mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Order.CauSeq.BigOperators

Goal state before:
pp: case hbc
α : Type u_1
β : Type u_2
inst✝⁴ : Field α
inst✝³ : LinearOrder α
inst✝² : IsStrictOrderedRing α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : ℕ → β
ha : IsCauSeq abs fun m ↦ ∑ n ∈ range m, abv (f n)
hb : IsCauSeq abv fun m ↦ ∑ 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 (∑ 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, |∑ 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 : ∑ m ∈ range K, ∑ k ∈ range (m + 1), f k * g (m - k) = ∑ m ∈ range K, ∑ n ∈ range (K - m), f m * g n
h₂ : (fun i ↦ ∑ k ∈ range (K - i), f i * g k) = fun i ↦ f i * ∑ k ∈ range (K - i), g k
h₃ :
  ∑ i ∈ range K, f i * ∑ k ∈ range (K - i), g k =
    ∑ i ∈ range K, f i * (∑ 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 = 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 :
  ∑ 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))
hsumltP : ∑ n ∈ range (max N M + 1), abv (f n) < P
i : ℕ
a✝ : i ∈ {i ∈ range K | max N M + 1 ≤ i}
⊢ abv (∑ k ∈ range (K - i), g k) + abv (∑ k ∈ range K, g k) ≤ Q + Q

Tactic:
]

Goal state after:
case hbc
α : Type u_1
β : Type u_2
inst✝⁴ : Field α
inst✝³ : LinearOrder α
inst✝² : IsStrictOrderedRing α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : ℕ → β
ha : IsCauSeq abs fun m ↦ ∑ n ∈ range m, abv (f n)
hb : IsCauSeq abv fun m ↦ ∑ 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 (∑ 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, |∑ 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 : ∑ m ∈ range K, ∑ k ∈ range (m + 1), f k * g (m - k) = ∑ m ∈ range K, ∑ n ∈ range (K - m), f m * g n
h₂ : (fun i ↦ ∑ k ∈ range (K - i), f i * g k) = fun i ↦ f i * ∑ k ∈ range (K - i), g k
h₃ :
  ∑ i ∈ range K, f i * ∑ k ∈ range (K - i), g k =
    ∑ i ∈ range K, f i * (∑ 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 = 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 :
  ∑ 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))
hsumltP : ∑ n ∈ range (max N M + 1), abv (f n) < P
i : ℕ
a✝ : i ∈ {i ∈ range K | max N M + 1 ≤ i}
⊢ abv (∑ k ∈ range (K - i), g k) + abv (∑ k ∈ range K, g k) ≤ Q + Q
Module: Mathlib.Analysis.Complex.Exponential

Goal state before:
pp: x : ℝ
h1 : 0 < x
h2 : x < 1
H : 0 < 1 - (1 + x + x ^ 2) * (1 - x)
⊢ ∑ x_1 ∈ range 0, x ^ x_1 / ↑x_1.factorial + x ^ 0 / ↑(Nat.factorial 0) + x ^ 1 / ↑(Nat.factorial 1) +
        x ^ (1 + 1) / ↑(1 + 1).factorial +
      x ^ (1 + 1 + 1) * (↑(1 + 1 + 1) + 1) / (↑(1 + 1 + 1).factorial * ↑(1 + 1 + 1)) ≤
    1 + x + x ^ 2

Tactic:
]

Goal state after:
x : ℝ
h1 : 0 < x
h2 : x < 1
H : 0 < 1 - (1 + x + x ^ 2) * (1 - x)
⊢ ∑ x_1 ∈ range 0, x ^ x_1 / ↑x_1.factorial + x ^ 0 / ↑(Nat.factorial 0) + x ^ 1 / ↑(Nat.factorial 1) +
        x ^ (1 + 1) / ↑(1 + 1).factorial +
      x ^ (1 + 1 + 1) * (↑(1 + 1 + 1) + 1) / (↑(1 + 1 + 1).factorial * ↑(1 + 1 + 1)) ≤
    1 + x + x ^ 2
Module: Mathlib.Analysis.Complex.Exponential

Goal state before:
pp: x : ℝ
h1 : 0 < x
h2 : x < 1
H : 0 < 1 - (1 + x + x ^ 2) * (1 - x)
⊢ (1 + x + x ^ 2) * (1 - x) < 1

pp: x : ℝ
h1 : 0 < x
h2 : x < 1
H : 0 < 1 - (1 + x + x ^ 2) * (1 - x)
⊢ 0 < 1 - x

Tactic:
]

Goal state after:
x : ℝ
h1 : 0 < x
h2 : x < 1
H : 0 < 1 - (1 + x + x ^ 2) * (1 - x)
⊢ (1 + x + x ^ 2) * (1 - x) < 1

x : ℝ
h1 : 0 < x
h2 : x < 1
H : 0 < 1 - (1 + x + x ^ 2) * (1 - x)
⊢ 0 < 1 - x
Module: Mathlib.Analysis.Complex.Exponential

