mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 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 ↦ P.leadingCoeff / Q.leadingCoeff * x ^ (↑P.natDegree - ↑Q.natDegree)

pp: case neg
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 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 ↦ P.leadingCoeff / Q.leadingCoeff * x ^ (↑P.natDegree - ↑Q.natDegree)

Tactic:
·

Goal state after:
case pos
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 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 ↦ P.leadingCoeff / Q.leadingCoeff * x ^ (↑P.natDegree - ↑Q.natDegree)
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : P.degree < Q.degree
hP : P = 0
⊢ Tendsto (fun x ↦ eval x P / eval x Q) atTop (𝓝 0)

pp: case neg
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : P.degree < Q.degree
hP : ¬P = 0
⊢ Tendsto (fun x ↦ eval x P / eval x Q) atTop (𝓝 0)

Tactic:
·

Goal state after:
case pos
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : P.degree < Q.degree
hP : P = 0
⊢ Tendsto (fun x ↦ eval x P / eval x Q) atTop (𝓝 0)
Module: Mathlib.Analysis.Polynomial.CauchyBound

Goal state before:
pp: case inr.inr
K : Type u_1
inst✝ : NormedDivisionRing K
p : K[X]
hp : p ≠ 0
a : K
h : ‖p.leadingCoeff‖₊ * ‖a‖₊ ^ p.natDegree = ‖∑ i ∈ range p.natDegree, p.coeff i * a ^ i‖₊
this : ‖a‖₊ ^ p.natDegree ≤ (p.cauchyBound - 1) * ∑ x ∈ range p.natDegree, ‖a‖₊ ^ x
ha✝ : ‖a‖₊ ≠ 1
ha : 1 < ‖a‖₊
⊢ ‖a‖₊ < p.cauchyBound

Tactic:
·

Goal state after:
case inr.inr
K : Type u_1
inst✝ : NormedDivisionRing K
p : K[X]
hp : p ≠ 0
a : K
h : ‖p.leadingCoeff‖₊ * ‖a‖₊ ^ p.natDegree = ‖∑ i ∈ range p.natDegree, p.coeff i * a ^ i‖₊
this : ‖a‖₊ ^ p.natDegree ≤ (p.cauchyBound - 1) * ∑ x ∈ range p.natDegree, ‖a‖₊ ^ x
ha✝ : ‖a‖₊ ≠ 1
ha : 1 < ‖a‖₊
⊢ ‖a‖₊ < p.cauchyBound
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hQ : Q ≠ 0
h : Tendsto (fun x ↦ eval x P / eval x Q) atTop (𝓝 0)
hPQ : P.leadingCoeff / Q.leadingCoeff = 0
⊢ P.degree < Q.degree

pp: case neg
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hQ : Q ≠ 0
h : Tendsto (fun x ↦ eval x P / eval x Q) atTop (𝓝 0)
hPQ : ¬P.leadingCoeff / Q.leadingCoeff = 0
⊢ P.degree < Q.degree

Tactic:
·

Goal state after:
case pos
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hQ : Q ≠ 0
h : Tendsto (fun x ↦ eval x P / eval x Q) atTop (𝓝 0)
hPQ : P.leadingCoeff / Q.leadingCoeff = 0
⊢ P.degree < Q.degree
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: case pos.inl
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hQ : Q ≠ 0
h : Tendsto (fun x ↦ eval x P / eval x Q) atTop (𝓝 0)
hP0 : P.leadingCoeff = 0
⊢ P.degree < Q.degree

pp: case pos.inr
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hQ : Q ≠ 0
h : Tendsto (fun x ↦ eval x P / eval x Q) atTop (𝓝 0)
hQ0 : Q.leadingCoeff = 0
⊢ P.degree < Q.degree

Tactic:
·

Goal state after:
case pos.inl
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hQ : Q ≠ 0
h : Tendsto (fun x ↦ eval x P / eval x Q) atTop (𝓝 0)
hP0 : P.leadingCoeff = 0
⊢ P.degree < Q.degree
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: case pos.inr
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hQ : Q ≠ 0
h : Tendsto (fun x ↦ eval x P / eval x Q) atTop (𝓝 0)
hQ0 : Q.leadingCoeff = 0
⊢ P.degree < Q.degree

