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
                      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
    refine IsLittleO.eventuallyLE ?_
    calc
      (fun (n:ℕ) => q (r i n) - q (b i * n))
          =O[atTop] fun n => (deriv q n) * (r i n - b i * 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)) =O[atTop] fun n => deriv q ↑n * (↑(r i n) - b i * ↑n)

Next tactic:
exact R.isBigO_apply_r_sub_b q h_diff_q
                (growsPolynomially_deriv_rpow_p_mul_one_sub_smoothingFn (p a b)) i
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
    refine IsLittleO.eventuallyLE ?_
    calc
      (fun (n:ℕ) => q (r i n) - q (b i * n))
          =O[atTop] fun n => (deriv q n) * (r i n - b i * n) := by
              exact R.isBigO_apply_r_sub_b q h_diff_q
                (growsPolynomially_deriv_rpow_p_mul_one_sub_smoothingFn (p a b)) i
        _ =o[atTop] fun n => (deriv q n) * (n / log n ^ 2) := 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 => deriv q ↑n * (↑(r i n) - b i * ↑n)) =o[atTop] fun n => deriv q ↑n * (↑n / log ↑n ^ 2)

Next tactic:
exact IsBigO.mul_isLittleO (isBigO_refl _ _) (R.dist_r_b i)
lemma rpow_p_mul_one_add_smoothingFn_ge :
    ∀ᶠ (n : ℕ) in atTop, ∀ i, (b i) ^ (p a b) * n ^ (p a b) * (1 + ε n)
      ≤ (r i n) ^ (p a b) * (1 + ε (r i 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
⊢ ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), b i ^ p a b * ↑n ^ p a b * (1 + ε ↑n) ≤ ↑(r i n) ^ p a b * (1 + ε ↑(r i n))

Next tactic:
rw [Filter.eventually_all]
lemma rpow_p_mul_one_add_smoothingFn_ge :
    ∀ᶠ (n : ℕ) in atTop, ∀ i, (b i) ^ (p a b) * n ^ (p a b) * (1 + ε n)
      ≤ (r i n) ^ (p a b) * (1 + ε (r i n)) := by
  rw [Filter.eventually_all]
  

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), ∀ᶠ (x : ℕ) in atTop, b i ^ p a b * ↑x ^ p a b * (1 + ε ↑x) ≤ ↑(r i x) ^ p a b * (1 + ε ↑(r i x))

Next tactic:
intro i
lemma rpow_p_mul_one_add_smoothingFn_ge :
    ∀ᶠ (n : ℕ) in atTop, ∀ i, (b i) ^ (p a b) * n ^ (p a b) * (1 + ε n)
      ≤ (r i n) ^ (p a b) * (1 + ε (r i n)) := by
  rw [Filter.eventually_all]
  intro i
  

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
⊢ ∀ᶠ (x : ℕ) in atTop, b i ^ p a b * ↑x ^ p a b * (1 + ε ↑x) ≤ ↑(r i x) ^ p a b * (1 + ε ↑(r i x))

Next tactic:
let q : ℝ → ℝ := fun x => x ^ (p a b) * (1 + ε x)
lemma rpow_p_mul_one_add_smoothingFn_ge :
    ∀ᶠ (n : ℕ) in atTop, ∀ i, (b i) ^ (p a b) * n ^ (p a b) * (1 + ε n)
      ≤ (r i n) ^ (p a b) * (1 + ε (r i n)) := by
  rw [Filter.eventually_all]
  intro i
  let q : ℝ → ℝ := fun x => x ^ (p a b) * (1 + ε x)
  

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)
⊢ ∀ᶠ (x : ℕ) in atTop, b i ^ p a b * ↑x ^ p a b * (1 + ε ↑x) ≤ ↑(r i x) ^ p a b * (1 + ε ↑(r i x))

Next tactic:
have h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1) := by
    refine DifferentiableOn.mul
        (DifferentiableOn.mono (differentiableOn_rpow_const _) fun z hz => ?_)
        differentiableOn_one_add_smoothingFn
    rw [Set.mem_compl_singleton_iff]
    exact ne_of_gt <| zero_lt_one.trans hz
lemma rpow_p_mul_one_add_smoothingFn_ge :
    ∀ᶠ (n : ℕ) in atTop, ∀ i, (b i) ^ (p a b) * n ^ (p a b) * (1 + ε n)
      ≤ (r i n) ^ (p a b) * (1 + ε (r i 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
    

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)
⊢ DifferentiableOn ℝ q (Set.Ioi 1)

Next tactic:
refine DifferentiableOn.mul
        (DifferentiableOn.mono (differentiableOn_rpow_const _) fun z hz => ?_)
        differentiableOn_one_add_smoothingFn
lemma rpow_p_mul_one_add_smoothingFn_ge :
    ∀ᶠ (n : ℕ) in atTop, ∀ i, (b i) ^ (p a b) * n ^ (p a b) * (1 + ε n)
      ≤ (r i n) ^ (p a b) * (1 + ε (r i 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_add_smoothingFn
    

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)
z : ℝ
hz : z ∈ Set.Ioi 1
⊢ z ∈ {0}ᶜ

