awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Analysis.SpecificLimits.Normed (hypothesis)
Goal:
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, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
⊢ [∃ a ∈ Set.Ioo (-R) R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x,
      ∃ a ∈ Set.Ioo 0 R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x,
      ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x,
      ∃ a ∈ Set.Ioo 0 R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x,
      ∃ a < R, ∃ (C : ℝ), (0 < C ∨ 0 < R) ∧ ∀ (n : ℕ), |f n| ≤ C * a ^ n,
      ∃ a ∈ Set.Ioo 0 R, ∃ C > 0, ∀ (n : ℕ), |f n| ≤ C * a ^ n, ∃ a < R, ∀ᶠ (n : ℕ) in atTop, |f n| ≤ a ^ n,
      ∃ a ∈ Set.Ioo 0 R, ∀ᶠ (n : ℕ) in atTop, |f n| ≤ a ^ n].TFAE
have tfae_2_to_1 : ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
  ∃ a ∈ Set.Ioo (-R) R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
Tactic: tfae_have 2 → 1 := fun ⟨a, ha, H⟩ ↦ ⟨a, B ha, H⟩
-- Lean 4 proof step from Mathlib.Analysis.SpecificLimits.Normed (goal)
Goal:
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, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
tfae_2_to_1 :
  ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo (-R) R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
a : ℝ
ha : a ∈ Set.Ioo (-R) R
H : IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
⊢ ∃ a ∈ Set.Ioo 0 R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
have rcases : ∀ (b : ℝ) (hab : |a| < b) (hbR : b < R), ∃ a ∈ Set.Ioo 0 R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
Tactic: rcases exists_between (abs_lt.2 ha) with ⟨b, hab, hbR⟩
-- Lean 4 proof step from Mathlib.Analysis.SpecificLimits.Normed (hypothesis)
Goal:
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, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
tfae_2_to_1 :
  ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo (-R) R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
⊢ [∃ a ∈ Set.Ioo (-R) R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x,
      ∃ a ∈ Set.Ioo 0 R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x,
      ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x,
      ∃ a ∈ Set.Ioo 0 R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x,
      ∃ a < R, ∃ (C : ℝ), (0 < C ∨ 0 < R) ∧ ∀ (n : ℕ), |f n| ≤ C * a ^ n,
      ∃ a ∈ Set.Ioo 0 R, ∃ C > 0, ∀ (n : ℕ), |f n| ≤ C * a ^ n, ∃ a < R, ∀ᶠ (n : ℕ) in atTop, |f n| ≤ a ^ n,
      ∃ a ∈ Set.Ioo 0 R, ∀ᶠ (n : ℕ) in atTop, |f n| ≤ a ^ n].TFAE
have tfae_3_to_2 : ∀ (a : ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
  ∃ a ∈ Set.Ioo 0 R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
Tactic: tfae_have 3 → 2
  | ⟨a, ha, H⟩ => by
    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)))⟩
-- Lean 4 proof step from Mathlib.Analysis.SpecificLimits.Normed (hypothesis)
Goal:
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, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
tfae_2_to_1 :
  ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo (-R) R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
tfae_3_to_2 :
  ∀ (a : ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo 0 R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
⊢ [∃ a ∈ Set.Ioo (-R) R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x,
      ∃ a ∈ Set.Ioo 0 R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x,
      ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x,
      ∃ a ∈ Set.Ioo 0 R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x,
      ∃ a < R, ∃ (C : ℝ), (0 < C ∨ 0 < R) ∧ ∀ (n : ℕ), |f n| ≤ C * a ^ n,
      ∃ a ∈ Set.Ioo 0 R, ∃ C > 0, ∀ (n : ℕ), |f n| ≤ C * a ^ n, ∃ a < R, ∀ᶠ (n : ℕ) in atTop, |f n| ≤ a ^ n,
