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

Proof steps:
state_before: f✝ f g : ℝ → ℝ
hf : GrowsPolynomially f
hg : GrowsPolynomially g
⊢ GrowsPolynomially fun x => f x * g x
state_after: f✝ f g : ℝ → ℝ
hf : GrowsPolynomially f
hg : GrowsPolynomially g
⊢ GrowsPolynomially fun x => |f x| * |g x|
tactic: suffices GrowsPolynomially fun x => |f x| * |g x| by
    cases eventually_atTop_nonneg_or_nonpos hf with
    | inl hf' =>
      cases eventually_atTop_nonneg_or_nonpos hg with
      | inl hg' =>
        have hmain : (fun x => f x * g x) =ᶠ[atTop] fun x => |f x| * |g x| := by
          filter_upwards [hf', hg'] with x hx₁ hx₂
          rw [abs_of_nonneg hx₁, abs_of_nonneg hx₂]
        rwa [iff_eventuallyEq hmain]
      | inr hg' =>
        have hmain : (fun x => f x * g x) =ᶠ[atTop] fun x => -|f x| * |g x| := by
          filter_upwards [hf', hg'] with x hx₁ hx₂
          simp [abs_of_nonneg hx₁, abs_of_nonpos hx₂]
        simp only [iff_eventuallyEq hmain, neg_mul]
        exact this.neg
    | inr hf' =>
      cases eventually_atTop_nonneg_or_nonpos hg with
      | inl hg' =>
        have hmain : (fun x => f x * g x) =ᶠ[atTop] fun x => -|f x| * |g x| := by
          filter_upwards [hf', hg'] with x hx₁ hx₂
          rw [abs_of_nonpos hx₁, abs_of_nonneg hx₂, neg_neg]
        simp only [iff_eventuallyEq hmain, neg_mul]
        exact this.neg
      | inr hg' =>
        have hmain : (fun x => f x * g x) =ᶠ[atTop] fun x => |f x| * |g x| := by
          filter_upwards [hf', hg'] with x hx₁ hx₂
          simp [abs_of_nonpos hx₁, abs_of_nonpos hx₂]
        simp only [iff_eventuallyEq hmain, neg_mul]
        exact this
state_before: f✝ f g : ℝ → ℝ
hf : GrowsPolynomially f
hg : GrowsPolynomially g
⊢ GrowsPolynomially fun x => |f x| * |g x|
state_after: f✝ f g : ℝ → ℝ
hf : GrowsPolynomially f
hg : GrowsPolynomially g
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| * |g x|) u ∈ Set.Icc (c₁ * (fun x => |f x| * |g x|) x) (c₂ * (fun x => |f x| * |g x|) x)
tactic: intro b hb
state_before: f✝ f g : ℝ → ℝ
hf : GrowsPolynomially f
hg : GrowsPolynomially g
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| * |g x|) u ∈ Set.Icc (c₁ * (fun x => |f x| * |g x|) x) (c₂ * (fun x => |f x| * |g x|) x)
state_after: f✝ f g : ℝ → ℝ
hf✝ : GrowsPolynomially f
hg : GrowsPolynomially g
b : ℝ
hb : b ∈ Set.Ioo 0 1
hf :
  ∃ c₁ > 0,
    ∃ c₂ > 0,
      ∀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| * |g x|) u ∈ Set.Icc (c₁ * (fun x => |f x| * |g x|) x) (c₂ * (fun x => |f x| * |g x|) x)
tactic: have hf := hf.abs b hb
state_before: f✝ f g : ℝ → ℝ
hf✝ : GrowsPolynomially f
hg : GrowsPolynomially g
b : ℝ
hb : b ∈ Set.Ioo 0 1
hf :
  ∃ c₁ > 0,
    ∃ c₂ > 0,
      ∀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| * |g x|) u ∈ Set.Icc (c₁ * (fun x => |f x| * |g x|) x) (c₂ * (fun x => |f x| * |g x|) x)
state_after: f✝ f g : ℝ → ℝ
hf✝ : GrowsPolynomially f
hg✝ : GrowsPolynomially g
b : ℝ
hb : b ∈ Set.Ioo 0 1
hf :
  ∃ c₁ > 0,
    ∃ c₂ > 0,
      ∀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)
hg :
  ∃ c₁ > 0,
    ∃ c₂ > 0,
      ∀ᶠ (x : ℝ) in atTop,
