-- Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean
lemma GrowsPolynomially.of_isTheta {f g : ℝ → ℝ} (hg : GrowsPolynomially g) (hf : f =Θ[atTop] g)
    (hf' : ∀ᶠ x in atTop, 0 ≤ f x) : GrowsPolynomially f

Proof state:
f g : ℝ → ℝ
hg✝ : GrowsPolynomially g
hf : f =Θ[atTop] g
hf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_pos : 0 < c₁
hf_lb : ∀ᶠ (x : ℝ) in atTop, c₁ * ‖g x‖ ≤ ‖f x‖
c₂ : ℝ
hc₂_pos : 0 < c₂
hf_ub : ∀ᶠ (x : ℝ) in atTop, ‖f x‖ ≤ c₂ * ‖g x‖
c₃ : ℝ
hc₃_pos : 0 < c₃
c₄ : ℝ
hc₄_pos : 0 < c₄
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)
h_lb_pos : 0 < c₁ * c₂⁻¹ * c₃
h_ub_pos : 0 < c₂ * c₄ * c₁⁻¹
c₂_cancel : c₂⁻¹ * c₂ = 1
c₁_cancel : c₁⁻¹ * c₁ = 1
x : ℝ
hf_pos : ∀ (y : ℝ), b * id x ≤ y → 0 ≤ f y
h_lb : ∀ (y : ℝ), b * id x ≤ y → c₁ * ‖g y‖ ≤ ‖f y‖
h_ub : ∀ (y : ℝ), b * id x ≤ y → ‖f y‖ ≤ c₂ * ‖g y‖
hg_bound✝ : ∀ (y : ℝ), b * id x ≤ y → ∀ u ∈ Set.Icc (b * y) y, ‖g u‖ ∈ Set.Icc (c₃ * ‖g y‖) (c₄ * ‖g y‖)
hx_pos : 0 ≤ x
u : ℝ
hu : u ∈ Set.Icc (b * x) x
hbx : b * x ≤ x
hg_bound : ∀ u ∈ Set.Icc (b * x) x, ‖g u‖ ∈ Set.Icc (c₃ * ‖g x‖) (c₄ * ‖g x‖)
⊢ c₁ * c₂⁻¹ * c₃ * f x ≤ f u

Tactic:
calc
c₁ * c₂⁻¹ * c₃ * f x ≤ c₁ * c₂⁻¹ * c₃ * (c₂ * ‖g x‖) := by
rw [← Real.norm_of_nonneg (hf_pos x hbx)]; gcongr; exact h_ub x hbx
_ = (c₂⁻¹ * c₂) * c₁ * (c₃ * ‖g x‖) := by ring
_ = c₁ * (c₃ * ‖g x‖) := by simp [c₂_cancel]
_ ≤ c₁ * ‖g u‖ := by gcongr; exact (hg_bound u hu).1
_ ≤ f u := by
rw [← Real.norm_of_nonneg (hf_pos u hu.1)]
exact h_lb u hu.1

Result:
no goals

Proof state:
f g : ℝ → ℝ
hg : GrowsPolynomially g
hf : f =Θ[atTop] g
hf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
hf_lb : ∃ c > 0, ∀ᶠ (x : ℝ) in atTop, c * ‖g x‖ ≤ ‖f x‖
hf_ub : ∃ c > 0, ∀ᶠ (x : ℝ) in atTop, ‖f x‖ ≤ c * ‖g x‖
⊢ ∃ c₁ > 0, ∃ c₂ > 0, ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)

Tactic:
obtain ⟨c₁, hc₁_pos : 0 < c₁, hf_lb⟩ := hf_lb

Result:
case intro.intro
f g : ℝ → ℝ
hg : GrowsPolynomially g
hf : f =Θ[atTop] g
hf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
hf_ub : ∃ c > 0, ∀ᶠ (x : ℝ) in atTop, ‖f x‖ ≤ c * ‖g x‖
c₁ : ℝ
hc₁_pos : 0 < c₁
hf_lb : ∀ᶠ (x : ℝ) in atTop, c₁ * ‖g x‖ ≤ ‖f x‖
⊢ ∃ c₁ > 0, ∃ c₂ > 0, ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)

