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https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.prod_mem_inf_of_mem_atInf | [353, 1] | [366, 13] | simp only [mem_prod_eq, mem_image, mem_union, mem_singleton_iff, mem_univ, true_and_iff,
Prod.ext_iff] at yt m ⊢ | case intro.intro.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
s : Set (X × ℂ)
x : X
f : s ∈ (𝓝 x).prod atInf
t : Set X
tx : t ∈ 𝓝 x
u : Set ℂ
ui : u ∈ atInf
sub : t ×ˢ u ⊆ s
y : X
z : 𝕊
yt : (y, z).1 ∈ t
m : (y, z).2 ∈ (fun z => ↑z) '' u ∪ {∞}
⊢ (y, z) ∈ (fun p => (p.1, ↑p.2)) '' s ∪ univ ×ˢ {∞} | case intro.intro.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
s : Set (X × ℂ)
x : X
f : s ∈ (𝓝 x).prod atInf
t : Set X
tx : t ∈ 𝓝 x
u : Set ℂ
ui : u ∈ atInf
sub : t ×ˢ u ⊆ s
y : X
z : 𝕊
yt : y ∈ t
m : (∃ x ∈ u, ↑x = z) ∨ z = ∞
⊢ (∃ x ∈ s, x.1 = y ∧ ↑x.2 = z) ∨ z = ∞ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
s : Set (X × ℂ)
x : X
f : s ∈ (𝓝 x).prod atInf
t : Set X
tx : t ∈ 𝓝 x
u : Set ℂ
ui : u ∈ atInf
sub : t ×ˢ u ⊆ s
y : X
z : 𝕊
yt : (y, z).1 ∈ t
m : (y, z).2 ∈ (fun z => ↑z) '' u ∪ {∞}
⊢ (y, z) ∈ (fun p => (p.1, ↑p.2)) '' s ∪ univ ×ˢ {∞}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.prod_mem_inf_of_mem_atInf | [353, 1] | [366, 13] | induction' z using OnePoint.rec with z | case intro.intro.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
s : Set (X × ℂ)
x : X
f : s ∈ (𝓝 x).prod atInf
t : Set X
tx : t ∈ 𝓝 x
u : Set ℂ
ui : u ∈ atInf
sub : t ×ˢ u ⊆ s
y : X
z : 𝕊
yt : y ∈ t
m : (∃ x ∈ u, ↑x = z) ∨ z = ∞
⊢ (∃ x ∈ s, x.1 = y ∧ ↑x.2 = z) ∨ z = ∞ | case intro.intro.intro.intro.h₁
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
s : Set (X × ℂ)
x : X
f : s ∈ (𝓝 x).prod atInf
t : Set X
tx : t ∈ 𝓝 x
u : Set ℂ
ui : u ∈ atInf
sub : t ×ˢ u ⊆ s
y : X
yt : y ∈ t
m : (∃ x ∈ u, ↑x = ∞) ∨ ∞ = ∞
⊢ (∃ x ∈ s, x.1 = y ∧ ↑x.2 = ∞) ∨ ∞ = ∞
case intro.intro.intro.intro.h₂
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
s : Set (X × ℂ)
x : X
f : s ∈ (𝓝 x).prod atInf
t : Set X
tx : t ∈ 𝓝 x
u : Set ℂ
ui : u ∈ atInf
sub : t ×ˢ u ⊆ s
y : X
yt : y ∈ t
z : ℂ
m : (∃ x ∈ u, ↑x = ↑z) ∨ ↑z = ∞
⊢ (∃ x ∈ s, x.1 = y ∧ ↑x.2 = ↑z) ∨ ↑z = ∞ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
s : Set (X × ℂ)
x : X
f : s ∈ (𝓝 x).prod atInf
t : Set X
tx : t ∈ 𝓝 x
u : Set ℂ
ui : u ∈ atInf
sub : t ×ˢ u ⊆ s
y : X
z : 𝕊
yt : y ∈ t
m : (∃ x ∈ u, ↑x = z) ∨ z = ∞
⊢ (∃ x ∈ s, x.1 = y ∧ ↑x.2 = z) ∨ z = ∞
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.prod_mem_inf_of_mem_atInf | [353, 1] | [366, 13] | simp only [eq_self_iff_true, or_true_iff] | case intro.intro.intro.intro.h₁
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
s : Set (X × ℂ)
x : X
f : s ∈ (𝓝 x).prod atInf
t : Set X
tx : t ∈ 𝓝 x
u : Set ℂ
ui : u ∈ atInf
sub : t ×ˢ u ⊆ s
y : X
yt : y ∈ t
m : (∃ x ∈ u, ↑x = ∞) ∨ ∞ = ∞
⊢ (∃ x ∈ s, x.1 = y ∧ ↑x.2 = ∞) ∨ ∞ = ∞ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.h₁
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
s : Set (X × ℂ)
x : X
f : s ∈ (𝓝 x).prod atInf
t : Set X
tx : t ∈ 𝓝 x
u : Set ℂ
ui : u ∈ atInf
sub : t ×ˢ u ⊆ s
y : X
yt : y ∈ t
m : (∃ x ∈ u, ↑x = ∞) ∨ ∞ = ∞
⊢ (∃ x ∈ s, x.1 = y ∧ ↑x.2 = ∞) ∨ ∞ = ∞
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.prod_mem_inf_of_mem_atInf | [353, 1] | [366, 13] | simp only [coe_eq_inf_iff, or_false_iff, coe_eq_coe] at m ⊢ | case intro.intro.intro.intro.h₂
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
s : Set (X × ℂ)
x : X
f : s ∈ (𝓝 x).prod atInf
t : Set X
tx : t ∈ 𝓝 x