        ∀ u ∈ Set.Icc (b * x) x, (fun x => |g x|) u ∈ Set.Icc (c₁ * (fun x => |g x|) x) (c₂ * (fun x => |g x|) x)
⊢ ∃ c₁ > 0,
    ∃ c₂ > 0,
      ∀ᶠ (x : ℝ) in atTop,
        ∀ u ∈ Set.Icc (b * x) x,
          (fun x => |f x| * |g x|) u ∈ Set.Icc (c₁ * (fun x => |f x| * |g x|) x) (c₂ * (fun x => |f x| * |g x|) x)
tactic: have hg := hg.abs b hb
state_before: f✝ f g : ℝ → ℝ
hf✝ : GrowsPolynomially f
hg✝ : GrowsPolynomially g
b : ℝ
hb : b ∈ Set.Ioo 0 1
hf :
  ∃ c₁ > 0,
    ∃ c₂ > 0,
      ∀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)
hg :
  ∃ c₁ > 0,
    ∃ c₂ > 0,
      ∀ᶠ (x : ℝ) in atTop,
        ∀ u ∈ Set.Icc (b * x) x, (fun x => |g x|) u ∈ Set.Icc (c₁ * (fun x => |g x|) x) (c₂ * (fun x => |g x|) x)
⊢ ∃ c₁ > 0,
    ∃ c₂ > 0,
      ∀ᶠ (x : ℝ) in atTop,
        ∀ u ∈ Set.Icc (b * x) x,
          (fun x => |f x| * |g x|) u ∈ Set.Icc (c₁ * (fun x => |f x| * |g x|) x) (c₂ * (fun x => |f x| * |g x|) x)
state_after: case intro.intro.intro.intro
f✝ f g : ℝ → ℝ
hf✝ : GrowsPolynomially f
hg✝ : GrowsPolynomially g
b : ℝ
hb : b ∈ Set.Ioo 0 1
hg :
  ∃ c₁ > 0,
    ∃ c₂ > 0,
      ∀ᶠ (x : ℝ) in atTop,
        ∀ u ∈ Set.Icc (b * x) x, (fun x => |g x|) u ∈ Set.Icc (c₁ * (fun x => |g x|) x) (c₂ * (fun x => |g x|) x)
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| * |g x|) u ∈ Set.Icc (c₁ * (fun x => |f x| * |g x|) x) (c₂ * (fun x => |f x| * |g x|) x)
tactic: obtain ⟨c₁, hc₁_mem, c₂, hc₂_mem, hf⟩ := hf
state_before: case intro.intro.intro.intro
f✝ f g : ℝ → ℝ
hf✝ : GrowsPolynomially f
hg✝ : GrowsPolynomially g
b : ℝ
hb : b ∈ Set.Ioo 0 1
hg :
  ∃ c₁ > 0,
    ∃ c₂ > 0,
      ∀ᶠ (x : ℝ) in atTop,
        ∀ u ∈ Set.Icc (b * x) x, (fun x => |g x|) u ∈ Set.Icc (c₁ * (fun x => |g x|) x) (c₂ * (fun x => |g x|) x)
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| * |g x|) u ∈ Set.Icc (c₁ * (fun x => |f x| * |g x|) x) (c₂ * (fun x => |f x| * |g x|) x)
state_after: case intro.intro.intro.intro.intro.intro.intro.intro
f✝ f g : ℝ → ℝ
hf✝ : GrowsPolynomially f
hg✝ : GrowsPolynomially g
b : ℝ
hb : b ∈ Set.Ioo 0 1
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)
c₃ : ℝ
hc₃_mem : c₃ > 0
c₄ : ℝ
hc₄_mem : c₄ > 0
hg :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |g x|) u ∈ Set.Icc (c₃ * (fun x => |g x|) x) (c₄ * (fun x => |g x|) x)
⊢ ∃ c₁ > 0,
    ∃ c₂ > 0,
      ∀ᶠ (x : ℝ) in atTop,
        ∀ u ∈ Set.Icc (b * x) x,
          (fun x => |f x| * |g x|) u ∈ Set.Icc (c₁ * (fun x => |f x| * |g x|) x) (c₂ * (fun x => |f x| * |g x|) x)
tactic: obtain ⟨c₃, hc₃_mem, c₄, hc₄_mem, hg⟩ := hg
state_before: case intro.intro.intro.intro.intro.intro.intro.intro
f✝ f g : ℝ → ℝ
hf✝ : GrowsPolynomially f
hg✝ : GrowsPolynomially g
b : ℝ
hb : b ∈ Set.Ioo 0 1
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)
c₃ : ℝ
hc₃_mem : c₃ > 0
c₄ : ℝ
hc₄_mem : c₄ > 0
hg :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |g x|) u ∈ Set.Icc (c₃ * (fun x => |g x|) x) (c₄ * (fun x => |g x|) x)
⊢ ∃ c₁ > 0,
    ∃ c₂ > 0,
      ∀ᶠ (x : ℝ) in atTop,
        ∀ u ∈ Set.Icc (b * x) x,
          (fun x => |f x| * |g x|) u ∈ Set.Icc (c₁ * (fun x => |f x| * |g x|) x) (c₂ * (fun x => |f x| * |g x|) x)
