l3lab/ntp-mathlib huggingface dataset:
protected lemma GrowsPolynomially.rpow (p : ℝ) (hf : GrowsPolynomially f)
    (hf_nonneg : ∀ᶠ x in atTop, 0 ≤ f x) : GrowsPolynomially fun x => (f x) ^ p := by
  intro b hb
  obtain ⟨c₁, (hc₁_mem : 0 < c₁), c₂, hc₂_mem, hfnew⟩ := hf b hb
  have hc₁p : 0 < c₁ ^ p := Real.rpow_pos_of_pos hc₁_mem _
  

Proof state:
case intro.intro.intro.intro
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
⊢ ∃ c₁ > 0,
    ∃ c₂ > 0,
      ∀ᶠ (x : ℝ) in atTop,
        ∀ u ∈ Set.Icc (b * x) x, (fun x => f x ^ p) u ∈ Set.Icc (c₁ * (fun x => f x ^ p) x) (c₂ * (fun x => f x ^ p) x)

Next tactic:
have hc₂p : 0 < c₂ ^ p := Real.rpow_pos_of_pos hc₂_mem _
protected lemma GrowsPolynomially.rpow (p : ℝ) (hf : GrowsPolynomially f)
    (hf_nonneg : ∀ᶠ x in atTop, 0 ≤ f x) : GrowsPolynomially fun x => (f x) ^ p := by
  intro b hb
  obtain ⟨c₁, (hc₁_mem : 0 < c₁), c₂, hc₂_mem, hfnew⟩ := hf b hb
  have hc₁p : 0 < c₁ ^ p := Real.rpow_pos_of_pos hc₁_mem _
  have hc₂p : 0 < c₂ ^ p := Real.rpow_pos_of_pos hc₂_mem _
  

Proof state:
case intro.intro.intro.intro
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
⊢ ∃ c₁ > 0,
    ∃ c₂ > 0,
      ∀ᶠ (x : ℝ) in atTop,
        ∀ u ∈ Set.Icc (b * x) x, (fun x => f x ^ p) u ∈ Set.Icc (c₁ * (fun x => f x ^ p) x) (c₂ * (fun x => f x ^ p) x)

Next tactic:
obtain _ | hp := le_or_lt 0 p
protected lemma GrowsPolynomially.rpow (p : ℝ) (hf : GrowsPolynomially f)
    (hf_nonneg : ∀ᶠ x in atTop, 0 ≤ f x) : GrowsPolynomially fun x => (f x) ^ p := by
  intro b hb
  obtain ⟨c₁, (hc₁_mem : 0 < c₁), c₂, hc₂_mem, hfnew⟩ := hf b hb
  have hc₁p : 0 < c₁ ^ p := Real.rpow_pos_of_pos hc₁_mem _
  have hc₂p : 0 < c₂ ^ p := Real.rpow_pos_of_pos hc₂_mem _
  obtain _ | hp := le_or_lt 0 p
  

Proof state:
case intro.intro.intro.intro.inl
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
h✝ : 0 ≤ p
⊢ ∃ c₁ > 0,
    ∃ c₂ > 0,
      ∀ᶠ (x : ℝ) in atTop,
        ∀ u ∈ Set.Icc (b * x) x, (fun x => f x ^ p) u ∈ Set.Icc (c₁ * (fun x => f x ^ p) x) (c₂ * (fun x => f x ^ p) x)
case intro.intro.intro.intro.inr
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
hp : p < 0
⊢ ∃ c₁ > 0,
    ∃ c₂ > 0,
      ∀ᶠ (x : ℝ) in atTop,
        ∀ u ∈ Set.Icc (b * x) x, (fun x => f x ^ p) u ∈ Set.Icc (c₁ * (fun x => f x ^ p) x) (c₂ * (fun x => f x ^ p) x)