u : Set ℂ
ui : u ∈ atInf
sub : t ×ˢ u ⊆ s
y : X
yt : y ∈ t
z : ℂ
m : (∃ x ∈ u, ↑x = ↑z) ∨ ↑z = ∞
⊢ (∃ x ∈ s, x.1 = y ∧ ↑x.2 = ↑z) ∨ ↑z = ∞ | case intro.intro.intro.intro.h₂
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
s : Set (X × ℂ)
x : X
f : s ∈ (𝓝 x).prod atInf
t : Set X
tx : t ∈ 𝓝 x
u : Set ℂ
ui : u ∈ atInf
sub : t ×ˢ u ⊆ s
y : X
yt : y ∈ t
z : ℂ
m : ∃ x ∈ u, x = z
⊢ ∃ x ∈ s, x.1 = y ∧ x.2 = z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.h₂
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
s : Set (X × ℂ)
x : X
f : s ∈ (𝓝 x).prod atInf
t : Set X
tx : t ∈ 𝓝 x
u : Set ℂ
ui : u ∈ atInf
sub : t ×ˢ u ⊆ s
y : X
yt : y ∈ t
z : ℂ
m : (∃ x ∈ u, ↑x = ↑z) ∨ ↑z = ∞
⊢ (∃ x ∈ s, x.1 = y ∧ ↑x.2 = ↑z) ∨ ↑z = ∞
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.prod_mem_inf_of_mem_atInf | [353, 1] | [366, 13] | rcases m with ⟨w, wu, wz⟩ | case intro.intro.intro.intro.h₂
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
s : Set (X × ℂ)
x : X
f : s ∈ (𝓝 x).prod atInf
t : Set X
tx : t ∈ 𝓝 x
u : Set ℂ
ui : u ∈ atInf
sub : t ×ˢ u ⊆ s
y : X
yt : y ∈ t
z : ℂ
m : ∃ x ∈ u, x = z
⊢ ∃ x ∈ s, x.1 = y ∧ x.2 = z | case intro.intro.intro.intro.h₂.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
s : Set (X × ℂ)
x : X
f : s ∈ (𝓝 x).prod atInf
t : Set X
tx : t ∈ 𝓝 x
u : Set ℂ
ui : u ∈ atInf
sub : t ×ˢ u ⊆ s
y : X
yt : y ∈ t
z w : ℂ
wu : w ∈ u
wz : w = z
⊢ ∃ x ∈ s, x.1 = y ∧ x.2 = z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.h₂
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
s : Set (X × ℂ)
x : X
f : s ∈ (𝓝 x).prod atInf
t : Set X
tx : t ∈ 𝓝 x
u : Set ℂ
ui : u ∈ atInf
sub : t ×ˢ u ⊆ s
y : X
yt : y ∈ t
z : ℂ
m : ∃ x ∈ u, x = z
⊢ ∃ x ∈ s, x.1 = y ∧ x.2 = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.prod_mem_inf_of_mem_atInf | [353, 1] | [366, 13] | refine ⟨⟨y, z⟩, sub (mk_mem_prod yt ?_), rfl, rfl⟩ | case intro.intro.intro.intro.h₂.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
s : Set (X × ℂ)
x : X
f : s ∈ (𝓝 x).prod atInf
t : Set X
tx : t ∈ 𝓝 x
u : Set ℂ
ui : u ∈ atInf
sub : t ×ˢ u ⊆ s
y : X
yt : y ∈ t
z w : ℂ
wu : w ∈ u
wz : w = z
⊢ ∃ x ∈ s, x.1 = y ∧ x.2 = z | case intro.intro.intro.intro.h₂.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
s : Set (X × ℂ)
x : X
f : s ∈ (𝓝 x).prod atInf
t : Set X
tx : t ∈ 𝓝 x
u : Set ℂ
ui : u ∈ atInf
sub : t ×ˢ u ⊆ s
y : X
yt : y ∈ t
z w : ℂ
wu : w ∈ u
wz : w = z
⊢ z ∈ u | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.h₂.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
s : Set (X × ℂ)
x : X
f : s ∈ (𝓝 x).prod atInf
t : Set X
tx : t ∈ 𝓝 x
u : Set ℂ
ui : u ∈ atInf
sub : t ×ˢ u ⊆ s
y : X
yt : y ∈ t
z w : ℂ
wu : w ∈ u
wz : w = z
⊢ ∃ x ∈ s, x.1 = y ∧ x.2 = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.prod_mem_inf_of_mem_atInf | [353, 1] | [366, 13] | rw [← wz] | case intro.intro.intro.intro.h₂.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
s : Set (X × ℂ)
x : X
f : s ∈ (𝓝 x).prod atInf
t : Set X
tx : t ∈ 𝓝 x
u : Set ℂ
ui : u ∈ atInf
sub : t ×ˢ u ⊆ s
y : X
yt : y ∈ t
z w : ℂ
wu : w ∈ u
wz : w = z
⊢ z ∈ u | case intro.intro.intro.intro.h₂.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
s : Set (X × ℂ)
x : X
f : s ∈ (𝓝 x).prod atInf
t : Set X
tx : t ∈ 𝓝 x
u : Set ℂ
ui : u ∈ atInf
sub : t ×ˢ u ⊆ s
y : X
yt : y ∈ t
z w : ℂ
wu : w ∈ u
wz : w = z
⊢ w ∈ u | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.h₂.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
s : Set (X × ℂ)
x : X
f : s ∈ (𝓝 x).prod atInf
t : Set X