Tactic:
·

Goal state after:
case pos.inr
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hQ : Q ≠ 0
h : Tendsto (fun x ↦ eval x P / eval x Q) atTop (𝓝 0)
hQ0 : Q.leadingCoeff = 0
⊢ P.degree < Q.degree
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hQ : Q ≠ 0
h : Tendsto (fun x ↦ eval x P / eval x Q) atTop (𝓝 0)
hPQ : ¬P.leadingCoeff / Q.leadingCoeff = 0
⊢ P.degree < Q.degree

Tactic:
·

Goal state after:
case neg
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hQ : Q ≠ 0
h : Tendsto (fun x ↦ eval x P / eval x Q) atTop (𝓝 0)
hPQ : ¬P.leadingCoeff / Q.leadingCoeff = 0
⊢ P.degree < Q.degree
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: case neg.inl
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hQ : Q ≠ 0
h✝ : Tendsto (fun x ↦ eval x P / eval x Q) atTop (𝓝 0)
hPQ : ¬P.leadingCoeff / Q.leadingCoeff = 0
h : ↑P.natDegree - ↑Q.natDegree = 0 ∧ P.leadingCoeff / Q.leadingCoeff = 0
⊢ P.degree < Q.degree

pp: case neg.inr
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hQ : Q ≠ 0
h✝ : Tendsto (fun x ↦ eval x P / eval x Q) atTop (𝓝 0)
hPQ : ¬P.leadingCoeff / Q.leadingCoeff = 0
h : ↑P.natDegree - ↑Q.natDegree < 0 ∧ 0 = 0
⊢ P.degree < Q.degree

Tactic:
·

Goal state after:
case neg.inl
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hQ : Q ≠ 0
h✝ : Tendsto (fun x ↦ eval x P / eval x Q) atTop (𝓝 0)
hPQ : ¬P.leadingCoeff / Q.leadingCoeff = 0
h : ↑P.natDegree - ↑Q.natDegree = 0 ∧ P.leadingCoeff / Q.leadingCoeff = 0
⊢ P.degree < Q.degree
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: case neg.inr
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hQ : Q ≠ 0
h✝ : Tendsto (fun x ↦ eval x P / eval x Q) atTop (𝓝 0)
hPQ : ¬P.leadingCoeff / Q.leadingCoeff = 0
h : ↑P.natDegree - ↑Q.natDegree < 0 ∧ 0 = 0
⊢ P.degree < Q.degree

Tactic:
·

Goal state after:
case neg.inr
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hQ : Q ≠ 0
h✝ : Tendsto (fun x ↦ eval x P / eval x Q) atTop (𝓝 0)
hPQ : ¬P.leadingCoeff / Q.leadingCoeff = 0
h : ↑P.natDegree - ↑Q.natDegree < 0 ∧ 0 = 0
⊢ P.degree < Q.degree
Module: Mathlib.Analysis.Polynomial.CauchyBound

Goal state before:
pp: case ha
K : Type u_1
inst✝ : NormedDivisionRing K
p : K[X]
hp : p ≠ 0
a : K
h : ‖p.leadingCoeff‖₊ * ‖a‖₊ ^ p.natDegree = ‖∑ i ∈ range p.natDegree, p.coeff i * a ^ i‖₊
this : ‖a‖₊ ^ p.natDegree ≤ (p.cauchyBound - 1) * ((‖a‖₊ ^ p.natDegree - 1) / (‖a‖₊ - 1))
ha✝ : ‖a‖₊ ≠ 1
ha : 1 < ‖a‖₊
⊢ 0 < p.cauchyBound - 1

pp: case hbc
K : Type u_1
inst✝ : NormedDivisionRing K
p : K[X]
hp : p ≠ 0
a : K
h : ‖p.leadingCoeff‖₊ * ‖a‖₊ ^ p.natDegree = ‖∑ i ∈ range p.natDegree, p.coeff i * a ^ i‖₊
this : ‖a‖₊ ^ p.natDegree ≤ (p.cauchyBound - 1) * ((‖a‖₊ ^ p.natDegree - 1) / (‖a‖₊ - 1))
ha✝ : ‖a‖₊ ≠ 1
ha : 1 < ‖a‖₊
⊢ ‖a‖₊ ^ p.natDegree - 1 < ‖a‖₊ ^ p.natDegree

