ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- AkraBazziRecurrence.isEquivalent_one_sub_smoothingFn_one
lemma isEquivalent_one_sub_smoothingFn_one : (fun x => 1 - ε x) ~[atTop] (fun _ => (1 : ℝ)) 
Goal: (fun x => 1 - AkraBazziRecurrence.smoothingFn x) ~[Filter.atTop] fun x => 1
-- AkraBazziRecurrence.growsPolynomially_one_sub_smoothingFn
lemma growsPolynomially_one_sub_smoothingFn : GrowsPolynomially fun x => 1 - ε x 
Goal: AkraBazziRecurrence.GrowsPolynomially fun x => 1 - AkraBazziRecurrence.smoothingFn x
-- AkraBazziRecurrence.growsPolynomially_one_add_smoothingFn
lemma growsPolynomially_one_add_smoothingFn : GrowsPolynomially fun x => 1 + ε x 
Goal: AkraBazziRecurrence.GrowsPolynomially fun x => 1 + AkraBazziRecurrence.smoothingFn x
-- AkraBazziRecurrence.eventually_one_sub_smoothingFn_gt_const_real
lemma eventually_one_sub_smoothingFn_gt_const_real (c : ℝ) (hc : c < 1) :
    ∀ᶠ (x : ℝ) in atTop, c < 1 - ε x 
Arguments: (c : ℝ) (hc : c < 1)
Goal: ∀ᶠ (x : ℝ) in Filter.atTop, c < 1 - AkraBazziRecurrence.smoothingFn x
-- AkraBazziRecurrence.eventually_one_sub_smoothingFn_gt_const
lemma eventually_one_sub_smoothingFn_gt_const (c : ℝ) (hc : c < 1) :
    ∀ᶠ (n : ℕ) in atTop, c < 1 - ε n 
Arguments: (c : ℝ) (hc : c < 1)
Goal: ∀ᶠ (n : ℕ) in Filter.atTop, c < 1 - AkraBazziRecurrence.smoothingFn ↑n
-- AkraBazziRecurrence.eventually_one_sub_smoothingFn_pos_real
lemma eventually_one_sub_smoothingFn_pos_real : ∀ᶠ (x : ℝ) in atTop, 0 < 1 - ε x 
Goal: ∀ᶠ (x : ℝ) in Filter.atTop, 0 < 1 - AkraBazziRecurrence.smoothingFn x
-- AkraBazziRecurrence.eventually_one_sub_smoothingFn_pos
lemma eventually_one_sub_smoothingFn_pos : ∀ᶠ (n : ℕ) in atTop, 0 < 1 - ε n 
Goal: ∀ᶠ (n : ℕ) in Filter.atTop, 0 < 1 - AkraBazziRecurrence.smoothingFn ↑n
-- AkraBazziRecurrence.eventually_one_sub_smoothingFn_nonneg
lemma eventually_one_sub_smoothingFn_nonneg : ∀ᶠ (n : ℕ) in atTop, 0 ≤ 1 - ε n 
Goal: ∀ᶠ (n : ℕ) in Filter.atTop, 0 ≤ 1 - AkraBazziRecurrence.smoothingFn ↑n
-- AkraBazziRecurrence.eventually_one_sub_smoothingFn_r_pos
lemma eventually_one_sub_smoothingFn_r_pos : ∀ᶠ (n : ℕ) in atTop, ∀ i, 0 < 1 - ε (r i n) 
Arguments: {α : Type u_1} [Fintype α] [Nonempty α] {T : ℕ → ℝ} {g : ℝ → ℝ} {a b : α → ℝ} {r : α → ℕ → ℕ} (R : AkraBazziRecurrence T g a b r)
Goal: ∀ᶠ (n : ℕ) in Filter.atTop, ∀ (i : α), 0 < 1 - AkraBazziRecurrence.smoothingFn ↑(r i n)
-- AkraBazziRecurrence.differentiableAt_smoothingFn
@[aesop safe apply]
lemma differentiableAt_smoothingFn {x : ℝ} (hx : 1 < x) : DifferentiableAt ℝ ε x 
Arguments: {x : ℝ} (hx : 1 < x)
Goal: DifferentiableAt ℝ AkraBazziRecurrence.smoothingFn x
-- AkraBazziRecurrence.differentiableAt_one_sub_smoothingFn
@[aesop safe apply]
lemma differentiableAt_one_sub_smoothingFn {x : ℝ} (hx : 1 < x) :
    DifferentiableAt ℝ (fun z => 1 - ε z) x 
Arguments: {x : ℝ} (hx : 1 < x)
Goal: DifferentiableAt ℝ (fun z => 1 - AkraBazziRecurrence.smoothingFn z) x
-- AkraBazziRecurrence.differentiableOn_one_sub_smoothingFn