Proof state:
case intro.intro.intro.intro
f g : ℝ → ℝ
hg✝ : GrowsPolynomially g
hf : f =Θ[atTop] g
hf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_pos : 0 < c₁
hf_lb : ∀ᶠ (x : ℝ) in atTop, c₁ * ‖g x‖ ≤ ‖f x‖
c₂ : ℝ
hc₂_pos : 0 < c₂
hf_ub : ∀ᶠ (x : ℝ) in atTop, ‖f x‖ ≤ c₂ * ‖g 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, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)

Tactic:
obtain ⟨c₃, hc₃_pos : 0 < c₃, hg⟩ := hg

Result:
case intro.intro.intro.intro.intro.intro
f g : ℝ → ℝ
hg✝ : GrowsPolynomially g
hf : f =Θ[atTop] g
hf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_pos : 0 < c₁
hf_lb : ∀ᶠ (x : ℝ) in atTop, c₁ * ‖g x‖ ≤ ‖f x‖
c₂ : ℝ
hc₂_pos : 0 < c₂
hf_ub : ∀ᶠ (x : ℝ) in atTop, ‖f x‖ ≤ c₂ * ‖g x‖
c₃ : ℝ
hc₃_pos : 0 < c₃
hg :
  ∃ c₂ > 0,
    ∀ᶠ (x : ℝ) in atTop,
      ∀ u ∈ Set.Icc (b * x) x, (fun x => ‖g x‖) u ∈ Set.Icc (c₃ * (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, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)

Proof state:
f g : ℝ → ℝ
hg✝ : GrowsPolynomially g
hf : f =Θ[atTop] g
hf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_pos : 0 < c₁
hf_lb : ∀ᶠ (x : ℝ) in atTop, c₁ * ‖g x‖ ≤ ‖f x‖
c₂ : ℝ
hc₂_pos : 0 < c₂
hf_ub : ∀ᶠ (x : ℝ) in atTop, ‖f x‖ ≤ c₂ * ‖g x‖
c₃ : ℝ
hc₃_pos : 0 < c₃
c₄ : ℝ
hc₄_pos : 0 < c₄
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)
h_lb_pos : 0 < c₁ * c₂⁻¹ * c₃
h_ub_pos : 0 < c₂ * c₄ * c₁⁻¹
c₂_cancel : c₂⁻¹ * c₂ = 1
c₁_cancel : c₁⁻¹ * c₁ = 1
x : ℝ
hf_pos : ∀ (y : ℝ), b * id x ≤ y → 0 ≤ f y
h_lb : ∀ (y : ℝ), b * id x ≤ y → c₁ * ‖g y‖ ≤ ‖f y‖
h_ub : ∀ (y : ℝ), b * id x ≤ y → ‖f y‖ ≤ c₂ * ‖g y‖
hg_bound✝ : ∀ (y : ℝ), b * id x ≤ y → ∀ u ∈ Set.Icc (b * y) y, ‖g u‖ ∈ Set.Icc (c₃ * ‖g y‖) (c₄ * ‖g y‖)
hx_pos : 0 ≤ x
u : ℝ
hu : u ∈ Set.Icc (b * x) x
hbx : b * x ≤ x
hg_bound : ∀ u ∈ Set.Icc (b * x) x, ‖g u‖ ∈ Set.Icc (c₃ * ‖g x‖) (c₄ * ‖g x‖)
⊢ ‖f u‖ ≤ c₂ * ‖g u‖

Tactic:
exact h_ub u hu.1

Result:
no goals

Proof state:
f g : ℝ → ℝ
hg : GrowsPolynomially g
hf : f =Θ[atTop] g
hf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ 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, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)

Tactic:
have hf_lb := isBigO_iff''.mp hf.isBigO_symm

Result:
f g : ℝ → ℝ
hg : GrowsPolynomially g
hf : f =Θ[atTop] g
hf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
hf_lb : ∃ c > 0, ∀ᶠ (x : ℝ) in atTop, c * ‖g x‖ ≤ ‖f x‖
⊢ ∃ c₁ > 0, ∃ c₂ > 0, ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)

