Theorem: Polynomial.div_tendsto_atTop_of_degree_gt'
File: Mathlib/Analysis/SpecialFunctions/Polynomials.lean

tactic: have hQ : Q ≠ 0 := fun h => by
  simp only [h, div_zero, leadingCoeff_zero] at hpos
  exact hpos.false
state_before: 𝕜 : Type u_1
inst✝¹ : NormedLinearOrderedField 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : Q.degree < P.degree
hpos : 0 < P.leadingCoeff / Q.leadingCoeff
⊢ Tendsto (fun x => eval x P / eval x Q) atTop atTop
state_after: 𝕜 : Type u_1
inst✝¹ : NormedLinearOrderedField 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : Q.degree < P.degree
hpos : 0 < P.leadingCoeff / Q.leadingCoeff
hQ : Q ≠ 0
⊢ Tendsto (fun x => eval x P / eval x Q) atTop atTop

tactic: rw [← natDegree_lt_natDegree_iff hQ] at hdeg
state_before: 𝕜 : Type u_1
inst✝¹ : NormedLinearOrderedField 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : Q.degree < P.degree
hpos : 0 < P.leadingCoeff / Q.leadingCoeff
hQ : Q ≠ 0
⊢ Tendsto (fun x => eval x P / eval x Q) atTop atTop
state_after: 𝕜 : Type u_1
inst✝¹ : NormedLinearOrderedField 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : Q.natDegree < P.natDegree
hpos : 0 < P.leadingCoeff / Q.leadingCoeff
hQ : Q ≠ 0
⊢ Tendsto (fun x => eval x P / eval x Q) atTop atTop

tactic: refine (isEquivalent_atTop_div P Q).symm.tendsto_atTop ?_
state_before: 𝕜 : Type u_1
inst✝¹ : NormedLinearOrderedField 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : Q.natDegree < P.natDegree
hpos : 0 < P.leadingCoeff / Q.leadingCoeff
hQ : Q ≠ 0
⊢ Tendsto (fun x => eval x P / eval x Q) atTop atTop
state_after: 𝕜 : Type u_1
inst✝¹ : NormedLinearOrderedField 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : Q.natDegree < P.natDegree
hpos : 0 < P.leadingCoeff / Q.leadingCoeff
hQ : Q ≠ 0
⊢ Tendsto (fun x => P.leadingCoeff / Q.leadingCoeff * x ^ (↑P.natDegree - ↑Q.natDegree)) atTop atTop

tactic: apply Tendsto.const_mul_atTop hpos
state_before: 𝕜 : Type u_1
inst✝¹ : NormedLinearOrderedField 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : Q.natDegree < P.natDegree
hpos : 0 < P.leadingCoeff / Q.leadingCoeff
hQ : Q ≠ 0
⊢ Tendsto (fun x => P.leadingCoeff / Q.leadingCoeff * x ^ (↑P.natDegree - ↑Q.natDegree)) atTop atTop
state_after: 𝕜 : Type u_1
inst✝¹ : NormedLinearOrderedField 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : Q.natDegree < P.natDegree
hpos : 0 < P.leadingCoeff / Q.leadingCoeff
hQ : Q ≠ 0
⊢ Tendsto (fun x => x ^ (↑P.natDegree - ↑Q.natDegree)) atTop atTop

tactic: apply tendsto_zpow_atTop_atTop
state_before: 𝕜 : Type u_1
inst✝¹ : NormedLinearOrderedField 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : Q.natDegree < P.natDegree
hpos : 0 < P.leadingCoeff / Q.leadingCoeff
hQ : Q ≠ 0
⊢ Tendsto (fun x => x ^ (↑P.natDegree - ↑Q.natDegree)) atTop atTop
state_after: case hn
𝕜 : Type u_1
inst✝¹ : NormedLinearOrderedField 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : Q.natDegree < P.natDegree
hpos : 0 < P.leadingCoeff / Q.leadingCoeff
hQ : Q ≠ 0
⊢ 0 < ↑P.natDegree - ↑Q.natDegree

tactic: omega
state_before: case hn
𝕜 : Type u_1
inst✝¹ : NormedLinearOrderedField 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : Q.natDegree < P.natDegree
hpos : 0 < P.leadingCoeff / Q.leadingCoeff
hQ : Q ≠ 0
⊢ 0 < ↑P.natDegree - ↑Q.natDegree
state_after: no goals

tactic: simp only [h, div_zero, leadingCoeff_zero] at hpos
state_before: 𝕜 : Type u_1
inst✝¹ : NormedLinearOrderedField 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : Q.degree < P.degree
hpos : 0 < P.leadingCoeff / Q.leadingCoeff
h : Q = 0
⊢ False
state_after: 𝕜 : Type u_1
inst✝¹ : NormedLinearOrderedField 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : Q.degree < P.degree
h : Q = 0
hpos : 0 < 0
⊢ False

tactic: exact hpos.false
state_before: 𝕜 : Type u_1
inst✝¹ : NormedLinearOrderedField 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : Q.degree < P.degree
h : Q = 0
hpos : 0 < 0
⊢ False
state_after: no goals