l3lab/ntp-mathlib huggingface dataset:
lemma rpow_p_mul_one_sub_smoothingFn_le :
    ∀ᶠ (n : ℕ) in atTop, ∀ i, (r i n) ^ (p a b) * (1 - ε (r i n))
      ≤ (b i) ^ (p a b) * n ^ (p a b) * (1 - ε n) := by
  rw [Filter.eventually_all]
  intro i
  let q : ℝ → ℝ := fun x => x ^ (p a b) * (1 - ε x)
  have h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1) := by
    refine DifferentiableOn.mul
      (DifferentiableOn.mono (differentiableOn_rpow_const _) fun z hz => ?_)
        differentiableOn_one_sub_smoothingFn
    rw [Set.mem_compl_singleton_iff]
    exact ne_of_gt <| zero_lt_one.trans hz
  have h_deriv_q : deriv q =O[atTop] fun x => x ^ ((p a b) - 1) := calc
    deriv q = deriv fun x => (fun z => z ^ (p a b)) x * (fun z => 1 - ε z) x := by rfl
          _ =ᶠ[atTop] fun x => deriv (fun z => z ^ (p a b)) x * (1 - ε x) +
                  x ^ (p a b) * deriv (fun z => 1 - ε z) x := by
              filter_upwards [eventually_ne_atTop 0, eventually_gt_atTop 1] with x hx hx'
              rw [deriv_mul] <;> aesop
          _ =O[atTop] fun x => x ^ ((p a b) - 1) := by
              refine IsBigO.add ?left ?right
              case left => calc
                (fun x => deriv (fun z => z ^ (p a b)) x * (1 - ε x))
                    =O[atTop] fun x => x ^ ((p a b) - 1) * (1 - ε x) := by
                      exact IsBigO.mul (isBigO_deriv_rpow_const_atTop (p a b)) (isBigO_refl _ _)
                  _ =O[atTop] fun x => x ^ ((p a b) - 1) * 1 := by
                      

Proof state:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : Nonempty α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
R : AkraBazziRecurrence T g a b r
i : α
q : ℝ → ℝ := fun x => x ^ p a b * (1 - ε x)
h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1)
⊢ (fun x => x ^ (p a b - 1) * (1 - ε x)) =O[atTop] fun x => x ^ (p a b - 1) * 1

Next tactic:
refine IsBigO.mul (isBigO_refl _ _)
                        isEquivalent_one_sub_smoothingFn_one.isBigO
lemma rpow_p_mul_one_sub_smoothingFn_le :
    ∀ᶠ (n : ℕ) in atTop, ∀ i, (r i n) ^ (p a b) * (1 - ε (r i n))
      ≤ (b i) ^ (p a b) * n ^ (p a b) * (1 - ε n) := by
  rw [Filter.eventually_all]
  intro i
  let q : ℝ → ℝ := fun x => x ^ (p a b) * (1 - ε x)
  have h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1) := by
    refine DifferentiableOn.mul
      (DifferentiableOn.mono (differentiableOn_rpow_const _) fun z hz => ?_)
        differentiableOn_one_sub_smoothingFn
    rw [Set.mem_compl_singleton_iff]
    exact ne_of_gt <| zero_lt_one.trans hz
  have h_deriv_q : deriv q =O[atTop] fun x => x ^ ((p a b) - 1) := calc
    deriv q = deriv fun x => (fun z => z ^ (p a b)) x * (fun z => 1 - ε z) x := by rfl
          _ =ᶠ[atTop] fun x => deriv (fun z => z ^ (p a b)) x * (1 - ε x) +
                  x ^ (p a b) * deriv (fun z => 1 - ε z) x := by
              filter_upwards [eventually_ne_atTop 0, eventually_gt_atTop 1] with x hx hx'
              rw [deriv_mul] <;> aesop
          _ =O[atTop] fun x => x ^ ((p a b) - 1) := by
              refine IsBigO.add ?left ?right
              case left => calc
                (fun x => deriv (fun z => z ^ (p a b)) x * (1 - ε x))
                    =O[atTop] fun x => x ^ ((p a b) - 1) * (1 - ε x) := by
                      exact IsBigO.mul (isBigO_deriv_rpow_const_atTop (p a b)) (isBigO_refl _ _)
                  _ =O[atTop] fun x => x ^ ((p a b) - 1) * 1 := by
                      refine IsBigO.mul (isBigO_refl _ _)
                        isEquivalent_one_sub_smoothingFn_one.isBigO
                  _ = fun x => x ^ ((p a b) - 1) := by 