tx : t ∈ 𝓝 x
u : Set ℂ
ui : u ∈ atInf
sub : t ×ˢ u ⊆ s
y : X
yt : y ∈ t
z w : ℂ
wu : w ∈ u
wz : w = z
⊢ z ∈ u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.prod_mem_inf_of_mem_atInf | [353, 1] | [366, 13] | exact wu | case intro.intro.intro.intro.h₂.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
s : Set (X × ℂ)
x : X
f : s ∈ (𝓝 x).prod atInf
t : Set X
tx : t ∈ 𝓝 x
u : Set ℂ
ui : u ∈ atInf
sub : t ×ˢ u ⊆ s
y : X
yt : y ∈ t
z w : ℂ
wu : w ∈ u
wz : w = z
⊢ w ∈ u | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.h₂.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
s : Set (X × ℂ)
x : X
f : s ∈ (𝓝 x).prod atInf
t : Set X
tx : t ∈ 𝓝 x
u : Set ℂ
ui : u ∈ atInf
sub : t ×ˢ u ⊆ s
y : X
yt : y ∈ t
z w : ℂ
wu : w ∈ u
wz : w = z
⊢ w ∈ u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_coe | [369, 1] | [373, 101] | rw [holomorphic_iff] | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
⊢ Holomorphic I I fun z => ↑z | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
⊢ (Continuous fun z => ↑z) ∧
∀ (x : ℂ), AnalyticAt ℂ (↑(extChartAt I ↑x) ∘ (fun z => ↑z) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
⊢ Holomorphic I I fun z => ↑z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_coe | [369, 1] | [373, 101] | use continuous_coe | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
⊢ (Continuous fun z => ↑z) ∧
∀ (x : ℂ), AnalyticAt ℂ (↑(extChartAt I ↑x) ∘ (fun z => ↑z) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) | case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
⊢ ∀ (x : ℂ), AnalyticAt ℂ (↑(extChartAt I ↑x) ∘ (fun z => ↑z) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
⊢ (Continuous fun z => ↑z) ∧
∀ (x : ℂ), AnalyticAt ℂ (↑(extChartAt I ↑x) ∘ (fun z => ↑z) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_coe | [369, 1] | [373, 101] | intro z | case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
⊢ ∀ (x : ℂ), AnalyticAt ℂ (↑(extChartAt I ↑x) ∘ (fun z => ↑z) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) | case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
⊢ AnalyticAt ℂ (↑(extChartAt I ↑z) ∘ (fun z => ↑z) ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) z) | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
⊢ ∀ (x : ℂ), AnalyticAt ℂ (↑(extChartAt I ↑x) ∘ (fun z => ↑z) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_coe | [369, 1] | [373, 101] | simp only [extChartAt_coe, extChartAt_eq_refl, PartialEquiv.refl_symm, PartialEquiv.refl_coe,
Function.comp_id, id_eq, Function.comp, PartialEquiv.invFun_as_coe] | case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
⊢ AnalyticAt ℂ (↑(extChartAt I ↑z) ∘ (fun z => ↑z) ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) z) | case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
⊢ AnalyticAt ℂ (fun x => ↑coePartialEquiv.symm ↑x) z | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
⊢ AnalyticAt ℂ (↑(extChartAt I ↑z) ∘ (fun z => ↑z) ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_coe | [369, 1] | [373, 101] | rw [← PartialEquiv.invFun_as_coe] | case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
⊢ AnalyticAt ℂ (fun x => ↑coePartialEquiv.symm ↑x) z | case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
⊢ AnalyticAt ℂ (fun x => coePartialEquiv.invFun ↑x) z | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
⊢ AnalyticAt ℂ (fun x => ↑coePartialEquiv.symm ↑x) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_coe | [369, 1] | [373, 101] | simp only [coePartialEquiv, toComplex_coe] | case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
