mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Complex.BranchLogRoot

Goal state before:
pp: case refine_1
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : LocallyPathConnectedSpace X
U : Set X
hUc : IsSimplyConnected U
hUo : IsOpen U
g : X → ℂ
hgc : ContinuousOn g U
hU₀ : 0 ∉ g '' U
n : ℕ
hn : n ≠ 0
f : X → ℂ
hfc : ContinuousOn f U
hf : EqOn (cexp ∘ f) g U
⊢ ContinuousOn (fun x ↦ cexp (f x / ↑n)) U

Tactic:
]

Goal state after:
case refine_1
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : LocallyPathConnectedSpace X
U : Set X
hUc : IsSimplyConnected U
hUo : IsOpen U
g : X → ℂ
hgc : ContinuousOn g U
hU₀ : 0 ∉ g '' U
n : ℕ
hn : n ≠ 0
f : X → ℂ
hfc : ContinuousOn f U
hf : EqOn (cexp ∘ f) g U
⊢ ContinuousOn (fun x ↦ cexp (f x / ↑n)) U
Module: Mathlib.Analysis.Complex.BranchLogRoot

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : LocallyPathConnectedSpace X
U : Set X
hUc : IsSimplyConnected U
hUo : IsOpen U
g : X → 𝔻
hgc : ContinuousOn g U
hU₀ : 0 ∉ g '' U
n : ℕ+
f : X → ℂ
hfc : ContinuousOn f U
hf : ∀ (x : X), f x ^ ↑n = (UnitDisc.coe ∘ g) x
x : X
⊢ ‖(UnitDisc.coe ∘ g) x‖ < 1

Tactic:
]

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : LocallyPathConnectedSpace X
U : Set X
hUc : IsSimplyConnected U
hUo : IsOpen U
g : X → 𝔻
hgc : ContinuousOn g U
hU₀ : 0 ∉ g '' U
n : ℕ+
f : X → ℂ
hfc : ContinuousOn f U
hf : ∀ (x : X), f x ^ ↑n = (UnitDisc.coe ∘ g) x
x : X
⊢ ‖(UnitDisc.coe ∘ g) x‖ < 1
Module: Mathlib.Analysis.Complex.Schwarz

Goal state before:
pp: case inr
E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
c z : E
R₁ R₂ : ℝ
n : ℕ
hd : DifferentiableOn ℂ f (ball c R₁)
h_maps : MapsTo f (ball c R₁) (closedBall (f c) R₂)
hn✝ : (fun x ↦ f x - f c) =o[𝓝 c] fun w ↦ ‖w - c‖ ^ n
hz : z ∈ ball c R₁
hR₁ : 0 < R₁
hR₂ : 0 ≤ R₂
hfne : f z ≠ f c
hne : z ≠ c
g : StrongDual ℂ F
hg : ‖g‖ = 1
hgf : g (f z) - g (f c) = ↑‖f z - f c‖
h : ℂ → ℂ := ⇑g ∘ f ∘ ⇑(lineMap c z)
hmaps_line : MapsTo (⇑(lineMap c z)) (ball 0 (R₁ / dist z c)) (ball c R₁)
hmaps : MapsTo h (ball 0 (R₁ / dist z c)) (closedBall (h 0) R₂)
hdiff : DifferentiableOn ℂ h (ball 0 (R₁ / dist z c))
hn : (fun x ↦ h x - h 0) =o[𝓝 0] fun w ↦ (w - 0) ^ n
hmem : 1 ∈ ball 0 (R₁ / dist z c)
⊢ dist (f z) (f c) ≤ R₂ * (dist z c / R₁) ^ (n + 1)

Tactic:
]

Goal state after:
case inr
E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
c z : E
R₁ R₂ : ℝ
n : ℕ
hd : DifferentiableOn ℂ f (ball c R₁)
h_maps : MapsTo f (ball c R₁) (closedBall (f c) R₂)
hn✝ : (fun x ↦ f x - f c) =o[𝓝 c] fun w ↦ ‖w - c‖ ^ n
hz : z ∈ ball c R₁
hR₁ : 0 < R₁
hR₂ : 0 ≤ R₂
hfne : f z ≠ f c
hne : z ≠ c
g : StrongDual ℂ F
hg : ‖g‖ = 1
hgf : g (f z) - g (f c) = ↑‖f z - f c‖
h : ℂ → ℂ := ⇑g ∘ f ∘ ⇑(lineMap c z)
hmaps_line : MapsTo (⇑(lineMap c z)) (ball 0 (R₁ / dist z c)) (ball c R₁)
hmaps : MapsTo h (ball 0 (R₁ / dist z c)) (closedBall (h 0) R₂)
hdiff : DifferentiableOn ℂ h (ball 0 (R₁ / dist z c))
hn : (fun x ↦ h x - h 0) =o[𝓝 0] fun w ↦ (w - 0) ^ n
hmem : 1 ∈ ball 0 (R₁ / dist z c)
⊢ dist (f z) (f c) ≤ R₂ * (dist z c / R₁) ^ (n + 1)
Module: Mathlib.Analysis.Complex.UnitDisc.Basic