Proof state:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : Nonempty α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
R : AkraBazziRecurrence T g a b r
i : α
q : ℝ → ℝ := fun x => x ^ p a b * (1 - ε x)
h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1)
⊢ (fun x => x ^ (p a b - 1) * 1) = fun x => x ^ (p a b - 1)

Next tactic:
ext
lemma rpow_p_mul_one_sub_smoothingFn_le :
    ∀ᶠ (n : ℕ) in atTop, ∀ i, (r i n) ^ (p a b) * (1 - ε (r i n))
      ≤ (b i) ^ (p a b) * n ^ (p a b) * (1 - ε n) := by
  rw [Filter.eventually_all]
  intro i
  let q : ℝ → ℝ := fun x => x ^ (p a b) * (1 - ε x)
  have h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1) := by
    refine DifferentiableOn.mul
      (DifferentiableOn.mono (differentiableOn_rpow_const _) fun z hz => ?_)
        differentiableOn_one_sub_smoothingFn
    rw [Set.mem_compl_singleton_iff]
    exact ne_of_gt <| zero_lt_one.trans hz
  have h_deriv_q : deriv q =O[atTop] fun x => x ^ ((p a b) - 1) := calc
    deriv q = deriv fun x => (fun z => z ^ (p a b)) x * (fun z => 1 - ε z) x := by rfl
          _ =ᶠ[atTop] fun x => deriv (fun z => z ^ (p a b)) x * (1 - ε x) +
                  x ^ (p a b) * deriv (fun z => 1 - ε z) x := by
              filter_upwards [eventually_ne_atTop 0, eventually_gt_atTop 1] with x hx hx'
              rw [deriv_mul] <;> aesop
          _ =O[atTop] fun x => x ^ ((p a b) - 1) := by
              refine IsBigO.add ?left ?right
              case left => calc
                (fun x => deriv (fun z => z ^ (p a b)) x * (1 - ε x))
                    =O[atTop] fun x => x ^ ((p a b) - 1) * (1 - ε x) := by
                      exact IsBigO.mul (isBigO_deriv_rpow_const_atTop (p a b)) (isBigO_refl _ _)
                  _ =O[atTop] fun x => x ^ ((p a b) - 1) * 1 := by
                      refine IsBigO.mul (isBigO_refl _ _)
                        isEquivalent_one_sub_smoothingFn_one.isBigO
                  _ = fun x => x ^ ((p a b) - 1) := by ext; 

Proof state:
case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : Nonempty α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
R : AkraBazziRecurrence T g a b r
i : α
q : ℝ → ℝ := fun x => x ^ p a b * (1 - ε x)
h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1)
x✝ : ℝ
⊢ x✝ ^ (p a b - 1) * 1 = x✝ ^ (p a b - 1)