⊢ AnalyticAt ℂ (fun x => coePartialEquiv.invFun ↑x) z | case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
⊢ AnalyticAt ℂ (fun x => x) z | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
⊢ AnalyticAt ℂ (fun x => coePartialEquiv.invFun ↑x) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_coe | [369, 1] | [373, 101] | apply analyticAt_id | case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
⊢ AnalyticAt ℂ (fun x => x) z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
⊢ AnalyticAt ℂ (fun x => x) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_toComplex | [376, 1] | [380, 22] | rw [holomorphicAt_iff] | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
⊢ HolomorphicAt I I OnePoint.toComplex ↑z | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
⊢ ContinuousAt OnePoint.toComplex ↑z ∧
AnalyticAt ℂ (↑(extChartAt I (↑z).toComplex) ∘ OnePoint.toComplex ∘ ↑(extChartAt I ↑z).symm) (↑(extChartAt I ↑z) ↑z) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
⊢ HolomorphicAt I I OnePoint.toComplex ↑z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_toComplex | [376, 1] | [380, 22] | use continuousAt_toComplex | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
⊢ ContinuousAt OnePoint.toComplex ↑z ∧
AnalyticAt ℂ (↑(extChartAt I (↑z).toComplex) ∘ OnePoint.toComplex ∘ ↑(extChartAt I ↑z).symm) (↑(extChartAt I ↑z) ↑z) | case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
⊢ AnalyticAt ℂ (↑(extChartAt I (↑z).toComplex) ∘ OnePoint.toComplex ∘ ↑(extChartAt I ↑z).symm) (↑(extChartAt I ↑z) ↑z) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
⊢ ContinuousAt OnePoint.toComplex ↑z ∧
AnalyticAt ℂ (↑(extChartAt I (↑z).toComplex) ∘ OnePoint.toComplex ∘ ↑(extChartAt I ↑z).symm) (↑(extChartAt I ↑z) ↑z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_toComplex | [376, 1] | [380, 22] | simp only [toComplex_coe, Function.comp, extChartAt_coe, extChartAt_eq_refl, PartialEquiv.refl_coe,
id, PartialEquiv.symm_symm, coePartialEquiv_apply, coePartialEquiv_symm_apply] | case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
⊢ AnalyticAt ℂ (↑(extChartAt I (↑z).toComplex) ∘ OnePoint.toComplex ∘ ↑(extChartAt I ↑z).symm) (↑(extChartAt I ↑z) ↑z) | case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
⊢ AnalyticAt ℂ (fun x => x) z | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
⊢ AnalyticAt ℂ (↑(extChartAt I (↑z).toComplex) ∘ OnePoint.toComplex ∘ ↑(extChartAt I ↑z).symm) (↑(extChartAt I ↑z) ↑z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_toComplex | [376, 1] | [380, 22] | apply analyticAt_id | case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
⊢ AnalyticAt ℂ (fun x => x) z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
⊢ AnalyticAt ℂ (fun x => x) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_inv | [383, 1] | [399, 63] | rw [holomorphic_iff] | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
⊢ Holomorphic I I fun z => z⁻¹ | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
⊢ (Continuous fun z => z⁻¹) ∧
∀ (x : 𝕊), AnalyticAt ℂ (↑(extChartAt I x⁻¹) ∘ (fun z => z⁻¹) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
⊢ Holomorphic I I fun z => z⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_inv | [383, 1] | [399, 63] | use continuous_inv | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
⊢ (Continuous fun z => z⁻¹) ∧