Goal state before:
pp: case inl
t : ℝ
ht' : t ≤ 0
⊢ t ≤ 0

Tactic:
]

Goal state after:
case inl
t : ℝ
ht' : t ≤ 0
⊢ t ≤ 0
Module: Mathlib.Analysis.Complex.Exponential

Goal state before:
pp: n : ℕ
t : ℝ
ht' : t ≤ ↑n
hn : n ≠ 0
⊢ rexp (-t) = rexp (-t)

pp: case h
n : ℕ
t : ℝ
ht' : t ≤ ↑n
hn : n ≠ 0
⊢ ↑n ≠ 0

Tactic:
]

Goal state after:
case h
n : ℕ
t : ℝ
ht' : t ≤ ↑n
hn : n ≠ 0
⊢ ↑n ≠ 0
Module: Mathlib.Analysis.Complex.Exponential

Goal state before:
pp: x c : ℝ
hc : 0 < c
⊢ c * x ≤ rexp (c * x)

Tactic:
]

Goal state after:
x c : ℝ
hc : 0 < c
⊢ c * x ≤ rexp (c * x)
Module: Mathlib.Analysis.Complex.Exponential

Goal state before:
pp: x : ℝ
hx : 0 < x
hx' : x < 2
⊢ (∑ m ∈ range 3, (x / 2) ^ m / ↑m.factorial + (x / 2) ^ 3 * (↑3 + 1) / (↑(Nat.factorial 3) * ↑3)) ^ 2 ≤ ?d

pp: case ha
x : ℝ
hx : 0 < x
hx' : x < 2
⊢ 0 ≤ rexp (x / 2)

Tactic:
]

Goal state after:
case ha
x : ℝ
hx : 0 < x
hx' : x < 2
⊢ 0 ≤ rexp (x / 2)
Module: Mathlib.Analysis.Complex.Exponential

Goal state before:
pp: x : ℝ
hx : 0 < x
hx' : x < 2
⊢ 0 <
    2 + x -
      (∑ m ∈ range 3, (x / 2) ^ m / ↑m.factorial + (x / 2) ^ 3 * (↑3 + 1) / (↑(Nat.factorial 3) * ↑3)) ^ 2 * (2 - x)

Tactic:
]

Goal state after:
x : ℝ
hx : 0 < x
hx' : x < 2
⊢ 0 <
    2 + x -
      (∑ m ∈ range 3, (x / 2) ^ m / ↑m.factorial + (x / 2) ^ 3 * (↑3 + 1) / (↑(Nat.factorial 3) * ↑3)) ^ 2 * (2 - x)
Module: Mathlib.Analysis.Asymptotics.Lemmas

Goal state before:
pp: case pos
E'' : Type u_9
F'' : Type u_10
inst✝¹ : NormedAddCommGroup E''
inst✝ : NormedAddCommGroup F''
f : ℕ → E''
g : ℕ → F''
h✝ : ∀ᶠ (x : ℕ) in atTop, ∀ (y : ℕ), x ≤ y → g y = 0 → f y = 0
hrec✝ :
  ∀ᶠ (n₀ : ℕ) in atTop,
    ∃ C₀, ∀ᶠ (n : ℕ) in atTop, ∀ C ≥ C₀, (∀ m ∈ Finset.Ico n₀ n, ‖f m‖ ≤ C * ‖g m‖) → ‖f n‖ ≤ C * ‖g n‖
n₀ : ℕ
h : ∀ (y : ℕ), n₀ ≤ y → g y = 0 → f y = 0
C₀ : ℝ
hrec : ∀ᶠ (x : ℕ) in atTop, ∀ (y : ℕ), x ≤ y → ∀ C ≥ C₀, (∀ m ∈ Finset.Ico n₀ y, ‖f m‖ ≤ C * ‖g m‖) → ‖f y‖ ≤ C * ‖g y‖
n₁ : ℕ
H₁ : n₀ ≤ n₁
H₂ : ∀ (y : ℕ), n₁ ≤ y → ∀ C ≥ C₀, (∀ m ∈ Finset.Ico n₀ y, ‖f m‖ ≤ C * ‖g m‖) → ‖f y‖ ≤ C * ‖g y‖
ubounds : Set ℝ := {C | ∀ m ∈ Finset.Icc n₀ n₁, ‖f m‖ ≤ C * ‖g m‖}
C₁ : ℝ := (Finset.Icc n₀ n₁).sup' ⋯ fun n ↦ ‖f n‖ / ‖g n‖
C₁_mem : C₁ ∈ ubounds
n : ℕ
h_ind : ∀ m < n, n₁ ≤ m → ‖f m‖ ≤ max C₀ C₁ * ‖g m‖
hn : n₁ ≤ n
m : ℕ
hm : m ∈ Finset.Ico n₀ n
hbase : m < n₁
hC₁ : C₁ ≤ max C₀ C₁
⊢ ‖f m‖ ≤ C₁ * ‖g m‖

