Theorem: AkraBazziRecurrence.GrowsPolynomially.inv
File: Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean

tactic: cases hf.eventually_atTop_zero_or_pos_or_neg with
| inl hf' =>
  refine fun b hb => ⟨1, by simp, 1, by simp, ?_⟩
  have hb_pos := hb.1
  filter_upwards [hf', (tendsto_id.const_mul_atTop hb_pos).eventually_forall_ge_atTop hf']
    with x hx hx'
  intro u hu
  simp only [hx, inv_zero, mul_zero, Set.Icc_self, Set.mem_singleton_iff, hx' u hu.1]
| inr hf_pos_or_neg =>
  suffices GrowsPolynomially fun x => |(f x)⁻¹| by
    cases hf_pos_or_neg with
    | inl hf' =>
      have hmain : (fun x => (f x)⁻¹) =ᶠ[atTop] fun x => |(f x)⁻¹| := by
        filter_upwards [hf'] with x hx₁
        rw [abs_of_nonneg (inv_nonneg_of_nonneg (le_of_lt hx₁))]
      rwa [iff_eventuallyEq hmain]
    | inr hf' =>
      have hmain : (fun x => (f x)⁻¹) =ᶠ[atTop] fun x => -|(f x)⁻¹| := by
        filter_upwards [hf'] with x hx₁
        simp [abs_of_nonpos (inv_nonpos.mpr (le_of_lt hx₁))]
      rw [iff_eventuallyEq hmain]
      exact this.neg
  have hf' : ∀ᶠ x in atTop, f x ≠ 0 := by
    cases hf_pos_or_neg with
    | inl H => filter_upwards [H] with _ hx; exact (ne_of_lt hx).symm
    | inr H => filter_upwards [H] with _ hx; exact (ne_of_gt hx).symm
  simp only [abs_inv]
  have hf := hf.abs
  intro b hb
  have hb_pos := hb.1
  obtain ⟨c₁, hc₁_mem, c₂, hc₂_mem, hf⟩ := hf b hb
  refine ⟨c₂⁻¹, by show 0 < c₂⁻¹; positivity, ?_⟩
  refine ⟨c₁⁻¹, by show 0 < c₁⁻¹; positivity, ?_⟩
  filter_upwards [hf, hf', (tendsto_id.const_mul_atTop hb_pos).eventually_forall_ge_atTop hf']
    with x hx hx' hx''
  intro u hu
  have h₁ : 0 < |f u| := by rw [abs_pos]; exact hx'' u hu.1
  refine ⟨?lb, ?ub⟩
  case lb =>
    rw [← mul_inv]
    gcongr
    exact (hx u hu).2
  case ub =>
    rw [← mul_inv]
    gcongr
    exact (hx u hu).1
state_before: f✝ f : ℝ → ℝ
hf : GrowsPolynomially f
⊢ GrowsPolynomially fun x => (f x)⁻¹
state_after: no goals

tactic: refine fun b hb => ⟨1, by simp, 1, by simp, ?_⟩
state_before: case inl
f✝ f : ℝ → ℝ
hf : GrowsPolynomially f
hf' : ∀ᶠ (x : ℝ) in atTop, f x = 0
⊢ GrowsPolynomially fun x => (f x)⁻¹
state_after: case inl
f✝ f : ℝ → ℝ
hf : GrowsPolynomially f
hf' : ∀ᶠ (x : ℝ) in atTop, f x = 0
b : ℝ
hb : b ∈ Set.Ioo 0 1
⊢ ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => (f x)⁻¹) u ∈ Set.Icc (1 * (fun x => (f x)⁻¹) x) (1 * (fun x => (f x)⁻¹) x)

tactic: have hb_pos := hb.1
state_before: case inl
f✝ f : ℝ → ℝ
hf : GrowsPolynomially f
hf' : ∀ᶠ (x : ℝ) in atTop, f x = 0
b : ℝ
hb : b ∈ Set.Ioo 0 1
⊢ ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => (f x)⁻¹) u ∈ Set.Icc (1 * (fun x => (f x)⁻¹) x) (1 * (fun x => (f x)⁻¹) x)
state_after: case inl
f✝ f : ℝ → ℝ
hf : GrowsPolynomially f
hf' : ∀ᶠ (x : ℝ) in atTop, f x = 0
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
⊢ ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => (f x)⁻¹) u ∈ Set.Icc (1 * (fun x => (f x)⁻¹) x) (1 * (fun x => (f x)⁻¹) x)

tactic: filter_upwards [hf', (tendsto_id.const_mul_atTop hb_pos).eventually_forall_ge_atTop hf']
  with x hx hx'
state_before: case inl
f✝ f : ℝ → ℝ
hf : GrowsPolynomially f
hf' : ∀ᶠ (x : ℝ) in atTop, f x = 0
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
⊢ ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => (f x)⁻¹) u ∈ Set.Icc (1 * (fun x => (f x)⁻¹) x) (1 * (fun x => (f x)⁻¹) x)
state_after: case h
f✝ f : ℝ → ℝ
hf : GrowsPolynomially f
hf' : ∀ᶠ (x : ℝ) in atTop, f x = 0
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
x : ℝ
hx : f x = 0
hx' : ∀ (y : ℝ), b * id x ≤ y → f y = 0
⊢ ∀ u ∈ Set.Icc (b * x) x, (f u)⁻¹ ∈ Set.Icc (1 * (f x)⁻¹) (1 * (f x)⁻¹)