lemma differentiableOn_one_sub_smoothingFn : DifferentiableOn ℝ (fun z => 1 - ε z) (Set.Ioi 1) 
Goal: DifferentiableOn ℝ (fun z => 1 - AkraBazziRecurrence.smoothingFn z) (Set.Ioi 1)
-- AkraBazziRecurrence.differentiableAt_one_add_smoothingFn
@[aesop safe apply]
lemma differentiableAt_one_add_smoothingFn {x : ℝ} (hx : 1 < x) :
    DifferentiableAt ℝ (fun z => 1 + ε z) x 
Arguments: {x : ℝ} (hx : 1 < x)
Goal: DifferentiableAt ℝ (fun z => 1 + AkraBazziRecurrence.smoothingFn z) x
-- AkraBazziRecurrence.differentiableOn_one_add_smoothingFn
lemma differentiableOn_one_add_smoothingFn : DifferentiableOn ℝ (fun z => 1 + ε z) (Set.Ioi 1) 
Goal: DifferentiableOn ℝ (fun z => 1 + AkraBazziRecurrence.smoothingFn z) (Set.Ioi 1)
-- AkraBazziRecurrence.deriv_smoothingFn
lemma deriv_smoothingFn {x : ℝ} (hx : 1 < x) : deriv ε x = -x⁻¹ / (log x ^ 2) 
Arguments: {x : ℝ} (hx : 1 < x)
Goal: deriv AkraBazziRecurrence.smoothingFn x = -x⁻¹ / Real.log x ^ 2
-- AkraBazziRecurrence.isLittleO_deriv_smoothingFn
lemma isLittleO_deriv_smoothingFn : deriv ε =o[atTop] fun x => x⁻¹ 
Goal: deriv AkraBazziRecurrence.smoothingFn =o[Filter.atTop] fun x => x⁻¹
-- AkraBazziRecurrence.eventually_deriv_one_sub_smoothingFn
lemma eventually_deriv_one_sub_smoothingFn :
    deriv (fun x => 1 - ε x) =ᶠ[atTop] fun x => x⁻¹ / (log x ^ 2) 
Goal: (deriv fun x => 1 - AkraBazziRecurrence.smoothingFn x) =ᶠ[Filter.atTop] fun x => x⁻¹ / Real.log x ^ 2
-- AkraBazziRecurrence.eventually_deriv_one_add_smoothingFn
lemma eventually_deriv_one_add_smoothingFn :
    deriv (fun x => 1 + ε x) =ᶠ[atTop] fun x => -x⁻¹ / (log x ^ 2) 
Goal: (deriv fun x => 1 + AkraBazziRecurrence.smoothingFn x) =ᶠ[Filter.atTop] fun x => -x⁻¹ / Real.log x ^ 2
-- AkraBazziRecurrence.isLittleO_deriv_one_sub_smoothingFn
lemma isLittleO_deriv_one_sub_smoothingFn :
    deriv (fun x => 1 - ε x) =o[atTop] fun (x : ℝ) => x⁻¹ 
Goal: (deriv fun x => 1 - AkraBazziRecurrence.smoothingFn x) =o[Filter.atTop] fun x => x⁻¹
-- AkraBazziRecurrence.isLittleO_deriv_one_add_smoothingFn
lemma isLittleO_deriv_one_add_smoothingFn :
    deriv (fun x => 1 + ε x) =o[atTop] fun (x : ℝ) => x⁻¹ 
Goal: (deriv fun x => 1 + AkraBazziRecurrence.smoothingFn x) =o[Filter.atTop] fun x => x⁻¹
-- AkraBazziRecurrence.eventually_one_add_smoothingFn_pos
lemma eventually_one_add_smoothingFn_pos : ∀ᶠ (n : ℕ) in atTop, 0 < 1 + ε n 
Goal: ∀ᶠ (n : ℕ) in Filter.atTop, 0 < 1 + AkraBazziRecurrence.smoothingFn ↑n
-- AkraBazziRecurrence.eventually_one_add_smoothingFn_r_pos
lemma eventually_one_add_smoothingFn_r_pos : ∀ᶠ (n : ℕ) in atTop, ∀ i, 0 < 1 + ε (r i n) 
Arguments: {α : Type u_1} [Fintype α] [Nonempty α] {T : ℕ → ℝ} {g : ℝ → ℝ} {a b : α → ℝ} {r : α → ℕ → ℕ} (R : AkraBazziRecurrence T g a b r)
Goal: ∀ᶠ (n : ℕ) in Filter.atTop, ∀ (i : α), 0 < 1 + AkraBazziRecurrence.smoothingFn ↑(r i n)
-- AkraBazziRecurrence.eventually_one_add_smoothingFn_nonneg
lemma eventually_one_add_smoothingFn_nonneg : ∀ᶠ (n : ℕ) in atTop, 0 ≤ 1 + ε n 
Goal: ∀ᶠ (n : ℕ) in Filter.atTop, 0 ≤ 1 + AkraBazziRecurrence.smoothingFn ↑n