Goal state before:
pp: z : 𝔻
⊢ ‖↑z‖ < 1

Tactic:
]

Goal state after:
z : 𝔻
⊢ ‖↑z‖ < 1
Module: Mathlib.Analysis.Complex.UnitDisc.Basic

Goal state before:
pp: z : 𝔻
⊢ ‖↑z‖ ^ 2 < 1

Tactic:
]

Goal state after:
z : 𝔻
⊢ ‖↑z‖ ^ 2 < 1
Module: Mathlib.Analysis.Complex.UnitDisc.Basic

Goal state before:
pp: z : 𝔻
n : ℕ+
⊢ ↑z ^ ↑n = ↑0 ↔ z = 0

Tactic:
]

Goal state after:
z : 𝔻
n : ℕ+
⊢ ↑z ^ ↑n = ↑0 ↔ z = 0
Module: Mathlib.Analysis.Complex.UnitDisc.Basic

Goal state before:
pp: z : 𝕔𝔻
⊢ ‖↑z‖ ≤ 1

Tactic:
]

Goal state after:
z : 𝕔𝔻
⊢ ‖↑z‖ ≤ 1
Module: Mathlib.Analysis.Complex.UnitDisc.Basic

Goal state before:
pp: z : 𝕔𝔻
⊢ ‖↑z‖ ^ 2 ≤ 1

Tactic:
by
  rw [sq_le_one_iff_abs_le_one, abs_norm]
  exact z.norm_le_one

Goal state after:
Module: Mathlib.Analysis.Complex.UnitDisc.Basic

Goal state before:
pp: z : 𝕔𝔻
⊢ ‖↑z‖ ^ 2 ≤ 1

Tactic:
]

Goal state after:
z : 𝕔𝔻
⊢ ‖↑z‖ ^ 2 ≤ 1
Module: Mathlib.Analysis.Meromorphic.IsolatedZeros

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
x : 𝕜
f : 𝕜 → E
hf : MeromorphicAt f x
h₁x : x ∈ U
h₂x : AccPt x (𝓟 U)
h : f =ᶠ[codiscreteWithin U] 0
⊢ ∃ᶠ (z : 𝕜) in 𝓝[≠] x, f z = 0

Tactic:
]

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
x : 𝕜
f : 𝕜 → E
hf : MeromorphicAt f x
h₁x : x ∈ U
h₂x : AccPt x (𝓟 U)
h : f =ᶠ[codiscreteWithin U] 0
⊢ ∃ᶠ (z : 𝕜) in 𝓝[≠] x, f z = 0
Module: Mathlib.Analysis.Meromorphic.IsolatedZeros

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : 𝕜
f g : 𝕜 → E
hf : MeromorphicAt f x
hg : MeromorphicAt g x
⊢ (∃ᶠ (z : 𝕜) in 𝓝[≠] x, f z = g z) ↔ ∃ᶠ (z : 𝕜) in 𝓝[≠] x, (f - g) z = 0

Tactic:
]

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : 𝕜
f g : 𝕜 → E
hf : MeromorphicAt f x
hg : MeromorphicAt g x
⊢ (∃ᶠ (z : 𝕜) in 𝓝[≠] x, f z = g z) ↔ ∃ᶠ (z : 𝕜) in 𝓝[≠] x, (f - g) z = 0
Module: Mathlib.Analysis.Complex.Schwarz

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
R : ℝ
f : E → F
z : E
hd : DifferentiableOn ℂ f (ball 0 R)
h_maps : MapsTo f (ball 0 R) (closedBall 0 R)
h₀ : f 0 = 0
hz : ‖z‖ < R
⊢ MapsTo f (ball 0 R) (closedBall 0 R)

Tactic:
]

Goal state after:
E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
R : ℝ
f : E → F
z : E
hd : DifferentiableOn ℂ f (ball 0 R)
h_maps : MapsTo f (ball 0 R) (closedBall 0 R)
h₀ : f 0 = 0
hz : ‖z‖ < R
⊢ MapsTo f (ball 0 R) (closedBall 0 R)
Module: Mathlib.Analysis.Complex.Schwarz