tactic: intro u hu
state_before: case h
f✝ f : ℝ → ℝ
hf : GrowsPolynomially f
hf' : ∀ᶠ (x : ℝ) in atTop, f x = 0
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
x : ℝ
hx : f x = 0
hx' : ∀ (y : ℝ), b * id x ≤ y → f y = 0
⊢ ∀ u ∈ Set.Icc (b * x) x, (f u)⁻¹ ∈ Set.Icc (1 * (f x)⁻¹) (1 * (f x)⁻¹)
state_after: case h
f✝ f : ℝ → ℝ
hf : GrowsPolynomially f
hf' : ∀ᶠ (x : ℝ) in atTop, f x = 0
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
x : ℝ
hx : f x = 0
hx' : ∀ (y : ℝ), b * id x ≤ y → f y = 0
u : ℝ
hu : u ∈ Set.Icc (b * x) x
⊢ (f u)⁻¹ ∈ Set.Icc (1 * (f x)⁻¹) (1 * (f x)⁻¹)

tactic: simp only [hx, inv_zero, mul_zero, Set.Icc_self, Set.mem_singleton_iff, hx' u hu.1]
state_before: case h
f✝ f : ℝ → ℝ
hf : GrowsPolynomially f
hf' : ∀ᶠ (x : ℝ) in atTop, f x = 0
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
x : ℝ
hx : f x = 0
hx' : ∀ (y : ℝ), b * id x ≤ y → f y = 0
u : ℝ
hu : u ∈ Set.Icc (b * x) x
⊢ (f u)⁻¹ ∈ Set.Icc (1 * (f x)⁻¹) (1 * (f x)⁻¹)
state_after: no goals

tactic: simp
state_before: f✝ f : ℝ → ℝ
hf : GrowsPolynomially f
hf' : ∀ᶠ (x : ℝ) in atTop, f x = 0
b : ℝ
hb : b ∈ Set.Ioo 0 1
⊢ 1 > 0
state_after: no goals

tactic: suffices GrowsPolynomially fun x => |(f x)⁻¹| by
  cases hf_pos_or_neg with
  | inl hf' =>
    have hmain : (fun x => (f x)⁻¹) =ᶠ[atTop] fun x => |(f x)⁻¹| := by
      filter_upwards [hf'] with x hx₁
      rw [abs_of_nonneg (inv_nonneg_of_nonneg (le_of_lt hx₁))]
    rwa [iff_eventuallyEq hmain]
  | inr hf' =>
    have hmain : (fun x => (f x)⁻¹) =ᶠ[atTop] fun x => -|(f x)⁻¹| := by
      filter_upwards [hf'] with x hx₁
      simp [abs_of_nonpos (inv_nonpos.mpr (le_of_lt hx₁))]
    rw [iff_eventuallyEq hmain]
    exact this.neg
state_before: case inr
f✝ f : ℝ → ℝ
hf : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
⊢ GrowsPolynomially fun x => (f x)⁻¹
state_after: case inr
f✝ f : ℝ → ℝ
hf : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
⊢ GrowsPolynomially fun x => |(f x)⁻¹|

tactic: have hf' : ∀ᶠ x in atTop, f x ≠ 0 := by
  cases hf_pos_or_neg with
  | inl H => filter_upwards [H] with _ hx; exact (ne_of_lt hx).symm
  | inr H => filter_upwards [H] with _ hx; exact (ne_of_gt hx).symm
state_before: case inr
f✝ f : ℝ → ℝ
hf : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
⊢ GrowsPolynomially fun x => |(f x)⁻¹|
state_after: case inr
f✝ f : ℝ → ℝ
hf : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
⊢ GrowsPolynomially fun x => |(f x)⁻¹|

tactic: simp only [abs_inv]
state_before: case inr
f✝ f : ℝ → ℝ
hf : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
⊢ GrowsPolynomially fun x => |(f x)⁻¹|
state_after: case inr
f✝ f : ℝ → ℝ
hf : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
⊢ GrowsPolynomially fun x => |f x|⁻¹

tactic: have hf := hf.abs
state_before: case inr
f✝ f : ℝ → ℝ
hf : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
⊢ GrowsPolynomially fun x => |f x|⁻¹
state_after: case inr
f✝ f : ℝ → ℝ
hf✝ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf : GrowsPolynomially fun x => |f x|
⊢ GrowsPolynomially fun x => |f x|⁻¹

tactic: intro b hb
state_before: case inr
f✝ f : ℝ → ℝ
hf✝ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf : GrowsPolynomially fun x => |f x|
⊢ GrowsPolynomially fun x => |f x|⁻¹
state_after: case inr
f✝ f : ℝ → ℝ
hf✝ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf : GrowsPolynomially fun x => |f x|
b : ℝ
hb : b ∈ Set.Ioo 0 1
⊢ ∃ c₁ > 0,
    ∃ c₂ > 0,
      ∀ᶠ (x : ℝ) in atTop,
        ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|⁻¹) u ∈ Set.Icc (c₁ * (fun x => |f x|⁻¹) x) (c₂ * (fun x => |f x|⁻¹) x)

tactic: have hb_pos := hb.1
state_before: case inr
f✝ f : ℝ → ℝ
hf✝ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf : GrowsPolynomially fun x => |f x|
b : ℝ
hb : b ∈ Set.Ioo 0 1
⊢ ∃ c₁ > 0,
    ∃ c₂ > 0,
      ∀ᶠ (x : ℝ) in atTop,
        ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|⁻¹) u ∈ Set.Icc (c₁ * (fun x => |f x|⁻¹) x) (c₂ * (fun x => |f x|⁻¹) x)
state_after: case inr
f✝ f : ℝ → ℝ
hf✝ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf : GrowsPolynomially fun x => |f x|
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
⊢ ∃ c₁ > 0,
    ∃ c₂ > 0,
      ∀ᶠ (x : ℝ) in atTop,
        ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|⁻¹) u ∈ Set.Icc (c₁ * (fun x => |f x|⁻¹) x) (c₂ * (fun x => |f x|⁻¹) x)

