Theorem: TFAE_exists_lt_isLittleO_pow
File: Mathlib/Analysis/SpecificLimits/Normed.lean

tactic: have A : Ico 0 R ⊆ Ioo (-R) R :=
  fun x hx ↦ ⟨(neg_lt_zero.2 (hx.1.trans_lt hx.2)).trans_le hx.1, hx.2⟩
state_before: α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
⊢ TFAE
    [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,
      ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,
      ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,
      ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,
      ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,
      ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]

tactic: have B : Ioo 0 R ⊆ Ioo (-R) R := Subset.trans Ioo_subset_Ico_self A
state_before: α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
⊢ TFAE
    [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,
      ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,
      ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,
      ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,
      ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,
      ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]

tactic: tfae_have 1 → 3
state_before: α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
⊢ TFAE
    [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,
      ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,
      ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,
      ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,
      ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,
      ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]

tactic: exact fun ⟨a, ha, H⟩ ↦ ⟨a, ha, H.isBigO⟩
state_before: case tfae_1_to_3
α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
⊢ (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x

α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
⊢ TFAE
    [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,
      ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,
      ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,
      ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,
      ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,
      ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]

tactic: tfae_have 2 → 1
state_before: α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
⊢ TFAE
    [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,
      ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,
      ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,
      ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,
      ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,
      ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]

tactic: exact fun ⟨a, ha, H⟩ ↦ ⟨a, B ha, H⟩
state_before: case tfae_2_to_1
α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
⊢ (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x

α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x
⊢ TFAE
    [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,
      ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,
      ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,
      ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,
      ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,
      ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]

tactic: tfae_have 3 → 2
state_before: α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x
⊢ TFAE
    [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,
      ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,
      ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,
      ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,
      ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,
      ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]

tactic: tfae_have 2 → 4
state_before: α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x
tfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x
⊢ TFAE
    [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,
      ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,
      ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,
      ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,
      ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,
      ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]

tactic: exact fun ⟨a, ha, H⟩ ↦ ⟨a, ha, H.isBigO⟩
state_before: case tfae_2_to_4
α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x
tfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x
⊢ (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x

α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x
tfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x
tfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x
⊢ TFAE
    [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,
      ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,
      ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,
      ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,
      ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,
      ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]

tactic: tfae_have 4 → 3
state_before: α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x
tfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x
tfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x
⊢ TFAE
    [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,
      ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,
      ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,
      ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,
      ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,
      ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]

tactic: exact fun ⟨a, ha, H⟩ ↦ ⟨a, B ha, H⟩
state_before: case tfae_4_to_3
α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x
tfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x
tfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x
⊢ (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x

α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x
tfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x
tfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
⊢ TFAE
    [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,
      ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,
      ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,
      ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,
      ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,
      ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]

tactic: tfae_have 4 → 6
state_before: α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x
tfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x
tfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
⊢ TFAE
    [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,
      ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,
      ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,
      ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,
      ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,
      ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]

tactic: tfae_have 6 → 5
state_before: α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x
tfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x
tfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_6 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →
    ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
⊢ TFAE
    [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,
      ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,
      ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,
      ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,
      ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,
      ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]

tactic: exact fun ⟨a, ha, C, H₀, H⟩ ↦ ⟨a, ha.2, C, Or.inl H₀, H⟩
state_before: case tfae_6_to_5
α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x
tfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x
tfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_6 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →
    ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
⊢ (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →
    ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n

α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x
tfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x
tfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_6 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →
    ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
tfae_6_to_5 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →
    ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
⊢ TFAE
    [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,
      ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,
      ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,
      ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,
      ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,
      ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]

tactic: tfae_have 5 → 3
state_before: α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x
tfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x
tfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_6 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →
    ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
tfae_6_to_5 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →
    ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
⊢ TFAE
    [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,
      ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,
      ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,
      ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,
      ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,
      ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]

tactic: tfae_have 2 → 8
state_before: α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x
tfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x
tfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_6 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →
    ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
tfae_6_to_5 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →
    ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
tfae_5_to_3 :
  (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
⊢ TFAE
    [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,
      ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,
      ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,
      ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,
      ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,
      ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]

tactic: tfae_have 8 → 7
state_before: α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x
tfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x
tfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_6 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →
    ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
tfae_6_to_5 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →
    ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
tfae_5_to_3 :
  (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_8 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n
⊢ TFAE
    [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,
      ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,
      ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,
      ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,
      ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,
      ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]

tactic: exact fun ⟨a, ha, H⟩ ↦ ⟨a, ha.2, H⟩
state_before: case tfae_8_to_7
α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x
tfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x
tfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_6 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →
    ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
tfae_6_to_5 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →
    ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
tfae_5_to_3 :
  (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_8 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n
⊢ (∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n) → ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n