∀ (x : 𝕊), AnalyticAt ℂ (↑(extChartAt I x⁻¹) ∘ (fun z => z⁻¹) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) | case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
⊢ ∀ (x : 𝕊), AnalyticAt ℂ (↑(extChartAt I x⁻¹) ∘ (fun z => z⁻¹) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
⊢ (Continuous fun z => z⁻¹) ∧
∀ (x : 𝕊), AnalyticAt ℂ (↑(extChartAt I x⁻¹) ∘ (fun z => z⁻¹) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_inv | [383, 1] | [399, 63] | intro z | case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
⊢ ∀ (x : 𝕊), AnalyticAt ℂ (↑(extChartAt I x⁻¹) ∘ (fun z => z⁻¹) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) | case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : 𝕊
⊢ AnalyticAt ℂ (↑(extChartAt I z⁻¹) ∘ (fun z => z⁻¹) ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) z) | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
⊢ ∀ (x : 𝕊), AnalyticAt ℂ (↑(extChartAt I x⁻¹) ∘ (fun z => z⁻¹) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_inv | [383, 1] | [399, 63] | induction' z using OnePoint.rec with z | case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : 𝕊
⊢ AnalyticAt ℂ (↑(extChartAt I z⁻¹) ∘ (fun z => z⁻¹) ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) z) | case right.h₁
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
⊢ AnalyticAt ℂ (↑(extChartAt I ∞⁻¹) ∘ (fun z => z⁻¹) ∘ ↑(extChartAt I ∞).symm) (↑(extChartAt I ∞) ∞)
case right.h₂
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
⊢ AnalyticAt ℂ (↑(extChartAt I (↑z)⁻¹) ∘ (fun z => z⁻¹) ∘ ↑(extChartAt I ↑z).symm) (↑(extChartAt I ↑z) ↑z) | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : 𝕊
⊢ AnalyticAt ℂ (↑(extChartAt I z⁻¹) ∘ (fun z => z⁻¹) ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_inv | [383, 1] | [399, 63] | simp only [inv_inf, extChartAt_inf, ← coe_zero, extChartAt_coe, Function.comp,
PartialEquiv.trans_apply, Equiv.toPartialEquiv_apply, invEquiv_apply, coePartialEquiv_symm_apply,
toComplex_coe, PartialEquiv.coe_trans_symm, PartialEquiv.symm_symm, coePartialEquiv_apply,
Equiv.toPartialEquiv_symm_apply, invEquiv_symm, inv_inv] | case right.h₁
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
⊢ AnalyticAt ℂ (↑(extChartAt I ∞⁻¹) ∘ (fun z => z⁻¹) ∘ ↑(extChartAt I ∞).symm) (↑(extChartAt I ∞) ∞) | case right.h₁
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
⊢ AnalyticAt ℂ (fun x => x) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case right.h₁
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
⊢ AnalyticAt ℂ (↑(extChartAt I ∞⁻¹) ∘ (fun z => z⁻¹) ∘ ↑(extChartAt I ∞).symm) (↑(extChartAt I ∞) ∞)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_inv | [383, 1] | [399, 63] | apply analyticAt_id | case right.h₁
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
⊢ AnalyticAt ℂ (fun x => x) 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right.h₁
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
⊢ AnalyticAt ℂ (fun x => x) 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_inv | [383, 1] | [399, 63] | simp only [extChartAt_coe, PartialEquiv.symm_symm, Function.comp, coePartialEquiv_apply,
coePartialEquiv_symm_apply, toComplex_coe] | case right.h₂
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
⊢ AnalyticAt ℂ (↑(extChartAt I (↑z)⁻¹) ∘ (fun z => z⁻¹) ∘ ↑(extChartAt I ↑z).symm) (↑(extChartAt I ↑z) ↑z) | case right.h₂
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
⊢ AnalyticAt ℂ (fun x => ↑(extChartAt I (↑z)⁻¹) (↑x)⁻¹) z | Please generate a tactic in lean4 to solve the state.