Next tactic:
rw [mul_one]
lemma rpow_p_mul_one_sub_smoothingFn_le :
    ∀ᶠ (n : ℕ) in atTop, ∀ i, (r i n) ^ (p a b) * (1 - ε (r i n))
      ≤ (b i) ^ (p a b) * n ^ (p a b) * (1 - ε n) := by
  rw [Filter.eventually_all]
  intro i
  let q : ℝ → ℝ := fun x => x ^ (p a b) * (1 - ε x)
  have h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1) := by
    refine DifferentiableOn.mul
      (DifferentiableOn.mono (differentiableOn_rpow_const _) fun z hz => ?_)
        differentiableOn_one_sub_smoothingFn
    rw [Set.mem_compl_singleton_iff]
    exact ne_of_gt <| zero_lt_one.trans hz
  have h_deriv_q : deriv q =O[atTop] fun x => x ^ ((p a b) - 1) := calc
    deriv q = deriv fun x => (fun z => z ^ (p a b)) x * (fun z => 1 - ε z) x := by rfl
          _ =ᶠ[atTop] fun x => deriv (fun z => z ^ (p a b)) x * (1 - ε x) +
                  x ^ (p a b) * deriv (fun z => 1 - ε z) x := by
              filter_upwards [eventually_ne_atTop 0, eventually_gt_atTop 1] with x hx hx'
              rw [deriv_mul] <;> aesop
          _ =O[atTop] fun x => x ^ ((p a b) - 1) := by
              refine IsBigO.add ?left ?right
              case left => calc
                (fun x => deriv (fun z => z ^ (p a b)) x * (1 - ε x))
                    =O[atTop] fun x => x ^ ((p a b) - 1) * (1 - ε x) := by
                      exact IsBigO.mul (isBigO_deriv_rpow_const_atTop (p a b)) (isBigO_refl _ _)
                  _ =O[atTop] fun x => x ^ ((p a b) - 1) * 1 := by
                      refine IsBigO.mul (isBigO_refl _ _)
                        isEquivalent_one_sub_smoothingFn_one.isBigO
                  _ = fun x => x ^ ((p a b) - 1) := by ext; rw [mul_one]
              

Proof state:
case right
α : Type u_1
inst✝¹ : Fintype α
inst✝ : Nonempty α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
R : AkraBazziRecurrence T g a b r
i : α
q : ℝ → ℝ := fun x => x ^ p a b * (1 - ε x)
h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1)
⊢ (fun x => x ^ p a b * deriv (fun z => 1 - ε z) x) =O[atTop] fun x => x ^ (p a b - 1)

Next tactic:
case right => calc
                (fun x => x ^ (p a b) * deriv (fun z => 1 - ε z) x)
                    =O[atTop] (fun x => x ^ (p a b) * x⁻¹) := by
                      exact IsBigO.mul (isBigO_refl _ _) isLittleO_deriv_one_sub_smoothingFn.isBigO
                  _ =ᶠ[atTop] fun x => x ^ ((p a b) - 1) := by
                      filter_upwards [eventually_gt_atTop 0] with x hx
                      rw [← Real.rpow_neg_one, ← Real.rpow_add hx, ← sub_eq_add_neg]
lemma rpow_p_mul_one_sub_smoothingFn_le :
    ∀ᶠ (n : ℕ) in atTop, ∀ i, (r i n) ^ (p a b) * (1 - ε (r i n))
      ≤ (b i) ^ (p a b) * n ^ (p a b) * (1 - ε n) := by
  rw [Filter.eventually_all]
  intro i
  let q : ℝ → ℝ := fun x => x ^ (p a b) * (1 - ε x)
  have h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1) := by
    refine DifferentiableOn.mul
      (DifferentiableOn.mono (differentiableOn_rpow_const _) fun z hz => ?_)
        differentiableOn_one_sub_smoothingFn
    rw [Set.mem_compl_singleton_iff]
    exact ne_of_gt <| zero_lt_one.trans hz
  have h_deriv_q : deriv q =O[atTop] fun x => x ^ ((p a b) - 1) := calc
    deriv q = deriv fun x => (fun z => z ^ (p a b)) x * (fun z => 1 - ε z) x := by rfl
          _ =ᶠ[atTop] fun x => deriv (fun z => z ^ (p a b)) x * (1 - ε x) +
                  x ^ (p a b) * deriv (fun z => 1 - ε z) x := by
              filter_upwards [eventually_ne_atTop 0, eventually_gt_atTop 1] with x hx hx'
              rw [deriv_mul] <;> aesop
          _ =O[atTop] fun x => x ^ ((p a b) - 1) := by
              refine IsBigO.add ?left ?right
              case left => calc
                (fun x => deriv (fun z => z ^ (p a b)) x * (1 - ε x))
                    =O[atTop] fun x => x ^ ((p a b) - 1) * (1 - ε x) := by
                      exact IsBigO.mul (isBigO_deriv_rpow_const_atTop (p a b)) (isBigO_refl _ _)
                  _ =O[atTop] fun x => x ^ ((p a b) - 1) * 1 := by
                      refine IsBigO.mul (isBigO_refl _ _)
                        isEquivalent_one_sub_smoothingFn_one.isBigO
                  _ = fun x => x ^ ((p a b) - 1) := by ext; rw [mul_one]
              