Next tactic:
case inl =>    -- 0 ≤ p
    refine ⟨c₁^p, hc₁p, ?_⟩
    refine ⟨c₂^p, hc₂p, ?_⟩
    filter_upwards [eventually_gt_atTop 0, hfnew, hf_nonneg,
        (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hf_nonneg]
        with x _ hf₁ hf_nonneg hf_nonneg₂
    intro u hu
    have fu_nonneg : 0 ≤ f u := hf_nonneg₂ u hu.1
    refine ⟨?lb, ?ub⟩
    case lb => calc
      c₁^p * (f x)^p = (c₁ * f x)^p := by rw [mul_rpow (le_of_lt hc₁_mem) hf_nonneg]
        _ ≤ _ := by gcongr; exact (hf₁ u hu).1
    case ub => calc
      (f u)^p ≤ (c₂ * f x)^p := by gcongr; exact (hf₁ u hu).2
        _ = _ := by rw [← mul_rpow (le_of_lt hc₂_mem) hf_nonneg]
protected lemma GrowsPolynomially.rpow (p : ℝ) (hf : GrowsPolynomially f)
    (hf_nonneg : ∀ᶠ x in atTop, 0 ≤ f x) : GrowsPolynomially fun x => (f x) ^ p := by
  intro b hb
  obtain ⟨c₁, (hc₁_mem : 0 < c₁), c₂, hc₂_mem, hfnew⟩ := hf b hb
  have hc₁p : 0 < c₁ ^ p := Real.rpow_pos_of_pos hc₁_mem _
  have hc₂p : 0 < c₂ ^ p := Real.rpow_pos_of_pos hc₂_mem _
  obtain _ | hp := le_or_lt 0 p
  

Proof state:
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
h✝ : 0 ≤ p
⊢ ∃ c₁ > 0,
    ∃ c₂ > 0,
      ∀ᶠ (x : ℝ) in atTop,
        ∀ u ∈ Set.Icc (b * x) x, (fun x => f x ^ p) u ∈ Set.Icc (c₁ * (fun x => f x ^ p) x) (c₂ * (fun x => f x ^ p) x)

Next tactic:
case inl =>    -- 0 ≤ p
    refine ⟨c₁^p, hc₁p, ?_⟩
    refine ⟨c₂^p, hc₂p, ?_⟩
    filter_upwards [eventually_gt_atTop 0, hfnew, hf_nonneg,
        (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hf_nonneg]
        with x _ hf₁ hf_nonneg hf_nonneg₂
    intro u hu
    have fu_nonneg : 0 ≤ f u := hf_nonneg₂ u hu.1
    refine ⟨?lb, ?ub⟩
    case lb => calc
      c₁^p * (f x)^p = (c₁ * f x)^p := by rw [mul_rpow (le_of_lt hc₁_mem) hf_nonneg]
        _ ≤ _ := by gcongr; exact (hf₁ u hu).1
    case ub => calc
      (f u)^p ≤ (c₂ * f x)^p := by gcongr; exact (hf₁ u hu).2
        _ = _ := by rw [← mul_rpow (le_of_lt hc₂_mem) hf_nonneg]
protected lemma GrowsPolynomially.rpow (p : ℝ) (hf : GrowsPolynomially f)
    (hf_nonneg : ∀ᶠ x in atTop, 0 ≤ f x) : GrowsPolynomially fun x => (f x) ^ p := by
  intro b hb
  obtain ⟨c₁, (hc₁_mem : 0 < c₁), c₂, hc₂_mem, hfnew⟩ := hf b hb
  have hc₁p : 0 < c₁ ^ p := Real.rpow_pos_of_pos hc₁_mem _
  have hc₂p : 0 < c₂ ^ p := Real.rpow_pos_of_pos hc₂_mem _
  obtain _ | hp := le_or_lt 0 p
  case inl =>    -- 0 ≤ p
    

Proof state:
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
h✝ : 0 ≤ p
⊢ ∃ c₁ > 0,
    ∃ c₂ > 0,
      ∀ᶠ (x : ℝ) in atTop,
        ∀ u ∈ Set.Icc (b * x) x, (fun x => f x ^ p) u ∈ Set.Icc (c₁ * (fun x => f x ^ p) x) (c₂ * (fun x => f x ^ p) x)

Next tactic:
refine ⟨c₁^p, hc₁p, ?_⟩
protected lemma GrowsPolynomially.rpow (p : ℝ) (hf : GrowsPolynomially f)
    (hf_nonneg : ∀ᶠ x in atTop, 0 ≤ f x) : GrowsPolynomially fun x => (f x) ^ p := by
  intro b hb
  obtain ⟨c₁, (hc₁_mem : 0 < c₁), c₂, hc₂_mem, hfnew⟩ := hf b hb
  have hc₁p : 0 < c₁ ^ p := Real.rpow_pos_of_pos hc₁_mem _
  have hc₂p : 0 < c₂ ^ p := Real.rpow_pos_of_pos hc₂_mem _
  obtain _ | hp := le_or_lt 0 p
  case inl =>    -- 0 ≤ p
    refine ⟨c₁^p, hc₁p, ?_⟩
    

