open List in
/-- Various statements equivalent to the fact that `f n` grows exponentially slower than `R ^ n`.

* 0: $f n = o(a ^ n)$ for some $-R < a < R$;
* 1: $f n = o(a ^ n)$ for some $0 < a < R$;
* 2: $f n = O(a ^ n)$ for some $-R < a < R$;
* 3: $f n = O(a ^ n)$ for some $0 < a < R$;
* 4: there exist `a < R` and `C` such that one of `C` and `R` is positive and $|f n| ≤ Ca^n$
     for all `n`;
* 5: there exists `0 < a < R` and a positive `C` such that $|f n| ≤ Ca^n$ for all `n`;
* 6: there exists `a < R` such that $|f n| ≤ a ^ n$ for sufficiently large `n`;
* 7: there exists `0 < a < R` such that $|f n| ≤ a ^ n$ for sufficiently large `n`.

NB: For backwards compatibility, if you add more items to the list, please append them at the end of
the list. -/
theorem TFAE_exists_lt_isLittleO_pow (f : ℕ → ℝ) (R : ℝ) :
    TFAE
      [∃ a ∈ Ioo (-R) R, f =o[atTop] (a ^ ·), ∃ a ∈ Ioo 0 R, f =o[atTop] (a ^ ·),
        ∃ a ∈ Ioo (-R) R, f =O[atTop] (a ^ ·), ∃ a ∈ Ioo 0 R, f =O[atTop] (a ^ ·),
        ∃ a < R, ∃ C : ℝ, (0 < C ∨ 0 < R) ∧ ∀ n, |f n| ≤ C * a ^ n,
        ∃ a ∈ Ioo 0 R, ∃ C > 0, ∀ n, |f n| ≤ C * a ^ n, ∃ a < R, ∀ᶠ n in atTop, |f n| ≤ a ^ n,
        ∃ a ∈ Ioo 0 R, ∀ᶠ n in atTop, |f n| ≤ a ^ n] := by
  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⟩
  have B : Ioo 0 R ⊆ Ioo (-R) R := Subset.trans Ioo_subset_Ico_self A
  -- First we prove that 1-4 are equivalent using 2 → 3 → 4, 1 → 3, and 2 → 1
  tfae_have 1 → 3
  exact fun ⟨a, ha, H⟩ ↦ ⟨a, ha, H.isBigO⟩
  tfae_have 2 → 1
  exact fun ⟨a, ha, H⟩ ↦ ⟨a, B ha, H⟩
  tfae_have 3 → 2
  · rintro ⟨a, ha, H⟩
    rcases exists_between (abs_lt.2 ha) with ⟨b, hab, hbR⟩
    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)))⟩
  tfae_have 2 → 4
  exact fun ⟨a, ha, H⟩ ↦ ⟨a, ha, H.isBigO⟩
  tfae_have 4 → 3
  exact fun ⟨a, ha, H⟩ ↦ ⟨a, B ha, H⟩
  -- Add 5 and 6 using 4 → 6 → 5 → 3
  tfae_have 4 → 6
  · rintro ⟨a, ha, H⟩
    rcases bound_of_isBigO_nat_atTop H with ⟨C, hC₀, hC⟩
    refine' ⟨a, ha, C, hC₀, fun n ↦ _⟩
    simpa only [Real.norm_eq_abs, abs_pow, abs_of_nonneg ha.1.le] using hC (pow_ne_zero n ha.1.ne')
  tfae_have 6 → 5
  exact fun ⟨a, ha, C, H₀, H⟩ ↦ ⟨a, ha.2, C, Or.inl H₀, H⟩
  tfae_have 5 → 3
  · rintro ⟨a, ha, C, h₀, H⟩
    rcases sign_cases_of_C_mul_pow_nonneg fun n ↦ (abs_nonneg _).trans (H n) with (rfl | ⟨hC₀, ha₀⟩)
    · obtain rfl : f = 0 := by
        ext n
        simpa using H n
      simp only [lt_irrefl, false_or_iff] at h₀
      exact ⟨0, ⟨neg_lt_zero.2 h₀, h₀⟩, isBigO_zero _ _⟩
    exact ⟨a, A ⟨ha₀, ha⟩,
      isBigO_of_le' _ fun n ↦ (H n).trans <| mul_le_mul_of_nonneg_left (le_abs_self _) hC₀.le⟩
  -- Add 7 and 8 using 2 → 8 → 7 → 3
  tfae_have 2 → 8
  · rintro ⟨a, ha, H⟩
    refine' ⟨a, ha, (H.def zero_lt_one).mono fun n hn ↦ _⟩
    rwa [Real.norm_eq_abs, Real.norm_eq_abs, one_mul, abs_pow, abs_of_pos ha.1] at hn
  tfae_have 8 → 7
  exact fun ⟨a, ha, H⟩ ↦ ⟨a, ha.2, H⟩
  tfae_have 7 → 3
  · rintro ⟨a, ha, H⟩
    have : 0 ≤ a := nonneg_of_eventually_pow_nonneg (H.mono fun n ↦ (abs_nonneg _).trans)
    refine' ⟨a, A ⟨this, ha⟩, IsBigO.of_bound 1 _⟩
    