Proof state:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : Nonempty α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
R : AkraBazziRecurrence T g a b r
i : α
q : ℝ → ℝ := fun x => x ^ p a b * (1 - ε x)
h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1)
⊢ (fun x => x ^ p a b * deriv (fun z => 1 - ε z) x) =O[atTop] fun x => x ^ (p a b - 1)

Next tactic:
case right => calc
                (fun x => x ^ (p a b) * deriv (fun z => 1 - ε z) x)
                    =O[atTop] (fun x => x ^ (p a b) * x⁻¹) := by
                      exact IsBigO.mul (isBigO_refl _ _) isLittleO_deriv_one_sub_smoothingFn.isBigO
                  _ =ᶠ[atTop] fun x => x ^ ((p a b) - 1) := by
                      filter_upwards [eventually_gt_atTop 0] with x hx
                      rw [← Real.rpow_neg_one, ← Real.rpow_add hx, ← sub_eq_add_neg]
lemma rpow_p_mul_one_sub_smoothingFn_le :
    ∀ᶠ (n : ℕ) in atTop, ∀ i, (r i n) ^ (p a b) * (1 - ε (r i n))
      ≤ (b i) ^ (p a b) * n ^ (p a b) * (1 - ε n) := by
  rw [Filter.eventually_all]
  intro i
  let q : ℝ → ℝ := fun x => x ^ (p a b) * (1 - ε x)
  have h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1) := by
    refine DifferentiableOn.mul
      (DifferentiableOn.mono (differentiableOn_rpow_const _) fun z hz => ?_)
        differentiableOn_one_sub_smoothingFn
    rw [Set.mem_compl_singleton_iff]
    exact ne_of_gt <| zero_lt_one.trans hz
  have h_deriv_q : deriv q =O[atTop] fun x => x ^ ((p a b) - 1) := calc
    deriv q = deriv fun x => (fun z => z ^ (p a b)) x * (fun z => 1 - ε z) x := by rfl
          _ =ᶠ[atTop] fun x => deriv (fun z => z ^ (p a b)) x * (1 - ε x) +
                  x ^ (p a b) * deriv (fun z => 1 - ε z) x := by
              filter_upwards [eventually_ne_atTop 0, eventually_gt_atTop 1] with x hx hx'
              rw [deriv_mul] <;> aesop
          _ =O[atTop] fun x => x ^ ((p a b) - 1) := by
              refine IsBigO.add ?left ?right
              case left => calc
                (fun x => deriv (fun z => z ^ (p a b)) x * (1 - ε x))
                    =O[atTop] fun x => x ^ ((p a b) - 1) * (1 - ε x) := by
                      exact IsBigO.mul (isBigO_deriv_rpow_const_atTop (p a b)) (isBigO_refl _ _)
                  _ =O[atTop] fun x => x ^ ((p a b) - 1) * 1 := by
                      refine IsBigO.mul (isBigO_refl _ _)
                        isEquivalent_one_sub_smoothingFn_one.isBigO
                  _ = fun x => x ^ ((p a b) - 1) := by ext; rw [mul_one]
              case right => 

Proof state:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : Nonempty α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
R : AkraBazziRecurrence T g a b r
i : α
q : ℝ → ℝ := fun x => x ^ p a b * (1 - ε x)
h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1)
⊢ (fun x => x ^ p a b * deriv (fun z => 1 - ε z) x) =O[atTop] fun x => x ^ (p a b - 1)