Tactic:
]

Goal state after:
case pos
E'' : Type u_9
F'' : Type u_10
inst✝¹ : NormedAddCommGroup E''
inst✝ : NormedAddCommGroup F''
f : ℕ → E''
g : ℕ → F''
h✝ : ∀ᶠ (x : ℕ) in atTop, ∀ (y : ℕ), x ≤ y → g y = 0 → f y = 0
hrec✝ :
  ∀ᶠ (n₀ : ℕ) in atTop,
    ∃ C₀, ∀ᶠ (n : ℕ) in atTop, ∀ C ≥ C₀, (∀ m ∈ Finset.Ico n₀ n, ‖f m‖ ≤ C * ‖g m‖) → ‖f n‖ ≤ C * ‖g n‖
n₀ : ℕ
h : ∀ (y : ℕ), n₀ ≤ y → g y = 0 → f y = 0
C₀ : ℝ
hrec : ∀ᶠ (x : ℕ) in atTop, ∀ (y : ℕ), x ≤ y → ∀ C ≥ C₀, (∀ m ∈ Finset.Ico n₀ y, ‖f m‖ ≤ C * ‖g m‖) → ‖f y‖ ≤ C * ‖g y‖
n₁ : ℕ
H₁ : n₀ ≤ n₁
H₂ : ∀ (y : ℕ), n₁ ≤ y → ∀ C ≥ C₀, (∀ m ∈ Finset.Ico n₀ y, ‖f m‖ ≤ C * ‖g m‖) → ‖f y‖ ≤ C * ‖g y‖
ubounds : Set ℝ := {C | ∀ m ∈ Finset.Icc n₀ n₁, ‖f m‖ ≤ C * ‖g m‖}
C₁ : ℝ := (Finset.Icc n₀ n₁).sup' ⋯ fun n ↦ ‖f n‖ / ‖g n‖
C₁_mem : C₁ ∈ ubounds
n : ℕ
h_ind : ∀ m < n, n₁ ≤ m → ‖f m‖ ≤ max C₀ C₁ * ‖g m‖
hn : n₁ ≤ n
m : ℕ
hm : m ∈ Finset.Ico n₀ n
hbase : m < n₁
hC₁ : C₁ ≤ max C₀ C₁
⊢ ‖f m‖ ≤ C₁ * ‖g m‖
Module: Mathlib.Analysis.Complex.Exponential

Goal state before:
pp: x : ℝ
⊢ ‖↑(Real.exp x)‖ = Real.exp x

Tactic:
]

Goal state after:
x : ℝ
⊢ ‖↑(Real.exp x)‖ = Real.exp x
Module: Mathlib.Algebra.Order.CauSeq.BigOperators

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁴ : Field α
inst✝³ : LinearOrder α
inst✝² : IsStrictOrderedRing α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : ℕ → β
ha : IsCauSeq abs fun m ↦ ∑ n ∈ range m, abv (f n)
hb : IsCauSeq abv fun m ↦ ∑ 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 (∑ 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, |∑ 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 : ∑ m ∈ range K, ∑ k ∈ range (m + 1), f k * g (m - k) = ∑ m ∈ range K, ∑ n ∈ range (K - m), f m * g n
h₂ : (fun i ↦ ∑ k ∈ range (K - i), f i * g k) = fun i ↦ f i * ∑ k ∈ range (K - i), g k
h₃ :
  ∑ i ∈ range K, f i * ∑ k ∈ range (K - i), g k =
    ∑ i ∈ range K, f i * (∑ 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 = 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 :
  ∑ 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))
hsumltP : ∑ n ∈ range (max N M + 1), abv (f n) < P
⊢ (∑ k ∈ range K, abv (f k) - ∑ k ∈ range (max N M + 1), abv (f k)) * (2 * Q) < ε / (4 * Q) * (2 * Q)

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
inst✝⁴ : Field α
inst✝³ : LinearOrder α
inst✝² : IsStrictOrderedRing α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : ℕ → β
ha : IsCauSeq abs fun m ↦ ∑ n ∈ range m, abv (f n)
hb : IsCauSeq abv fun m ↦ ∑ 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 (∑ 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, |∑ 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 : ∑ m ∈ range K, ∑ k ∈ range (m + 1), f k * g (m - k) = ∑ m ∈ range K, ∑ n ∈ range (K - m), f m * g n
h₂ : (fun i ↦ ∑ k ∈ range (K - i), f i * g k) = fun i ↦ f i * ∑ k ∈ range (K - i), g k
h₃ :
  ∑ i ∈ range K, f i * ∑ k ∈ range (K - i), g k =
    ∑ i ∈ range K, f i * (∑ 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 = 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 :
  ∑ 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))
hsumltP : ∑ n ∈ range (max N M + 1), abv (f n) < P
⊢ (∑ k ∈ range K, abv (f k) - ∑ k ∈ range (max N M + 1), abv (f k)) * (2 * Q) < ε / (4 * Q) * (2 * Q)
Module: Mathlib.Algebra.Order.CauSeq.BigOperators

