case pos
𝕜 : Type u_1
inst✝¹ : NormedLinearOrderedField 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hP : P = 0
⊢ (fun x => eval x P / eval x Q) ~[atTop] fun x =>
    leadingCoeff P / leadingCoeff Q * x ^ (↑(natDegree P) - ↑(natDegree Q))

case neg
𝕜 : Type u_1
inst✝¹ : NormedLinearOrderedField 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hP : ¬P = 0
⊢ (fun x => eval x P / eval x Q) ~[atTop] fun x =>
    leadingCoeff P / leadingCoeff Q * x ^ (↑(natDegree P) - ↑(natDegree Q))
𝕜 : Type u_1
inst✝¹ : NormedLinearOrderedField 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
⊢ (fun x => eval x P / eval x Q) ~[atTop] fun x =>
    leadingCoeff P / leadingCoeff Q * x ^ (↑(natDegree P) - ↑(natDegree Q))
by_cases hP : P = 0
case pos
𝕜 : Type u_1
inst✝¹ : NormedLinearOrderedField 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hP : ¬P = 0
hQ : Q = 0
⊢ (fun x => eval x P / eval x Q) ~[atTop] fun x =>
    leadingCoeff P / leadingCoeff Q * x ^ (↑(natDegree P) - ↑(natDegree Q))

case neg
𝕜 : Type u_1
inst✝¹ : NormedLinearOrderedField 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hP : ¬P = 0
hQ : ¬Q = 0
⊢ (fun x => eval x P / eval x Q) ~[atTop] fun x =>
    leadingCoeff P / leadingCoeff Q * x ^ (↑(natDegree P) - ↑(natDegree Q))
case neg
𝕜 : Type u_1
inst✝¹ : NormedLinearOrderedField 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hP : ¬P = 0
⊢ (fun x => eval x P / eval x Q) ~[atTop] fun x =>
    leadingCoeff P / leadingCoeff Q * x ^ (↑(natDegree P) - ↑(natDegree Q))
by_cases hQ : Q = 0
case neg
𝕜 : Type u_1
inst✝¹ : NormedLinearOrderedField 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hP : ¬P = 0
hQ : ¬Q = 0
x : 𝕜
hx : 0 < x
⊢ (fun x => leadingCoeff P * x ^ natDegree P / (leadingCoeff Q * x ^ natDegree Q)) x =
    (fun x => leadingCoeff P / leadingCoeff Q * x ^ (↑(natDegree P) - ↑(natDegree Q))) x
case neg
𝕜 : Type u_1
inst✝¹ : NormedLinearOrderedField 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hP : ¬P = 0
hQ : ¬Q = 0
⊢ (fun x => eval x P / eval x Q) ~[atTop] fun x =>
    leadingCoeff P / leadingCoeff Q * x ^ (↑(natDegree P) - ↑(natDegree Q))
refine'
  (P.isEquivalent_atTop_lead.symm.div Q.isEquivalent_atTop_lead.symm).symm.trans
    (EventuallyEq.isEquivalent ((eventually_gt_atTop 0).mono fun x hx => _))
no goals
case neg
𝕜 : Type u_1
inst✝¹ : NormedLinearOrderedField 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hP : ¬P = 0
hQ : ¬Q = 0
x : 𝕜
hx : 0 < x
⊢ (fun x => leadingCoeff P * x ^ natDegree P / (leadingCoeff Q * x ^ natDegree Q)) x =
    (fun x => leadingCoeff P / leadingCoeff Q * x ^ (↑(natDegree P) - ↑(natDegree Q))) x
simp [← div_mul_div_comm, hP, hQ, zpow_sub₀ hx.ne.symm]
no goals
case pos
𝕜 : Type u_1
inst✝¹ : NormedLinearOrderedField 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hP : P = 0
⊢ (fun x => eval x P / eval x Q) ~[atTop] fun x =>
    leadingCoeff P / leadingCoeff Q * x ^ (↑(natDegree P) - ↑(natDegree Q))
simp [hP, IsEquivalent.refl]
no goals
case pos
𝕜 : Type u_1
inst✝¹ : NormedLinearOrderedField 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hP : ¬P = 0
hQ : Q = 0
⊢ (fun x => eval x P / eval x Q) ~[atTop] fun x =>
    leadingCoeff P / leadingCoeff Q * x ^ (↑(natDegree P) - ↑(natDegree Q))
simp [hQ, IsEquivalent.refl]