Tactic:
·

Goal state after:
case ha
K : Type u_1
inst✝ : NormedDivisionRing K
p : K[X]
hp : p ≠ 0
a : K
h : ‖p.leadingCoeff‖₊ * ‖a‖₊ ^ p.natDegree = ‖∑ i ∈ range p.natDegree, p.coeff i * a ^ i‖₊
this : ‖a‖₊ ^ p.natDegree ≤ (p.cauchyBound - 1) * ((‖a‖₊ ^ p.natDegree - 1) / (‖a‖₊ - 1))
ha✝ : ‖a‖₊ ≠ 1
ha : 1 < ‖a‖₊
⊢ 0 < p.cauchyBound - 1
Module: Mathlib.Analysis.Polynomial.Fourier

Goal state before:
pp: case inl
p : ℂ[X]
this :
  ∑' (i : ℤ), ‖if 0 ≤ i then p.coeff i.natAbs else 0‖ ^ 2 =
    ∫ (t : AddCircle (2 * π)), ‖↑↑((ContinuousMap.toLp 2 haarAddCircle ℂ) (toAddCircle p)) t‖ ^ 2 ∂haarAddCircle
b : ℤ
hb : b ∉ Finset.map { toFun := Nat.cast, inj' := ⋯ } p.support
h : 0 ≤ b
⊢ ‖if 0 ≤ b then p.coeff b.natAbs else 0‖ ^ 2 = 0

pp: case inr
p : ℂ[X]
this :
  ∑' (i : ℤ), ‖if 0 ≤ i then p.coeff i.natAbs else 0‖ ^ 2 =
    ∫ (t : AddCircle (2 * π)), ‖↑↑((ContinuousMap.toLp 2 haarAddCircle ℂ) (toAddCircle p)) t‖ ^ 2 ∂haarAddCircle
b : ℤ
hb : b ∉ Finset.map { toFun := Nat.cast, inj' := ⋯ } p.support
h : b < 0
⊢ ‖if 0 ≤ b then p.coeff b.natAbs else 0‖ ^ 2 = 0

Tactic:
·

Goal state after:
case inl
p : ℂ[X]
this :
  ∑' (i : ℤ), ‖if 0 ≤ i then p.coeff i.natAbs else 0‖ ^ 2 =
    ∫ (t : AddCircle (2 * π)), ‖↑↑((ContinuousMap.toLp 2 haarAddCircle ℂ) (toAddCircle p)) t‖ ^ 2 ∂haarAddCircle
b : ℤ
hb : b ∉ Finset.map { toFun := Nat.cast, inj' := ⋯ } p.support
h : 0 ≤ b
⊢ ‖if 0 ≤ b then p.coeff b.natAbs else 0‖ ^ 2 = 0
Module: Mathlib.Analysis.Polynomial.Fourier

Goal state before:
pp: case inr
p : ℂ[X]
this :
  ∑' (i : ℤ), ‖if 0 ≤ i then p.coeff i.natAbs else 0‖ ^ 2 =
    ∫ (t : AddCircle (2 * π)), ‖↑↑((ContinuousMap.toLp 2 haarAddCircle ℂ) (toAddCircle p)) t‖ ^ 2 ∂haarAddCircle
b : ℤ
hb : b ∉ Finset.map { toFun := Nat.cast, inj' := ⋯ } p.support
h : b < 0
⊢ ‖if 0 ≤ b then p.coeff b.natAbs else 0‖ ^ 2 = 0

Tactic:
·

Goal state after:
case inr
p : ℂ[X]
this :
  ∑' (i : ℤ), ‖if 0 ≤ i then p.coeff i.natAbs else 0‖ ^ 2 =
    ∫ (t : AddCircle (2 * π)), ‖↑↑((ContinuousMap.toLp 2 haarAddCircle ℂ) (toAddCircle p)) t‖ ^ 2 ∂haarAddCircle
b : ℤ
hb : b ∉ Finset.map { toFun := Nat.cast, inj' := ⋯ } p.support
h : b < 0
⊢ ‖if 0 ≤ b then p.coeff b.natAbs else 0‖ ^ 2 = 0
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : Q.degree < P.degree
hQ : Q ≠ 0
h : 0 ≤ P.leadingCoeff / Q.leadingCoeff
⊢ Tendsto (fun x ↦ |eval x P / eval x Q|) atTop atTop

pp: case neg
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : Q.degree < P.degree
hQ : Q ≠ 0
h : P.leadingCoeff / Q.leadingCoeff < 0
⊢ Tendsto (fun x ↦ |eval x P / eval x Q|) atTop atTop