Next tactic:
calc
                (fun x => x ^ (p a b) * deriv (fun z => 1 - ε z) x)
                    =O[atTop] (fun x => x ^ (p a b) * x⁻¹) := by
                      exact IsBigO.mul (isBigO_refl _ _) isLittleO_deriv_one_sub_smoothingFn.isBigO
                  _ =ᶠ[atTop] fun x => x ^ ((p a b) - 1) := by
                      filter_upwards [eventually_gt_atTop 0] with x hx
                      rw [← Real.rpow_neg_one, ← Real.rpow_add hx, ← sub_eq_add_neg]
lemma rpow_p_mul_one_sub_smoothingFn_le :
    ∀ᶠ (n : ℕ) in atTop, ∀ i, (r i n) ^ (p a b) * (1 - ε (r i n))
      ≤ (b i) ^ (p a b) * n ^ (p a b) * (1 - ε n) := by
  rw [Filter.eventually_all]
  intro i
  let q : ℝ → ℝ := fun x => x ^ (p a b) * (1 - ε x)
  have h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1) := by
    refine DifferentiableOn.mul
      (DifferentiableOn.mono (differentiableOn_rpow_const _) fun z hz => ?_)
        differentiableOn_one_sub_smoothingFn
    rw [Set.mem_compl_singleton_iff]
    exact ne_of_gt <| zero_lt_one.trans hz
  have h_deriv_q : deriv q =O[atTop] fun x => x ^ ((p a b) - 1) := calc
    deriv q = deriv fun x => (fun z => z ^ (p a b)) x * (fun z => 1 - ε z) x := by rfl
          _ =ᶠ[atTop] fun x => deriv (fun z => z ^ (p a b)) x * (1 - ε x) +
                  x ^ (p a b) * deriv (fun z => 1 - ε z) x := by
              filter_upwards [eventually_ne_atTop 0, eventually_gt_atTop 1] with x hx hx'
              rw [deriv_mul] <;> aesop
          _ =O[atTop] fun x => x ^ ((p a b) - 1) := by
              refine IsBigO.add ?left ?right
              case left => calc
                (fun x => deriv (fun z => z ^ (p a b)) x * (1 - ε x))
                    =O[atTop] fun x => x ^ ((p a b) - 1) * (1 - ε x) := by
                      exact IsBigO.mul (isBigO_deriv_rpow_const_atTop (p a b)) (isBigO_refl _ _)
                  _ =O[atTop] fun x => x ^ ((p a b) - 1) * 1 := by
                      refine IsBigO.mul (isBigO_refl _ _)
                        isEquivalent_one_sub_smoothingFn_one.isBigO
                  _ = fun x => x ^ ((p a b) - 1) := by ext; rw [mul_one]
              case right => calc
                (fun x => x ^ (p a b) * deriv (fun z => 1 - ε z) x)
                    =O[atTop] (fun x => x ^ (p a b) * x⁻¹) := by
                      

Proof state:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : Nonempty α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
R : AkraBazziRecurrence T g a b r
i : α
q : ℝ → ℝ := fun x => x ^ p a b * (1 - ε x)
h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1)
⊢ (fun x => x ^ p a b * deriv (fun z => 1 - ε z) x) =O[atTop] fun x => x ^ p a b * x⁻¹

Next tactic:
exact IsBigO.mul (isBigO_refl _ _) isLittleO_deriv_one_sub_smoothingFn.isBigO
lemma rpow_p_mul_one_sub_smoothingFn_le :
    ∀ᶠ (n : ℕ) in atTop, ∀ i, (r i n) ^ (p a b) * (1 - ε (r i n))
      ≤ (b i) ^ (p a b) * n ^ (p a b) * (1 - ε n) := by
  rw [Filter.eventually_all]
  intro i
  let q : ℝ → ℝ := fun x => x ^ (p a b) * (1 - ε x)
  have h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1) := by
    refine DifferentiableOn.mul
      (DifferentiableOn.mono (differentiableOn_rpow_const _) fun z hz => ?_)
        differentiableOn_one_sub_smoothingFn
    rw [Set.mem_compl_singleton_iff]
    exact ne_of_gt <| zero_lt_one.trans hz
  have h_deriv_q : deriv q =O[atTop] fun x => x ^ ((p a b) - 1) := calc
    deriv q = deriv fun x => (fun z => z ^ (p a b)) x * (fun z => 1 - ε z) x := by rfl
          _ =ᶠ[atTop] fun x => deriv (fun z => z ^ (p a b)) x * (1 - ε x) +
                  x ^ (p a b) * deriv (fun z => 1 - ε z) x := by
              filter_upwards [eventually_ne_atTop 0, eventually_gt_atTop 1] with x hx hx'
              rw [deriv_mul] <;> aesop
          _ =O[atTop] fun x => x ^ ((p a b) - 1) := by
              refine IsBigO.add ?left ?right
              case left => calc
                (fun x => deriv (fun z => z ^ (p a b)) x * (1 - ε x))
                    =O[atTop] fun x => x ^ ((p a b) - 1) * (1 - ε x) := by
                      exact IsBigO.mul (isBigO_deriv_rpow_const_atTop (p a b)) (isBigO_refl _ _)
                  _ =O[atTop] fun x => x ^ ((p a b) - 1) * 1 := by
                      refine IsBigO.mul (isBigO_refl _ _)
                        isEquivalent_one_sub_smoothingFn_one.isBigO
                  _ = fun x => x ^ ((p a b) - 1) := by ext; rw [mul_one]
              case right => calc
                (fun x => x ^ (p a b) * deriv (fun z => 1 - ε z) x)
                    =O[atTop] (fun x => x ^ (p a b) * x⁻¹) := by
                      exact IsBigO.mul (isBigO_refl _ _) isLittleO_deriv_one_sub_smoothingFn.isBigO
                  _ =ᶠ[atTop] fun x => x ^ ((p a b) - 1) := by
                      