Proof state:
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
h✝ : 0 ≤ p
⊢ ∃ c₂ > 0,
    ∀ᶠ (x : ℝ) in atTop,
      ∀ u ∈ Set.Icc (b * x) x,
        (fun x => f x ^ p) u ∈ Set.Icc (c₁ ^ p * (fun x => f x ^ p) x) (c₂ * (fun x => f x ^ p) x)

Next tactic:
refine ⟨c₂^p, hc₂p, ?_⟩
protected lemma GrowsPolynomially.rpow (p : ℝ) (hf : GrowsPolynomially f)
    (hf_nonneg : ∀ᶠ x in atTop, 0 ≤ f x) : GrowsPolynomially fun x => (f x) ^ p := by
  intro b hb
  obtain ⟨c₁, (hc₁_mem : 0 < c₁), c₂, hc₂_mem, hfnew⟩ := hf b hb
  have hc₁p : 0 < c₁ ^ p := Real.rpow_pos_of_pos hc₁_mem _
  have hc₂p : 0 < c₂ ^ p := Real.rpow_pos_of_pos hc₂_mem _
  obtain _ | hp := le_or_lt 0 p
  case inl =>    -- 0 ≤ p
    refine ⟨c₁^p, hc₁p, ?_⟩
    refine ⟨c₂^p, hc₂p, ?_⟩
    

Proof state:
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
h✝ : 0 ≤ p
⊢ ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x,
      (fun x => f x ^ p) u ∈ Set.Icc (c₁ ^ p * (fun x => f x ^ p) x) (c₂ ^ p * (fun x => f x ^ p) x)

Next tactic:
filter_upwards [eventually_gt_atTop 0, hfnew, hf_nonneg,
        (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hf_nonneg]
        with x _ hf₁ hf_nonneg hf_nonneg₂
protected lemma GrowsPolynomially.rpow (p : ℝ) (hf : GrowsPolynomially f)
    (hf_nonneg : ∀ᶠ x in atTop, 0 ≤ f x) : GrowsPolynomially fun x => (f x) ^ p := by
  intro b hb
  obtain ⟨c₁, (hc₁_mem : 0 < c₁), c₂, hc₂_mem, hfnew⟩ := hf b hb
  have hc₁p : 0 < c₁ ^ p := Real.rpow_pos_of_pos hc₁_mem _
  have hc₂p : 0 < c₂ ^ p := Real.rpow_pos_of_pos hc₂_mem _
  obtain _ | hp := le_or_lt 0 p
  case inl =>    -- 0 ≤ p
    refine ⟨c₁^p, hc₁p, ?_⟩
    refine ⟨c₂^p, hc₂p, ?_⟩
    filter_upwards [eventually_gt_atTop 0, hfnew, hf_nonneg,
        (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hf_nonneg]
        with x _ hf₁ hf_nonneg hf_nonneg₂
    

Proof state:
case h
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg✝ : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
h✝ : 0 ≤ p
x : ℝ
a✝ : 0 < x
hf₁ : ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hf_nonneg : 0 ≤ f x
hf_nonneg₂ : ∀ (y : ℝ), b * id x ≤ y → 0 ≤ f y
⊢ ∀ u ∈ Set.Icc (b * x) x, f u ^ p ∈ Set.Icc (c₁ ^ p * f x ^ p) (c₂ ^ p * f x ^ p)

Next tactic:
intro u hu
protected lemma GrowsPolynomially.rpow (p : ℝ) (hf : GrowsPolynomially f)
    (hf_nonneg : ∀ᶠ x in atTop, 0 ≤ f x) : GrowsPolynomially fun x => (f x) ^ p := by
  intro b hb
  obtain ⟨c₁, (hc₁_mem : 0 < c₁), c₂, hc₂_mem, hfnew⟩ := hf b hb
  have hc₁p : 0 < c₁ ^ p := Real.rpow_pos_of_pos hc₁_mem _
  have hc₂p : 0 < c₂ ^ p := Real.rpow_pos_of_pos hc₂_mem _
  obtain _ | hp := le_or_lt 0 p
  case inl =>    -- 0 ≤ p
    refine ⟨c₁^p, hc₁p, ?_⟩
    refine ⟨c₂^p, hc₂p, ?_⟩
    filter_upwards [eventually_gt_atTop 0, hfnew, hf_nonneg,
        (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hf_nonneg]
        with x _ hf₁ hf_nonneg hf_nonneg₂
    intro u hu
    