tactic: obtain ⟨c₁, hc₁_mem, c₂, hc₂_mem, hf⟩ := hf b hb
state_before: case inr
f✝ f : ℝ → ℝ
hf✝ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf : GrowsPolynomially fun x => |f x|
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
⊢ ∃ c₁ > 0,
    ∃ c₂ > 0,
      ∀ᶠ (x : ℝ) in atTop,
        ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|⁻¹) u ∈ Set.Icc (c₁ * (fun x => |f x|⁻¹) x) (c₂ * (fun x => |f x|⁻¹) x)
state_after: case inr.intro.intro.intro.intro
f✝ f : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf✝ : GrowsPolynomially fun x => |f x|
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|) u ∈ Set.Icc (c₁ * (fun x => |f x|) x) (c₂ * (fun x => |f x|) x)
⊢ ∃ c₁ > 0,
    ∃ c₂ > 0,
      ∀ᶠ (x : ℝ) in atTop,
        ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|⁻¹) u ∈ Set.Icc (c₁ * (fun x => |f x|⁻¹) x) (c₂ * (fun x => |f x|⁻¹) x)

tactic: refine ⟨c₂⁻¹, by show 0 < c₂⁻¹; positivity, ?_⟩
state_before: case inr.intro.intro.intro.intro
f✝ f : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf✝ : GrowsPolynomially fun x => |f x|
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|) u ∈ Set.Icc (c₁ * (fun x => |f x|) x) (c₂ * (fun x => |f x|) x)
⊢ ∃ c₁ > 0,
    ∃ c₂ > 0,
      ∀ᶠ (x : ℝ) in atTop,
        ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|⁻¹) u ∈ Set.Icc (c₁ * (fun x => |f x|⁻¹) x) (c₂ * (fun x => |f x|⁻¹) x)
state_after: case inr.intro.intro.intro.intro
f✝ f : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf✝ : GrowsPolynomially fun x => |f x|
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|) u ∈ Set.Icc (c₁ * (fun x => |f x|) x) (c₂ * (fun x => |f x|) x)
⊢ ∃ c₂_1 > 0,
    ∀ᶠ (x : ℝ) in atTop,
      ∀ u ∈ Set.Icc (b * x) x,
        (fun x => |f x|⁻¹) u ∈ Set.Icc (c₂⁻¹ * (fun x => |f x|⁻¹) x) (c₂_1 * (fun x => |f x|⁻¹) x)

tactic: refine ⟨c₁⁻¹, by show 0 < c₁⁻¹; positivity, ?_⟩
state_before: case inr.intro.intro.intro.intro
f✝ f : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf✝ : GrowsPolynomially fun x => |f x|
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|) u ∈ Set.Icc (c₁ * (fun x => |f x|) x) (c₂ * (fun x => |f x|) x)
⊢ ∃ c₂_1 > 0,
    ∀ᶠ (x : ℝ) in atTop,
      ∀ u ∈ Set.Icc (b * x) x,
        (fun x => |f x|⁻¹) u ∈ Set.Icc (c₂⁻¹ * (fun x => |f x|⁻¹) x) (c₂_1 * (fun x => |f x|⁻¹) x)
state_after: case inr.intro.intro.intro.intro
f✝ f : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf✝ : GrowsPolynomially fun x => |f x|
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|) u ∈ Set.Icc (c₁ * (fun x => |f x|) x) (c₂ * (fun x => |f x|) x)
⊢ ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|⁻¹) u ∈ Set.Icc (c₂⁻¹ * (fun x => |f x|⁻¹) x) (c₁⁻¹ * (fun x => |f x|⁻¹) x)

tactic: filter_upwards [hf, hf', (tendsto_id.const_mul_atTop hb_pos).eventually_forall_ge_atTop hf']
  with x hx hx' hx''
state_before: case inr.intro.intro.intro.intro
f✝ f : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf✝ : GrowsPolynomially fun x => |f x|
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|) u ∈ Set.Icc (c₁ * (fun x => |f x|) x) (c₂ * (fun x => |f x|) x)
⊢ ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|⁻¹) u ∈ Set.Icc (c₂⁻¹ * (fun x => |f x|⁻¹) x) (c₁⁻¹ * (fun x => |f x|⁻¹) x)
state_after: case h
f✝ f : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf✝ : GrowsPolynomially fun x => |f x|
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|) u ∈ Set.Icc (c₁ * (fun x => |f x|) x) (c₂ * (fun x => |f x|) x)
x : ℝ
hx : ∀ u ∈ Set.Icc (b * x) x, |f u| ∈ Set.Icc (c₁ * |f x|) (c₂ * |f x|)
hx' : f x ≠ 0
hx'' : ∀ (y : ℝ), b * id x ≤ y → f y ≠ 0
⊢ ∀ u ∈ Set.Icc (b * x) x, |f u|⁻¹ ∈ Set.Icc (c₂⁻¹ * |f x|⁻¹) (c₁⁻¹ * |f x|⁻¹)