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R₁ R₂ : ℝ
f : ℂ → E
c : ℂ
hd : DifferentiableOn ℂ f (ball c R₁)
h_maps : MapsTo f (ball c R₁) (closedBall (f c) R₂)
h₀ : 0 < R₁
⊢ ‖fderiv ℂ f c‖ ≤ R₂ / R₁

Tactic:
]

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R₁ R₂ : ℝ
f : ℂ → E
c : ℂ
hd : DifferentiableOn ℂ f (ball c R₁)
h_maps : MapsTo f (ball c R₁) (closedBall (f c) R₂)
h₀ : 0 < R₁
⊢ ‖fderiv ℂ f c‖ ≤ R₂ / R₁
Module: Mathlib.Analysis.Meromorphic.IsolatedZeros

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
x : 𝕜
f g : 𝕜 → E
hf : MeromorphicAt f x
hg : MeromorphicAt g x
h₁x : x ∈ U
h₂x : AccPt x (𝓟 U)
h : f - g =ᶠ[codiscreteWithin U] 0
⊢ f - g =ᶠ[𝓝[≠] x] 0

Tactic:
]

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
x : 𝕜
f g : 𝕜 → E
hf : MeromorphicAt f x
hg : MeromorphicAt g x
h₁x : x ∈ U
h₂x : AccPt x (𝓟 U)
h : f - g =ᶠ[codiscreteWithin U] 0
⊢ f - g =ᶠ[𝓝[≠] x] 0
Module: Mathlib.Analysis.Meromorphic.IsolatedZeros

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f g : 𝕜 → E
hf : MeromorphicOn f U
hg : MeromorphicOn g U
hU : Preperfect U
h : f =ᶠ[codiscreteWithin U] g
⊢ ∃ t, IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t ∧ U ⊆ t ∧ ∀ x ∈ t, f =ᶠ[𝓝[≠] x] g

Tactic:
]

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f g : 𝕜 → E
hf : MeromorphicOn f U
hg : MeromorphicOn g U
hU : Preperfect U
h : f =ᶠ[codiscreteWithin U] g
⊢ ∃ t, IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t ∧ U ⊆ t ∧ ∀ x ∈ t, f =ᶠ[𝓝[≠] x] g
Module: Mathlib.Analysis.Complex.Schwarz

Goal state before:
pp: case inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R₁ R₂ : ℝ
f : ℂ → E
c z : ℂ
hd : DifferentiableOn ℂ f (ball c R₁)
h_maps : MapsTo f (ball c R₁) (closedBall (f c) R₂)
hz : z ∈ ball c R₁
hne : z ≠ c
⊢ dist (f z) (f c) ≤ R₂ / R₁ * dist z c

pp: case inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R₁ R₂ : ℝ
f : ℂ → E
c z : ℂ
hd : DifferentiableOn ℂ f (ball c R₁)
h_maps : MapsTo f (ball c R₁) (closedBall (f c) R₂)
hz : z ∈ ball c R₁
hne : z ≠ c
⊢ 0 < dist z c

Tactic:
]

Goal state after:
case inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R₁ R₂ : ℝ
f : ℂ → E
c z : ℂ
hd : DifferentiableOn ℂ f (ball c R₁)
h_maps : MapsTo f (ball c R₁) (closedBall (f c) R₂)
hz : z ∈ ball c R₁
hne : z ≠ c
⊢ dist (f z) (f c) ≤ R₂ / R₁ * dist z c

case inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R₁ R₂ : ℝ
f : ℂ → E
c z : ℂ
hd : DifferentiableOn ℂ f (ball c R₁)
h_maps : MapsTo f (ball c R₁) (closedBall (f c) R₂)
hz : z ∈ ball c R₁
hne : z ≠ c
⊢ 0 < dist z c
Module: Mathlib.Analysis.Meromorphic.IsolatedZeros

Goal state before:
pp: case h.right
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f g : 𝕜 → E
hf : MeromorphicOn f U
hg : MeromorphicOn g U
hU : Preperfect U
h : f =ᶠ[codiscreteWithin U] g
x : 𝕜
hx : x ∈ U
⊢ f =ᶠ[𝓝[≠] x] g

Tactic:
]

