
    HAdd.hAdd (α := β) (∑ i ∈ range K, f i * HSub.hSub (α := β) (∑ k ∈ range (K - i), g k) (∑ k ∈ range K, g k))
      (∑ i ∈ range K, f i * ∑ k ∈ range K, g k)
two_mul_two : 4 = HMul.hMul (α := α) 2 2
hQ0 : Q ≠ 0
h2Q0 : 2 * Q ≠ 0
hε : ε / (2 * P) * P + ε / (4 * Q) * (2 * Q) = ε
hNMK : max N M + 1 < K
hKN : N < K
⊢ LE.le.{u_1} (α := α)
    (∑ i ∈ range (max N M + 1), abv (f i) * abv (HSub.hSub (α := β) (∑ k ∈ range (K - i), g k) (∑ k ∈ range K, g k)))
    (∑ i ∈ range (max N M + 1), abv (f i) * (ε / (2 * P)))

Tactic:
have hsumlesum :
      (∑ i ∈ range (max N M + 1),
        abv (f i) * abv ((∑ k ∈ range (K - i), g k) - ∑ k ∈ range K, g k)) ≤
      ∑ i ∈ range (max N M + 1), abv (f i) * (ε / (2 * P)) := by
    gcongr with m hmJ
    refine le_of_lt <| hN (K - m) (le_tsub_of_add_le_left <| hK.trans' ?_) K hKN.le
    rw [two_mul]
    gcongr
    · exact (mem_range.1 hmJ).le
    · exact Nat.le_succ_of_le (le_max_left _ _)
-- Module: Mathlib.Algebra.Order.CauSeq.BigOperators
Goal:
α : Type u_1
β : Type u_2
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
inst_3 : Ring β
abv : β → α
inst_4 : IsAbsoluteValue abv
f g : ℕ → β
ha : IsCauSeq.{u_1, u_1} (α := α) abs fun (m : ℕ) ↦ ∑ n ∈ range m, abv (f n)
hb : IsCauSeq abv fun (m : ℕ) ↦ ∑ n ∈ range m, g n
ε : α
ε0 : 0 < ε
P : α
hP : ∀ (i : ℕ), |∑ n ∈ range i, abv (f n)| < P
Q : α
hQ : ∀ (i : ℕ), abv (∑ n ∈ range i, g n) < Q
hP0 : 0 < P
hPε0 : 0 < ε / (2 * P)
N : ℕ
hN : ∀ j ≥ N, ∀ k ≥ N, abv (HSub.hSub (α := β) (∑ n ∈ range j, g n) (∑ n ∈ range k, g n)) < ε / (2 * P)
hQε0 : 0 < ε / (4 * Q)
M : ℕ
hM : ∀ j ≥ M, ∀ k ≥ M, |HSub.hSub (α := α) (∑ n ∈ range j, abv (f n)) (∑ n ∈ range k, abv (f n))| < ε / (4 * Q)
K : ℕ
hK : K ≥ 2 * (max N M + 1)
h₁ :
  Eq.{u_2 + 1} (α := β) (∑ m ∈ range K, ∑ k ∈ range (m + 1), f k * g (m - k))
    (∑ m ∈ range K, ∑ n ∈ range (K - m), f m * g n)
h₂ :
  Eq.{u_2 + 1} (α := (i : ℕ) → β) (fun (i : ℕ) ↦ ∑ k ∈ range (K - i), f i * g k) fun (i : ℕ) ↦
    f i * ∑ k ∈ range (K - i), g k
h₃ :
  ∑ i ∈ range K, f i * ∑ k ∈ range (K - i), g k =
    HAdd.hAdd (α := β) (∑ i ∈ range K, f i * HSub.hSub (α := β) (∑ k ∈ range (K - i), g k) (∑ k ∈ range K, g k))
      (∑ i ∈ range K, f i * ∑ k ∈ range K, g k)
two_mul_two : 4 = HMul.hMul (α := α) 2 2
hQ0 : Q ≠ 0
h2Q0 : 2 * Q ≠ 0
hε : ε / (2 * P) * P + ε / (4 * Q) * (2 * Q) = ε
hNMK : max N M + 1 < K
hKN : N < K
hsumlesum :
  LE.le.{u_1} (α := α)
    (∑ i ∈ range (max N M + 1), abv (f i) * abv (HSub.hSub (α := β) (∑ k ∈ range (K - i), g k) (∑ k ∈ range K, g k)))
    (∑ i ∈ range (max N M + 1), abv (f i) * (ε / (2 * P)))
⊢ ∑ n ∈ range (max N M + 1), abv (f n) < P