Goal state before:
pp: α : Type u_1
inst✝³ : Field α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
inst✝ : Archimedean α
f : ℕ → α
a : α
m : ℕ
ham : ∀ n ≥ m, |f n| ≤ a
hnm✝ : ∀ n ≥ m, f n.succ ≤ f n
ε : α
ε0 : ε > 0
k : ℕ
hk : a ≤ k • ε
n : ℕ
hnm : n ≥ m
⊢ a + a < k • ε + (k • ε + ε)

Tactic:
]

Goal state after:
α : Type u_1
inst✝³ : Field α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
inst✝ : Archimedean α
f : ℕ → α
a : α
m : ℕ
ham : ∀ n ≥ m, |f n| ≤ a
hnm✝ : ∀ n ≥ m, f n.succ ≤ f n
ε : α
ε0 : ε > 0
k : ℕ
hk : a ≤ k • ε
n : ℕ
hnm : n ≥ m
⊢ a + a < k • ε + (k • ε + ε)
Module: Mathlib.Analysis.Asymptotics.Lemmas

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
E : Type u_3
inst✝¹ : Norm E
F : Type u_4
inst✝ : Norm F
e : OpenPartialHomeomorph α β
b : β
hb : b ∈ e.target
f : β → E
g : β → F
C : ℝ
h : IsBigOWith C (𝓝 b) f g
this : Tendsto (↑e) (𝓝 (↑e.symm b)) (𝓝 b)
⊢ Tendsto (↑e) (𝓝 (↑e.symm b)) (𝓝 b)

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
E : Type u_3
inst✝¹ : Norm E
F : Type u_4
inst✝ : Norm F
e : OpenPartialHomeomorph α β
b : β
hb : b ∈ e.target
f : β → E
g : β → F
C : ℝ
h : IsBigOWith C (𝓝 b) f g
this : Tendsto (↑e) (𝓝 (↑e.symm b)) (𝓝 b)
⊢ Tendsto (↑e) (𝓝 (↑e.symm b)) (𝓝 b)
Module: Mathlib.Analysis.Asymptotics.Lemmas

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝² : TopologicalSpace α
s : Set α
f : α → E
c : F
inst✝¹ : SeminormedAddGroup E
inst✝ : Norm F
hf : ContinuousOn f s
hs : IsCompact s
hc : ‖c‖ ≠ 0
⊢ ∀ x ∈ s, ‖f x‖ ≤ sSup (Norm.norm '' f '' s)

Tactic:
]

Goal state after:
α : Type u_1
E : Type u_2
F : Type u_3
inst✝² : TopologicalSpace α
s : Set α
f : α → E
c : F
inst✝¹ : SeminormedAddGroup E
inst✝ : Norm F
hf : ContinuousOn f s
hs : IsCompact s
hc : ‖c‖ ≠ 0
⊢ ∀ x ∈ s, ‖f x‖ ≤ sSup (Norm.norm '' f '' s)
Module: Mathlib.Algebra.Order.CauSeq.BigOperators

Goal state before:
pp: α : Type u_1
inst✝³ : Field α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
inst✝ : Archimedean α
f : ℕ → α
a : α
m : ℕ
ham : ∀ n ≥ m, |f n| ≤ a
hnm : ∀ n ≥ m, f n.succ ≤ f n
ε : α
ε0 : ε > 0
k : ℕ
hk : a ≤ k • ε
h : ∃ l, ∀ n ≥ m, a - l • ε < f n
l : ℕ := Nat.find h
hl : ∀ n ≥ m, f n > a - l • ε
hl0 : l ≠ 0
i : ℕ
hi : i ≥ m ∧ f i ≤ a - (Nat.find h).pred • ε
⊢ ∃ i, ∀ j ≥ i, |f j - f i| < ε

Tactic:
]

Goal state after:
α : Type u_1
inst✝³ : Field α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
inst✝ : Archimedean α
f : ℕ → α
a : α
m : ℕ
ham : ∀ n ≥ m, |f n| ≤ a
hnm : ∀ n ≥ m, f n.succ ≤ f n
ε : α
ε0 : ε > 0
k : ℕ
hk : a ≤ k • ε
h : ∃ l, ∀ n ≥ m, a - l • ε < f n
l : ℕ := Nat.find h
hl : ∀ n ≥ m, f n > a - l • ε
hl0 : l ≠ 0
i : ℕ
hi : i ≥ m ∧ f i ≤ a - (Nat.find h).pred • ε
⊢ ∃ i, ∀ j ≥ i, |f j - f i| < ε
Module: Mathlib.Analysis.Asymptotics.Lemmas