Proof state:
case h
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg✝ : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
h✝ : 0 ≤ p
x : ℝ
a✝ : 0 < x
hf₁ : ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hf_nonneg : 0 ≤ f x
hf_nonneg₂ : ∀ (y : ℝ), b * id x ≤ y → 0 ≤ f y
u : ℝ
hu : u ∈ Set.Icc (b * x) x
⊢ f u ^ p ∈ Set.Icc (c₁ ^ p * f x ^ p) (c₂ ^ p * f x ^ p)

Next tactic:
have fu_nonneg : 0 ≤ f u := hf_nonneg₂ u hu.1
protected lemma GrowsPolynomially.rpow (p : ℝ) (hf : GrowsPolynomially f)
    (hf_nonneg : ∀ᶠ x in atTop, 0 ≤ f x) : GrowsPolynomially fun x => (f x) ^ p := by
  intro b hb
  obtain ⟨c₁, (hc₁_mem : 0 < c₁), c₂, hc₂_mem, hfnew⟩ := hf b hb
  have hc₁p : 0 < c₁ ^ p := Real.rpow_pos_of_pos hc₁_mem _
  have hc₂p : 0 < c₂ ^ p := Real.rpow_pos_of_pos hc₂_mem _
  obtain _ | hp := le_or_lt 0 p
  case inl =>    -- 0 ≤ p
    refine ⟨c₁^p, hc₁p, ?_⟩
    refine ⟨c₂^p, hc₂p, ?_⟩
    filter_upwards [eventually_gt_atTop 0, hfnew, hf_nonneg,
        (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hf_nonneg]
        with x _ hf₁ hf_nonneg hf_nonneg₂
    intro u hu
    have fu_nonneg : 0 ≤ f u := hf_nonneg₂ u hu.1
    

Proof state:
case h
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg✝ : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
h✝ : 0 ≤ p
x : ℝ
a✝ : 0 < x
hf₁ : ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hf_nonneg : 0 ≤ f x
hf_nonneg₂ : ∀ (y : ℝ), b * id x ≤ y → 0 ≤ f y
u : ℝ
hu : u ∈ Set.Icc (b * x) x
fu_nonneg : 0 ≤ f u
⊢ f u ^ p ∈ Set.Icc (c₁ ^ p * f x ^ p) (c₂ ^ p * f x ^ p)

Next tactic:
refine ⟨?lb, ?ub⟩
protected lemma GrowsPolynomially.rpow (p : ℝ) (hf : GrowsPolynomially f)
    (hf_nonneg : ∀ᶠ x in atTop, 0 ≤ f x) : GrowsPolynomially fun x => (f x) ^ p := by
  intro b hb
  obtain ⟨c₁, (hc₁_mem : 0 < c₁), c₂, hc₂_mem, hfnew⟩ := hf b hb
  have hc₁p : 0 < c₁ ^ p := Real.rpow_pos_of_pos hc₁_mem _
  have hc₂p : 0 < c₂ ^ p := Real.rpow_pos_of_pos hc₂_mem _
  obtain _ | hp := le_or_lt 0 p
  case inl =>    -- 0 ≤ p
    refine ⟨c₁^p, hc₁p, ?_⟩
    refine ⟨c₂^p, hc₂p, ?_⟩
    filter_upwards [eventually_gt_atTop 0, hfnew, hf_nonneg,
        (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hf_nonneg]
        with x _ hf₁ hf_nonneg hf_nonneg₂
    intro u hu
    have fu_nonneg : 0 ≤ f u := hf_nonneg₂ u hu.1
    refine ⟨?lb, ?ub⟩
    