tactic: intro u hu
state_before: case h
f✝ f : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf✝ : GrowsPolynomially fun x => |f x|
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|) u ∈ Set.Icc (c₁ * (fun x => |f x|) x) (c₂ * (fun x => |f x|) x)
x : ℝ
hx : ∀ u ∈ Set.Icc (b * x) x, |f u| ∈ Set.Icc (c₁ * |f x|) (c₂ * |f x|)
hx' : f x ≠ 0
hx'' : ∀ (y : ℝ), b * id x ≤ y → f y ≠ 0
⊢ ∀ u ∈ Set.Icc (b * x) x, |f u|⁻¹ ∈ Set.Icc (c₂⁻¹ * |f x|⁻¹) (c₁⁻¹ * |f x|⁻¹)
state_after: case h
f✝ f : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf✝ : GrowsPolynomially fun x => |f x|
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|) u ∈ Set.Icc (c₁ * (fun x => |f x|) x) (c₂ * (fun x => |f x|) x)
x : ℝ
hx : ∀ u ∈ Set.Icc (b * x) x, |f u| ∈ Set.Icc (c₁ * |f x|) (c₂ * |f x|)
hx' : f x ≠ 0
hx'' : ∀ (y : ℝ), b * id x ≤ y → f y ≠ 0
u : ℝ
hu : u ∈ Set.Icc (b * x) x
⊢ |f u|⁻¹ ∈ Set.Icc (c₂⁻¹ * |f x|⁻¹) (c₁⁻¹ * |f x|⁻¹)

tactic: have h₁ : 0 < |f u| := by rw [abs_pos]; exact hx'' u hu.1
state_before: case h
f✝ f : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf✝ : GrowsPolynomially fun x => |f x|
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|) u ∈ Set.Icc (c₁ * (fun x => |f x|) x) (c₂ * (fun x => |f x|) x)
x : ℝ
hx : ∀ u ∈ Set.Icc (b * x) x, |f u| ∈ Set.Icc (c₁ * |f x|) (c₂ * |f x|)
hx' : f x ≠ 0
hx'' : ∀ (y : ℝ), b * id x ≤ y → f y ≠ 0
u : ℝ
hu : u ∈ Set.Icc (b * x) x
⊢ |f u|⁻¹ ∈ Set.Icc (c₂⁻¹ * |f x|⁻¹) (c₁⁻¹ * |f x|⁻¹)
state_after: case h
f✝ f : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf✝ : GrowsPolynomially fun x => |f x|
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|) u ∈ Set.Icc (c₁ * (fun x => |f x|) x) (c₂ * (fun x => |f x|) x)
x : ℝ
hx : ∀ u ∈ Set.Icc (b * x) x, |f u| ∈ Set.Icc (c₁ * |f x|) (c₂ * |f x|)
hx' : f x ≠ 0
hx'' : ∀ (y : ℝ), b * id x ≤ y → f y ≠ 0
u : ℝ
hu : u ∈ Set.Icc (b * x) x
h₁ : 0 < |f u|
⊢ |f u|⁻¹ ∈ Set.Icc (c₂⁻¹ * |f x|⁻¹) (c₁⁻¹ * |f x|⁻¹)

tactic: refine ⟨?lb, ?ub⟩
state_before: case h
f✝ f : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf✝ : GrowsPolynomially fun x => |f x|
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|) u ∈ Set.Icc (c₁ * (fun x => |f x|) x) (c₂ * (fun x => |f x|) x)
x : ℝ
hx : ∀ u ∈ Set.Icc (b * x) x, |f u| ∈ Set.Icc (c₁ * |f x|) (c₂ * |f x|)
hx' : f x ≠ 0
hx'' : ∀ (y : ℝ), b * id x ≤ y → f y ≠ 0
u : ℝ
hu : u ∈ Set.Icc (b * x) x
h₁ : 0 < |f u|
⊢ |f u|⁻¹ ∈ Set.Icc (c₂⁻¹ * |f x|⁻¹) (c₁⁻¹ * |f x|⁻¹)
state_after: case lb
f✝ f : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf✝ : GrowsPolynomially fun x => |f x|
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|) u ∈ Set.Icc (c₁ * (fun x => |f x|) x) (c₂ * (fun x => |f x|) x)
x : ℝ
hx : ∀ u ∈ Set.Icc (b * x) x, |f u| ∈ Set.Icc (c₁ * |f x|) (c₂ * |f x|)
hx' : f x ≠ 0
hx'' : ∀ (y : ℝ), b * id x ≤ y → f y ≠ 0
u : ℝ
hu : u ∈ Set.Icc (b * x) x
h₁ : 0 < |f u|
⊢ c₂⁻¹ * |f x|⁻¹ ≤ |f u|⁻¹

case ub
f✝ f : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf✝ : GrowsPolynomially fun x => |f x|
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|) u ∈ Set.Icc (c₁ * (fun x => |f x|) x) (c₂ * (fun x => |f x|) x)
x : ℝ
hx : ∀ u ∈ Set.Icc (b * x) x, |f u| ∈ Set.Icc (c₁ * |f x|) (c₂ * |f x|)
hx' : f x ≠ 0
hx'' : ∀ (y : ℝ), b * id x ≤ y → f y ≠ 0
u : ℝ
hu : u ∈ Set.Icc (b * x) x
h₁ : 0 < |f u|
⊢ |f u|⁻¹ ≤ c₁⁻¹ * |f x|⁻¹

tactic: case lb =>
  rw [← mul_inv]
  gcongr
  exact (hx u hu).2
state_before: case lb
f✝ f : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf✝ : GrowsPolynomially fun x => |f x|
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|) u ∈ Set.Icc (c₁ * (fun x => |f x|) x) (c₂ * (fun x => |f x|) x)
x : ℝ
hx : ∀ u ∈ Set.Icc (b * x) x, |f u| ∈ Set.Icc (c₁ * |f x|) (c₂ * |f x|)
hx' : f x ≠ 0
hx'' : ∀ (y : ℝ), b * id x ≤ y → f y ≠ 0
u : ℝ
hu : u ∈ Set.Icc (b * x) x
h₁ : 0 < |f u|
⊢ c₂⁻¹ * |f x|⁻¹ ≤ |f u|⁻¹