Tactic:
·

Goal state after:
case pos
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : Q.degree < P.degree
hQ : Q ≠ 0
h : 0 ≤ P.leadingCoeff / Q.leadingCoeff
⊢ Tendsto (fun x ↦ |eval x P / eval x Q|) atTop atTop
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : Q.degree < P.degree
hQ : Q ≠ 0
h : P.leadingCoeff / Q.leadingCoeff < 0
⊢ Tendsto (fun x ↦ |eval x P / eval x Q|) atTop atTop

Tactic:
·

Goal state after:
case neg
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : Q.degree < P.degree
hQ : Q ≠ 0
h : P.leadingCoeff / Q.leadingCoeff < 0
⊢ Tendsto (fun x ↦ |eval x P / eval x Q|) atTop atTop
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hP : P = 0
⊢ (fun x ↦ eval x P / eval x Q) ~[atBot] fun x ↦ P.leadingCoeff / Q.leadingCoeff * x ^ (↑P.natDegree - ↑Q.natDegree)

pp: case neg
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hP : ¬P = 0
⊢ (fun x ↦ eval x P / eval x Q) ~[atBot] fun x ↦ P.leadingCoeff / Q.leadingCoeff * x ^ (↑P.natDegree - ↑Q.natDegree)

Tactic:
·

Goal state after:
case pos
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hP : P = 0
⊢ (fun x ↦ eval x P / eval x Q) ~[atBot] fun x ↦ P.leadingCoeff / Q.leadingCoeff * x ^ (↑P.natDegree - ↑Q.natDegree)
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hP : ¬P = 0
hQ : Q = 0
⊢ (fun x ↦ eval x P / eval x Q) ~[atBot] fun x ↦ P.leadingCoeff / Q.leadingCoeff * x ^ (↑P.natDegree - ↑Q.natDegree)

pp: case neg
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hP : ¬P = 0
hQ : ¬Q = 0
⊢ (fun x ↦ eval x P / eval x Q) ~[atBot] fun x ↦ P.leadingCoeff / Q.leadingCoeff * x ^ (↑P.natDegree - ↑Q.natDegree)

Tactic:
·

Goal state after:
case pos
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hP : ¬P = 0
hQ : Q = 0
⊢ (fun x ↦ eval x P / eval x Q) ~[atBot] fun x ↦ P.leadingCoeff / Q.leadingCoeff * x ^ (↑P.natDegree - ↑Q.natDegree)
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
h : P.degree < Q.degree
hp : P = 0
⊢ (fun x ↦ eval x P) =o[atTop] fun x ↦ eval x Q

pp: case neg
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
h : P.degree < Q.degree
hp : ¬P = 0
⊢ (fun x ↦ eval x P) =o[atTop] fun x ↦ eval x Q

Tactic:
·

Goal state after:
case pos
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
h : P.degree < Q.degree
hp : P = 0
⊢ (fun x ↦ eval x P) =o[atTop] fun x ↦ eval x Q
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
h : P.degree < Q.degree
hp : ¬P = 0
⊢ (fun x ↦ eval x P) =o[atTop] fun x ↦ eval x Q

Tactic:
·

Goal state after:
case neg
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
h : P.degree < Q.degree
hp : ¬P = 0
⊢ (fun x ↦ eval x P) =o[atTop] fun x ↦ eval x Q
Module: Mathlib.Analysis.ODE.PicardLindelof