Tactic:
have hsumltP : (∑ n ∈ range (max N M + 1), abv (f n)) < P :=
    calc
      (∑ n ∈ range (max N M + 1), abv (f n)) = |∑ n ∈ range (max N M + 1), abv (f n)| :=
        Eq.symm (abs_of_nonneg (sum_nonneg fun x _ ↦ abv_nonneg abv (f x)))
      _ < P := hP (max N M + 1)
-- Module: Mathlib.Algebra.Order.CauSeq.BigOperators
Goal:
α : Type u_1
β : Type u_2
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
inst_3 : Ring β
abv : β → α
inst_4 : IsAbsoluteValue abv
f g : ℕ → β
ha : IsCauSeq.{u_1, u_1} (α := α) abs fun (m : ℕ) ↦ ∑ n ∈ range m, abv (f n)
hb : IsCauSeq abv fun (m : ℕ) ↦ ∑ n ∈ range m, g n
ε : α
ε0 : 0 < ε
P : α
hP : ∀ (i : ℕ), |∑ n ∈ range i, abv (f n)| < P
Q : α
hQ : ∀ (i : ℕ), abv (∑ n ∈ range i, g n) < Q
hP0 : 0 < P
hPε0 : 0 < ε / (2 * P)
N : ℕ
hN : ∀ j ≥ N, ∀ k ≥ N, abv (HSub.hSub (α := β) (∑ n ∈ range j, g n) (∑ n ∈ range k, g n)) < ε / (2 * P)
hQε0 : 0 < ε / (4 * Q)
M : ℕ
hM : ∀ j ≥ M, ∀ k ≥ M, |HSub.hSub (α := α) (∑ n ∈ range j, abv (f n)) (∑ n ∈ range k, abv (f n))| < ε / (4 * Q)
K : ℕ
hK : K ≥ 2 * (max N M + 1)
h₁ :
  Eq.{u_2 + 1} (α := β) (∑ m ∈ range K, ∑ k ∈ range (m + 1), f k * g (m - k))
    (∑ m ∈ range K, ∑ n ∈ range (K - m), f m * g n)
h₂ :
  Eq.{u_2 + 1} (α := (i : ℕ) → β) (fun (i : ℕ) ↦ ∑ k ∈ range (K - i), f i * g k) fun (i : ℕ) ↦
    f i * ∑ k ∈ range (K - i), g k
h₃ :
  ∑ i ∈ range K, f i * ∑ k ∈ range (K - i), g k =
    HAdd.hAdd (α := β) (∑ i ∈ range K, f i * HSub.hSub (α := β) (∑ k ∈ range (K - i), g k) (∑ k ∈ range K, g k))
      (∑ i ∈ range K, f i * ∑ k ∈ range K, g k)
two_mul_two : 4 = HMul.hMul (α := α) 2 2
hQ0 : Q ≠ 0
h2Q0 : 2 * Q ≠ 0
hε : ε / (2 * P) * P + ε / (4 * Q) * (2 * Q) = ε
hNMK : max N M + 1 < K
hKN : N < K
hsumlesum :
  LE.le.{u_1} (α := α)
    (∑ i ∈ range (max N M + 1), abv (f i) * abv (HSub.hSub (α := β) (∑ k ∈ range (K - i), g k) (∑ k ∈ range K, g k)))
    (∑ i ∈ range (max N M + 1), abv (f i) * (ε / (2 * P)))
hsumltP : ∑ n ∈ range (max N M + 1), abv (f n) < P
rw : abv (∑ i ∈ range K, f i * HSub.hSub (α := β) (∑ k ∈ range (K - i), g k) (∑ k ∈ range K, g k)) < ε
⊢ abv
      (HMul.hMul (α := β) (∑ k ∈ range K, f k) (∑ k ∈ range K, g k) -
        ∑ n ∈ range K, ∑ m ∈ range (n + 1), f m * g (n - m)) <
    ε