Next tactic:
rw [Set.mem_compl_singleton_iff]
lemma rpow_p_mul_one_add_smoothingFn_ge :
    ∀ᶠ (n : ℕ) in atTop, ∀ i, (b i) ^ (p a b) * n ^ (p a b) * (1 + ε n)
      ≤ (r i n) ^ (p a b) * (1 + ε (r i 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_add_smoothingFn
    rw [Set.mem_compl_singleton_iff]
    

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)
z : ℝ
hz : z ∈ Set.Ioi 1
⊢ z ≠ 0

Next tactic:
exact ne_of_gt <| zero_lt_one.trans hz
lemma rpow_p_mul_one_add_smoothingFn_ge :
    ∀ᶠ (n : ℕ) in atTop, ∀ i, (b i) ^ (p a b) * n ^ (p a b) * (1 + ε n)
      ≤ (r i n) ^ (p a b) * (1 + ε (r i 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_add_smoothingFn
    rw [Set.mem_compl_singleton_iff]
    exact ne_of_gt <| zero_lt_one.trans hz
  

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)
⊢ ∀ᶠ (x : ℕ) in atTop, b i ^ p a b * ↑x ^ p a b * (1 + ε ↑x) ≤ ↑(r i x) ^ p a b * (1 + ε ↑(r i x))

Next tactic:
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 :=
                        IsBigO.mul (isBigO_refl _ _) isEquivalent_one_add_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_add_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_add_smoothingFn_ge :
    ∀ᶠ (n : ℕ) in atTop, ∀ i, (b i) ^ (p a b) * n ^ (p a b) * (1 + ε n)
      ≤ (r i n) ^ (p a b) * (1 + ε (r i 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_add_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 

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)
⊢ deriv q = deriv fun x => (fun z => z ^ p a b) x * (fun z => 1 + ε z) x

Next tactic:
rfl
lemma rpow_p_mul_one_add_smoothingFn_ge :
    ∀ᶠ (n : ℕ) in atTop, ∀ i, (b i) ^ (p a b) * n ^ (p a b) * (1 + ε n)
      ≤ (r i n) ^ (p a b) * (1 + ε (r i 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_add_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
                

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)
⊢ (deriv fun x => (fun z => z ^ p a b) x * (fun z => 1 + ε z) x) =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

Next tactic:
filter_upwards [eventually_ne_atTop 0, eventually_gt_atTop 1] with x hx hx'
lemma rpow_p_mul_one_add_smoothingFn_ge :
    ∀ᶠ (n : ℕ) in atTop, ∀ i, (b i) ^ (p a b) * n ^ (p a b) * (1 + ε n)
      ≤ (r i n) ^ (p a b) * (1 + ε (r i 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_add_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'
                

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 : x ≠ 0
hx' : 1 < x
⊢ deriv (fun x => x ^ p a b * (1 + ε x)) x =
    deriv (fun z => z ^ p a b) x * (1 + ε x) + x ^ p a b * deriv (fun z => 1 + ε z) x

Next tactic:
rw [deriv_mul]
lemma rpow_p_mul_one_add_smoothingFn_ge :
    ∀ᶠ (n : ℕ) in atTop, ∀ i, (b i) ^ (p a b) * n ^ (p a b) * (1 + ε n)
      ≤ (r i n) ^ (p a b) * (1 + ε (r i 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_add_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] <;> 

Proof state:
case h.hc
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 : x ≠ 0
hx' : 1 < x
⊢ DifferentiableAt ℝ (fun x => x ^ p a b) x

Next tactic:
aesop
lemma rpow_p_mul_one_add_smoothingFn_ge :
    ∀ᶠ (n : ℕ) in atTop, ∀ i, (b i) ^ (p a b) * n ^ (p a b) * (1 + ε n)
      ≤ (r i n) ^ (p a b) * (1 + ε (r i 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_add_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] <;> 

Proof state:
case h.hd
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 : x ≠ 0
hx' : 1 < x
⊢ DifferentiableAt ℝ (fun x => 1 + ε x) x

Next tactic:
aesop
lemma rpow_p_mul_one_add_smoothingFn_ge :
    ∀ᶠ (n : ℕ) in atTop, ∀ i, (b i) ^ (p a b) * n ^ (p a b) * (1 + ε n)
      ≤ (r i n) ^ (p a b) * (1 + ε (r i 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_add_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
                

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 => deriv (fun z => z ^ p a b) x * (1 + ε x) + x ^ p a b * deriv (fun z => 1 + ε z) x) =O[atTop] fun x =>
    x ^ (p a b - 1)

Next tactic:
refine IsBigO.add ?left ?right
lemma rpow_p_mul_one_add_smoothingFn_ge :
    ∀ᶠ (n : ℕ) in atTop, ∀ i, (b i) ^ (p a b) * n ^ (p a b) * (1 + ε n)
      ≤ (r i n) ^ (p a b) * (1 + ε (r i 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_add_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
                

Proof state:
case left
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 => deriv (fun z => z ^ p a b) x * (1 + ε x)) =O[atTop] fun x => x ^ (p a b - 1)
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 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 :=
                        IsBigO.mul (isBigO_refl _ _) isEquivalent_one_add_smoothingFn_one.isBigO
                    _ = fun x => x ^ ((p a b) - 1) := by ext; rw [mul_one]