α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x
tfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x
tfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_6 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →
    ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
tfae_6_to_5 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →
    ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
tfae_5_to_3 :
  (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_8 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n
tfae_8_to_7 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n) → ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n
⊢ TFAE
    [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,
      ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,
      ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,
      ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,
      ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,
      ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]

tactic: tfae_have 7 → 3
state_before: α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x
tfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x
tfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_6 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →
    ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
tfae_6_to_5 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →
    ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
tfae_5_to_3 :
  (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_8 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n
tfae_8_to_7 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n) → ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n
⊢ TFAE
    [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,
      ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,
      ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,
      ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,
      ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,
      ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]

tactic: tfae_have 6 → 7
state_before: α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x
tfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x
tfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_6 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →
    ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
tfae_6_to_5 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →
    ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
tfae_5_to_3 :
  (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_8 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n
tfae_8_to_7 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n) → ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n
tfae_7_to_3 :
  (∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
⊢ TFAE
    [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,
      ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,
      ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,
      ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,
      ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,
      ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]

tactic: tfae_finish
state_before: α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x
tfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x
tfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_6 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →
    ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
tfae_6_to_5 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →
    ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
tfae_5_to_3 :
  (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_8 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n
tfae_8_to_7 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n) → ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n
tfae_7_to_3 :
  (∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_6_to_7 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →
    ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n
⊢ TFAE
    [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,
      ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,
      ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,
      ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,
      ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,
      ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]

tactic: rintro ⟨a, ha, H⟩
state_before: case tfae_3_to_2
α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x
⊢ (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x

tactic: rcases exists_between (abs_lt.2 ha) with ⟨b, hab, hbR⟩
state_before: case tfae_3_to_2.intro.intro
α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x
a : ℝ
ha : a ∈ Set.Ioo (-R) R
H : f =O[atTop] fun x => a ^ x
⊢ ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x

tactic: exact ⟨b, ⟨(abs_nonneg a).trans_lt hab, hbR⟩,
  H.trans_isLittleO (isLittleO_pow_pow_of_abs_lt_left (hab.trans_le (le_abs_self b)))⟩
state_before: case tfae_3_to_2.intro.intro.intro.intro
α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x
a : ℝ
ha : a ∈ Set.Ioo (-R) R
H : f =O[atTop] fun x => a ^ x
b : ℝ
hab : abs a < b
hbR : b < R
⊢ ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x

tactic: rintro ⟨a, ha, H⟩
state_before: case tfae_4_to_6
α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x
tfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x
tfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
⊢ (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →
    ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n

tactic: rcases bound_of_isBigO_nat_atTop H with ⟨C, hC₀, hC⟩
state_before: case tfae_4_to_6.intro.intro
α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x
tfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x
tfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
a : ℝ
ha : a ∈ Set.Ioo 0 R
H : f =O[atTop] fun x => a ^ x
⊢ ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n

tactic: refine' ⟨a, ha, C, hC₀, fun n ↦ _⟩
state_before: case tfae_4_to_6.intro.intro.intro.intro
α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x
tfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x
tfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
a : ℝ
ha : a ∈ Set.Ioo 0 R
H : f =O[atTop] fun x => a ^ x
C : ℝ
hC₀ : C > 0
hC : ∀ ⦃x : ℕ⦄, a ^ x ≠ 0 → ‖f x‖ ≤ C * ‖a ^ x‖
⊢ ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n

tactic: simpa only [Real.norm_eq_abs, abs_pow, abs_of_nonneg ha.1.le] using hC (pow_ne_zero n ha.1.ne')
state_before: case tfae_4_to_6.intro.intro.intro.intro
α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x
tfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x
tfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
a : ℝ
ha : a ∈ Set.Ioo 0 R
H : f =O[atTop] fun x => a ^ x
C : ℝ
hC₀ : C > 0
hC : ∀ ⦃x : ℕ⦄, a ^ x ≠ 0 → ‖f x‖ ≤ C * ‖a ^ x‖
n : ℕ
⊢ abs (f n) ≤ C * a ^ n

tactic: rintro ⟨a, ha, C, h₀, H⟩
state_before: case tfae_5_to_3
α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x
tfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x
tfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_6 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →
    ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
tfae_6_to_5 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →
    ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
⊢ (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x