case ub
f✝ f : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf✝ : GrowsPolynomially fun x => |f x|
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|) u ∈ Set.Icc (c₁ * (fun x => |f x|) x) (c₂ * (fun x => |f x|) x)
x : ℝ
hx : ∀ u ∈ Set.Icc (b * x) x, |f u| ∈ Set.Icc (c₁ * |f x|) (c₂ * |f x|)
hx' : f x ≠ 0
hx'' : ∀ (y : ℝ), b * id x ≤ y → f y ≠ 0
u : ℝ
hu : u ∈ Set.Icc (b * x) x
h₁ : 0 < |f u|
⊢ |f u|⁻¹ ≤ c₁⁻¹ * |f x|⁻¹
state_after: case ub
f✝ f : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf✝ : GrowsPolynomially fun x => |f x|
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|) u ∈ Set.Icc (c₁ * (fun x => |f x|) x) (c₂ * (fun x => |f x|) x)
x : ℝ
hx : ∀ u ∈ Set.Icc (b * x) x, |f u| ∈ Set.Icc (c₁ * |f x|) (c₂ * |f x|)
hx' : f x ≠ 0
hx'' : ∀ (y : ℝ), b * id x ≤ y → f y ≠ 0
u : ℝ
hu : u ∈ Set.Icc (b * x) x
h₁ : 0 < |f u|
⊢ |f u|⁻¹ ≤ c₁⁻¹ * |f x|⁻¹

tactic: case ub =>
  rw [← mul_inv]
  gcongr
  exact (hx u hu).1
state_before: case ub
f✝ f : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf✝ : GrowsPolynomially fun x => |f x|
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|) u ∈ Set.Icc (c₁ * (fun x => |f x|) x) (c₂ * (fun x => |f x|) x)
x : ℝ
hx : ∀ u ∈ Set.Icc (b * x) x, |f u| ∈ Set.Icc (c₁ * |f x|) (c₂ * |f x|)
hx' : f x ≠ 0
hx'' : ∀ (y : ℝ), b * id x ≤ y → f y ≠ 0
u : ℝ
hu : u ∈ Set.Icc (b * x) x
h₁ : 0 < |f u|
⊢ |f u|⁻¹ ≤ c₁⁻¹ * |f x|⁻¹
state_after: no goals

tactic: cases hf_pos_or_neg with
| inl hf' =>
  have hmain : (fun x => (f x)⁻¹) =ᶠ[atTop] fun x => |(f x)⁻¹| := by
    filter_upwards [hf'] with x hx₁
    rw [abs_of_nonneg (inv_nonneg_of_nonneg (le_of_lt hx₁))]
  rwa [iff_eventuallyEq hmain]
| inr hf' =>
  have hmain : (fun x => (f x)⁻¹) =ᶠ[atTop] fun x => -|(f x)⁻¹| := by
    filter_upwards [hf'] with x hx₁
    simp [abs_of_nonpos (inv_nonpos.mpr (le_of_lt hx₁))]
  rw [iff_eventuallyEq hmain]
  exact this.neg
state_before: f✝ f : ℝ → ℝ
hf : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
this : GrowsPolynomially fun x => |(f x)⁻¹|
⊢ GrowsPolynomially fun x => (f x)⁻¹
state_after: no goals

tactic: have hmain : (fun x => (f x)⁻¹) =ᶠ[atTop] fun x => |(f x)⁻¹| := by
  filter_upwards [hf'] with x hx₁
  rw [abs_of_nonneg (inv_nonneg_of_nonneg (le_of_lt hx₁))]
state_before: case inl
f✝ f : ℝ → ℝ
hf : GrowsPolynomially f
this : GrowsPolynomially fun x => |(f x)⁻¹|
hf' : ∀ᶠ (x : ℝ) in atTop, 0 < f x
⊢ GrowsPolynomially fun x => (f x)⁻¹
state_after: case inl
f✝ f : ℝ → ℝ
hf : GrowsPolynomially f
this : GrowsPolynomially fun x => |(f x)⁻¹|
hf' : ∀ᶠ (x : ℝ) in atTop, 0 < f x
hmain : (fun x => (f x)⁻¹) =ᶠ[atTop] fun x => |(f x)⁻¹|
⊢ GrowsPolynomially fun x => (f x)⁻¹

tactic: rwa [iff_eventuallyEq hmain]
state_before: case inl
f✝ f : ℝ → ℝ
hf : GrowsPolynomially f
this : GrowsPolynomially fun x => |(f x)⁻¹|
hf' : ∀ᶠ (x : ℝ) in atTop, 0 < f x
hmain : (fun x => (f x)⁻¹) =ᶠ[atTop] fun x => |(f x)⁻¹|
⊢ GrowsPolynomially fun x => (f x)⁻¹
state_after: no goals

tactic: filter_upwards [hf'] with x hx₁
state_before: f✝ f : ℝ → ℝ
hf : GrowsPolynomially f
this : GrowsPolynomially fun x => |(f x)⁻¹|
hf' : ∀ᶠ (x : ℝ) in atTop, 0 < f x
⊢ (fun x => (f x)⁻¹) =ᶠ[atTop] fun x => |(f x)⁻¹|
state_after: case h
f✝ f : ℝ → ℝ
hf : GrowsPolynomially f
this : GrowsPolynomially fun x => |(f x)⁻¹|
hf' : ∀ᶠ (x : ℝ) in atTop, 0 < f x
x : ℝ
hx₁ : 0 < f x
⊢ (f x)⁻¹ = |(f x)⁻¹|