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : ℝ → E → E
tmin tmax : ℝ
t₀ : ↑(Icc tmin tmax)
x₀ : E
a r L K : ℝ≥0
inst✝ : CompleteSpace E
hf : IsPicardLindelof f t₀ x₀ a r L K
m : ℕ
C : ℝ≥0
h : ∀ (x : E) (hx : x ∈ closedBall x₀ ↑r), ContractingWith C (next hf hx)^[m]
L' : ℝ := ⋯
hL' : 0 ≤ L'
x y : E
hx : x ∈ closedBall x₀ ↑r
hy : y ∈ closedBall x₀ ↑r
α β : FunSpace t₀ x₀ r L
hα : IsFixedPt (next hf hx) α
hβ : IsFixedPt (next hf hy) β
⊢ Filter.Tendsto (fun x ↦ dist α ((next hf hy)^[m]^[x] α)) Filter.atTop (𝓝 (dist α β))

pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : ℝ → E → E
tmin tmax : ℝ
t₀ : ↑(Icc tmin tmax)
x₀ : E
a r L K : ℝ≥0
inst✝ : CompleteSpace E
hf : IsPicardLindelof f t₀ x₀ a r L K
m : ℕ
C : ℝ≥0
h : ∀ (x : E) (hx : x ∈ closedBall x₀ ↑r), ContractingWith C (next hf hx)^[m]
L' : ℝ := ⋯
hL' : 0 ≤ L'
x y : E
hx : x ∈ closedBall x₀ ↑r
hy : y ∈ closedBall x₀ ↑r
α β : FunSpace t₀ x₀ r L
hα : IsFixedPt (next hf hx) α
hβ : IsFixedPt (next hf hy) β
⊢ Filter.Tendsto
    (fun x ↦
      ((∑ i ∈ Finset.range m, (↑K * max (tmax - ↑t₀) (↑t₀ - tmin)) ^ i / ↑i !) * ∑ i ∈ Finset.range x, ↑C ^ i) *
        dist α (next hf hy α))
    Filter.atTop (𝓝 (L' * dist x y))

Tactic:
·

Goal state after:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : ℝ → E → E
tmin tmax : ℝ
t₀ : ↑(Icc tmin tmax)
x₀ : E
a r L K : ℝ≥0
inst✝ : CompleteSpace E
hf : IsPicardLindelof f t₀ x₀ a r L K
m : ℕ
C : ℝ≥0
h : ∀ (x : E) (hx : x ∈ closedBall x₀ ↑r), ContractingWith C (next hf hx)^[m]
L' : ℝ := ⋯
hL' : 0 ≤ L'
x y : E
hx : x ∈ closedBall x₀ ↑r
hy : y ∈ closedBall x₀ ↑r
α β : FunSpace t₀ x₀ r L
hα : IsFixedPt (next hf hx) α
hβ : IsFixedPt (next hf hy) β
⊢ Filter.Tendsto (fun x ↦ dist α ((next hf hy)^[m]^[x] α)) Filter.atTop (𝓝 (dist α β))
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
h : P.degree ≤ Q.degree
hp : P = 0
⊢ (fun x ↦ eval x P) =O[atTop] fun x ↦ eval x Q

pp: case neg
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
h : P.degree ≤ Q.degree
hp : ¬P = 0
⊢ (fun x ↦ eval x P) =O[atTop] fun x ↦ eval x Q

Tactic:
·

Goal state after:
case pos
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
h : P.degree ≤ Q.degree
hp : P = 0
⊢ (fun x ↦ eval x P) =O[atTop] fun x ↦ eval x Q
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
h : P.degree ≤ Q.degree
hp : ¬P = 0
⊢ (fun x ↦ eval x P) =O[atTop] fun x ↦ eval x Q

Tactic:
·

Goal state after:
case neg
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
h : P.degree ≤ Q.degree
hp : ¬P = 0
⊢ (fun x ↦ eval x P) =O[atTop] fun x ↦ eval x Q
Module: Mathlib.RingTheory.PowerSeries.GaussNorm

Goal state before:
pp: case mp
R : Type u_1
inst✝ : Semiring R
v : R → ℝ
c : ℝ
f : R⟦X⟧
h : HasGaussNorm v c f
x✝ : ℝ
⊢ (∃ y, v ((coeff y) f) * c ^ y = x✝) → ∃ y, v ((MvPowerSeries.coeff y) f) * c ^ y PUnit.unit = x✝

pp: case mpr
R : Type u_1
inst✝ : Semiring R
v : R → ℝ
c : ℝ
f : R⟦X⟧
h : HasGaussNorm v c f
x✝ : ℝ
⊢ (∃ y, v ((MvPowerSeries.coeff y) f) * c ^ y PUnit.unit = x✝) → ∃ y, v ((coeff y) f) * c ^ y = x✝