Proof state:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : Nonempty α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
R : AkraBazziRecurrence T g a b r
i : α
q : ℝ → ℝ := fun x => x ^ p a b * (1 - ε x)
h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1)
⊢ (fun x => x ^ p a b * x⁻¹) =ᶠ[atTop] fun x => x ^ (p a b - 1)

Next tactic:
filter_upwards [eventually_gt_atTop 0] with x hx
lemma rpow_p_mul_one_sub_smoothingFn_le :
    ∀ᶠ (n : ℕ) in atTop, ∀ i, (r i n) ^ (p a b) * (1 - ε (r i n))
      ≤ (b i) ^ (p a b) * n ^ (p a b) * (1 - ε n) := by
  rw [Filter.eventually_all]
  intro i
  let q : ℝ → ℝ := fun x => x ^ (p a b) * (1 - ε x)
  have h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1) := by
    refine DifferentiableOn.mul
      (DifferentiableOn.mono (differentiableOn_rpow_const _) fun z hz => ?_)
        differentiableOn_one_sub_smoothingFn
    rw [Set.mem_compl_singleton_iff]
    exact ne_of_gt <| zero_lt_one.trans hz
  have h_deriv_q : deriv q =O[atTop] fun x => x ^ ((p a b) - 1) := calc
    deriv q = deriv fun x => (fun z => z ^ (p a b)) x * (fun z => 1 - ε z) x := by rfl
          _ =ᶠ[atTop] fun x => deriv (fun z => z ^ (p a b)) x * (1 - ε x) +
                  x ^ (p a b) * deriv (fun z => 1 - ε z) x := by
              filter_upwards [eventually_ne_atTop 0, eventually_gt_atTop 1] with x hx hx'
              rw [deriv_mul] <;> aesop
          _ =O[atTop] fun x => x ^ ((p a b) - 1) := by
              refine IsBigO.add ?left ?right
              case left => calc
                (fun x => deriv (fun z => z ^ (p a b)) x * (1 - ε x))
                    =O[atTop] fun x => x ^ ((p a b) - 1) * (1 - ε x) := by
                      exact IsBigO.mul (isBigO_deriv_rpow_const_atTop (p a b)) (isBigO_refl _ _)
                  _ =O[atTop] fun x => x ^ ((p a b) - 1) * 1 := by
                      refine IsBigO.mul (isBigO_refl _ _)
                        isEquivalent_one_sub_smoothingFn_one.isBigO
                  _ = fun x => x ^ ((p a b) - 1) := by ext; rw [mul_one]
              case right => calc
                (fun x => x ^ (p a b) * deriv (fun z => 1 - ε z) x)
                    =O[atTop] (fun x => x ^ (p a b) * x⁻¹) := by
                      exact IsBigO.mul (isBigO_refl _ _) isLittleO_deriv_one_sub_smoothingFn.isBigO
                  _ =ᶠ[atTop] fun x => x ^ ((p a b) - 1) := by
                      filter_upwards [eventually_gt_atTop 0] with x hx
                      