-- AkraBazziRecurrence.strictAntiOn_smoothingFn
lemma strictAntiOn_smoothingFn : StrictAntiOn ε (Set.Ioi 1) 
Goal: StrictAntiOn AkraBazziRecurrence.smoothingFn (Set.Ioi 1)
-- AkraBazziRecurrence.strictMonoOn_one_sub_smoothingFn
lemma strictMonoOn_one_sub_smoothingFn :
    StrictMonoOn (fun (x : ℝ) => (1 : ℝ) - ε x) (Set.Ioi 1) 
Goal: StrictMonoOn (fun x => 1 - AkraBazziRecurrence.smoothingFn x) (Set.Ioi 1)
-- AkraBazziRecurrence.strictAntiOn_one_add_smoothingFn
lemma strictAntiOn_one_add_smoothingFn : StrictAntiOn (fun (x : ℝ) => (1 : ℝ) + ε x) (Set.Ioi 1) 
Goal: StrictAntiOn (fun x => 1 + AkraBazziRecurrence.smoothingFn x) (Set.Ioi 1)
-- AkraBazziRecurrence.isEquivalent_smoothingFn_sub_self
lemma isEquivalent_smoothingFn_sub_self (i : α) :
    (fun (n : ℕ) => ε (b i * n) - ε n) ~[atTop] fun n => -log (b i) / (log n)^2 
Arguments: {α : Type u_1} [Fintype α] [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)
Goal: (fun n => AkraBazziRecurrence.smoothingFn (b i * ↑n) - AkraBazziRecurrence.smoothingFn ↑n) ~[Filter.atTop] fun n =>
  -Real.log (b i) / Real.log ↑n ^ 2
-- AkraBazziRecurrence.isTheta_smoothingFn_sub_self
lemma isTheta_smoothingFn_sub_self (i : α) :
    (fun (n : ℕ) => ε (b i * n) - ε n) =Θ[atTop] fun n => 1 / (log n)^2 
Arguments: {α : Type u_1} [Fintype α] [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)
Goal: (fun n => AkraBazziRecurrence.smoothingFn (b i * ↑n) - AkraBazziRecurrence.smoothingFn ↑n) =Θ[Filter.atTop] fun n =>
  1 / Real.log ↑n ^ 2
-- AkraBazziRecurrence.continuous_sumCoeffsExp
@[continuity]
lemma continuous_sumCoeffsExp : Continuous (fun (p : ℝ) => ∑ i, a i * (b i) ^ p) 
Arguments: {α : Type u_1} [Fintype α] [Nonempty α] {T : ℕ → ℝ} {g : ℝ → ℝ} {a b : α → ℝ} {r : α → ℕ → ℕ} (R : AkraBazziRecurrence T g a b r)
Goal: Continuous fun p => ∑ i : α, a i * b i ^ p
-- AkraBazziRecurrence.strictAnti_sumCoeffsExp
lemma strictAnti_sumCoeffsExp : StrictAnti (fun (p : ℝ) => ∑ i, a i * (b i) ^ p) 
Arguments: {α : Type u_1} [Fintype α] [Nonempty α] {T : ℕ → ℝ} {g : ℝ → ℝ} {a b : α → ℝ} {r : α → ℕ → ℕ} (R : AkraBazziRecurrence T g a b r)
Goal: StrictAnti fun p => ∑ i : α, a i * b i ^ p
-- AkraBazziRecurrence.tendsto_zero_sumCoeffsExp
lemma tendsto_zero_sumCoeffsExp : Tendsto (fun (p : ℝ) => ∑ i, a i * (b i) ^ p) atTop (𝓝 0) 
Arguments: {α : Type u_1} [Fintype α] [Nonempty α] {T : ℕ → ℝ} {g : ℝ → ℝ} {a b : α → ℝ} {r : α → ℕ → ℕ} (R : AkraBazziRecurrence T g a b r)
Goal: Filter.Tendsto (fun p => ∑ i : α, a i * b i ^ p) Filter.atTop (𝓝 0)
-- AkraBazziRecurrence.tendsto_atTop_sumCoeffsExp
lemma tendsto_atTop_sumCoeffsExp : Tendsto (fun (p : ℝ) => ∑ i, a i * (b i) ^ p) atBot atTop 
Arguments: {α : Type u_1} [Fintype α] [Nonempty α] {T : ℕ → ℝ} {g : ℝ → ℝ} {a b : α → ℝ} {r : α → ℕ → ℕ} (R : AkraBazziRecurrence T g a b r)
Goal: Filter.Tendsto (fun p => ∑ i : α, a i * b i ^ p) Filter.atBot Filter.atTop
-- AkraBazziRecurrence.one_mem_range_sumCoeffsExp