Tactic:
rw [h₁, h₂, h₃, sum_mul, ← sub_sub, sub_right_comm, sub_self, zero_sub, abv_neg abv]

Premises used: sub_self, sub_right_comm, sub_sub, Finset.sum_mul, zero_sub, IsAbsoluteValue.abv_neg
-- Module: Mathlib.Algebra.Order.CauSeq.BigOperators
Goal:
α : Type u_1
β : Type u_2
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
inst_3 : Ring β
abv : β → α
inst_4 : IsAbsoluteValue abv
f g : ℕ → β
ha : IsCauSeq.{u_1, u_1} (α := α) abs fun (m : ℕ) ↦ ∑ n ∈ range m, abv (f n)
hb : IsCauSeq abv fun (m : ℕ) ↦ ∑ n ∈ range m, g n
ε : α
ε0 : 0 < ε
P : α
hP : ∀ (i : ℕ), |∑ n ∈ range i, abv (f n)| < P
Q : α
hQ : ∀ (i : ℕ), abv (∑ n ∈ range i, g n) < Q
hP0 : 0 < P
hPε0 : 0 < ε / (2 * P)
N : ℕ
hN : ∀ j ≥ N, ∀ k ≥ N, abv (HSub.hSub (α := β) (∑ n ∈ range j, g n) (∑ n ∈ range k, g n)) < ε / (2 * P)
hQε0 : 0 < ε / (4 * Q)
M : ℕ
hM : ∀ j ≥ M, ∀ k ≥ M, |HSub.hSub (α := α) (∑ n ∈ range j, abv (f n)) (∑ n ∈ range k, abv (f n))| < ε / (4 * Q)
K : ℕ
hK : K ≥ 2 * (max N M + 1)
h₁ :
  Eq.{u_2 + 1} (α := β) (∑ m ∈ range K, ∑ k ∈ range (m + 1), f k * g (m - k))
    (∑ m ∈ range K, ∑ n ∈ range (K - m), f m * g n)
h₂ :
  Eq.{u_2 + 1} (α := (i : ℕ) → β) (fun (i : ℕ) ↦ ∑ k ∈ range (K - i), f i * g k) fun (i : ℕ) ↦
    f i * ∑ k ∈ range (K - i), g k
h₃ :
  ∑ i ∈ range K, f i * ∑ k ∈ range (K - i), g k =
    HAdd.hAdd (α := β) (∑ i ∈ range K, f i * HSub.hSub (α := β) (∑ k ∈ range (K - i), g k) (∑ k ∈ range K, g k))
      (∑ i ∈ range K, f i * ∑ k ∈ range K, g k)
two_mul_two : 4 = HMul.hMul (α := α) 2 2
hQ0 : Q ≠ 0
h2Q0 : 2 * Q ≠ 0
hε : ε / (2 * P) * P + ε / (4 * Q) * (2 * Q) = ε
hNMK : max N M + 1 < K
hKN : N < K
hsumlesum :
  LE.le.{u_1} (α := α)
    (∑ i ∈ range (max N M + 1), abv (f i) * abv (HSub.hSub (α := β) (∑ k ∈ range (K - i), g k) (∑ k ∈ range K, g k)))
    (∑ i ∈ range (max N M + 1), abv (f i) * (ε / (2 * P)))
hsumltP : ∑ n ∈ range (max N M + 1), abv (f n) < P
⊢ abv (∑ i ∈ range K, f i * HSub.hSub (α := β) (∑ k ∈ range (K - i), g k) (∑ k ∈ range K, g k)) < ε

Tactic:
lt_of_le_of_lt (IsAbsoluteValue.abv_sum _ _ _) ?_

Premises used: lt_of_le_of_lt, IsAbsoluteValue.abv_sum