Tactic:
·

Goal state after:
case mp
R : Type u_1
inst✝ : Semiring R
v : R → ℝ
c : ℝ
f : R⟦X⟧
h : HasGaussNorm v c f
x✝ : ℝ
⊢ (∃ y, v ((coeff y) f) * c ^ y = x✝) → ∃ y, v ((MvPowerSeries.coeff y) f) * c ^ y PUnit.unit = x✝
Module: Mathlib.RingTheory.PowerSeries.GaussNorm

Goal state before:
pp: case mpr
R : Type u_1
inst✝ : Semiring R
v : R → ℝ
c : ℝ
f : R⟦X⟧
h : HasGaussNorm v c f
x✝ : ℝ
⊢ (∃ y, v ((MvPowerSeries.coeff y) f) * c ^ y PUnit.unit = x✝) → ∃ y, v ((coeff y) f) * c ^ y = x✝

Tactic:
·

Goal state after:
case mpr
R : Type u_1
inst✝ : Semiring R
v : R → ℝ
c : ℝ
f : R⟦X⟧
h : HasGaussNorm v c f
x✝ : ℝ
⊢ (∃ y, v ((MvPowerSeries.coeff y) f) * c ^ y PUnit.unit = x✝) → ∃ y, v ((coeff y) f) * c ^ y = x✝
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: case neg.inl
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
h✝ : P.degree ≤ Q.degree
hp : ¬P = 0
hq : Q ≠ 0
hPQ : ∀ᶠ (x : 𝕜) in atTop, eval x Q = 0 → eval x P = 0
h : P.degree < Q.degree
⊢ (fun x ↦ eval x P) =O[atTop] fun x ↦ eval x Q

pp: case neg.inr
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
h✝ : P.degree ≤ Q.degree
hp : ¬P = 0
hq : Q ≠ 0
hPQ : ∀ᶠ (x : 𝕜) in atTop, eval x Q = 0 → eval x P = 0
h : P.degree = Q.degree
⊢ (fun x ↦ eval x P) =O[atTop] fun x ↦ eval x Q

Tactic:
·

Goal state after:
case neg.inl
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
h✝ : P.degree ≤ Q.degree
hp : ¬P = 0
hq : Q ≠ 0
hPQ : ∀ᶠ (x : 𝕜) in atTop, eval x Q = 0 → eval x P = 0
h : P.degree < Q.degree
⊢ (fun x ↦ eval x P) =O[atTop] fun x ↦ eval x Q
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: case neg.inr
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
h✝ : P.degree ≤ Q.degree
hp : ¬P = 0
hq : Q ≠ 0
hPQ : ∀ᶠ (x : 𝕜) in atTop, eval x Q = 0 → eval x P = 0
h : P.degree = Q.degree
⊢ (fun x ↦ eval x P) =O[atTop] fun x ↦ eval x Q

Tactic:
·

Goal state after:
case neg.inr
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
h✝ : P.degree ≤ Q.degree
hp : ¬P = 0
hq : Q ≠ 0
hPQ : ∀ᶠ (x : 𝕜) in atTop, eval x Q = 0 → eval x P = 0
h : P.degree = Q.degree
⊢ (fun x ↦ eval x P) =O[atTop] fun x ↦ eval x Q
Module: Mathlib.Analysis.ODE.PicardLindelof

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : ℝ → E → E
tmin tmax : ℝ
t₀ : ↑(Icc tmin tmax)
x₀ : E
a r L K : ℝ≥0
inst✝ : CompleteSpace E
hf : IsPicardLindelof f t₀ x₀ a r L K
m : ℕ
C : ℝ≥0
h : ∀ (x : E) (hx : x ∈ closedBall x₀ ↑r), ContractingWith C (next hf hx)^[m]
L' : ℝ := ⋯
hL' : 0 ≤ L'
x y : E
hx : x ∈ closedBall x₀ ↑r
hy : y ∈ closedBall x₀ ↑r
α β : FunSpace t₀ x₀ r L
hα : IsFixedPt (next hf hx) α
hβ : IsFixedPt (next hf hy) β
⊢ Filter.Tendsto
    (fun x ↦
      ((∑ i ∈ Finset.range m, (↑K * max (tmax - ↑t₀) (↑t₀ - tmin)) ^ i / ↑i !) * ∑ i ∈ Finset.range x, ↑C ^ i) *
        dist α (next hf hy α))
    Filter.atTop (𝓝 (L' * dist x y))