lemma one_mem_range_sumCoeffsExp : 1 ∈ Set.range (fun (p : ℝ) => ∑ i, a i * (b i) ^ p) 
Arguments: {α : Type u_1} [Fintype α] [Nonempty α] {T : ℕ → ℝ} {g : ℝ → ℝ} {a b : α → ℝ} {r : α → ℕ → ℕ} (R : AkraBazziRecurrence T g a b r)
Goal: 1 ∈ Set.range fun p => ∑ i : α, a i * b i ^ p
-- AkraBazziRecurrence.injective_sumCoeffsExp
lemma injective_sumCoeffsExp : Function.Injective (fun (p : ℝ) => ∑ i, a i * (b i) ^ p) 
Arguments: {α : Type u_1} [Fintype α] [Nonempty α] {T : ℕ → ℝ} {g : ℝ → ℝ} {a b : α → ℝ} {r : α → ℕ → ℕ} (R : AkraBazziRecurrence T g a b r)
Goal: Function.Injective fun p => ∑ i : α, a i * b i ^ p
-- AkraBazziRecurrence.p_def
noncomputable irreducible_def p : ℝ 
Arguments: {α : Type u_2} [Fintype α] (a b : α → ℝ)
Goal: AkraBazziRecurrence.p a b = Function.invFun (fun p => ∑ i : α, a i * b i ^ p) 1
-- AkraBazziRecurrence.sumCoeffsExp_p_eq_one
@[simp]
lemma sumCoeffsExp_p_eq_one : ∑ i, a i * (b i) ^ p a b = 1 
Arguments: {α : Type u_1} [Fintype α] [Nonempty α] {T : ℕ → ℝ} {g : ℝ → ℝ} {a b : α → ℝ} {r : α → ℕ → ℕ} (R : AkraBazziRecurrence T g a b r)
Goal: ∑ i : α, a i * b i ^ AkraBazziRecurrence.p a b = 1
-- AkraBazziRecurrence.sumTransform_def
lemma sumTransform_def {p : ℝ} {g : ℝ → ℝ} {n₀ n : ℕ} :
    sumTransform p g n₀ n = n^p * ∑ u ∈ Finset.Ico n₀ n, g u / u^(p + 1) 
Arguments: {p : ℝ} {g : ℝ → ℝ} {n₀ n : ℕ}
Goal: AkraBazziRecurrence.sumTransform p g n₀ n = ↑n ^ p * ∑ u ∈ Finset.Ico n₀ n, g ↑u / ↑u ^ (p + 1)
-- AkraBazziRecurrence.asympBound_def
lemma asympBound_def {n : ℕ} : asympBound g a b n = n ^ p a b + sumTransform (p a b) g 0 n 
Arguments: {α : Type u_1} [Fintype α] (g : ℝ → ℝ) (a b : α → ℝ) {n : ℕ}
Goal: AkraBazziRecurrence.asympBound g a b n =
  ↑n ^ AkraBazziRecurrence.p a b + AkraBazziRecurrence.sumTransform (AkraBazziRecurrence.p a b) g 0 n
-- AkraBazziRecurrence.asympBound_def'
lemma asympBound_def' {n : ℕ} :
    asympBound g a b n = n ^ p a b * (1 + (∑ u ∈ range n, g u / u ^ (p a b + 1))) 
Arguments: {α : Type u_1} [Fintype α] {g : ℝ → ℝ} {a b : α → ℝ} {n : ℕ}
Goal: AkraBazziRecurrence.asympBound g a b n =
  ↑n ^ AkraBazziRecurrence.p a b * (1 + ∑ u ∈ Finset.range n, g ↑u / ↑u ^ (AkraBazziRecurrence.p a b + 1))
-- AkraBazziRecurrence.asympBound_pos
lemma asympBound_pos (n : ℕ) (hn : 0 < n) : 0 < asympBound g a b n 
Arguments: {α : Type u_1} [Fintype α] [Nonempty α] {T : ℕ → ℝ} {g : ℝ → ℝ} {a b : α → ℝ} {r : α → ℕ → ℕ} (R : AkraBazziRecurrence T g a b r) (n : ℕ) (hn : 0 < n)
Goal: 0 < AkraBazziRecurrence.asympBound g a b n
-- AkraBazziRecurrence.eventually_asympBound_pos
lemma eventually_asympBound_pos : ∀ᶠ (n : ℕ) in atTop, 0 < asympBound g a b n 
Arguments: {α : Type u_1} [Fintype α] [Nonempty α] {T : ℕ → ℝ} {g : ℝ → ℝ} {a b : α → ℝ} {r : α → ℕ → ℕ} (R : AkraBazziRecurrence T g a b r)
Goal: ∀ᶠ (n : ℕ) in Filter.atTop, 0 < AkraBazziRecurrence.asympBound g a b n
-- AkraBazziRecurrence.eventually_asympBound_r_pos
lemma eventually_asympBound_r_pos : ∀ᶠ (n : ℕ) in atTop, ∀ i, 0 < asympBound g a b (r i n) 