Goal state after:
case h.right
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f g : 𝕜 → E
hf : MeromorphicOn f U
hg : MeromorphicOn g U
hU : Preperfect U
h : f =ᶠ[codiscreteWithin U] g
x : 𝕜
hx : x ∈ U
⊢ f =ᶠ[𝓝[≠] x] g
Module: Mathlib.Analysis.Meromorphic.IsolatedZeros

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f g : 𝕜 → E
hf : MeromorphicOn f U
hg : MeromorphicOn g U
h : f =ᶠ[codiscreteWithin U] g
⊢ ∀ x ∈ U, {x | deriv f x = deriv g x} ∪ Uᶜ ∈ 𝓝[≠] x

Tactic:
]

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f g : 𝕜 → E
hf : MeromorphicOn f U
hg : MeromorphicOn g U
h : f =ᶠ[codiscreteWithin U] g
⊢ ∀ x ∈ U, {x | deriv f x = deriv g x} ∪ Uᶜ ∈ 𝓝[≠] x
Module: Mathlib.Analysis.Complex.Schwarz

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
R₁ R₂ : ℝ
f : ℂ → E
c z₀ : ℂ
inst✝ : StrictConvexSpace ℝ E
hd : DifferentiableOn ℂ f (ball c R₁)
h_maps : MapsTo f (ball c R₁) (closedBall (f c) R₂)
h_z₀ : z₀ ∈ ball c R₁
h_eq : ‖dslope f c z₀‖ = R₂ / R₁
e : E →L[ℂ] UniformSpace.Completion E := UniformSpace.Completion.toComplL
g : ℂ → UniformSpace.Completion E := dslope (⇑e ∘ f) c
z : ℂ
hz : z ∈ ball c R₁
h_R₁ : 0 < R₁
w : ℂ
⊢ e (dslope f c w) = e (dslope f c w)

pp: case H
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
R₁ R₂ : ℝ
f : ℂ → E
c z₀ : ℂ
inst✝ : StrictConvexSpace ℝ E
hd : DifferentiableOn ℂ f (ball c R₁)
h_maps : MapsTo f (ball c R₁) (closedBall (f c) R₂)
h_z₀ : z₀ ∈ ball c R₁
h_eq : ‖dslope f c z₀‖ = R₂ / R₁
e : E →L[ℂ] UniformSpace.Completion E := UniformSpace.Completion.toComplL
g : ℂ → UniformSpace.Completion E := dslope (⇑e ∘ f) c
z : ℂ
hz : z ∈ ball c R₁
h_R₁ : 0 < R₁
w : ℂ
⊢ c = w → DifferentiableAt ℂ f c

Tactic:
]

Goal state after:
case H
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
R₁ R₂ : ℝ
f : ℂ → E
c z₀ : ℂ
inst✝ : StrictConvexSpace ℝ E
hd : DifferentiableOn ℂ f (ball c R₁)
h_maps : MapsTo f (ball c R₁) (closedBall (f c) R₂)
h_z₀ : z₀ ∈ ball c R₁
h_eq : ‖dslope f c z₀‖ = R₂ / R₁
e : E →L[ℂ] UniformSpace.Completion E := UniformSpace.Completion.toComplL
g : ℂ → UniformSpace.Completion E := dslope (⇑e ∘ f) c
z : ℂ
hz : z ∈ ball c R₁
h_R₁ : 0 < R₁
w : ℂ
⊢ c = w → DifferentiableAt ℂ f c
Module: Mathlib.Analysis.Meromorphic.IsolatedZeros

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f g : 𝕜 → E
hf : MeromorphicOn f U
hg : MeromorphicOn g U
h : f =ᶠ[codiscreteWithin U] g
x : 𝕜
hx : x ∈ U
hacc : ∀ᶠ (x : 𝕜) in 𝓝[≠] x, x ∉ U
⊢ {x | deriv f x = deriv g x} ∪ Uᶜ ∈ 𝓝[≠] x

Tactic:
]