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝² : TopologicalSpace α
s : Set α
f : α → E
inst✝¹ : NormedAddGroup E
inst✝ : SeminormedAddGroup F
hf : ContinuousOn f s
hs : IsCompact s
hC : ∀ i ∈ s, f i ≠ 0
c : F
x : α
hx : x ∈ s
⊢ ‖c‖ ≤ ‖c‖ * (‖f x‖ / sInf (Norm.norm '' f '' s))

Tactic:
]

Goal state after:
α : Type u_1
E : Type u_2
F : Type u_3
inst✝² : TopologicalSpace α
s : Set α
f : α → E
inst✝¹ : NormedAddGroup E
inst✝ : SeminormedAddGroup F
hf : ContinuousOn f s
hs : IsCompact s
hC : ∀ i ∈ s, f i ≠ 0
c : F
x : α
hx : x ∈ s
⊢ ‖c‖ ≤ ‖c‖ * (‖f x‖ / sInf (Norm.norm '' f '' s))
Module: Mathlib.Analysis.Asymptotics.Lemmas

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝² : TopologicalSpace α
s : Set α
f : α → E
inst✝¹ : NormedAddGroup E
inst✝ : SeminormedAddGroup F
hf : ContinuousOn f s
hC : ∀ i ∈ s, f i ≠ 0
c : F
x✝ : α
hx✝ : x✝ ∈ s
hs : IsCompact ((fun a ↦ ‖a‖) '' f '' s)
h_sInf : IsGLB ((fun a ↦ ‖a‖) '' f '' s) (sInf ((fun a ↦ ‖a‖) '' f '' s))
w✝ : E
hnormCx : (fun a ↦ ‖a‖) w✝ = sInf ((fun a ↦ ‖a‖) '' f '' s)
x : α
hx : x ∈ s
hCx : f x = w✝
⊢ 0 < (fun a ↦ ‖a‖) (f x)

Tactic:
]

Goal state after:
α : Type u_1
E : Type u_2
F : Type u_3
inst✝² : TopologicalSpace α
s : Set α
f : α → E
inst✝¹ : NormedAddGroup E
inst✝ : SeminormedAddGroup F
hf : ContinuousOn f s
hC : ∀ i ∈ s, f i ≠ 0
c : F
x✝ : α
hx✝ : x✝ ∈ s
hs : IsCompact ((fun a ↦ ‖a‖) '' f '' s)
h_sInf : IsGLB ((fun a ↦ ‖a‖) '' f '' s) (sInf ((fun a ↦ ‖a‖) '' f '' s))
w✝ : E
hnormCx : (fun a ↦ ‖a‖) w✝ = sInf ((fun a ↦ ‖a‖) '' f '' s)
x : α
hx : x ∈ s
hCx : f x = w✝
⊢ 0 < (fun a ↦ ‖a‖) (f x)
Module: Mathlib.Analysis.Asymptotics.Lemmas

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝³ : NormedDivisionRing 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
l : Filter ι
ε : ι → 𝕜
f : ι → E
hε : ε =o[l] fun _x ↦ 1
hf : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l (norm ∘ f)
⊢ (ε • f) =o[l] fun _x ↦ 1

Tactic:
]

Goal state after:
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝³ : NormedDivisionRing 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
l : Filter ι
ε : ι → 𝕜
f : ι → E
hε : ε =o[l] fun _x ↦ 1
hf : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l (norm ∘ f)
⊢ (ε • f) =o[l] fun _x ↦ 1
Module: Mathlib.Algebra.Order.CauSeq.BigOperators

Goal state before:
pp: α : Type u_1
inst✝³ : Field α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
inst✝ : Archimedean α
f : ℕ → α
a : α
m : ℕ
ham : ∀ n ≥ m, |f n| ≤ a
hnm : ∀ n ≥ m, f n.succ ≤ f n
ε : α
ε0 : ε > 0
k : ℕ
hk : a ≤ k • ε
h : ∃ l, ∀ n ≥ m, a - l • ε < f n
l : ℕ := Nat.find h
hl : ∀ n ≥ m, f n > a - l • ε
hl0 : l ≠ 0
i : ℕ
hi : i ≥ m ∧ f i ≤ a - (Nat.find h).pred • ε
j : ℕ
hj : j ≥ i
hfij : f j ≤ f i
⊢ f i < f j + ε

Tactic:
]