Arguments: {α : Type u_1} [Fintype α] [Nonempty α] {T : ℕ → ℝ} {g : ℝ → ℝ} {a b : α → ℝ} {r : α → ℕ → ℕ} (R : AkraBazziRecurrence T g a b r)
Goal: ∀ᶠ (n : ℕ) in Filter.atTop, ∀ (i : α), 0 < AkraBazziRecurrence.asympBound g a b (r i n)
-- AkraBazziRecurrence.eventually_atTop_sumTransform_le
lemma eventually_atTop_sumTransform_le :
    ∃ c > 0, ∀ᶠ (n : ℕ) in atTop, ∀ i, sumTransform (p a b) g (r i n) n ≤ c * g n 
Arguments: {α : Type u_1} [Fintype α] [Nonempty α] {T : ℕ → ℝ} {g : ℝ → ℝ} {a b : α → ℝ} {r : α → ℕ → ℕ} (R : AkraBazziRecurrence T g a b r)
Goal: ∃ c > 0,
  ∀ᶠ (n : ℕ) in Filter.atTop,
    ∀ (i : α), AkraBazziRecurrence.sumTransform (AkraBazziRecurrence.p a b) g (r i n) n ≤ c * g ↑n
-- AkraBazziRecurrence.eventually_atTop_sumTransform_ge
lemma eventually_atTop_sumTransform_ge :
    ∃ c > 0, ∀ᶠ (n : ℕ) in atTop, ∀ i, c * g n ≤ sumTransform (p a b) g (r i n) n 
Arguments: {α : Type u_1} [Fintype α] [Nonempty α] {T : ℕ → ℝ} {g : ℝ → ℝ} {a b : α → ℝ} {r : α → ℕ → ℕ} (R : AkraBazziRecurrence T g a b r)
Goal: ∃ c > 0,
  ∀ᶠ (n : ℕ) in Filter.atTop,
    ∀ (i : α), c * g ↑n ≤ AkraBazziRecurrence.sumTransform (AkraBazziRecurrence.p a b) g (r i n) n
-- AkraBazziRecurrence.isBigO_apply_r_sub_b
lemma isBigO_apply_r_sub_b (q : ℝ → ℝ) (hq_diff : DifferentiableOn ℝ q (Set.Ioi 1))
    (hq_poly : GrowsPolynomially fun x => ‖deriv q x‖) (i : α) :
    (fun n => q (r i n) - q (b i * n)) =O[atTop] fun n => (deriv q n) * (r i n - b i * n) 
Arguments: {α : Type u_1} [Fintype α] [Nonempty α] {T : ℕ → ℝ} {g : ℝ → ℝ} {a b : α → ℝ} {r : α → ℕ → ℕ} (R : AkraBazziRecurrence T g a b r) (q : ℝ → ℝ) (hq_diff : DifferentiableOn ℝ q (Set.Ioi 1)) (hq_poly : AkraBazziRecurrence.GrowsPolynomially fun x => ‖deriv q x‖) (i : α)
Goal: (fun n => q ↑(r i n) - q (b i * ↑n)) =O[Filter.atTop] fun n => deriv q ↑n * (↑(r i n) - b i * ↑n)
-- AkraBazziRecurrence.eventually_deriv_rpow_p_mul_one_sub_smoothingFn
lemma eventually_deriv_rpow_p_mul_one_sub_smoothingFn (p : ℝ) :
    deriv (fun z => z ^ p * (1 - ε z))
      =ᶠ[atTop] fun z => p * z ^ (p-1) * (1 - ε z) + z ^ (p-1) / (log z ^ 2) 
Arguments: (p : ℝ)
Goal: (deriv fun z => z ^ p * (1 - AkraBazziRecurrence.smoothingFn z)) =ᶠ[Filter.atTop] fun z =>
  p * z ^ (p - 1) * (1 - AkraBazziRecurrence.smoothingFn z) + z ^ (p - 1) / Real.log z ^ 2
-- AkraBazziRecurrence.eventually_deriv_rpow_p_mul_one_add_smoothingFn
lemma eventually_deriv_rpow_p_mul_one_add_smoothingFn (p : ℝ) :
    deriv (fun z => z ^ p * (1 + ε z))
      =ᶠ[atTop] fun z => p * z ^ (p-1) * (1 + ε z) - z ^ (p-1) / (log z ^ 2) 
Arguments: (p : ℝ)
Goal: (deriv fun z => z ^ p * (1 + AkraBazziRecurrence.smoothingFn z)) =ᶠ[Filter.atTop] fun z =>
  p * z ^ (p - 1) * (1 + AkraBazziRecurrence.smoothingFn z) - z ^ (p - 1) / Real.log z ^ 2
-- AkraBazziRecurrence.isEquivalent_deriv_rpow_p_mul_one_sub_smoothingFn
lemma isEquivalent_deriv_rpow_p_mul_one_sub_smoothingFn {p : ℝ} (hp : p ≠ 0) :
    deriv (fun z => z ^ p * (1 - ε z)) ~[atTop] fun z => p * z ^ (p-1) 
Arguments: {p : ℝ} (hp : p ≠ 0)