Goal state after:
case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f g : 𝕜 → E
hf : MeromorphicOn f U
hg : MeromorphicOn g U
h : f =ᶠ[codiscreteWithin U] g
x : 𝕜
hx : x ∈ U
hacc : ∀ᶠ (x : 𝕜) in 𝓝[≠] x, x ∉ U
⊢ {x | deriv f x = deriv g x} ∪ Uᶜ ∈ 𝓝[≠] x
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: case e'_1
E : Type u_1
X : Type u_2
A : Type u_3
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
p : E → X
f : C(↑I × A, X)
g : ↑I × A → E
g_lifts : p ∘ g = ⇑f
cont_0 : Continuous[inst✝, inst✝²] fun x ↦ g (0, x)
a : A
cont_a : Continuous[_, inst✝²] fun x ↦ g (x, a)
q : E → OpenPartialHomeomorph E X
mem_source : ∀ (x : E), x ∈ (q x).source
hpq : ∀ (x : E), p = ↑(q x)
t : ℕ → ↑I
t_0 : t 0 = 0
t_mono : Monotone t
t_sub : ∀ (n : ℕ), ∃ i, Set.Icc (t n) (t (n + 1)) ⊆ (fun x ↦ g (x, a)) ⁻¹' (q i).source
n_max : ℕ
h_max : ∀ m ≥ n_max, t m = 1
this :
  ∀ (n : ℕ),
    ∃ N,
      a ∈ N ∧
        IsOpen[inst✝] N ∧
          ∃ g',
            ContinuousOn g' (Set.Icc 0 (t n) ×ˢ N) ∧
              p ∘ g' = ⇑f ∧ (∀ (a : A), g' (0, a) = g (0, a)) ∧ ∀ t' ≤ t n, g' (t', a) = g (t', a)
N : Set A
haN : a ∈ N
N_open : IsOpen[inst✝] N
hN :
  ∃ g',
    ContinuousOn g' (Set.Icc 0 1 ×ˢ N) ∧
      p ∘ g' = ⇑f ∧ (∀ (a : A), g' (0, a) = g (0, a)) ∧ ∀ t' ≤ 1, g' (t', a) = g (t', a)
x✝ : ↑I × A → E
⊢ ∀ (x : ↑I), x ∈ Set.Icc 0 1

Tactic:
]

Goal state after:
case e'_1
E : Type u_1
X : Type u_2
A : Type u_3
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
p : E → X
f : C(↑I × A, X)
g : ↑I × A → E
g_lifts : p ∘ g = ⇑f
cont_0 : Continuous[inst✝, inst✝²] fun x ↦ g (0, x)
a : A
cont_a : Continuous[_, inst✝²] fun x ↦ g (x, a)
q : E → OpenPartialHomeomorph E X
mem_source : ∀ (x : E), x ∈ (q x).source
hpq : ∀ (x : E), p = ↑(q x)
t : ℕ → ↑I
t_0 : t 0 = 0
t_mono : Monotone t
t_sub : ∀ (n : ℕ), ∃ i, Set.Icc (t n) (t (n + 1)) ⊆ (fun x ↦ g (x, a)) ⁻¹' (q i).source
n_max : ℕ
h_max : ∀ m ≥ n_max, t m = 1
this :
  ∀ (n : ℕ),
    ∃ N,
      a ∈ N ∧
        IsOpen[inst✝] N ∧
          ∃ g',
            ContinuousOn g' (Set.Icc 0 (t n) ×ˢ N) ∧
              p ∘ g' = ⇑f ∧ (∀ (a : A), g' (0, a) = g (0, a)) ∧ ∀ t' ≤ t n, g' (t', a) = g (t', a)
N : Set A
haN : a ∈ N
N_open : IsOpen[inst✝] N
hN :
  ∃ g',
    ContinuousOn g' (Set.Icc 0 1 ×ˢ N) ∧
      p ∘ g' = ⇑f ∧ (∀ (a : A), g' (0, a) = g (0, a)) ∧ ∀ t' ≤ 1, g' (t', a) = g (t', a)
x✝ : ↑I × A → E
⊢ ∀ (x : ↑I), x ∈ Set.Icc 0 1
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: case refine_1
E : Type u_1
X : Type u_2
A : Type u_3
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
p : E → X
f : C(↑I × A, X)
g : ↑I × A → E
g_lifts : p ∘ g = ⇑f
cont_0 : Continuous[inst✝, inst✝²] fun x ↦ g (0, x)
a : A
cont_a : Continuous[_, inst✝²] fun x ↦ g (x, a)
q : E → OpenPartialHomeomorph E X
mem_source : ∀ (x : E), x ∈ (q x).source
hpq : ∀ (x : E), p = ↑(q x)
t : ℕ → ↑I
t_0 : t 0 = 0
t_mono : Monotone t
t_sub : ∀ (n : ℕ), ∃ i, Set.Icc (t n) (t (n + 1)) ⊆ (fun x ↦ g (x, a)) ⁻¹' (q i).source
n_max : ℕ
h_max : ∀ m ≥ n_max, t m = 1
ta : ↑I × A
x✝ : ta ∈ Set.Icc 0 (t Nat.zero) ×ˢ Set.univ
ht : ta.1 = 0
right✝ : ta.2 ∈ Set.univ
⊢ g ta = ((fun x ↦ g (0, x)) ∘ Prod.snd) ta