STATE:
case right.h₂
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
⊢ AnalyticAt ℂ (↑(extChartAt I (↑z)⁻¹) ∘ (fun z => z⁻¹) ∘ ↑(extChartAt I ↑z).symm) (↑(extChartAt I ↑z) ↑z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_inv | [383, 1] | [399, 63] | by_cases z0 : z = 0 | case right.h₂
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
⊢ AnalyticAt ℂ (fun x => ↑(extChartAt I (↑z)⁻¹) (↑x)⁻¹) z | case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
z0 : z = 0
⊢ AnalyticAt ℂ (fun x => ↑(extChartAt I (↑z)⁻¹) (↑x)⁻¹) z
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
z0 : ¬z = 0
⊢ AnalyticAt ℂ (fun x => ↑(extChartAt I (↑z)⁻¹) (↑x)⁻¹) z | Please generate a tactic in lean4 to solve the state.
STATE:
case right.h₂
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
⊢ AnalyticAt ℂ (fun x => ↑(extChartAt I (↑z)⁻¹) (↑x)⁻¹) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_inv | [383, 1] | [399, 63] | simp only [z0, coe_zero, extChartAt_inf, PartialEquiv.trans_apply, coePartialEquiv_symm_apply,
invEquiv_apply, Equiv.toPartialEquiv_apply, inv_zero', inv_inv, toComplex_coe] | case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
z0 : z = 0
⊢ AnalyticAt ℂ (fun x => ↑(extChartAt I (↑z)⁻¹) (↑x)⁻¹) z | case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
z0 : z = 0
⊢ AnalyticAt ℂ (fun x => x) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
z0 : z = 0
⊢ AnalyticAt ℂ (fun x => ↑(extChartAt I (↑z)⁻¹) (↑x)⁻¹) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_inv | [383, 1] | [399, 63] | apply analyticAt_id | case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
z0 : z = 0
⊢ AnalyticAt ℂ (fun x => x) 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
z0 : z = 0
⊢ AnalyticAt ℂ (fun x => x) 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_inv | [383, 1] | [399, 63] | simp only [inv_coe z0, extChartAt_coe, coePartialEquiv_symm_apply] | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
z0 : ¬z = 0
⊢ AnalyticAt ℂ (fun x => ↑(extChartAt I (↑z)⁻¹) (↑x)⁻¹) z | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
z0 : ¬z = 0
⊢ AnalyticAt ℂ (fun x => (↑x)⁻¹.toComplex) z | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
z0 : ¬z = 0
⊢ AnalyticAt ℂ (fun x => ↑(extChartAt I (↑z)⁻¹) (↑x)⁻¹) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_inv | [383, 1] | [399, 63] | refine ((analyticAt_id _ _).inv z0).congr ?_ | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
z0 : ¬z = 0
⊢ AnalyticAt ℂ (fun x => (↑x)⁻¹.toComplex) z | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
z0 : ¬z = 0
⊢ (𝓝 z).EventuallyEq (fun x => (id x)⁻¹) fun x => (↑x)⁻¹.toComplex | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
z0 : ¬z = 0
⊢ AnalyticAt ℂ (fun x => (↑x)⁻¹.toComplex) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_inv | [383, 1] | [399, 63] | refine (continuousAt_id.eventually_ne z0).mp (eventually_of_forall fun w w0 ↦ ?_) | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
z0 : ¬z = 0
⊢ (𝓝 z).EventuallyEq (fun x => (id x)⁻¹) fun x => (↑x)⁻¹.toComplex | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
z0 : ¬z = 0
w : ℂ
w0 : id w ≠ 0
⊢ (fun x => (id x)⁻¹) w = (fun x => (↑x)⁻¹.toComplex) w | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
z0 : ¬z = 0
⊢ (𝓝 z).EventuallyEq (fun x => (id x)⁻¹) fun x => (↑x)⁻¹.toComplex
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_inv | [383, 1] | [399, 63] | rw [id] at w0 | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
z0 : ¬z = 0
w : ℂ
w0 : id w ≠ 0
⊢ (fun x => (id x)⁻¹) w = (fun x => (↑x)⁻¹.toComplex) w | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
z0 : ¬z = 0
w : ℂ
w0 : w ≠ 0
⊢ (fun x => (id x)⁻¹) w = (fun x => (↑x)⁻¹.toComplex) w | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
z0 : ¬z = 0
w : ℂ
w0 : id w ≠ 0
⊢ (fun x => (id x)⁻¹) w = (fun x => (↑x)⁻¹.toComplex) w
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_inv | [383, 1] | [399, 63] | simp only [inv_coe w0, toComplex_coe, id] | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
z0 : ¬z = 0
w : ℂ
w0 : w ≠ 0