Proof state:
case tfae_7_to_3.intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
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 ∈ Set.Ioo (-R) R, f =o[atTop] fun x => a ^ x) → ∃ a ∈ Set.Ioo (-R) R, f =O[atTop] fun x => a ^ x
tfae_2_to_1 : (∃ a ∈ Set.Ioo 0 R, f =o[atTop] fun x => a ^ x) → ∃ a ∈ Set.Ioo (-R) R, f =o[atTop] fun x => a ^ x
tfae_3_to_2 : (∃ a ∈ Set.Ioo (-R) R, f =O[atTop] fun x => a ^ x) → ∃ a ∈ Set.Ioo 0 R, f =o[atTop] fun x => a ^ x
tfae_2_to_4 : (∃ a ∈ Set.Ioo 0 R, f =o[atTop] fun x => a ^ x) → ∃ a ∈ Set.Ioo 0 R, f =O[atTop] fun x => a ^ x
tfae_4_to_3 : (∃ a ∈ Set.Ioo 0 R, f =O[atTop] fun x => a ^ x) → ∃ a ∈ Set.Ioo (-R) R, f =O[atTop] fun x => a ^ x
tfae_4_to_6 : (∃ a ∈ Set.Ioo 0 R, f =O[atTop] fun x => a ^ x) → ∃ a ∈ Set.Ioo 0 R, ∃ C > 0, ∀ (n : ℕ), |f n| ≤ C * a ^ n
tfae_6_to_5 :
  (∃ a ∈ Set.Ioo 0 R, ∃ C > 0, ∀ (n : ℕ), |f n| ≤ C * a ^ n) →
    ∃ a < R, ∃ C, (0 < C ∨ 0 < R) ∧ ∀ (n : ℕ), |f n| ≤ C * a ^ n
tfae_5_to_3 :
  (∃ a < R, ∃ C, (0 < C ∨ 0 < R) ∧ ∀ (n : ℕ), |f n| ≤ C * a ^ n) → ∃ a ∈ Set.Ioo (-R) R, f =O[atTop] fun x => a ^ x
tfae_2_to_8 : (∃ a ∈ Set.Ioo 0 R, f =o[atTop] fun x => a ^ x) → ∃ a ∈ Set.Ioo 0 R, ∀ᶠ (n : ℕ) in atTop, |f n| ≤ a ^ n
tfae_8_to_7 : (∃ a ∈ Set.Ioo 0 R, ∀ᶠ (n : ℕ) in atTop, |f n| ≤ a ^ n) → ∃ a < R, ∀ᶠ (n : ℕ) in atTop, |f n| ≤ a ^ n
a : ℝ
ha : a < R
H : ∀ᶠ (n : ℕ) in atTop, |f n| ≤ a ^ n
this : 0 ≤ a
⊢ ∀ᶠ (x : ℕ) in atTop, ‖f x‖ ≤ 1 * ‖a ^ x‖

Next tactic:
simpa only [Real.norm_eq_abs, one_mul, abs_pow, abs_of_nonneg this]