Proof state:
case lb
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg✝ : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
h✝ : 0 ≤ p
x : ℝ
a✝ : 0 < x
hf₁ : ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hf_nonneg : 0 ≤ f x
hf_nonneg₂ : ∀ (y : ℝ), b * id x ≤ y → 0 ≤ f y
u : ℝ
hu : u ∈ Set.Icc (b * x) x
fu_nonneg : 0 ≤ f u
⊢ c₁ ^ p * f x ^ p ≤ f u ^ p
case ub
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg✝ : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
h✝ : 0 ≤ p
x : ℝ
a✝ : 0 < x
hf₁ : ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hf_nonneg : 0 ≤ f x
hf_nonneg₂ : ∀ (y : ℝ), b * id x ≤ y → 0 ≤ f y
u : ℝ
hu : u ∈ Set.Icc (b * x) x
fu_nonneg : 0 ≤ f u
⊢ f u ^ p ≤ c₂ ^ p * f x ^ p

Next tactic:
case lb => calc
      c₁^p * (f x)^p = (c₁ * f x)^p := by rw [mul_rpow (le_of_lt hc₁_mem) hf_nonneg]
        _ ≤ _ := by gcongr; exact (hf₁ u hu).1
protected lemma GrowsPolynomially.rpow (p : ℝ) (hf : GrowsPolynomially f)
    (hf_nonneg : ∀ᶠ x in atTop, 0 ≤ f x) : GrowsPolynomially fun x => (f x) ^ p := by
  intro b hb
  obtain ⟨c₁, (hc₁_mem : 0 < c₁), c₂, hc₂_mem, hfnew⟩ := hf b hb
  have hc₁p : 0 < c₁ ^ p := Real.rpow_pos_of_pos hc₁_mem _
  have hc₂p : 0 < c₂ ^ p := Real.rpow_pos_of_pos hc₂_mem _
  obtain _ | hp := le_or_lt 0 p
  case inl =>    -- 0 ≤ p
    refine ⟨c₁^p, hc₁p, ?_⟩
    refine ⟨c₂^p, hc₂p, ?_⟩
    filter_upwards [eventually_gt_atTop 0, hfnew, hf_nonneg,
        (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hf_nonneg]
        with x _ hf₁ hf_nonneg hf_nonneg₂
    intro u hu
    have fu_nonneg : 0 ≤ f u := hf_nonneg₂ u hu.1
    refine ⟨?lb, ?ub⟩
    

Proof state:
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg✝ : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
h✝ : 0 ≤ p
x : ℝ
a✝ : 0 < x
hf₁ : ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hf_nonneg : 0 ≤ f x
hf_nonneg₂ : ∀ (y : ℝ), b * id x ≤ y → 0 ≤ f y
u : ℝ
hu : u ∈ Set.Icc (b * x) x
fu_nonneg : 0 ≤ f u
⊢ c₁ ^ p * f x ^ p ≤ f u ^ p

Next tactic:
case lb => calc
      c₁^p * (f x)^p = (c₁ * f x)^p := by rw [mul_rpow (le_of_lt hc₁_mem) hf_nonneg]
        _ ≤ _ := by gcongr; exact (hf₁ u hu).1
protected lemma GrowsPolynomially.rpow (p : ℝ) (hf : GrowsPolynomially f)
    (hf_nonneg : ∀ᶠ x in atTop, 0 ≤ f x) : GrowsPolynomially fun x => (f x) ^ p := by
  intro b hb
  obtain ⟨c₁, (hc₁_mem : 0 < c₁), c₂, hc₂_mem, hfnew⟩ := hf b hb
  have hc₁p : 0 < c₁ ^ p := Real.rpow_pos_of_pos hc₁_mem _
  have hc₂p : 0 < c₂ ^ p := Real.rpow_pos_of_pos hc₂_mem _
  obtain _ | hp := le_or_lt 0 p
  case inl =>    -- 0 ≤ p
    refine ⟨c₁^p, hc₁p, ?_⟩
    refine ⟨c₂^p, hc₂p, ?_⟩
    filter_upwards [eventually_gt_atTop 0, hfnew, hf_nonneg,
        (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hf_nonneg]
        with x _ hf₁ hf_nonneg hf_nonneg₂
    intro u hu
    have fu_nonneg : 0 ≤ f u := hf_nonneg₂ u hu.1
    refine ⟨?lb, ?ub⟩
    case lb => 