Goal state after:
α : Type u_1
inst✝³ : Field α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
inst✝ : Archimedean α
f : ℕ → α
a : α
m : ℕ
ham : ∀ n ≥ m, |f n| ≤ a
hnm : ∀ n ≥ m, f n.succ ≤ f n
ε : α
ε0 : ε > 0
k : ℕ
hk : a ≤ k • ε
h : ∃ l, ∀ n ≥ m, a - l • ε < f n
l : ℕ := Nat.find h
hl : ∀ n ≥ m, f n > a - l • ε
hl0 : l ≠ 0
i : ℕ
hi : i ≥ m ∧ f i ≤ a - (Nat.find h).pred • ε
j : ℕ
hj : j ≥ i
hfij : f j ≤ f i
⊢ f i < f j + ε
Module: Mathlib.Analysis.SpecialFunctions.Exp

Goal state before:
pp: x z : ℂ
hz : ‖z‖ ≤ 1
⊢ cexp x * cexp z - cexp x - z * cexp x = cexp x * (cexp z - 1 - z)

Tactic:
]

Goal state after:
x z : ℂ
hz : ‖z‖ ≤ 1
⊢ cexp x * cexp z - cexp x - z * cexp x = cexp x * (cexp z - 1 - z)
Module: Mathlib.Analysis.SpecialFunctions.Exp

Goal state before:
pp: r : ℝ
hr_nonneg : 0 ≤ r
hr_le : r ≤ 1
x y : ℂ
hyx : ‖y - x‖ < r
hy_eq : y = x + (y - x)
⊢ ‖y - x‖ * ‖y - x‖ ≤ r * ‖y - x‖

Tactic:
]

Goal state after:
r : ℝ
hr_nonneg : 0 ≤ r
hr_le : r ≤ 1
x y : ℂ
hyx : ‖y - x‖ < r
hy_eq : y = x + (y - x)
⊢ ‖y - x‖ * ‖y - x‖ ≤ r * ‖y - x‖
Module: Mathlib.Analysis.SpecialFunctions.Exp

Goal state before:
pp: r : ℝ
hr_nonneg : 0 ≤ r
hr_le : r ≤ 1
x y : ℂ
hyx : ‖y - x‖ < r
hy_eq : y = x + (y - x)
hyx_sq_le : ‖y - x‖ ^ 2 ≤ r * ‖y - x‖
z : ℂ
hz : ‖z‖ ≤ 1
this : ‖cexp (x + z) - cexp x - z • cexp x‖ ≤ ‖cexp x‖ * ‖z‖ ^ 2
⊢ ‖cexp (x + z) - cexp x‖ - ‖z • cexp x‖ ≤ ‖cexp x‖ * ‖z‖ ^ 2

Tactic:
]

Goal state after:
r : ℝ
hr_nonneg : 0 ≤ r
hr_le : r ≤ 1
x y : ℂ
hyx : ‖y - x‖ < r
hy_eq : y = x + (y - x)
hyx_sq_le : ‖y - x‖ ^ 2 ≤ r * ‖y - x‖
z : ℂ
hz : ‖z‖ ≤ 1
this : ‖cexp (x + z) - cexp x - z • cexp x‖ ≤ ‖cexp x‖ * ‖z‖ ^ 2
⊢ ‖cexp (x + z) - cexp x‖ - ‖z • cexp x‖ ≤ ‖cexp x‖ * ‖z‖ ^ 2
Module: Mathlib.Analysis.SpecialFunctions.Exp

Goal state before:
pp: r : ℝ
hr_nonneg : 0 ≤ r
hr_le : r ≤ 1
x y : ℂ
hyx : ‖y - x‖ < r
hy_eq : y = x + (y - x)
hyx_sq_le : ‖y - x‖ ^ 2 ≤ r * ‖y - x‖
h_sq : ∀ (z : ℂ), ‖z‖ ≤ 1 → ‖cexp (x + z) - cexp x‖ ≤ ‖z‖ * ‖cexp x‖ + ‖cexp x‖ * ‖z‖ ^ 2
⊢ ‖cexp (x + (y - x)) - cexp x‖ = ‖cexp (x + (y - x)) - cexp x‖

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecificLimits.Normed

Goal state before:
pp: α : Type u_1
inst✝² : NormedRing α
inst✝¹ : NormSMulClass ℤ α
inst✝ : Nontrivial α
⊢ Tendsto (fun x ↦ ‖↑x‖) atTop atTop

Tactic:
]

Goal state after:
α : Type u_1
inst✝² : NormedRing α
inst✝¹ : NormSMulClass ℤ α
inst✝ : Nontrivial α
⊢ Tendsto (fun x ↦ ‖↑x‖) atTop atTop
Module: Mathlib.Analysis.SpecialFunctions.Exp