state_after: case intro.intro.intro.intro.intro.intro.intro.intro
f✝ f g : ℝ → ℝ
hf✝ : GrowsPolynomially f
hg✝ : GrowsPolynomially g
b : ℝ
hb : b ∈ Set.Ioo 0 1
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)
c₃ : ℝ
hc₃_mem : c₃ > 0
c₄ : ℝ
hc₄_mem : c₄ > 0
hg :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |g x|) u ∈ Set.Icc (c₃ * (fun x => |g x|) x) (c₄ * (fun x => |g x|) x)
⊢ ∃ c₂ > 0,
    ∀ᶠ (x : ℝ) in atTop,
      ∀ u ∈ Set.Icc (b * x) x,
        (fun x => |f x| * |g x|) u ∈ Set.Icc (c₁ * c₃ * (fun x => |f x| * |g x|) x) (c₂ * (fun x => |f x| * |g x|) x)
tactic: refine ⟨c₁ * c₃, by show 0 < c₁ * c₃; positivity, ?_⟩
state_before: case intro.intro.intro.intro.intro.intro.intro.intro
f✝ f g : ℝ → ℝ
hf✝ : GrowsPolynomially f
hg✝ : GrowsPolynomially g
b : ℝ
hb : b ∈ Set.Ioo 0 1
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)
c₃ : ℝ
hc₃_mem : c₃ > 0
c₄ : ℝ
hc₄_mem : c₄ > 0
hg :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |g x|) u ∈ Set.Icc (c₃ * (fun x => |g x|) x) (c₄ * (fun x => |g x|) x)
⊢ ∃ c₂ > 0,
    ∀ᶠ (x : ℝ) in atTop,
      ∀ u ∈ Set.Icc (b * x) x,
        (fun x => |f x| * |g x|) u ∈ Set.Icc (c₁ * c₃ * (fun x => |f x| * |g x|) x) (c₂ * (fun x => |f x| * |g x|) x)
state_after: case intro.intro.intro.intro.intro.intro.intro.intro
f✝ f g : ℝ → ℝ
hf✝ : GrowsPolynomially f
hg✝ : GrowsPolynomially g
b : ℝ
hb : b ∈ Set.Ioo 0 1
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)
c₃ : ℝ
hc₃_mem : c₃ > 0
c₄ : ℝ
hc₄_mem : c₄ > 0
hg :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |g x|) u ∈ Set.Icc (c₃ * (fun x => |g x|) x) (c₄ * (fun x => |g x|) x)
⊢ ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x,
      (fun x => |f x| * |g x|) u ∈ Set.Icc (c₁ * c₃ * (fun x => |f x| * |g x|) x) (c₂ * c₄ * (fun x => |f x| * |g x|) x)
tactic: refine ⟨c₂ * c₄, by show 0 < c₂ * c₄; positivity, ?_⟩
state_before: case intro.intro.intro.intro.intro.intro.intro.intro
f✝ f g : ℝ → ℝ
hf✝ : GrowsPolynomially f
hg✝ : GrowsPolynomially g
b : ℝ
hb : b ∈ Set.Ioo 0 1
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)
c₃ : ℝ
hc₃_mem : c₃ > 0
c₄ : ℝ
hc₄_mem : c₄ > 0
hg :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |g x|) u ∈ Set.Icc (c₃ * (fun x => |g x|) x) (c₄ * (fun x => |g x|) x)
⊢ ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x,
      (fun x => |f x| * |g x|) u ∈ Set.Icc (c₁ * c₃ * (fun x => |f x| * |g x|) x) (c₂ * c₄ * (fun x => |f x| * |g x|) x)
state_after: case h
f✝ f g : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hg✝¹ : GrowsPolynomially g
b : ℝ
hb : b ∈ Set.Ioo 0 1
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)
c₃ : ℝ
hc₃_mem : c₃ > 0
c₄ : ℝ
hc₄_mem : c₄ > 0
hg✝ :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |g x|) u ∈ Set.Icc (c₃ * (fun x => |g x|) x) (c₄ * (fun x => |g x|) x)
x : ℝ
hf : ∀ u ∈ Set.Icc (b * x) x, |f u| ∈ Set.Icc (c₁ * |f x|) (c₂ * |f x|)
hg : ∀ u ∈ Set.Icc (b * x) x, |g u| ∈ Set.Icc (c₃ * |g x|) (c₄ * |g x|)
⊢ ∀ u ∈ Set.Icc (b * x) x, |f u| * |g u| ∈ Set.Icc (c₁ * c₃ * (|f x| * |g x|)) (c₂ * c₄ * (|f x| * |g x|))
tactic: filter_upwards [hf, hg] with x hf hg
state_before: case h