      ∃ a ∈ Set.Ioo 0 R, ∀ᶠ (n : ℕ) in atTop, |f n| ≤ a ^ n].TFAE
have tfae_2_to_4 : ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
  ∃ a ∈ Set.Ioo 0 R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
Tactic: tfae_have 2 → 4 := fun ⟨a, ha, H⟩ ↦ ⟨a, ha, H.isBigO⟩
-- Lean 4 proof step from Mathlib.Analysis.SpecificLimits.Normed (hypothesis)
Goal:
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, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
tfae_2_to_1 :
  ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo (-R) R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
tfae_3_to_2 :
  ∀ (a : ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo 0 R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
tfae_2_to_4 :
  ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo 0 R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
⊢ [∃ a ∈ Set.Ioo (-R) R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x,
      ∃ a ∈ Set.Ioo 0 R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x,
      ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x,
      ∃ a ∈ Set.Ioo 0 R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x,
      ∃ a < R, ∃ (C : ℝ), (0 < C ∨ 0 < R) ∧ ∀ (n : ℕ), |f n| ≤ C * a ^ n,
      ∃ a ∈ Set.Ioo 0 R, ∃ C > 0, ∀ (n : ℕ), |f n| ≤ C * a ^ n, ∃ a < R, ∀ᶠ (n : ℕ) in atTop, |f n| ≤ a ^ n,
      ∃ a ∈ Set.Ioo 0 R, ∀ᶠ (n : ℕ) in atTop, |f n| ≤ a ^ n].TFAE
have tfae_4_to_3 : ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
  ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
Tactic: tfae_have 4 → 3 := fun ⟨a, ha, H⟩ ↦ ⟨a, B ha, H⟩
-- Lean 4 proof step from Mathlib.Analysis.SpecificLimits.Normed (goal)
Goal:
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, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
tfae_2_to_1 :
  ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo (-R) R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
tfae_3_to_2 :
  ∀ (a : ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo 0 R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
tfae_2_to_4 :
  ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo 0 R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
tfae_4_to_3 :
  ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
a : ℝ
ha : a ∈ Set.Ioo 0 R
H : IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
⊢ ∃ a ∈ Set.Ioo 0 R, ∃ C > 0, ∀ (n : ℕ), |f n| ≤ C * a ^ n
have rcases : ∀ (C : ℝ) (hC₀ : C > 0) (hC : ∀ ⦃x : ℕ⦄, a ^ x ≠ 0 → ‖f x‖ ≤ C * ‖a ^ x‖),
  ∃ a ∈ Set.Ioo 0 R, ∃ C > 0, ∀ (n : ℕ), |f n| ≤ C * a ^ n
Tactic: rcases bound_of_isBigO_nat_atTop H with ⟨C, hC₀, hC⟩
-- Lean 4 proof step from Mathlib.Analysis.SpecificLimits.Normed (goal)
Goal:
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, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
tfae_2_to_1 :
  ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo (-R) R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
tfae_3_to_2 :
  ∀ (a : ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo 0 R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
tfae_2_to_4 :
  ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo 0 R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
tfae_4_to_3 :
  ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
a : ℝ
ha : a ∈ Set.Ioo 0 R
H : IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
C : ℝ
hC₀ : C > 0
hC : ∀ ⦃x : ℕ⦄, a ^ x ≠ 0 → ‖f x‖ ≤ C * ‖a ^ x‖
⊢ ∃ a ∈ Set.Ioo 0 R, ∃ C > 0, ∀ (n : ℕ), |f n| ≤ C * a ^ n
have refine : ∀ (n : ℕ), |f n| ≤ C * a ^ n
Tactic: refine ⟨a, ha, C, hC₀, fun n ↦ ?_⟩
-- Lean 4 proof step from Mathlib.Analysis.SpecificLimits.Normed (hypothesis)
Goal:
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, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