Goal state before:
pp: r : ℝ
hr_nonneg : 0 ≤ r
hr_le : r ≤ 1
x y : ℂ
hyx : ‖y - x‖ < r
hy_eq : y = x + (y - x)
hyx_sq_le : ‖y - x‖ ^ 2 ≤ r * ‖y - x‖
h_sq : ∀ (z : ℂ), ‖z‖ ≤ 1 → ‖cexp (x + z) - cexp x‖ ≤ ‖z‖ * ‖cexp x‖ + ‖cexp x‖ * ‖z‖ ^ 2
⊢ ‖y - x‖ * ‖cexp x‖ + ‖cexp x‖ * (r * ‖y - x‖) ≤ ‖y - x‖ * ‖cexp x‖ + ‖cexp x‖ * (r * ‖y - x‖)

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecificLimits.Normed

Goal state before:
pp: α : Type u_1
inst✝² : NormedRing α
inst✝¹ : NormSMulClass ℤ α
inst✝ : Nontrivial α
⊢ Tendsto (fun x ↦ ‖↑x‖) (atBot ⊔ atTop) atTop

Tactic:
]

Goal state after:
α : Type u_1
inst✝² : NormedRing α
inst✝¹ : NormSMulClass ℤ α
inst✝ : Nontrivial α
⊢ Tendsto (fun x ↦ ‖↑x‖) (atBot ⊔ atTop) atTop
Module: Mathlib.Analysis.SpecialFunctions.Exp

Goal state before:
pp: r : ℝ
hr_nonneg : 0 ≤ r
hr_le : r ≤ 1
x y : ℂ
hyx : ‖y - x‖ < r
hy_eq : y = x + (y - x)
hyx_sq_le : ‖y - x‖ ^ 2 ≤ r * ‖y - x‖
h_sq : ∀ (z : ℂ), ‖z‖ ≤ 1 → ‖cexp (x + z) - cexp x‖ ≤ ‖z‖ * ‖cexp x‖ + ‖cexp x‖ * ‖z‖ ^ 2
⊢ ‖y - x‖ * ‖cexp x‖ + ‖cexp x‖ * (r * ‖y - x‖) = (1 + r) * ‖cexp x‖ * ‖y - x‖

Tactic:
by ring

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Exp

Goal state before:
pp: case inr
n : ℕ
hn : 0 < n
⊢ ∃ i,
    0 < i ∧
      ∀ ⦃x : ℂ⦄, x ∈ {y | ‖y‖ < i} → ‖cexp x - ∑ i ∈ Finset.range n, x ^ i / ↑i !‖ ≤ ↑n.succ / (↑n ! * ↑n) * ‖x ^ n‖

Tactic:
]

Goal state after:
case inr
n : ℕ
hn : 0 < n
⊢ ∃ i,
    0 < i ∧
      ∀ ⦃x : ℂ⦄, x ∈ {y | ‖y‖ < i} → ‖cexp x - ∑ i ∈ Finset.range n, x ^ i / ↑i !‖ ≤ ↑n.succ / (↑n ! * ↑n) * ‖x ^ n‖
Module: Mathlib.Analysis.SpecialFunctions.Exp

Goal state before:
pp: a : ℝ
this : ∀ (a : ℂ), ∀ ε > 0, ∀ᶠ (x : ℂ) in 𝓝 a, dist ((cexp - 1) x) ((cexp - 1) a) < ε
⊢ ∀ ε > 0, ∃ δ > 0, ∀ x ∈ {x | x.re ≤ a}, ∀ y ∈ {x | x.re ≤ a}, dist x y < δ → dist (cexp x) (cexp y) < ε

Tactic:
]

Goal state after:
a : ℝ
this : ∀ (a : ℂ), ∀ ε > 0, ∀ᶠ (x : ℂ) in 𝓝 a, dist ((cexp - 1) x) ((cexp - 1) a) < ε
⊢ ∀ ε > 0, ∃ δ > 0, ∀ x ∈ {x | x.re ≤ a}, ∀ y ∈ {x | x.re ≤ a}, dist x y < δ → dist (cexp x) (cexp y) < ε
Module: Mathlib.Analysis.SpecialFunctions.Exp

Goal state before:
pp: a ε : ℝ
hε : ε > 0
this : ∀ (a : ℂ) (ε : ℝ), 0 < ε → ∀ᶠ (x : ℂ) in 𝓝 a, dist (cexp x) (cexp a) < ε
ha : 0 < ε / (2 * Real.exp a)
H : ∃ ε_1 > 0, ∀ ⦃y : ℂ⦄, dist y 0 < ε_1 → dist (cexp y) (cexp 0) < ε / (2 * Real.exp a)
⊢ ∃ δ > 0, ∀ x ∈ {x | x.re ≤ a}, ∀ y ∈ {x | x.re ≤ a}, dist x y < δ → dist (cexp x) (cexp y) < ε

Tactic:
]

