awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.BigOperators (goal)
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 + N ≤ max N M + 1 + (max N M + 1)
have h : m ≤ max N M + 1
Tactic: gcongr
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.BigOperators (goal)
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
⊢ m + N ≤ max N M + 1 + (max N M + 1)
have h₄ : N ≤ max N M + 1
Tactic: gcongr
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.BigOperators (hypothesis)
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
⊢ 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
have 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)))
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 _ _)
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.BigOperators (hypothesis)
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)))
⊢ 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
have hsumltP : ∑ 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)
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.BigOperators (goal)
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
      (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
have rw : abv (∑ i ∈ range K, f i * HSub.hSub (α := β) (∑ k ∈ range (K - i), g k) (∑ k ∈ range K, g k)) < ε
Tactic: rw [h₁, h₂, h₃, sum_mul, ← sub_sub, sub_right_comm, sub_self, zero_sub, abv_neg abv]
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.BigOperators (goal)
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
have refine : ∑ i ∈ range K, abv (f i * HSub.hSub (α := β) (∑ k ∈ range (K - i), g k) (∑ k ∈ range K, g k)) < ε
Tactic: refine lt_of_le_of_lt (IsAbsoluteValue.abv_sum _ _ _) ?_
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.BigOperators (hypothesis)
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
this :
  ∑ 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)) +
      HSub.hSub (α := α)
        (∑ i ∈ range K, 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) * abv (HSub.hSub (α := β) (∑ k ∈ range (K - i), g k) (∑ k ∈ range K, g k))) <
    ε / (2 * P) * P + ε / (4 * Q) * (2 * Q)
⊢ ∑ i ∈ range K, abv (f i * HSub.hSub (α := β) (∑ k ∈ range (K - i), g k) (∑ k ∈ range K, g k)) < ε
have this₁ : ∑ 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)) +
    HSub.hSub (α := α)
      (∑ i ∈ range K, 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) * abv (HSub.hSub (α := β) (∑ k ∈ range (K - i), g k) (∑ k ∈ range K, g k))) <
  ε
Tactic: rw [hε] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.BigOperators (goal)
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
⊢ ∑ i ∈ range K, abv (f i * HSub.hSub (α := β) (∑ k ∈ range (K - i), g k) (∑ k ∈ range K, g k)) < ε
have i : ∑ 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)) +
    HSub.hSub (α := α)
      (∑ i ∈ range K, 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) * abv (HSub.hSub (α := β) (∑ k ∈ range (K - i), g k) (∑ k ∈ range K, g k))) <
  ε / (2 * P) * P + ε / (4 * Q) * (2 * Q)
Tactic: suffices
    (∑ i ∈ range (max N M + 1),
          abv (f i) * abv ((∑ k ∈ range (K - i), g k) - ∑ k ∈ range K, g k)) +
        ((∑ i ∈ range K, abv (f i) * abv ((∑ k ∈ range (K - i), g k) - ∑ k ∈ range K, g k)) -
          ∑ i ∈ range (max N M + 1),
            abv (f i) * abv ((∑ k ∈ range (K - i), g k) - ∑ k ∈ range K, g k)) <
      ε / (2 * P) * P + ε / (4 * Q) * (2 * Q) by
    rw [hε] at this
    simpa [abv_mul abv] using this
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.BigOperators (goal)
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
⊢ ∑ 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)) +
      HSub.hSub (α := α)
        (∑ i ∈ range K, 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) * abv (HSub.hSub (α := β) (∑ k ∈ range (K - i), g k) (∑ k ∈ range K, g k))) <
    ε / (2 * P) * P + ε / (4 * Q) * (2 * Q)
have h₄ : ∑ 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)) <
  ε / (2 * P) * P
Tactic: gcongr
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.BigOperators (goal)
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
h₄ :
  ∑ 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)) <
    ε / (2 * P) * P
⊢ ∑ 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)) +
      HSub.hSub (α := α)
        (∑ i ∈ range K, 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) * abv (HSub.hSub (α := β) (∑ k ∈ range (K - i), g k) (∑ k ∈ range K, g k))) <
    ε / (2 * P) * P + ε / (4 * Q) * (2 * Q)
have h₅ : HSub.hSub (α := α) (∑ i ∈ range K, 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) * abv (HSub.hSub (α := β) (∑ k ∈ range (K - i), g k) (∑ k ∈ range K, g k))) <
  ε / (4 * Q) * (2 * Q)
Tactic: gcongr
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.BigOperators (goal)
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
⊢ ∑ i ∈ range (max N M + 1), abv (f i) * (ε / (2 * P)) < ε / (2 * P) * P
have rw : ε / (2 * P) * ∑ i ∈ range (max N M + 1), abv (f i) < ε / (2 * P) * P
Tactic: rw [← sum_mul, mul_comm]