Goal: (deriv fun z => z ^ p * (1 - AkraBazziRecurrence.smoothingFn z)) ~[Filter.atTop] fun z => p * z ^ (p - 1)
-- AkraBazziRecurrence.isEquivalent_deriv_rpow_p_mul_one_add_smoothingFn
lemma isEquivalent_deriv_rpow_p_mul_one_add_smoothingFn {p : ℝ} (hp : p ≠ 0) :
    deriv (fun z => z ^ p * (1 + ε z)) ~[atTop] fun z => p * z ^ (p-1) 
Arguments: {p : ℝ} (hp : p ≠ 0)
Goal: (deriv fun z => z ^ p * (1 + AkraBazziRecurrence.smoothingFn z)) ~[Filter.atTop] fun z => p * z ^ (p - 1)
-- AkraBazziRecurrence.isTheta_deriv_rpow_p_mul_one_sub_smoothingFn
lemma isTheta_deriv_rpow_p_mul_one_sub_smoothingFn {p : ℝ} (hp : p ≠ 0) :
    (fun x => ‖deriv (fun z => z ^ p * (1 - ε z)) x‖) =Θ[atTop] fun z => z ^ (p-1) 
Arguments: {p : ℝ} (hp : p ≠ 0)
Goal: (fun x => ‖deriv (fun z => z ^ p * (1 - AkraBazziRecurrence.smoothingFn z)) x‖) =Θ[Filter.atTop] fun z => z ^ (p - 1)
-- AkraBazziRecurrence.isTheta_deriv_rpow_p_mul_one_add_smoothingFn
lemma isTheta_deriv_rpow_p_mul_one_add_smoothingFn {p : ℝ} (hp : p ≠ 0) :
    (fun x => ‖deriv (fun z => z ^ p * (1 + ε z)) x‖) =Θ[atTop] fun z => z ^ (p-1) 
Arguments: {p : ℝ} (hp : p ≠ 0)
Goal: (fun x => ‖deriv (fun z => z ^ p * (1 + AkraBazziRecurrence.smoothingFn z)) x‖) =Θ[Filter.atTop] fun z => z ^ (p - 1)
-- AkraBazziRecurrence.growsPolynomially_deriv_rpow_p_mul_one_sub_smoothingFn
lemma growsPolynomially_deriv_rpow_p_mul_one_sub_smoothingFn (p : ℝ) :
    GrowsPolynomially fun x => ‖deriv (fun z => z ^ p * (1 - ε z)) x‖ 
Arguments: (p : ℝ)
Goal: AkraBazziRecurrence.GrowsPolynomially fun x => ‖deriv (fun z => z ^ p * (1 - AkraBazziRecurrence.smoothingFn z)) x‖
-- AkraBazziRecurrence.growsPolynomially_deriv_rpow_p_mul_one_add_smoothingFn
lemma growsPolynomially_deriv_rpow_p_mul_one_add_smoothingFn (p : ℝ) :
    GrowsPolynomially fun x => ‖deriv (fun z => z ^ p * (1 + ε z)) x‖ 
Arguments: (p : ℝ)
Goal: AkraBazziRecurrence.GrowsPolynomially fun x => ‖deriv (fun z => z ^ p * (1 + AkraBazziRecurrence.smoothingFn z)) x‖
-- AkraBazziRecurrence.rpow_p_mul_one_sub_smoothingFn_le
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) 
Arguments: {α : Type u_1} [Fintype α] [Nonempty α] {T : ℕ → ℝ} {g : ℝ → ℝ} {a b : α → ℝ} {r : α → ℕ → ℕ} (R : AkraBazziRecurrence T g a b r)
Goal: ∀ᶠ (n : ℕ) in Filter.atTop,
  ∀ (i : α),
    ↑(r i n) ^ AkraBazziRecurrence.p a b * (1 - AkraBazziRecurrence.smoothingFn ↑(r i n)) ≤
      b i ^ AkraBazziRecurrence.p a b * ↑n ^ AkraBazziRecurrence.p a b * (1 - AkraBazziRecurrence.smoothingFn ↑n)
-- AkraBazziRecurrence.rpow_p_mul_one_add_smoothingFn_ge
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)) 
Arguments: {α : Type u_1} [Fintype α] [Nonempty α] {T : ℕ → ℝ} {g : ℝ → ℝ} {a b : α → ℝ} {r : α → ℕ → ℕ} (R : AkraBazziRecurrence T g a b r)
Goal: ∀ᶠ (n : ℕ) in Filter.atTop,
  ∀ (i : α),
    b i ^ AkraBazziRecurrence.p a b * ↑n ^ AkraBazziRecurrence.p a b * (1 + AkraBazziRecurrence.smoothingFn ↑n) ≤
      ↑(r i n) ^ AkraBazziRecurrence.p a b * (1 + AkraBazziRecurrence.smoothingFn ↑(r i n))