Proof state:
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg✝ : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
h✝ : 0 ≤ p
x : ℝ
a✝ : 0 < x
hf₁ : ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hf_nonneg : 0 ≤ f x
hf_nonneg₂ : ∀ (y : ℝ), b * id x ≤ y → 0 ≤ f y
u : ℝ
hu : u ∈ Set.Icc (b * x) x
fu_nonneg : 0 ≤ f u
⊢ c₁ ^ p * f x ^ p ≤ f u ^ p

Next tactic:
calc
      c₁^p * (f x)^p = (c₁ * f x)^p := by rw [mul_rpow (le_of_lt hc₁_mem) hf_nonneg]
        _ ≤ _ := by gcongr; exact (hf₁ u hu).1
protected lemma GrowsPolynomially.rpow (p : ℝ) (hf : GrowsPolynomially f)
    (hf_nonneg : ∀ᶠ x in atTop, 0 ≤ f x) : GrowsPolynomially fun x => (f x) ^ p := by
  intro b hb
  obtain ⟨c₁, (hc₁_mem : 0 < c₁), c₂, hc₂_mem, hfnew⟩ := hf b hb
  have hc₁p : 0 < c₁ ^ p := Real.rpow_pos_of_pos hc₁_mem _
  have hc₂p : 0 < c₂ ^ p := Real.rpow_pos_of_pos hc₂_mem _
  obtain _ | hp := le_or_lt 0 p
  case inl =>    -- 0 ≤ p
    refine ⟨c₁^p, hc₁p, ?_⟩
    refine ⟨c₂^p, hc₂p, ?_⟩
    filter_upwards [eventually_gt_atTop 0, hfnew, hf_nonneg,
        (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hf_nonneg]
        with x _ hf₁ hf_nonneg hf_nonneg₂
    intro u hu
    have fu_nonneg : 0 ≤ f u := hf_nonneg₂ u hu.1
    refine ⟨?lb, ?ub⟩
    case lb => calc
      c₁^p * (f x)^p = (c₁ * f x)^p := by 

Proof state:
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg✝ : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
h✝ : 0 ≤ p
x : ℝ
a✝ : 0 < x
hf₁ : ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hf_nonneg : 0 ≤ f x
hf_nonneg₂ : ∀ (y : ℝ), b * id x ≤ y → 0 ≤ f y
u : ℝ
hu : u ∈ Set.Icc (b * x) x
fu_nonneg : 0 ≤ f u
⊢ c₁ ^ p * f x ^ p = (c₁ * f x) ^ p

Next tactic:
rw [mul_rpow (le_of_lt hc₁_mem) hf_nonneg]
protected lemma GrowsPolynomially.rpow (p : ℝ) (hf : GrowsPolynomially f)
    (hf_nonneg : ∀ᶠ x in atTop, 0 ≤ f x) : GrowsPolynomially fun x => (f x) ^ p := by
  intro b hb
  obtain ⟨c₁, (hc₁_mem : 0 < c₁), c₂, hc₂_mem, hfnew⟩ := hf b hb
  have hc₁p : 0 < c₁ ^ p := Real.rpow_pos_of_pos hc₁_mem _
  have hc₂p : 0 < c₂ ^ p := Real.rpow_pos_of_pos hc₂_mem _
  obtain _ | hp := le_or_lt 0 p
  case inl =>    -- 0 ≤ p
    refine ⟨c₁^p, hc₁p, ?_⟩
    refine ⟨c₂^p, hc₂p, ?_⟩
    filter_upwards [eventually_gt_atTop 0, hfnew, hf_nonneg,
        (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hf_nonneg]
        with x _ hf₁ hf_nonneg hf_nonneg₂
    intro u hu
    have fu_nonneg : 0 ≤ f u := hf_nonneg₂ u hu.1
    refine ⟨?lb, ?ub⟩
    case lb => calc
      c₁^p * (f x)^p = (c₁ * f x)^p := by rw [mul_rpow (le_of_lt hc₁_mem) hf_nonneg]
        _ ≤ _ := by 

Proof state:
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg✝ : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
h✝ : 0 ≤ p
x : ℝ
a✝ : 0 < x
hf₁ : ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hf_nonneg : 0 ≤ f x
hf_nonneg₂ : ∀ (y : ℝ), b * id x ≤ y → 0 ≤ f y
u : ℝ
hu : u ∈ Set.Icc (b * x) x
fu_nonneg : 0 ≤ f u
⊢ (c₁ * f x) ^ p ≤ f u ^ p