Proof state:
case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : Nonempty α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
R : AkraBazziRecurrence T g a b r
i : α
q : ℝ → ℝ := fun x => x ^ p a b * (1 - ε x)
h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1)
x : ℝ
hx : 0 < x
⊢ x ^ p a b * x⁻¹ = x ^ (p a b - 1)

Next tactic:
rw [← Real.rpow_neg_one, ← Real.rpow_add hx, ← sub_eq_add_neg]
lemma rpow_p_mul_one_sub_smoothingFn_le :
    ∀ᶠ (n : ℕ) in atTop, ∀ i, (r i n) ^ (p a b) * (1 - ε (r i n))
      ≤ (b i) ^ (p a b) * n ^ (p a b) * (1 - ε n) := by
  rw [Filter.eventually_all]
  intro i
  let q : ℝ → ℝ := fun x => x ^ (p a b) * (1 - ε x)
  have h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1) := by
    refine DifferentiableOn.mul
      (DifferentiableOn.mono (differentiableOn_rpow_const _) fun z hz => ?_)
        differentiableOn_one_sub_smoothingFn
    rw [Set.mem_compl_singleton_iff]
    exact ne_of_gt <| zero_lt_one.trans hz
  have h_deriv_q : deriv q =O[atTop] fun x => x ^ ((p a b) - 1) := calc
    deriv q = deriv fun x => (fun z => z ^ (p a b)) x * (fun z => 1 - ε z) x := by rfl
          _ =ᶠ[atTop] fun x => deriv (fun z => z ^ (p a b)) x * (1 - ε x) +
                  x ^ (p a b) * deriv (fun z => 1 - ε z) x := by
              filter_upwards [eventually_ne_atTop 0, eventually_gt_atTop 1] with x hx hx'
              rw [deriv_mul] <;> aesop
          _ =O[atTop] fun x => x ^ ((p a b) - 1) := by
              refine IsBigO.add ?left ?right
              case left => calc
                (fun x => deriv (fun z => z ^ (p a b)) x * (1 - ε x))
                    =O[atTop] fun x => x ^ ((p a b) - 1) * (1 - ε x) := by
                      exact IsBigO.mul (isBigO_deriv_rpow_const_atTop (p a b)) (isBigO_refl _ _)
                  _ =O[atTop] fun x => x ^ ((p a b) - 1) * 1 := by
                      refine IsBigO.mul (isBigO_refl _ _)
                        isEquivalent_one_sub_smoothingFn_one.isBigO
                  _ = fun x => x ^ ((p a b) - 1) := by ext; rw [mul_one]
              case right => calc
                (fun x => x ^ (p a b) * deriv (fun z => 1 - ε z) x)
                    =O[atTop] (fun x => x ^ (p a b) * x⁻¹) := by
                      exact IsBigO.mul (isBigO_refl _ _) isLittleO_deriv_one_sub_smoothingFn.isBigO
                  _ =ᶠ[atTop] fun x => x ^ ((p a b) - 1) := by
                      filter_upwards [eventually_gt_atTop 0] with x hx
                      rw [← Real.rpow_neg_one, ← Real.rpow_add hx, ← sub_eq_add_neg]
  have h_main_norm : (fun (n:ℕ) => ‖q (r i n) - q (b i * n)‖)
      ≤ᶠ[atTop] fun (n:ℕ) => ‖(b i) ^ (p a b) * n ^ (p a b) * (ε (b i * n) - ε n)‖ := by
    

Proof state:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : Nonempty α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
R : AkraBazziRecurrence T g a b r
i : α
q : ℝ → ℝ := fun x => x ^ p a b * (1 - ε x)
h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1)
h_deriv_q : deriv q =O[atTop] fun x => x ^ (p a b - 1)
⊢ (fun n => ‖q ↑(r i n) - q (b i * ↑n)‖) ≤ᶠ[atTop] fun n => ‖b i ^ p a b * ↑n ^ p a b * (ε (b i * ↑n) - ε ↑n)‖

Next tactic:
refine IsLittleO.eventuallyLE ?_