Goal state after:
a ε : ℝ
hε : ε > 0
this : ∀ (a : ℂ) (ε : ℝ), 0 < ε → ∀ᶠ (x : ℂ) in 𝓝 a, dist (cexp x) (cexp a) < ε
ha : 0 < ε / (2 * Real.exp a)
H : ∃ ε_1 > 0, ∀ ⦃y : ℂ⦄, dist y 0 < ε_1 → dist (cexp y) (cexp 0) < ε / (2 * Real.exp a)
⊢ ∃ δ > 0, ∀ x ∈ {x | x.re ≤ a}, ∀ y ∈ {x | x.re ≤ a}, dist x y < δ → dist (cexp x) (cexp y) < ε
Module: Mathlib.Algebra.Order.CauSeq.BigOperators

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
inst✝⁶ : Field α
inst✝⁵ : LinearOrder α
inst✝⁴ : IsStrictOrderedRing α
inst✝³ : Ring β
abv : β → α
inst✝² : IsAbsoluteValue abv
inst✝¹ : Archimedean α
inst✝ : Nontrivial β
x : β
hx1 : abv x < 1
hx1' : abv x ≠ 1
this : 0 < 1 - abv x
n : ℕ
a✝ : n ≥ 0
⊢ (1 - abv x ^ n) / (1 - abv x) ≤ 1 / (1 - abv x)

pp: case refine_1
α : Type u_1
β : Type u_2
inst✝⁶ : Field α
inst✝⁵ : LinearOrder α
inst✝⁴ : IsStrictOrderedRing α
inst✝³ : Ring β
abv : β → α
inst✝² : IsAbsoluteValue abv
inst✝¹ : Archimedean α
inst✝ : Nontrivial β
x : β
hx1 : abv x < 1
hx1' : abv x ≠ 1
this : 0 < 1 - abv x
n : ℕ
a✝ : n ≥ 0
⊢ 0 ≤ (1 - abv x ^ n) / (1 - abv x)

Tactic:
]

Goal state after:
case refine_1
α : Type u_1
β : Type u_2
inst✝⁶ : Field α
inst✝⁵ : LinearOrder α
inst✝⁴ : IsStrictOrderedRing α
inst✝³ : Ring β
abv : β → α
inst✝² : IsAbsoluteValue abv
inst✝¹ : Archimedean α
inst✝ : Nontrivial β
x : β
hx1 : abv x < 1
hx1' : abv x ≠ 1
this : 0 < 1 - abv x
n : ℕ
a✝ : n ≥ 0
⊢ (1 - abv x ^ n) / (1 - abv x) ≤ 1 / (1 - abv x)

case refine_1
α : Type u_1
β : Type u_2
inst✝⁶ : Field α
inst✝⁵ : LinearOrder α
inst✝⁴ : IsStrictOrderedRing α
inst✝³ : Ring β
abv : β → α
inst✝² : IsAbsoluteValue abv
inst✝¹ : Archimedean α
inst✝ : Nontrivial β
x : β
hx1 : abv x < 1
hx1' : abv x ≠ 1
this : 0 < 1 - abv x
n : ℕ
a✝ : n ≥ 0
⊢ 0 ≤ (1 - abv x ^ n) / (1 - abv x)
Module: Mathlib.Analysis.SpecialFunctions.Exp

Goal state before:
pp: a ε : ℝ
hε : ε > 0
this : ∀ (a : ℂ) (ε : ℝ), 0 < ε → ∀ᶠ (x : ℂ) in 𝓝 a, dist (cexp x) (cexp a) < ε
ha : 0 < ε / (2 * Real.exp a)
δ : ℝ
hδ : δ > 0 ∧ ∀ ⦃y : ℂ⦄, dist y 0 < δ → dist (cexp y) 1 < ε / (2 * Real.exp a)
x : ℂ
a✝ : x ∈ {x | x.re ≤ a}
y : ℂ
hy : y ∈ {x | x.re ≤ a}
hxy : ‖x - y‖ < δ
h3 : ‖cexp (x - y) - 1‖ < ε / (2 * Real.exp a)
⊢ ‖cexp x - cexp y‖ < ε

Tactic:
]

Goal state after:
a ε : ℝ
hε : ε > 0
this : ∀ (a : ℂ) (ε : ℝ), 0 < ε → ∀ᶠ (x : ℂ) in 𝓝 a, dist (cexp x) (cexp a) < ε
ha : 0 < ε / (2 * Real.exp a)
δ : ℝ
hδ : δ > 0 ∧ ∀ ⦃y : ℂ⦄, dist y 0 < δ → dist (cexp y) 1 < ε / (2 * Real.exp a)
x : ℂ
a✝ : x ∈ {x | x.re ≤ a}
y : ℂ
hy : y ∈ {x | x.re ≤ a}
hxy : ‖x - y‖ < δ
h3 : ‖cexp (x - y) - 1‖ < ε / (2 * Real.exp a)
⊢ ‖cexp x - cexp y‖ < ε