⊢ (fun x => (id x)⁻¹) w = (fun x => (↑x)⁻¹.toComplex) w | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
z : ℂ
z0 : ¬z = 0
w : ℂ
w0 : w ≠ 0
⊢ (fun x => (id x)⁻¹) w = (fun x => (↑x)⁻¹.toComplex) w
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuousAt_fill_coe | [426, 1] | [428, 69] | simp only [OnePoint.continuousAt_coe, Function.comp, fill_coe, fc] | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → X
y : X
fc : ContinuousAt f z
⊢ ContinuousAt (fill f y) ↑z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → X
y : X
fc : ContinuousAt f z
⊢ ContinuousAt (fill f y) ↑z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuousAt_fill_inf | [431, 1] | [434, 63] | simp only [OnePoint.continuousAt_infty', lift_inf, Filter.coclosedCompact_eq_cocompact, ←
atInf_eq_cocompact, Function.comp, fill_coe, fill_inf, fi] | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → X
y : X
fi : Tendsto f atInf (𝓝 y)
⊢ ContinuousAt (fill f y) ∞ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → X
y : X
fi : Tendsto f atInf (𝓝 y)
⊢ ContinuousAt (fill f y) ∞
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_fill | [437, 1] | [441, 48] | rw [continuous_iff_continuousAt] | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → X
y : X
fc : Continuous f
fi : Tendsto f atInf (𝓝 y)
⊢ Continuous (fill f y) | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → X
y : X
fc : Continuous f
fi : Tendsto f atInf (𝓝 y)
⊢ ∀ (x : 𝕊), ContinuousAt (fill f y) x | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → X
y : X
fc : Continuous f
fi : Tendsto f atInf (𝓝 y)
⊢ Continuous (fill f y)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_fill | [437, 1] | [441, 48] | intro z | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → X
y : X
fc : Continuous f
fi : Tendsto f atInf (𝓝 y)
⊢ ∀ (x : 𝕊), ContinuousAt (fill f y) x | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → X
y : X
fc : Continuous f
fi : Tendsto f atInf (𝓝 y)
z : 𝕊
⊢ ContinuousAt (fill f y) z | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → X
y : X
fc : Continuous f
fi : Tendsto f atInf (𝓝 y)
⊢ ∀ (x : 𝕊), ContinuousAt (fill f y) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_fill | [437, 1] | [441, 48] | induction z using OnePoint.rec | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → X
y : X
fc : Continuous f
fi : Tendsto f atInf (𝓝 y)
z : 𝕊
⊢ ContinuousAt (fill f y) z | case h₁
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → X
y : X
fc : Continuous f
fi : Tendsto f atInf (𝓝 y)
⊢ ContinuousAt (fill f y) ∞
case h₂
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → X
y : X
fc : Continuous f
fi : Tendsto f atInf (𝓝 y)
x✝ : ℂ
⊢ ContinuousAt (fill f y) ↑x✝ | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → X
y : X
fc : Continuous f
fi : Tendsto f atInf (𝓝 y)
z : 𝕊
⊢ ContinuousAt (fill f y) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_fill | [437, 1] | [441, 48] | exact continuousAt_fill_inf fi | case h₁
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → X
y : X
fc : Continuous f
fi : Tendsto f atInf (𝓝 y)
⊢ ContinuousAt (fill f y) ∞ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → X
y : X
fc : Continuous f
fi : Tendsto f atInf (𝓝 y)
⊢ ContinuousAt (fill f y) ∞
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_fill | [437, 1] | [441, 48] | exact continuousAt_fill_coe fc.continuousAt | case h₂
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → X
y : X
fc : Continuous f
fi : Tendsto f atInf (𝓝 y)
x✝ : ℂ
⊢ ContinuousAt (fill f y) ↑x✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → X
y : X
fc : Continuous f
fi : Tendsto f atInf (𝓝 y)
x✝ : ℂ
⊢ ContinuousAt (fill f y) ↑x✝
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_coe | [444, 1] | [451, 28] | have e : (fun x : 𝕊 ↦ f x.toComplex) =ᶠ[𝓝 ↑z] fill f y := by
simp only [OnePoint.nhds_coe_eq, Filter.EventuallyEq, Filter.eventually_map, toComplex_coe,
fill_coe, eq_self_iff_true, Filter.eventually_true] | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : HolomorphicAt I I f z
⊢ HolomorphicAt I I (fill f y) ↑z | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : HolomorphicAt I I f z