-- AkraBazziRecurrence.base_nonempty
lemma base_nonempty {n : ℕ} (hn : 0 < n) : (Finset.Ico (⌊b (min_bi b) / 2 * n⌋₊) n).Nonempty 
Arguments: {α : Type u_1} [Fintype α] [Nonempty α] {T : ℕ → ℝ} {g : ℝ → ℝ} {a b : α → ℝ} {r : α → ℕ → ℕ} (R : AkraBazziRecurrence T g a b r) {n : ℕ} (hn : 0 < n)
Goal: (Finset.Ico ⌊b (AkraBazziRecurrence.min_bi b) / 2 * ↑n⌋₊ n).Nonempty
-- AkraBazziRecurrence.T_isBigO_smoothingFn_mul_asympBound
lemma T_isBigO_smoothingFn_mul_asympBound :
    T =O[atTop] (fun n => (1 - ε n) * asympBound g a b n) 
Arguments: {α : Type u_1} [Fintype α] [Nonempty α] {T : ℕ → ℝ} {g : ℝ → ℝ} {a b : α → ℝ} {r : α → ℕ → ℕ} (R : AkraBazziRecurrence T g a b r)
Goal: T =O[Filter.atTop] fun n => (1 - AkraBazziRecurrence.smoothingFn ↑n) * AkraBazziRecurrence.asympBound g a b n
-- AkraBazziRecurrence.smoothingFn_mul_asympBound_isBigO_T
lemma smoothingFn_mul_asympBound_isBigO_T :
    (fun (n : ℕ) => (1 + ε n) * asympBound g a b n) =O[atTop] T 
Arguments: {α : Type u_1} [Fintype α] [Nonempty α] {T : ℕ → ℝ} {g : ℝ → ℝ} {a b : α → ℝ} {r : α → ℕ → ℕ} (R : AkraBazziRecurrence T g a b r)
Goal: (fun n => (1 + AkraBazziRecurrence.smoothingFn ↑n) * AkraBazziRecurrence.asympBound g a b n) =O[Filter.atTop] T
-- AkraBazziRecurrence.isBigO_asympBound
theorem isBigO_asympBound : T =O[atTop] asympBound g a b 
Arguments: {α : Type u_1} [Fintype α] [Nonempty α] {T : ℕ → ℝ} {g : ℝ → ℝ} {a b : α → ℝ} {r : α → ℕ → ℕ} (R : AkraBazziRecurrence T g a b r)
Goal: T =O[Filter.atTop] AkraBazziRecurrence.asympBound g a b
-- AkraBazziRecurrence.isBigO_symm_asympBound
theorem isBigO_symm_asympBound : asympBound g a b =O[atTop] T 
Arguments: {α : Type u_1} [Fintype α] [Nonempty α] {T : ℕ → ℝ} {g : ℝ → ℝ} {a b : α → ℝ} {r : α → ℕ → ℕ} (R : AkraBazziRecurrence T g a b r)
Goal: AkraBazziRecurrence.asympBound g a b =O[Filter.atTop] T
-- AkraBazziRecurrence.isTheta_asympBound
theorem isTheta_asympBound : T =Θ[atTop] asympBound g a b 
Arguments: {α : Type u_1} [Fintype α] [Nonempty α] {T : ℕ → ℝ} {g : ℝ → ℝ} {a b : α → ℝ} {r : α → ℕ → ℕ} (R : AkraBazziRecurrence T g a b r)
Goal: T =Θ[Filter.atTop] AkraBazziRecurrence.asympBound g a b
-- cyclotomic_comp_X_add_one_isEisensteinAt
theorem cyclotomic_comp_X_add_one_isEisensteinAt [hp : Fact p.Prime] :
    ((cyclotomic p ℤ).comp (X + 1)).IsEisensteinAt 𝓟 
Arguments: (p : ℕ) [hp : Fact (Nat.Prime p)]
Goal: ((Polynomial.cyclotomic p ℤ).comp (Polynomial.X + 1)).IsEisensteinAt (Submodule.span ℤ {↑p})
-- cyclotomic_prime_pow_comp_X_add_one_isEisensteinAt
theorem cyclotomic_prime_pow_comp_X_add_one_isEisensteinAt [hp : Fact p.Prime] (n : ℕ) :
    ((cyclotomic (p ^ (n + 1)) ℤ).comp (X + 1)).IsEisensteinAt 𝓟 
Arguments: (p : ℕ) [hp : Fact (Nat.Prime p)] (n : ℕ)
Goal: ((Polynomial.cyclotomic (p ^ (n + 1)) ℤ).comp (Polynomial.X + 1)).IsEisensteinAt (Submodule.span ℤ {↑p})
-- dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_isEisensteinAt
theorem dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_isEisensteinAt {B : PowerBasis K L}