tactic: rw [abs_of_nonneg (inv_nonneg_of_nonneg (le_of_lt hx₁))]
state_before: case h
f✝ f : ℝ → ℝ
hf : GrowsPolynomially f
this : GrowsPolynomially fun x => |(f x)⁻¹|
hf' : ∀ᶠ (x : ℝ) in atTop, 0 < f x
x : ℝ
hx₁ : 0 < f x
⊢ (f x)⁻¹ = |(f x)⁻¹|
state_after: no goals

tactic: have hmain : (fun x => (f x)⁻¹) =ᶠ[atTop] fun x => -|(f x)⁻¹| := by
  filter_upwards [hf'] with x hx₁
  simp [abs_of_nonpos (inv_nonpos.mpr (le_of_lt hx₁))]
state_before: case inr
f✝ f : ℝ → ℝ
hf : GrowsPolynomially f
this : GrowsPolynomially fun x => |(f x)⁻¹|
hf' : ∀ᶠ (x : ℝ) in atTop, f x < 0
⊢ GrowsPolynomially fun x => (f x)⁻¹
state_after: case inr
f✝ f : ℝ → ℝ
hf : GrowsPolynomially f
this : GrowsPolynomially fun x => |(f x)⁻¹|
hf' : ∀ᶠ (x : ℝ) in atTop, f x < 0
hmain : (fun x => (f x)⁻¹) =ᶠ[atTop] fun x => -|(f x)⁻¹|
⊢ GrowsPolynomially fun x => (f x)⁻¹

tactic: rw [iff_eventuallyEq hmain]
state_before: case inr
f✝ f : ℝ → ℝ
hf : GrowsPolynomially f
this : GrowsPolynomially fun x => |(f x)⁻¹|
hf' : ∀ᶠ (x : ℝ) in atTop, f x < 0
hmain : (fun x => (f x)⁻¹) =ᶠ[atTop] fun x => -|(f x)⁻¹|
⊢ GrowsPolynomially fun x => (f x)⁻¹
state_after: case inr
f✝ f : ℝ → ℝ
hf : GrowsPolynomially f
this : GrowsPolynomially fun x => |(f x)⁻¹|
hf' : ∀ᶠ (x : ℝ) in atTop, f x < 0
hmain : (fun x => (f x)⁻¹) =ᶠ[atTop] fun x => -|(f x)⁻¹|
⊢ GrowsPolynomially fun x => -|(f x)⁻¹|

tactic: exact this.neg
state_before: case inr
f✝ f : ℝ → ℝ
hf : GrowsPolynomially f
this : GrowsPolynomially fun x => |(f x)⁻¹|
hf' : ∀ᶠ (x : ℝ) in atTop, f x < 0
hmain : (fun x => (f x)⁻¹) =ᶠ[atTop] fun x => -|(f x)⁻¹|
⊢ GrowsPolynomially fun x => -|(f x)⁻¹|
state_after: no goals

tactic: filter_upwards [hf'] with x hx₁
state_before: f✝ f : ℝ → ℝ
hf : GrowsPolynomially f
this : GrowsPolynomially fun x => |(f x)⁻¹|
hf' : ∀ᶠ (x : ℝ) in atTop, f x < 0
⊢ (fun x => (f x)⁻¹) =ᶠ[atTop] fun x => -|(f x)⁻¹|
state_after: case h
f✝ f : ℝ → ℝ
hf : GrowsPolynomially f
this : GrowsPolynomially fun x => |(f x)⁻¹|
hf' : ∀ᶠ (x : ℝ) in atTop, f x < 0
x : ℝ
hx₁ : f x < 0
⊢ (f x)⁻¹ = -|(f x)⁻¹|

tactic: simp [abs_of_nonpos (inv_nonpos.mpr (le_of_lt hx₁))]
state_before: case h
f✝ f : ℝ → ℝ
hf : GrowsPolynomially f
this : GrowsPolynomially fun x => |(f x)⁻¹|
hf' : ∀ᶠ (x : ℝ) in atTop, f x < 0
x : ℝ
hx₁ : f x < 0
⊢ (f x)⁻¹ = -|(f x)⁻¹|
state_after: no goals

tactic: cases hf_pos_or_neg with
| inl H => filter_upwards [H] with _ hx; exact (ne_of_lt hx).symm
| inr H => filter_upwards [H] with _ hx; exact (ne_of_gt hx).symm
state_before: f✝ f : ℝ → ℝ
hf : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
⊢ ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
state_after: no goals

tactic: filter_upwards [H] with _ hx
state_before: case inl
f✝ f : ℝ → ℝ
hf : GrowsPolynomially f
H : ∀ᶠ (x : ℝ) in atTop, 0 < f x
⊢ ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
state_after: case h
f✝ f : ℝ → ℝ
hf : GrowsPolynomially f
H : ∀ᶠ (x : ℝ) in atTop, 0 < f x
a✝ : ℝ
hx : 0 < f a✝
⊢ f a✝ ≠ 0

tactic: exact (ne_of_lt hx).symm
state_before: case h
f✝ f : ℝ → ℝ
hf : GrowsPolynomially f
H : ∀ᶠ (x : ℝ) in atTop, 0 < f x
a✝ : ℝ
hx : 0 < f a✝
⊢ f a✝ ≠ 0
state_after: no goals

tactic: filter_upwards [H] with _ hx
state_before: case inr
f✝ f : ℝ → ℝ
hf : GrowsPolynomially f
H : ∀ᶠ (x : ℝ) in atTop, f x < 0
⊢ ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
state_after: case h
f✝ f : ℝ → ℝ
hf : GrowsPolynomially f
H : ∀ᶠ (x : ℝ) in atTop, f x < 0
a✝ : ℝ
hx : f a✝ < 0
⊢ f a✝ ≠ 0