Tactic:
]

Goal state after:
case refine_1
E : Type u_1
X : Type u_2
A : Type u_3
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
p : E → X
f : C(↑I × A, X)
g : ↑I × A → E
g_lifts : p ∘ g = ⇑f
cont_0 : Continuous[inst✝, inst✝²] fun x ↦ g (0, x)
a : A
cont_a : Continuous[_, inst✝²] fun x ↦ g (x, a)
q : E → OpenPartialHomeomorph E X
mem_source : ∀ (x : E), x ∈ (q x).source
hpq : ∀ (x : E), p = ↑(q x)
t : ℕ → ↑I
t_0 : t 0 = 0
t_mono : Monotone t
t_sub : ∀ (n : ℕ), ∃ i, Set.Icc (t n) (t (n + 1)) ⊆ (fun x ↦ g (x, a)) ⁻¹' (q i).source
n_max : ℕ
h_max : ∀ m ≥ n_max, t m = 1
ta : ↑I × A
x✝ : ta ∈ Set.Icc 0 (t Nat.zero) ×ˢ Set.univ
ht : ta.1 = 0
right✝ : ta.2 ∈ Set.univ
⊢ g ta = ((fun x ↦ g (0, x)) ∘ Prod.snd) ta
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: case refine_1
E : Type u_1
X : Type u_2
A : Type u_3
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
p : E → X
f : C(↑I × A, X)
g : ↑I × A → E
g_lifts : p ∘ g = ⇑f
cont_0 : Continuous[inst✝, inst✝²] fun x ↦ g (0, x)
a : A
cont_a : Continuous[_, inst✝²] fun x ↦ g (x, a)
q : E → OpenPartialHomeomorph E X
mem_source : ∀ (x : E), x ∈ (q x).source
hpq : ∀ (x : E), p = ↑(q x)
t : ℕ → ↑I
t_0 : t 0 = 0
t_mono : Monotone t
t_sub : ∀ (n : ℕ), ∃ i, Set.Icc (t n) (t (n + 1)) ⊆ (fun x ↦ g (x, a)) ⁻¹' (q i).source
n_max : ℕ
h_max : ∀ m ≥ n_max, t m = 1
ta : ↑I × A
x✝ : ta ∈ Set.Icc 0 (t Nat.zero) ×ˢ Set.univ
ht : ta.1 = 0
right✝ : ta.2 ∈ Set.univ
⊢ g (0, ta.2) = ((fun x ↦ g (0, x)) ∘ Prod.snd) (0, ta.2)

Tactic:
]

Goal state after:
case refine_1
E : Type u_1
X : Type u_2
A : Type u_3
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
p : E → X
f : C(↑I × A, X)
g : ↑I × A → E
g_lifts : p ∘ g = ⇑f
cont_0 : Continuous[inst✝, inst✝²] fun x ↦ g (0, x)
a : A
cont_a : Continuous[_, inst✝²] fun x ↦ g (x, a)
q : E → OpenPartialHomeomorph E X
mem_source : ∀ (x : E), x ∈ (q x).source
hpq : ∀ (x : E), p = ↑(q x)
t : ℕ → ↑I
t_0 : t 0 = 0
t_mono : Monotone t
t_sub : ∀ (n : ℕ), ∃ i, Set.Icc (t n) (t (n + 1)) ⊆ (fun x ↦ g (x, a)) ⁻¹' (q i).source
n_max : ℕ
h_max : ∀ m ≥ n_max, t m = 1
ta : ↑I × A
x✝ : ta ∈ Set.Icc 0 (t Nat.zero) ×ˢ Set.univ
ht : ta.1 = 0
right✝ : ta.2 ∈ Set.univ
⊢ g (0, ta.2) = ((fun x ↦ g (0, x)) ∘ Prod.snd) (0, ta.2)
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: E : Type u_1
X : Type u_2
A : Type u_3
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
p : E → X
f : C(↑I × A, X)
g : ↑I × A → E
g_lifts : p ∘ g = ⇑f
cont_0 : Continuous[inst✝, inst✝²] fun x ↦ g (0, x)
a : A
cont_a : Continuous[_, inst✝²] fun x ↦ g (x, a)
q : E → OpenPartialHomeomorph E X
mem_source : ∀ (x : E), x ∈ (q x).source
hpq : ∀ (x : E), p = ↑(q x)
t : ℕ → ↑I
t_0 : t 0 = 0
t_mono : Monotone t
t_sub : ∀ (n : ℕ), ∃ i, Set.Icc (t n) (t (n + 1)) ⊆ (fun x ↦ g (x, a)) ⁻¹' (q i).source
n_max : ℕ
h_max : ∀ m ≥ n_max, t m = 1
n : ℕ
x✝ :
  ∃ N,
    a ∈ N ∧
      IsOpen[inst✝] N ∧
        ∃ g',
          ContinuousOn g' (Set.Icc 0 (t n) ×ˢ N) ∧
            p ∘ g' = ⇑f ∧ (∀ (a : A), g' (0, a) = g (0, a)) ∧ ∀ t' ≤ t n, g' (t', a) = g (t', a)
N : Set A
haN : a ∈ N
N_open : IsOpen[inst✝] N
g' : ↑I × A → E
cont_g' : ContinuousOn g' (Set.Icc 0 (t n) ×ˢ N)
g'_lifts : p ∘ g' = ⇑f
g'_0 : ∀ (a : A), g' (0, a) = g (0, a)
g'_a : ∀ t' ≤ t n, g' (t', a) = g (t', a)
e : E
h_sub : Set.Icc (t n) (t (n + 1)) ⊆ (fun x ↦ g (x, a)) ⁻¹' (q e).source
t0 : ↑I
a' : A
ht : (t0, a').1 ∈ Set.Icc (t n) (t (n + 1))
ha : (t0, a').2 = a
⊢ (t0, a') ∈ ⇑f ⁻¹' (q e).target