tfae_2_to_1 :
  ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo (-R) R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
tfae_3_to_2 :
  ∀ (a : ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo 0 R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
tfae_2_to_4 :
  ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo 0 R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
tfae_4_to_3 :
  ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
⊢ [∃ a ∈ Set.Ioo (-R) R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x,
      ∃ a ∈ Set.Ioo 0 R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x,
      ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x,
      ∃ a ∈ Set.Ioo 0 R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x,
      ∃ a < R, ∃ (C : ℝ), (0 < C ∨ 0 < R) ∧ ∀ (n : ℕ), |f n| ≤ C * a ^ n,
      ∃ a ∈ Set.Ioo 0 R, ∃ C > 0, ∀ (n : ℕ), |f n| ≤ C * a ^ n, ∃ a < R, ∀ᶠ (n : ℕ) in atTop, |f n| ≤ a ^ n,
      ∃ a ∈ Set.Ioo 0 R, ∀ᶠ (n : ℕ) in atTop, |f n| ≤ a ^ n].TFAE
have tfae_4_to_6 : ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
  ∃ a ∈ Set.Ioo 0 R, ∃ C > 0, ∀ (n : ℕ), |f n| ≤ C * a ^ n
Tactic: tfae_have 4 → 6
  | ⟨a, ha, H⟩ => by
    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')
-- Lean 4 proof step from Mathlib.Analysis.SpecificLimits.Normed (hypothesis)
Goal:
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, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
tfae_2_to_1 :
  ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo (-R) R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
tfae_3_to_2 :
  ∀ (a : ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo 0 R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
tfae_2_to_4 :
  ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo 0 R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
tfae_4_to_3 :
  ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
tfae_4_to_6 :
  ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo 0 R, ∃ C > 0, ∀ (n : ℕ), |f n| ≤ C * a ^ n
⊢ [∃ a ∈ Set.Ioo (-R) R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x,
      ∃ a ∈ Set.Ioo 0 R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x,
      ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x,
      ∃ a ∈ Set.Ioo 0 R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x,
      ∃ a < R, ∃ (C : ℝ), (0 < C ∨ 0 < R) ∧ ∀ (n : ℕ), |f n| ≤ C * a ^ n,
      ∃ a ∈ Set.Ioo 0 R, ∃ C > 0, ∀ (n : ℕ), |f n| ≤ C * a ^ n, ∃ a < R, ∀ᶠ (n : ℕ) in atTop, |f n| ≤ a ^ n,
      ∃ a ∈ Set.Ioo 0 R, ∀ᶠ (n : ℕ) in atTop, |f n| ≤ a ^ n].TFAE
have tfae_6_to_5 : ∀ (a : ∃ a ∈ Set.Ioo 0 R, ∃ C > 0, ∀ (n : ℕ), |f n| ≤ C * a ^ n),
  ∃ a < R, ∃ (C : ℝ), (0 < C ∨ 0 < R) ∧ ∀ (n : ℕ), |f n| ≤ C * a ^ n
Tactic: tfae_have 6 → 5 := fun ⟨a, ha, C, H₀, H⟩ ↦ ⟨a, ha.2, C, Or.inl H₀, H⟩
-- Lean 4 proof step from Mathlib.Analysis.SpecificLimits.Normed (goal)
Goal:
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, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
tfae_2_to_1 :
  ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo (-R) R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
tfae_3_to_2 :
  ∀ (a : ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo 0 R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
tfae_2_to_4 :
  ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo 0 R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
tfae_4_to_3 :
  ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
tfae_4_to_6 :
  ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo 0 R, ∃ C > 0, ∀ (n : ℕ), |f n| ≤ C * a ^ n
tfae_6_to_5 :
  ∀ (a : ∃ a ∈ Set.Ioo 0 R, ∃ C > 0, ∀ (n : ℕ), |f n| ≤ C * a ^ n),