tactic: exact (ne_of_gt hx).symm
state_before: case h
f✝ f : ℝ → ℝ
hf : GrowsPolynomially f
H : ∀ᶠ (x : ℝ) in atTop, f x < 0
a✝ : ℝ
hx : f a✝ < 0
⊢ f a✝ ≠ 0
state_after: no goals

tactic: show 0 < c₂⁻¹
state_before: f✝ f : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf✝ : GrowsPolynomially fun x => |f x|
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|) u ∈ Set.Icc (c₁ * (fun x => |f x|) x) (c₂ * (fun x => |f x|) x)
⊢ c₂⁻¹ > 0
state_after: f✝ f : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf✝ : GrowsPolynomially fun x => |f x|
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|) u ∈ Set.Icc (c₁ * (fun x => |f x|) x) (c₂ * (fun x => |f x|) x)
⊢ 0 < c₂⁻¹

tactic: positivity
state_before: f✝ f : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf✝ : GrowsPolynomially fun x => |f x|
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|) u ∈ Set.Icc (c₁ * (fun x => |f x|) x) (c₂ * (fun x => |f x|) x)
⊢ 0 < c₂⁻¹
state_after: no goals

tactic: show 0 < c₁⁻¹
state_before: f✝ f : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf✝ : GrowsPolynomially fun x => |f x|
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|) u ∈ Set.Icc (c₁ * (fun x => |f x|) x) (c₂ * (fun x => |f x|) x)
⊢ c₁⁻¹ > 0
state_after: f✝ f : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf✝ : GrowsPolynomially fun x => |f x|
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|) u ∈ Set.Icc (c₁ * (fun x => |f x|) x) (c₂ * (fun x => |f x|) x)
⊢ 0 < c₁⁻¹

tactic: positivity
state_before: f✝ f : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf✝ : GrowsPolynomially fun x => |f x|
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|) u ∈ Set.Icc (c₁ * (fun x => |f x|) x) (c₂ * (fun x => |f x|) x)
⊢ 0 < c₁⁻¹
state_after: no goals

tactic: rw [abs_pos]
state_before: f✝ f : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf✝ : GrowsPolynomially fun x => |f x|
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|) u ∈ Set.Icc (c₁ * (fun x => |f x|) x) (c₂ * (fun x => |f x|) x)
x : ℝ
hx : ∀ u ∈ Set.Icc (b * x) x, |f u| ∈ Set.Icc (c₁ * |f x|) (c₂ * |f x|)
hx' : f x ≠ 0
hx'' : ∀ (y : ℝ), b * id x ≤ y → f y ≠ 0
u : ℝ
hu : u ∈ Set.Icc (b * x) x
⊢ 0 < |f u|
state_after: f✝ f : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf✝ : GrowsPolynomially fun x => |f x|
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|) u ∈ Set.Icc (c₁ * (fun x => |f x|) x) (c₂ * (fun x => |f x|) x)
x : ℝ
hx : ∀ u ∈ Set.Icc (b * x) x, |f u| ∈ Set.Icc (c₁ * |f x|) (c₂ * |f x|)
hx' : f x ≠ 0
hx'' : ∀ (y : ℝ), b * id x ≤ y → f y ≠ 0
u : ℝ
hu : u ∈ Set.Icc (b * x) x
⊢ f u ≠ 0

tactic: exact hx'' u hu.1
state_before: f✝ f : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf✝ : GrowsPolynomially fun x => |f x|
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|) u ∈ Set.Icc (c₁ * (fun x => |f x|) x) (c₂ * (fun x => |f x|) x)
x : ℝ
hx : ∀ u ∈ Set.Icc (b * x) x, |f u| ∈ Set.Icc (c₁ * |f x|) (c₂ * |f x|)
hx' : f x ≠ 0
hx'' : ∀ (y : ℝ), b * id x ≤ y → f y ≠ 0
u : ℝ
hu : u ∈ Set.Icc (b * x) x
⊢ f u ≠ 0
state_after: no goals

tactic: rw [← mul_inv]
state_before: f✝ f : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf✝ : GrowsPolynomially fun x => |f x|
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|) u ∈ Set.Icc (c₁ * (fun x => |f x|) x) (c₂ * (fun x => |f x|) x)
x : ℝ
hx : ∀ u ∈ Set.Icc (b * x) x, |f u| ∈ Set.Icc (c₁ * |f x|) (c₂ * |f x|)
hx' : f x ≠ 0
hx'' : ∀ (y : ℝ), b * id x ≤ y → f y ≠ 0
u : ℝ
hu : u ∈ Set.Icc (b * x) x
h₁ : 0 < |f u|
⊢ c₂⁻¹ * |f x|⁻¹ ≤ |f u|⁻¹
state_after: f✝ f : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf✝ : GrowsPolynomially fun x => |f x|
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|) u ∈ Set.Icc (c₁ * (fun x => |f x|) x) (c₂ * (fun x => |f x|) x)
x : ℝ
hx : ∀ u ∈ Set.Icc (b * x) x, |f u| ∈ Set.Icc (c₁ * |f x|) (c₂ * |f x|)
hx' : f x ≠ 0
hx'' : ∀ (y : ℝ), b * id x ≤ y → f y ≠ 0
u : ℝ
hu : u ∈ Set.Icc (b * x) x
h₁ : 0 < |f u|
⊢ (c₂ * |f x|)⁻¹ ≤ |f u|⁻¹