Tactic:
·

Goal state after:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : ℝ → E → E
tmin tmax : ℝ
t₀ : ↑(Icc tmin tmax)
x₀ : E
a r L K : ℝ≥0
inst✝ : CompleteSpace E
hf : IsPicardLindelof f t₀ x₀ a r L K
m : ℕ
C : ℝ≥0
h : ∀ (x : E) (hx : x ∈ closedBall x₀ ↑r), ContractingWith C (next hf hx)^[m]
L' : ℝ := ⋯
hL' : 0 ≤ L'
x y : E
hx : x ∈ closedBall x₀ ↑r
hy : y ∈ closedBall x₀ ↑r
α β : FunSpace t₀ x₀ r L
hα : IsFixedPt (next hf hx) α
hβ : IsFixedPt (next hf hy) β
⊢ Filter.Tendsto
    (fun x ↦
      ((∑ i ∈ Finset.range m, (↑K * max (tmax - ↑t₀) (↑t₀ - tmin)) ^ i / ↑i !) * ∑ i ∈ Finset.range x, ↑C ^ i) *
        dist α (next hf hy α))
    Filter.atTop (𝓝 (L' * dist x y))
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: case pos
R : Type u_2
inst✝¹ : NormedRing R
inst✝ : NormMulClass R
P : R[X]
h : P = 0
⊢ (fun x ↦ eval x P) ~[cobounded R] fun x ↦ P.leadingCoeff * x ^ P.natDegree

pp: case neg
R : Type u_2
inst✝¹ : NormedRing R
inst✝ : NormMulClass R
P : R[X]
h : ¬P = 0
⊢ (fun x ↦ eval x P) ~[cobounded R] fun x ↦ P.leadingCoeff * x ^ P.natDegree

Tactic:
·

Goal state after:
case pos
R : Type u_2
inst✝¹ : NormedRing R
inst✝ : NormMulClass R
P : R[X]
h : P = 0
⊢ (fun x ↦ eval x P) ~[cobounded R] fun x ↦ P.leadingCoeff * x ^ P.natDegree
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: case neg
R : Type u_2
inst✝¹ : NormedRing R
inst✝ : NormMulClass R
P : R[X]
h : ¬P = 0
⊢ (fun x ↦ eval x P) ~[cobounded R] fun x ↦ P.leadingCoeff * x ^ P.natDegree

Tactic:
·

Goal state after:
case neg
R : Type u_2
inst✝¹ : NormedRing R
inst✝ : NormMulClass R
P : R[X]
h : ¬P = 0
⊢ (fun x ↦ eval x P) ~[cobounded R] fun x ↦ P.leadingCoeff * x ^ P.natDegree
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: case pos
R : Type u_2
inst✝¹ : NormedRing R
inst✝ : NormMulClass R
P Q : R[X]
h : P.degree < Q.degree
hP : P = 0
⊢ (fun x ↦ eval x P) =o[cobounded R] fun x ↦ eval x Q

pp: case neg
R : Type u_2
inst✝¹ : NormedRing R
inst✝ : NormMulClass R
P Q : R[X]
h : P.degree < Q.degree
hP : ¬P = 0
⊢ (fun x ↦ eval x P) =o[cobounded R] fun x ↦ eval x Q

Tactic:
·

Goal state after:
case pos
R : Type u_2
inst✝¹ : NormedRing R
inst✝ : NormMulClass R
P Q : R[X]
h : P.degree < Q.degree
hP : P = 0
⊢ (fun x ↦ eval x P) =o[cobounded R] fun x ↦ eval x Q
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: case neg
R : Type u_2
inst✝¹ : NormedRing R
inst✝ : NormMulClass R
P Q : R[X]
h : P.degree < Q.degree
hP : ¬P = 0
⊢ (fun x ↦ eval x P) =o[cobounded R] fun x ↦ eval x Q

Tactic:
·

Goal state after:
case neg
R : Type u_2
inst✝¹ : NormedRing R
inst✝ : NormMulClass R
P Q : R[X]
h : P.degree < Q.degree
hP : ¬P = 0
⊢ (fun x ↦ eval x P) =o[cobounded R] fun x ↦ eval x Q