tactic: rcases sign_cases_of_C_mul_pow_nonneg fun n ↦ (abs_nonneg _).trans (H n) with (rfl | ⟨hC₀, ha₀⟩)
state_before: case tfae_5_to_3.intro.intro.intro.intro
α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x
tfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x
tfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_6 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →
    ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
tfae_6_to_5 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →
    ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
a : ℝ
ha : a < R
C : ℝ
h₀ : 0 < C ∨ 0 < R
H : ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
⊢ ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x

tactic: exact ⟨a, A ⟨ha₀, ha⟩,
  isBigO_of_le' _ fun n ↦ (H n).trans <| mul_le_mul_of_nonneg_left (le_abs_self _) hC₀.le⟩
state_before: case tfae_5_to_3.intro.intro.intro.intro.inr.intro
α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x
tfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x
tfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_6 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →
    ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
tfae_6_to_5 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →
    ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
a : ℝ
ha : a < R
C : ℝ
h₀ : 0 < C ∨ 0 < R
H : ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
hC₀ : 0 < C
ha₀ : 0 ≤ a
⊢ ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x

tactic: obtain rfl : f = 0 := by
  ext n
  simpa using H n
state_before: case tfae_5_to_3.intro.intro.intro.intro.inl
α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x
tfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x
tfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_6 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →
    ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
tfae_6_to_5 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →
    ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
a : ℝ
ha : a < R
h₀ : 0 < 0 ∨ 0 < R
H : ∀ (n : ℕ), abs (f n) ≤ 0 * a ^ n
⊢ ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x

tactic: simp only [lt_irrefl, false_or_iff] at h₀
state_before: case tfae_5_to_3.intro.intro.intro.intro.inl
α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
a : ℝ
ha : a < R
h₀ : 0 < 0 ∨ 0 < R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ 0 =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ 0 =O[atTop] fun x => a ^ x
tfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ 0 =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ 0 =o[atTop] fun x => a ^ x
tfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ 0 =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ 0 =o[atTop] fun x => a ^ x
tfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ 0 =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ 0 =O[atTop] fun x => a ^ x
tfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ 0 =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ 0 =O[atTop] fun x => a ^ x
tfae_4_to_6 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ 0 =O[atTop] fun x => a ^ x) →
    ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (OfNat.ofNat 0 n) ≤ C * a ^ n
tfae_6_to_5 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (OfNat.ofNat 0 n) ≤ C * a ^ n) →
    ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (OfNat.ofNat 0 n) ≤ C * a ^ n
H : ∀ (n : ℕ), abs (OfNat.ofNat 0 n) ≤ 0 * a ^ n
⊢ ∃ a, a ∈ Set.Ioo (-R) R ∧ 0 =O[atTop] fun x => a ^ x

tactic: exact ⟨0, ⟨neg_lt_zero.2 h₀, h₀⟩, isBigO_zero _ _⟩
state_before: case tfae_5_to_3.intro.intro.intro.intro.inl
α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
a : ℝ
ha : a < R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ 0 =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ 0 =O[atTop] fun x => a ^ x
tfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ 0 =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ 0 =o[atTop] fun x => a ^ x
tfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ 0 =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ 0 =o[atTop] fun x => a ^ x
tfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ 0 =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ 0 =O[atTop] fun x => a ^ x
tfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ 0 =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ 0 =O[atTop] fun x => a ^ x
tfae_4_to_6 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ 0 =O[atTop] fun x => a ^ x) →
    ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (OfNat.ofNat 0 n) ≤ C * a ^ n
tfae_6_to_5 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (OfNat.ofNat 0 n) ≤ C * a ^ n) →
    ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (OfNat.ofNat 0 n) ≤ C * a ^ n
H : ∀ (n : ℕ), abs (OfNat.ofNat 0 n) ≤ 0 * a ^ n
h₀ : 0 < R
⊢ ∃ a, a ∈ Set.Ioo (-R) R ∧ 0 =O[atTop] fun x => a ^ x

tactic: ext n
state_before: α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x
tfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x
tfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_6 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →
    ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
tfae_6_to_5 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →
    ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
a : ℝ
ha : a < R
h₀ : 0 < 0 ∨ 0 < R
H : ∀ (n : ℕ), abs (f n) ≤ 0 * a ^ n
⊢ f = 0