Next tactic:
gcongr
protected lemma GrowsPolynomially.rpow (p : ℝ) (hf : GrowsPolynomially f)
    (hf_nonneg : ∀ᶠ x in atTop, 0 ≤ f x) : GrowsPolynomially fun x => (f x) ^ p := by
  intro b hb
  obtain ⟨c₁, (hc₁_mem : 0 < c₁), c₂, hc₂_mem, hfnew⟩ := hf b hb
  have hc₁p : 0 < c₁ ^ p := Real.rpow_pos_of_pos hc₁_mem _
  have hc₂p : 0 < c₂ ^ p := Real.rpow_pos_of_pos hc₂_mem _
  obtain _ | hp := le_or_lt 0 p
  case inl =>    -- 0 ≤ p
    refine ⟨c₁^p, hc₁p, ?_⟩
    refine ⟨c₂^p, hc₂p, ?_⟩
    filter_upwards [eventually_gt_atTop 0, hfnew, hf_nonneg,
        (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hf_nonneg]
        with x _ hf₁ hf_nonneg hf_nonneg₂
    intro u hu
    have fu_nonneg : 0 ≤ f u := hf_nonneg₂ u hu.1
    refine ⟨?lb, ?ub⟩
    case lb => calc
      c₁^p * (f x)^p = (c₁ * f x)^p := by rw [mul_rpow (le_of_lt hc₁_mem) hf_nonneg]
        _ ≤ _ := by gcongr; 

Proof state:
case h₁
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg✝ : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
h✝ : 0 ≤ p
x : ℝ
a✝ : 0 < x
hf₁ : ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hf_nonneg : 0 ≤ f x
hf_nonneg₂ : ∀ (y : ℝ), b * id x ≤ y → 0 ≤ f y
u : ℝ
hu : u ∈ Set.Icc (b * x) x
fu_nonneg : 0 ≤ f u
⊢ c₁ * f x ≤ f u

Next tactic:
exact (hf₁ u hu).1
protected lemma GrowsPolynomially.rpow (p : ℝ) (hf : GrowsPolynomially f)
    (hf_nonneg : ∀ᶠ x in atTop, 0 ≤ f x) : GrowsPolynomially fun x => (f x) ^ p := by
  intro b hb
  obtain ⟨c₁, (hc₁_mem : 0 < c₁), c₂, hc₂_mem, hfnew⟩ := hf b hb
  have hc₁p : 0 < c₁ ^ p := Real.rpow_pos_of_pos hc₁_mem _
  have hc₂p : 0 < c₂ ^ p := Real.rpow_pos_of_pos hc₂_mem _
  obtain _ | hp := le_or_lt 0 p
  case inl =>    -- 0 ≤ p
    refine ⟨c₁^p, hc₁p, ?_⟩
    refine ⟨c₂^p, hc₂p, ?_⟩
    filter_upwards [eventually_gt_atTop 0, hfnew, hf_nonneg,
        (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hf_nonneg]
        with x _ hf₁ hf_nonneg hf_nonneg₂
    intro u hu
    have fu_nonneg : 0 ≤ f u := hf_nonneg₂ u hu.1
    refine ⟨?lb, ?ub⟩
    case lb => calc
      c₁^p * (f x)^p = (c₁ * f x)^p := by rw [mul_rpow (le_of_lt hc₁_mem) hf_nonneg]
        _ ≤ _ := by gcongr; exact (hf₁ u hu).1
    

Proof state:
case ub
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg✝ : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
h✝ : 0 ≤ p
x : ℝ
a✝ : 0 < x
hf₁ : ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hf_nonneg : 0 ≤ f x
hf_nonneg₂ : ∀ (y : ℝ), b * id x ≤ y → 0 ≤ f y
u : ℝ
hu : u ∈ Set.Icc (b * x) x
fu_nonneg : 0 ≤ f u
⊢ f u ^ p ≤ c₂ ^ p * f x ^ p

Next tactic:
case ub => calc
      (f u)^p ≤ (c₂ * f x)^p := by gcongr; exact (hf₁ u hu).2
        _ = _ := by rw [← mul_rpow (le_of_lt hc₂_mem) hf_nonneg]