Proof state:
f g : ℝ → ℝ
hg✝ : GrowsPolynomially g
hf : f =Θ[atTop] g
hf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_pos : 0 < c₁
hf_lb : ∀ᶠ (x : ℝ) in atTop, c₁ * ‖g x‖ ≤ ‖f x‖
c₂ : ℝ
hc₂_pos : 0 < c₂
hf_ub : ∀ᶠ (x : ℝ) in atTop, ‖f x‖ ≤ c₂ * ‖g x‖
c₃ : ℝ
hc₃_pos : 0 < c₃
c₄ : ℝ
hc₄_pos : 0 < c₄
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)
h_lb_pos : 0 < c₁ * c₂⁻¹ * c₃
h_ub_pos : 0 < c₂ * c₄ * c₁⁻¹
c₂_cancel : c₂⁻¹ * c₂ = 1
c₁_cancel : c₁⁻¹ * c₁ = 1
x : ℝ
hf_pos : ∀ (y : ℝ), b * id x ≤ y → 0 ≤ f y
h_lb : ∀ (y : ℝ), b * id x ≤ y → c₁ * ‖g y‖ ≤ ‖f y‖
h_ub : ∀ (y : ℝ), b * id x ≤ y → ‖f y‖ ≤ c₂ * ‖g y‖
hg_bound✝ : ∀ (y : ℝ), b * id x ≤ y → ∀ u ∈ Set.Icc (b * y) y, ‖g u‖ ∈ Set.Icc (c₃ * ‖g y‖) (c₄ * ‖g y‖)
hx_pos : 0 ≤ x
u : ℝ
hu : u ∈ Set.Icc (b * x) x
hbx : b * x ≤ x
hg_bound : ∀ u ∈ Set.Icc (b * x) x, ‖g u‖ ∈ Set.Icc (c₃ * ‖g x‖) (c₄ * ‖g x‖)
⊢ c₁ * ‖g u‖ ≤ ‖f u‖

Tactic:
exact h_lb u hu.1

Result:
no goals

Proof state:
case h
f g : ℝ → ℝ
hg✝ : GrowsPolynomially g
hf : f =Θ[atTop] g
hf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_pos : 0 < c₁
hf_lb : ∀ᶠ (x : ℝ) in atTop, c₁ * ‖g x‖ ≤ ‖f x‖
c₂ : ℝ
hc₂_pos : 0 < c₂
hf_ub : ∀ᶠ (x : ℝ) in atTop, ‖f x‖ ≤ c₂ * ‖g x‖
c₃ : ℝ
hc₃_pos : 0 < c₃
c₄ : ℝ
hc₄_pos : 0 < c₄
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)
h_lb_pos : 0 < c₁ * c₂⁻¹ * c₃
h_ub_pos : 0 < c₂ * c₄ * c₁⁻¹
c₂_cancel : c₂⁻¹ * c₂ = 1
c₁_cancel : c₁⁻¹ * c₁ = 1
x : ℝ
hf_pos : ∀ (y : ℝ), b * id x ≤ y → 0 ≤ f y
h_lb : ∀ (y : ℝ), b * id x ≤ y → c₁ * ‖g y‖ ≤ ‖f y‖
h_ub : ∀ (y : ℝ), b * id x ≤ y → ‖f y‖ ≤ c₂ * ‖g y‖
hg_bound✝ : ∀ (y : ℝ), b * id x ≤ y → ∀ u ∈ Set.Icc (b * y) y, ‖g u‖ ∈ Set.Icc (c₃ * ‖g y‖) (c₄ * ‖g y‖)
hx_pos : 0 ≤ x
u : ℝ
hu : u ∈ Set.Icc (b * x) x
hbx : b * x ≤ x
hg_bound : ∀ u ∈ Set.Icc (b * x) x, ‖g u‖ ∈ Set.Icc (c₃ * ‖g x‖) (c₄ * ‖g x‖)
⊢ ‖g u‖ ≤ c₄ * ‖g x‖

Tactic:
exact (hg_bound u hu).2

Result:
no goals