tactic: simpa using H n
state_before: case h
α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x
tfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x
tfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_6 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →
    ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
tfae_6_to_5 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →
    ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
a : ℝ
ha : a < R
h₀ : 0 < 0 ∨ 0 < R
H : ∀ (n : ℕ), abs (f n) ≤ 0 * a ^ n
n : ℕ
⊢ f n = OfNat.ofNat 0 n

tactic: rintro ⟨a, ha, H⟩
state_before: case tfae_2_to_8
α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x
tfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x
tfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_6 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →
    ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
tfae_6_to_5 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →
    ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
tfae_5_to_3 :
  (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
⊢ (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n

tactic: refine' ⟨a, ha, (H.def zero_lt_one).mono fun n hn ↦ _⟩
state_before: case tfae_2_to_8.intro.intro
α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x
tfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x
tfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_6 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →
    ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
tfae_6_to_5 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →
    ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
tfae_5_to_3 :
  (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
a : ℝ
ha : a ∈ Set.Ioo 0 R
H : f =o[atTop] fun x => a ^ x
⊢ ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n

tactic: rwa [Real.norm_eq_abs, Real.norm_eq_abs, one_mul, abs_pow, abs_of_pos ha.1] at hn
state_before: case tfae_2_to_8.intro.intro
α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x
tfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x
tfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_6 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →
    ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
tfae_6_to_5 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →
    ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
tfae_5_to_3 :
  (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
a : ℝ
ha : a ∈ Set.Ioo 0 R
H : f =o[atTop] fun x => a ^ x
n : ℕ
hn : ‖f n‖ ≤ 1 * ‖a ^ n‖
⊢ abs (f n) ≤ a ^ n

tactic: rintro ⟨a, ha, H⟩
state_before: case tfae_7_to_3
α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x
tfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x
tfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_6 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →
    ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
tfae_6_to_5 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →
    ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
tfae_5_to_3 :
  (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_8 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n
tfae_8_to_7 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n) → ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n
⊢ (∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x

tactic: have : 0 ≤ a := nonneg_of_eventually_pow_nonneg (H.mono fun n ↦ (abs_nonneg _).trans)
state_before: case tfae_7_to_3.intro.intro
α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x
tfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x
tfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_6 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →
    ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
tfae_6_to_5 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →
    ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
tfae_5_to_3 :
  (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_8 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n
tfae_8_to_7 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n) → ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n
a : ℝ
ha : a < R
H : ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n
⊢ ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x

tactic: refine' ⟨a, A ⟨this, ha⟩, IsBigO.of_bound 1 _⟩
state_before: case tfae_7_to_3.intro.intro
α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x
tfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x
tfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_6 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →
    ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
tfae_6_to_5 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →
    ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
tfae_5_to_3 :
  (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_8 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n
tfae_8_to_7 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n) → ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n
a : ℝ
ha : a < R
H : ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n
this : 0 ≤ a
⊢ ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x

tactic: simpa only [Real.norm_eq_abs, one_mul, abs_pow, abs_of_nonneg this]
state_before: case tfae_7_to_3.intro.intro
α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x
tfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x
tfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_6 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →
    ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
tfae_6_to_5 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →
    ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
tfae_5_to_3 :
  (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_8 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n
tfae_8_to_7 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n) → ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n
a : ℝ
ha : a < R
H : ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n
this : 0 ≤ a
⊢ ∀ᶠ (x : ℕ) in atTop, ‖f x‖ ≤ 1 * ‖a ^ x‖

tactic: exact fun h ↦  tfae_8_to_7 <| tfae_2_to_8 <| tfae_3_to_2 <| tfae_5_to_3 <| tfae_6_to_5 h
state_before: case tfae_6_to_7
α : Type ?u.73187
β : Type ?u.73190
ι : Type ?u.73193
f : ℕ → ℝ
R : ℝ
A : Set.Ico 0 R ⊆ Set.Ioo (-R) R
B : Set.Ioo 0 R ⊆ Set.Ioo (-R) R
tfae_1_to_3 :
  (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x
tfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x
tfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_4_to_6 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →
    ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
tfae_6_to_5 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →
    ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n
tfae_5_to_3 :
  (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
tfae_2_to_8 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n
tfae_8_to_7 :
  (∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n) → ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n
tfae_7_to_3 :
  (∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x
⊢ (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →
    ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n