Tactic:
]

Goal state after:
E : Type u_1
X : Type u_2
A : Type u_3
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
p : E → X
f : C(↑I × A, X)
g : ↑I × A → E
g_lifts : p ∘ g = ⇑f
cont_0 : Continuous[inst✝, inst✝²] fun x ↦ g (0, x)
a : A
cont_a : Continuous[_, inst✝²] fun x ↦ g (x, a)
q : E → OpenPartialHomeomorph E X
mem_source : ∀ (x : E), x ∈ (q x).source
hpq : ∀ (x : E), p = ↑(q x)
t : ℕ → ↑I
t_0 : t 0 = 0
t_mono : Monotone t
t_sub : ∀ (n : ℕ), ∃ i, Set.Icc (t n) (t (n + 1)) ⊆ (fun x ↦ g (x, a)) ⁻¹' (q i).source
n_max : ℕ
h_max : ∀ m ≥ n_max, t m = 1
n : ℕ
x✝ :
  ∃ N,
    a ∈ N ∧
      IsOpen[inst✝] N ∧
        ∃ g',
          ContinuousOn g' (Set.Icc 0 (t n) ×ˢ N) ∧
            p ∘ g' = ⇑f ∧ (∀ (a : A), g' (0, a) = g (0, a)) ∧ ∀ t' ≤ t n, g' (t', a) = g (t', a)
N : Set A
haN : a ∈ N
N_open : IsOpen[inst✝] N
g' : ↑I × A → E
cont_g' : ContinuousOn g' (Set.Icc 0 (t n) ×ˢ N)
g'_lifts : p ∘ g' = ⇑f
g'_0 : ∀ (a : A), g' (0, a) = g (0, a)
g'_a : ∀ t' ≤ t n, g' (t', a) = g (t', a)
e : E
h_sub : Set.Icc (t n) (t (n + 1)) ⊆ (fun x ↦ g (x, a)) ⁻¹' (q e).source
t0 : ↑I
a' : A
ht : (t0, a').1 ∈ Set.Icc (t n) (t (n + 1))
ha : (t0, a').2 = a
⊢ (t0, a') ∈ ⇑f ⁻¹' (q e).target
Module: Mathlib.Analysis.Meromorphic.Divisor

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
U : Set 𝕜
z : 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
hf : AnalyticOnNhd 𝕜 f U
hz : z ∈ U
⊢ (ENat.map Nat.cast (analyticOrderAt f z)).untop₀ = (ENat.map Nat.cast (analyticOrderAt f z)).untop₀