    ∃ a < R, ∃ (C : ℝ), (0 < C ∨ 0 < R) ∧ ∀ (n : ℕ), |f n| ≤ C * a ^ n
a : ℝ
ha : a < R
C : ℝ
h₀ : 0 < C ∨ 0 < R
H : ∀ (n : ℕ), |f n| ≤ C * a ^ n
⊢ ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
have inl : ∀ (h₀ : LT.lt (α := ℝ) 0 0 ∨ 0 < R) (H : ∀ (n : ℕ), |f n| ≤ 0 * a ^ n),
  ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
Tactic: rcases sign_cases_of_C_mul_pow_nonneg fun n ↦ (abs_nonneg _).trans (H n) with (rfl | ⟨hC₀, ha₀⟩)
Cleared hypotheses: C, h₀, H
-- Lean 4 proof step from Mathlib.Analysis.SpecificLimits.Normed (goal)
Goal:
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, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
tfae_2_to_1 :
  ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo (-R) R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
tfae_3_to_2 :
  ∀ (a : ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo 0 R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
tfae_2_to_4 :
  ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo 0 R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
tfae_4_to_3 :
  ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
tfae_4_to_6 :
  ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo 0 R, ∃ C > 0, ∀ (n : ℕ), |f n| ≤ C * a ^ n
tfae_6_to_5 :
  ∀ (a : ∃ a ∈ Set.Ioo 0 R, ∃ C > 0, ∀ (n : ℕ), |f n| ≤ C * a ^ n),
    ∃ a < R, ∃ (C : ℝ), (0 < C ∨ 0 < R) ∧ ∀ (n : ℕ), |f n| ≤ C * a ^ n
a : ℝ
ha : a < R
C : ℝ
h₀ : 0 < C ∨ 0 < R
H : ∀ (n : ℕ), |f n| ≤ C * a ^ n
inl :
  ∀ (h₀ : LT.lt (α := ℝ) 0 0 ∨ 0 < R) (H : ∀ (n : ℕ), |f n| ≤ 0 * a ^ n),
    ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
⊢ ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
have inr : ∀ (hC₀ : 0 < C) (ha₀ : 0 ≤ a), ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
Tactic: rcases sign_cases_of_C_mul_pow_nonneg fun n ↦ (abs_nonneg _).trans (H n) with (rfl | ⟨hC₀, ha₀⟩)
-- Lean 4 proof step from Mathlib.Analysis.SpecificLimits.Normed (goal)
Goal:
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, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
tfae_2_to_1 :
  ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo (-R) R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
tfae_3_to_2 :
  ∀ (a : ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo 0 R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
tfae_2_to_4 :
  ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo 0 R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
tfae_4_to_3 :
  ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
tfae_4_to_6 :
  ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo 0 R, ∃ C > 0, ∀ (n : ℕ), |f n| ≤ C * a ^ n
tfae_6_to_5 :
  ∀ (a : ∃ a ∈ Set.Ioo 0 R, ∃ C > 0, ∀ (n : ℕ), |f n| ≤ C * a ^ n),
    ∃ a < R, ∃ (C : ℝ), (0 < C ∨ 0 < R) ∧ ∀ (n : ℕ), |f n| ≤ C * a ^ n
a : ℝ
ha : a < R
h₀ : LT.lt (α := ℝ) 0 0 ∨ 0 < R
H : ∀ (n : ℕ), |f n| ≤ 0 * a ^ n
⊢ f = 0
have h : ∀ (n : ℕ), f n = 0 n
Tactic: ext n
-- Lean 4 proof step from Mathlib.Analysis.SpecificLimits.Normed (goal)
Goal:
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, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
tfae_2_to_1 :
  ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo (-R) R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
tfae_3_to_2 :
  ∀ (a : ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo 0 R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
tfae_2_to_4 :
  ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsLittleO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo 0 R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
tfae_4_to_3 :
  ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
tfae_4_to_6 :
  ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo 0 R, ∃ C > 0, ∀ (n : ℕ), |f n| ≤ C * a ^ n