    (hp : Prime p) (hBint : IsIntegral R B.gen) {z : L} {Q : R[X]} (hQ : aeval B.gen Q = p • z)
    (hzint : IsIntegral R z) (hei : (minpoly R B.gen).IsEisensteinAt 𝓟) : p ∣ Q.coeff 0 
Arguments: {R : Type u} {K : Type v} {L : Type z} {p : R} [CommRing R] [Field K] [Field L] [Algebra K L] [Algebra R L] [Algebra R K] [IsScalarTower R K L] [Algebra.IsSeparable K L] [IsDomain R] [IsFractionRing R K] [IsIntegrallyClosed R] {B : PowerBasis K L} (hp : Prime p) (hBint : IsIntegral R B.gen) {z : L} {Q : R[X]} (hQ : (Polynomial.aeval B.gen) Q = p • z) (hzint : IsIntegral R z) (hei : (minpoly R B.gen).IsEisensteinAt (Submodule.span R {p}))
Goal: p ∣ Q.coeff 0
-- mem_adjoin_of_dvd_coeff_of_dvd_aeval
theorem mem_adjoin_of_dvd_coeff_of_dvd_aeval {A B : Type*} [CommSemiring A] [CommRing B]
    [Algebra A B] [NoZeroSMulDivisors A B] {Q : A[X]} {p : A} {x z : B} (hp : p ≠ 0)
    (hQ : ∀ i ∈ range (Q.natDegree + 1), p ∣ Q.coeff i) (hz : aeval x Q = p • z) :
    z ∈ adjoin A ({x} : Set B) 
Arguments: {A : Type u_1} {B : Type u_2} [CommSemiring A] [CommRing B] [Algebra A B] [NoZeroSMulDivisors A B] {Q : A[X]} {p : A} {x z : B} (hp : p ≠ 0) (hQ : ∀ i ∈ Finset.range (Q.natDegree + 1), p ∣ Q.coeff i) (hz : (Polynomial.aeval x) Q = p • z)
Goal: z ∈ Algebra.adjoin A {x}
-- mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt
theorem mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt {B : PowerBasis K L}
    (hp : Prime p) (hBint : IsIntegral R B.gen) {z : L} (hzint : IsIntegral R z)
    (hz : p • z ∈ adjoin R ({B.gen} : Set L)) (hei : (minpoly R B.gen).IsEisensteinAt 𝓟) :
    z ∈ adjoin R ({B.gen} : Set L) 
Arguments: {R : Type u} {K : Type v} {L : Type z} {p : R} [CommRing R] [Field K] [Field L] [Algebra K L] [Algebra R L] [Algebra R K] [IsScalarTower R K L] [Algebra.IsSeparable K L] [IsDomain R] [IsFractionRing R K] [IsIntegrallyClosed R] {B : PowerBasis K L} (hp : Prime p) (hBint : IsIntegral R B.gen) {z : L} (hzint : IsIntegral R z) (hz : p • z ∈ Algebra.adjoin R {B.gen}) (hei : (minpoly R B.gen).IsEisensteinAt (Submodule.span R {p}))
Goal: z ∈ Algebra.adjoin R {B.gen}
-- mem_adjoin_of_smul_prime_pow_smul_of_minpoly_isEisensteinAt
theorem mem_adjoin_of_smul_prime_pow_smul_of_minpoly_isEisensteinAt {B : PowerBasis K L}
    (hp : Prime p) (hBint : IsIntegral R B.gen) {n : ℕ} {z : L} (hzint : IsIntegral R z)
    (hz : p ^ n • z ∈ adjoin R ({B.gen} : Set L)) (hei : (minpoly R B.gen).IsEisensteinAt 𝓟) :
    z ∈ adjoin R ({B.gen} : Set L) 
Arguments: {R : Type u} {K : Type v} {L : Type z} {p : R} [CommRing R] [Field K] [Field L] [Algebra K L] [Algebra R L] [Algebra R K] [IsScalarTower R K L] [Algebra.IsSeparable K L] [IsDomain R] [IsFractionRing R K] [IsIntegrallyClosed R] {B : PowerBasis K L} (hp : Prime p) (hBint : IsIntegral R B.gen) {n : ℕ} {z : L} (hzint : IsIntegral R z) (hz : p ^ n • z ∈ Algebra.adjoin R {B.gen}) (hei : (minpoly R B.gen).IsEisensteinAt (Submodule.span R {p}))
Goal: z ∈ Algebra.adjoin R {B.gen}