tactic: gcongr
state_before: f✝ f : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf✝ : GrowsPolynomially fun x => |f x|
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|) u ∈ Set.Icc (c₁ * (fun x => |f x|) x) (c₂ * (fun x => |f x|) x)
x : ℝ
hx : ∀ u ∈ Set.Icc (b * x) x, |f u| ∈ Set.Icc (c₁ * |f x|) (c₂ * |f x|)
hx' : f x ≠ 0
hx'' : ∀ (y : ℝ), b * id x ≤ y → f y ≠ 0
u : ℝ
hu : u ∈ Set.Icc (b * x) x
h₁ : 0 < |f u|
⊢ (c₂ * |f x|)⁻¹ ≤ |f u|⁻¹
state_after: case h
f✝ f : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf✝ : GrowsPolynomially fun x => |f x|
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|) u ∈ Set.Icc (c₁ * (fun x => |f x|) x) (c₂ * (fun x => |f x|) x)
x : ℝ
hx : ∀ u ∈ Set.Icc (b * x) x, |f u| ∈ Set.Icc (c₁ * |f x|) (c₂ * |f x|)
hx' : f x ≠ 0
hx'' : ∀ (y : ℝ), b * id x ≤ y → f y ≠ 0
u : ℝ
hu : u ∈ Set.Icc (b * x) x
h₁ : 0 < |f u|
⊢ |f u| ≤ c₂ * |f x|

tactic: exact (hx u hu).2
state_before: case h
f✝ f : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf✝ : GrowsPolynomially fun x => |f x|
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|) u ∈ Set.Icc (c₁ * (fun x => |f x|) x) (c₂ * (fun x => |f x|) x)
x : ℝ
hx : ∀ u ∈ Set.Icc (b * x) x, |f u| ∈ Set.Icc (c₁ * |f x|) (c₂ * |f x|)
hx' : f x ≠ 0
hx'' : ∀ (y : ℝ), b * id x ≤ y → f y ≠ 0
u : ℝ
hu : u ∈ Set.Icc (b * x) x
h₁ : 0 < |f u|
⊢ |f u| ≤ c₂ * |f x|
state_after: no goals

tactic: rw [← mul_inv]
state_before: f✝ f : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf✝ : GrowsPolynomially fun x => |f x|
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|) u ∈ Set.Icc (c₁ * (fun x => |f x|) x) (c₂ * (fun x => |f x|) x)
x : ℝ
hx : ∀ u ∈ Set.Icc (b * x) x, |f u| ∈ Set.Icc (c₁ * |f x|) (c₂ * |f x|)
hx' : f x ≠ 0
hx'' : ∀ (y : ℝ), b * id x ≤ y → f y ≠ 0
u : ℝ
hu : u ∈ Set.Icc (b * x) x
h₁ : 0 < |f u|
⊢ |f u|⁻¹ ≤ c₁⁻¹ * |f x|⁻¹
state_after: f✝ f : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf✝ : GrowsPolynomially fun x => |f x|
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|) u ∈ Set.Icc (c₁ * (fun x => |f x|) x) (c₂ * (fun x => |f x|) x)
x : ℝ
hx : ∀ u ∈ Set.Icc (b * x) x, |f u| ∈ Set.Icc (c₁ * |f x|) (c₂ * |f x|)
hx' : f x ≠ 0
hx'' : ∀ (y : ℝ), b * id x ≤ y → f y ≠ 0
u : ℝ
hu : u ∈ Set.Icc (b * x) x
h₁ : 0 < |f u|
⊢ |f u|⁻¹ ≤ (c₁ * |f x|)⁻¹

tactic: gcongr
state_before: f✝ f : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf✝ : GrowsPolynomially fun x => |f x|
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|) u ∈ Set.Icc (c₁ * (fun x => |f x|) x) (c₂ * (fun x => |f x|) x)
x : ℝ
hx : ∀ u ∈ Set.Icc (b * x) x, |f u| ∈ Set.Icc (c₁ * |f x|) (c₂ * |f x|)
hx' : f x ≠ 0
hx'' : ∀ (y : ℝ), b * id x ≤ y → f y ≠ 0
u : ℝ
hu : u ∈ Set.Icc (b * x) x
h₁ : 0 < |f u|
⊢ |f u|⁻¹ ≤ (c₁ * |f x|)⁻¹
state_after: case h
f✝ f : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf✝ : GrowsPolynomially fun x => |f x|
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|) u ∈ Set.Icc (c₁ * (fun x => |f x|) x) (c₂ * (fun x => |f x|) x)
x : ℝ
hx : ∀ u ∈ Set.Icc (b * x) x, |f u| ∈ Set.Icc (c₁ * |f x|) (c₂ * |f x|)
hx' : f x ≠ 0
hx'' : ∀ (y : ℝ), b * id x ≤ y → f y ≠ 0
u : ℝ
hu : u ∈ Set.Icc (b * x) x
h₁ : 0 < |f u|
⊢ c₁ * |f x| ≤ |f u|

tactic: exact (hx u hu).1
state_before: case h
f✝ f : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf✝ : GrowsPolynomially fun x => |f x|
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|) u ∈ Set.Icc (c₁ * (fun x => |f x|) x) (c₂ * (fun x => |f x|) x)
x : ℝ
hx : ∀ u ∈ Set.Icc (b * x) x, |f u| ∈ Set.Icc (c₁ * |f x|) (c₂ * |f x|)
hx' : f x ≠ 0
hx'' : ∀ (y : ℝ), b * id x ≤ y → f y ≠ 0
u : ℝ
hu : u ∈ Set.Icc (b * x) x
h₁ : 0 < |f u|
⊢ c₁ * |f x| ≤ |f u|
state_after: no goals