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: E : Type u_1
X : Type u_2
A : Type u_3
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
p : E → X
f : C(↑I × A, X)
g : ↑I × A → E
g_lifts : p ∘ g = ⇑f
cont_0 : Continuous[inst✝, inst✝²] fun x ↦ g (0, x)
a : A
cont_a : Continuous[_, inst✝²] fun x ↦ g (x, a)
q : E → OpenPartialHomeomorph E X
mem_source : ∀ (x : E), x ∈ (q x).source
hpq : ∀ (x : E), p = ↑(q x)
t : ℕ → ↑I
t_0 : t 0 = 0
t_mono : Monotone t
t_sub : ∀ (n : ℕ), ∃ i, Set.Icc (t n) (t (n + 1)) ⊆ (fun x ↦ g (x, a)) ⁻¹' (q i).source
n_max : ℕ
h_max : ∀ m ≥ n_max, t m = 1
n : ℕ
x✝ :
  ∃ N,
    a ∈ N ∧
      IsOpen[inst✝] N ∧
        ∃ g',
          ContinuousOn g' (Set.Icc 0 (t n) ×ˢ N) ∧
            p ∘ g' = ⇑f ∧ (∀ (a : A), g' (0, a) = g (0, a)) ∧ ∀ t' ≤ t n, g' (t', a) = g (t', a)
N : Set A
haN : a ∈ N
N_open : IsOpen[inst✝] N
g' : ↑I × A → E
cont_g' : ContinuousOn g' (Set.Icc 0 (t n) ×ˢ N)
g'_lifts : p ∘ g' = ⇑f
g'_0 : ∀ (a : A), g' (0, a) = g (0, a)
g'_a : ∀ t' ≤ t n, g' (t', a) = g (t', a)
e : E
h_sub : Set.Icc (t n) (t (n + 1)) ⊆ (fun x ↦ g (x, a)) ⁻¹' (q e).source
t0 : ↑I
a' : A
ht : (t0, a').1 ∈ Set.Icc (t n) (t (n + 1))
ha : a' = a
⊢ (t0, a) ∈ ↑(q e) ∘ g ⁻¹' (q e).target

Tactic:
]

Goal state after:
E : Type u_1
X : Type u_2
A : Type u_3
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
p : E → X
f : C(↑I × A, X)
g : ↑I × A → E
g_lifts : p ∘ g = ⇑f
cont_0 : Continuous[inst✝, inst✝²] fun x ↦ g (0, x)
a : A
cont_a : Continuous[_, inst✝²] fun x ↦ g (x, a)
q : E → OpenPartialHomeomorph E X
mem_source : ∀ (x : E), x ∈ (q x).source
hpq : ∀ (x : E), p = ↑(q x)
t : ℕ → ↑I
t_0 : t 0 = 0
t_mono : Monotone t
t_sub : ∀ (n : ℕ), ∃ i, Set.Icc (t n) (t (n + 1)) ⊆ (fun x ↦ g (x, a)) ⁻¹' (q i).source
n_max : ℕ
h_max : ∀ m ≥ n_max, t m = 1
n : ℕ
x✝ :
  ∃ N,
    a ∈ N ∧
      IsOpen[inst✝] N ∧
        ∃ g',
          ContinuousOn g' (Set.Icc 0 (t n) ×ˢ N) ∧
            p ∘ g' = ⇑f ∧ (∀ (a : A), g' (0, a) = g (0, a)) ∧ ∀ t' ≤ t n, g' (t', a) = g (t', a)
N : Set A
haN : a ∈ N
N_open : IsOpen[inst✝] N
g' : ↑I × A → E
cont_g' : ContinuousOn g' (Set.Icc 0 (t n) ×ˢ N)
g'_lifts : p ∘ g' = ⇑f
g'_0 : ∀ (a : A), g' (0, a) = g (0, a)
g'_a : ∀ t' ≤ t n, g' (t', a) = g (t', a)
e : E
h_sub : Set.Icc (t n) (t (n + 1)) ⊆ (fun x ↦ g (x, a)) ⁻¹' (q e).source
t0 : ↑I
a' : A
ht : (t0, a').1 ∈ Set.Icc (t n) (t (n + 1))
ha : a' = a
⊢ (t0, a) ∈ ↑(q e) ∘ g ⁻¹' (q e).target
Module: Mathlib.Analysis.Meromorphic.Divisor

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
U : Set 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
V : Set 𝕜
hf : MeromorphicOn f U
hU : IsCompact U
hV : V ⊆ U
b : 𝕜
hb : b ∈ (divisor f V).support
⊢ ¬(meromorphicOrderAt f b).untop₀ = 0

Tactic:
]

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
U : Set 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
V : Set 𝕜
hf : MeromorphicOn f U
hU : IsCompact U
hV : V ⊆ U
b : 𝕜
hb : b ∈ (divisor f V).support
⊢ ¬(meromorphicOrderAt f b).untop₀ = 0