tfae_6_to_5 :
  ∀ (a : ∃ a ∈ Set.Ioo 0 R, ∃ C > 0, ∀ (n : ℕ), |f n| ≤ C * a ^ n),
    ∃ a < R, ∃ (C : ℝ), (0 < C ∨ 0 < R) ∧ ∀ (n : ℕ), |f n| ≤ C * a ^ n
a : ℝ
ha : a < R
h₀ : LT.lt (α := ℝ) 0 0 ∨ 0 < R
H : ∀ (n : ℕ), |f n| ≤ 0 * a ^ n
⊢ ∃ a ∈ Set.Ioo (-R) R, IsBigO (F := ℝ) atTop f fun (x : ℕ) ↦ a ^ x
have inl : ∀
  (tfae_1_to_3 :
    ∀ (a : ∃ a ∈ Set.Ioo (-R) R, IsLittleO (E := ℝ) (F := ℝ) atTop 0 fun (x : ℕ) ↦ a ^ x),
      ∃ a ∈ Set.Ioo (-R) R, IsBigO (E := ℝ) (F := ℝ) atTop 0 fun (x : ℕ) ↦ a ^ x)
  (tfae_2_to_1 :
    ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsLittleO (E := ℝ) (F := ℝ) atTop 0 fun (x : ℕ) ↦ a ^ x),
      ∃ a ∈ Set.Ioo (-R) R, IsLittleO (E := ℝ) (F := ℝ) atTop 0 fun (x : ℕ) ↦ a ^ x)
  (tfae_3_to_2 :
    ∀ (a : ∃ a ∈ Set.Ioo (-R) R, IsBigO (E := ℝ) (F := ℝ) atTop 0 fun (x : ℕ) ↦ a ^ x),
      ∃ a ∈ Set.Ioo 0 R, IsLittleO (E := ℝ) (F := ℝ) atTop 0 fun (x : ℕ) ↦ a ^ x)
  (tfae_2_to_4 :
    ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsLittleO (E := ℝ) (F := ℝ) atTop 0 fun (x : ℕ) ↦ a ^ x),
      ∃ a ∈ Set.Ioo 0 R, IsBigO (E := ℝ) (F := ℝ) atTop 0 fun (x : ℕ) ↦ a ^ x)
  (tfae_4_to_3 :
    ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsBigO (E := ℝ) (F := ℝ) atTop 0 fun (x : ℕ) ↦ a ^ x),
      ∃ a ∈ Set.Ioo (-R) R, IsBigO (E := ℝ) (F := ℝ) atTop 0 fun (x : ℕ) ↦ a ^ x)
  (tfae_4_to_6 :
    ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsBigO (E := ℝ) (F := ℝ) atTop 0 fun (x : ℕ) ↦ a ^ x),
      ∃ a ∈ Set.Ioo 0 R, ∃ C > 0, ∀ (n : ℕ), |0 n| ≤ C * a ^ n)
  (tfae_6_to_5 :
    ∀ (a : ∃ a ∈ Set.Ioo 0 R, ∃ C > 0, ∀ (n : ℕ), |0 n| ≤ C * a ^ n),
      ∃ a < R, ∃ (C : ℝ), (0 < C ∨ 0 < R) ∧ ∀ (n : ℕ), |0 n| ≤ C * a ^ n)
  (H : ∀ (n : ℕ), |0 n| ≤ 0 * a ^ n), ∃ a ∈ Set.Ioo (-R) R, IsBigO (E := ℝ) (F := ℝ) atTop 0 fun (x : ℕ) ↦ a ^ x
Tactic: obtain rfl : f = 0 := by
        ext n
        simpa using H n
Cleared hypotheses: f, tfae_1_to_3, tfae_2_to_1, tfae_3_to_2, tfae_2_to_4, tfae_4_to_3, tfae_4_to_6, tfae_6_to_5, H
-- Lean 4 proof step from Mathlib.Analysis.SpecificLimits.Normed (hypothesis)
Goal:
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 : LT.lt (α := ℝ) 0 0 ∨ 0 < R
tfae_1_to_3 :
  ∀ (a : ∃ a ∈ Set.Ioo (-R) R, IsLittleO (E := ℝ) (F := ℝ) atTop 0 fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo (-R) R, IsBigO (E := ℝ) (F := ℝ) atTop 0 fun (x : ℕ) ↦ a ^ x
tfae_2_to_1 :
  ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsLittleO (E := ℝ) (F := ℝ) atTop 0 fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo (-R) R, IsLittleO (E := ℝ) (F := ℝ) atTop 0 fun (x : ℕ) ↦ a ^ x
tfae_3_to_2 :
  ∀ (a : ∃ a ∈ Set.Ioo (-R) R, IsBigO (E := ℝ) (F := ℝ) atTop 0 fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo 0 R, IsLittleO (E := ℝ) (F := ℝ) atTop 0 fun (x : ℕ) ↦ a ^ x
tfae_2_to_4 :
  ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsLittleO (E := ℝ) (F := ℝ) atTop 0 fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo 0 R, IsBigO (E := ℝ) (F := ℝ) atTop 0 fun (x : ℕ) ↦ a ^ x
tfae_4_to_3 :
  ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsBigO (E := ℝ) (F := ℝ) atTop 0 fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo (-R) R, IsBigO (E := ℝ) (F := ℝ) atTop 0 fun (x : ℕ) ↦ a ^ x
tfae_4_to_6 :
  ∀ (a : ∃ a ∈ Set.Ioo 0 R, IsBigO (E := ℝ) (F := ℝ) atTop 0 fun (x : ℕ) ↦ a ^ x),
    ∃ a ∈ Set.Ioo 0 R, ∃ C > 0, ∀ (n : ℕ), |0 n| ≤ C * a ^ n
tfae_6_to_5 :
  ∀ (a : ∃ a ∈ Set.Ioo 0 R, ∃ C > 0, ∀ (n : ℕ), |0 n| ≤ C * a ^ n),
    ∃ a < R, ∃ (C : ℝ), (0 < C ∨ 0 < R) ∧ ∀ (n : ℕ), |0 n| ≤ C * a ^ n
H : ∀ (n : ℕ), |0 n| ≤ 0 * a ^ n
⊢ ∃ a ∈ Set.Ioo (-R) R, IsBigO (E := ℝ) (F := ℝ) atTop 0 fun (x : ℕ) ↦ a ^ x
have h₁ : 0 < R
Tactic: simp only [lt_irrefl, false_or] at h₀
Cleared hypotheses: h₀