f✝ f g : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hg✝¹ : GrowsPolynomially g
b : ℝ
hb : b ∈ Set.Ioo 0 1
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)
c₃ : ℝ
hc₃_mem : c₃ > 0
c₄ : ℝ
hc₄_mem : c₄ > 0
hg✝ :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |g x|) u ∈ Set.Icc (c₃ * (fun x => |g x|) x) (c₄ * (fun x => |g x|) x)
x : ℝ
hf : ∀ u ∈ Set.Icc (b * x) x, |f u| ∈ Set.Icc (c₁ * |f x|) (c₂ * |f x|)
hg : ∀ u ∈ Set.Icc (b * x) x, |g u| ∈ Set.Icc (c₃ * |g x|) (c₄ * |g x|)
⊢ ∀ u ∈ Set.Icc (b * x) x, |f u| * |g u| ∈ Set.Icc (c₁ * c₃ * (|f x| * |g x|)) (c₂ * c₄ * (|f x| * |g x|))
state_after: case h
f✝ f g : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hg✝¹ : GrowsPolynomially g
b : ℝ
hb : b ∈ Set.Ioo 0 1
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)
c₃ : ℝ
hc₃_mem : c₃ > 0
c₄ : ℝ
hc₄_mem : c₄ > 0
hg✝ :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |g x|) u ∈ Set.Icc (c₃ * (fun x => |g x|) x) (c₄ * (fun x => |g x|) x)
x : ℝ
hf : ∀ u ∈ Set.Icc (b * x) x, |f u| ∈ Set.Icc (c₁ * |f x|) (c₂ * |f x|)
hg : ∀ u ∈ Set.Icc (b * x) x, |g u| ∈ Set.Icc (c₃ * |g x|) (c₄ * |g x|)
u : ℝ
hu : u ∈ Set.Icc (b * x) x
⊢ |f u| * |g u| ∈ Set.Icc (c₁ * c₃ * (|f x| * |g x|)) (c₂ * c₄ * (|f x| * |g x|))
tactic: intro u hu
state_before: case h
f✝ f g : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hg✝¹ : GrowsPolynomially g
b : ℝ
hb : b ∈ Set.Ioo 0 1
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)
c₃ : ℝ
hc₃_mem : c₃ > 0
c₄ : ℝ
hc₄_mem : c₄ > 0
hg✝ :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |g x|) u ∈ Set.Icc (c₃ * (fun x => |g x|) x) (c₄ * (fun x => |g x|) x)
x : ℝ
hf : ∀ u ∈ Set.Icc (b * x) x, |f u| ∈ Set.Icc (c₁ * |f x|) (c₂ * |f x|)
hg : ∀ u ∈ Set.Icc (b * x) x, |g u| ∈ Set.Icc (c₃ * |g x|) (c₄ * |g x|)
u : ℝ
hu : u ∈ Set.Icc (b * x) x
⊢ |f u| * |g u| ∈ Set.Icc (c₁ * c₃ * (|f x| * |g x|)) (c₂ * c₄ * (|f x| * |g x|))
state_after: case lb
f✝ f g : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hg✝¹ : GrowsPolynomially g
b : ℝ
hb : b ∈ Set.Ioo 0 1
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)
c₃ : ℝ
hc₃_mem : c₃ > 0
c₄ : ℝ
hc₄_mem : c₄ > 0
hg✝ :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |g x|) u ∈ Set.Icc (c₃ * (fun x => |g x|) x) (c₄ * (fun x => |g x|) x)
x : ℝ
hf : ∀ u ∈ Set.Icc (b * x) x, |f u| ∈ Set.Icc (c₁ * |f x|) (c₂ * |f x|)
hg : ∀ u ∈ Set.Icc (b * x) x, |g u| ∈ Set.Icc (c₃ * |g x|) (c₄ * |g x|)
u : ℝ
hu : u ∈ Set.Icc (b * x) x
⊢ c₁ * c₃ * (|f x| * |g x|) ≤ |f u| * |g u|

case ub
f✝ f g : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hg✝¹ : GrowsPolynomially g
b : ℝ
hb : b ∈ Set.Ioo 0 1
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)
c₃ : ℝ
hc₃_mem : c₃ > 0
c₄ : ℝ
hc₄_mem : c₄ > 0
hg✝ :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => |g x|) u ∈ Set.Icc (c₃ * (fun x => |g x|) x) (c₄ * (fun x => |g x|) x)
x : ℝ
hf : ∀ u ∈ Set.Icc (b * x) x, |f u| ∈ Set.Icc (c₁ * |f x|) (c₂ * |f x|)
hg : ∀ u ∈ Set.Icc (b * x) x, |g u| ∈ Set.Icc (c₃ * |g x|) (c₄ * |g x|)
u : ℝ
hu : u ∈ Set.Icc (b * x) x
⊢ |f u| * |g u| ≤ c₂ * c₄ * (|f x| * |g x|)
tactic: refine ⟨?lb, ?ub⟩