e : (𝓝 ↑z).EventuallyEq (fun x => f x.toComplex) (fill f y)
⊢ HolomorphicAt I I (fill f y) ↑z | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : HolomorphicAt I I f z
⊢ HolomorphicAt I I (fill f y) ↑z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_coe | [444, 1] | [451, 28] | refine HolomorphicAt.congr ?_ e | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : HolomorphicAt I I f z
e : (𝓝 ↑z).EventuallyEq (fun x => f x.toComplex) (fill f y)
⊢ HolomorphicAt I I (fill f y) ↑z | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : HolomorphicAt I I f z
e : (𝓝 ↑z).EventuallyEq (fun x => f x.toComplex) (fill f y)
⊢ HolomorphicAt I I (fun x => f x.toComplex) ↑z | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : HolomorphicAt I I f z
e : (𝓝 ↑z).EventuallyEq (fun x => f x.toComplex) (fill f y)
⊢ HolomorphicAt I I (fill f y) ↑z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_coe | [444, 1] | [451, 28] | refine fa.comp_of_eq holomorphicAt_toComplex ?_ | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : HolomorphicAt I I f z
e : (𝓝 ↑z).EventuallyEq (fun x => f x.toComplex) (fill f y)
⊢ HolomorphicAt I I (fun x => f x.toComplex) ↑z | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : HolomorphicAt I I f z
e : (𝓝 ↑z).EventuallyEq (fun x => f x.toComplex) (fill f y)
⊢ (↑z).toComplex = z | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : HolomorphicAt I I f z
e : (𝓝 ↑z).EventuallyEq (fun x => f x.toComplex) (fill f y)
⊢ HolomorphicAt I I (fun x => f x.toComplex) ↑z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_coe | [444, 1] | [451, 28] | simp only [toComplex_coe] | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : HolomorphicAt I I f z
e : (𝓝 ↑z).EventuallyEq (fun x => f x.toComplex) (fill f y)
⊢ (↑z).toComplex = z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : HolomorphicAt I I f z
e : (𝓝 ↑z).EventuallyEq (fun x => f x.toComplex) (fill f y)
⊢ (↑z).toComplex = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_coe | [444, 1] | [451, 28] | simp only [OnePoint.nhds_coe_eq, Filter.EventuallyEq, Filter.eventually_map, toComplex_coe,
fill_coe, eq_self_iff_true, Filter.eventually_true] | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : HolomorphicAt I I f z
⊢ (𝓝 ↑z).EventuallyEq (fun x => f x.toComplex) (fill f y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : HolomorphicAt I I f z
⊢ (𝓝 ↑z).EventuallyEq (fun x => f x.toComplex) (fill f y)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_inf | [454, 1] | [492, 53] | rw [holomorphicAt_iff] | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ HolomorphicAt I I (fill f y) ∞ | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ ContinuousAt (fill f y) ∞ ∧
AnalyticAt ℂ (↑(extChartAt I (fill f y ∞)) ∘ fill f y ∘ ↑(extChartAt I ∞).symm) (↑(extChartAt I ∞) ∞) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ HolomorphicAt I I (fill f y) ∞
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_inf | [454, 1] | [492, 53] | use continuousAt_fill_inf fi | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ ContinuousAt (fill f y) ∞ ∧
AnalyticAt ℂ (↑(extChartAt I (fill f y ∞)) ∘ fill f y ∘ ↑(extChartAt I ∞).symm) (↑(extChartAt I ∞) ∞) | case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ AnalyticAt ℂ (↑(extChartAt I (fill f y ∞)) ∘ fill f y ∘ ↑(extChartAt I ∞).symm) (↑(extChartAt I ∞) ∞) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ ContinuousAt (fill f y) ∞ ∧
AnalyticAt ℂ (↑(extChartAt I (fill f y ∞)) ∘ fill f y ∘ ↑(extChartAt I ∞).symm) (↑(extChartAt I ∞) ∞)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_inf | [454, 1] | [492, 53] | simp only [Function.comp, extChartAt, PartialHomeomorph.extend, fill, rec_inf,
modelWithCornersSelf_partialEquiv, PartialEquiv.trans_refl, chartAt_inf,
PartialHomeomorph.symm_toPartialEquiv, PartialEquiv.symm_symm, PartialHomeomorph.toFun_eq_coe,
invCoePartialHomeomorph_apply, PartialHomeomorph.coe_coe_symm, invCoePartialHomeomorph_symm_apply,
inv_inf, toComplex_zero] | case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ AnalyticAt ℂ (↑(extChartAt I (fill f y ∞)) ∘ fill f y ∘ ↑(extChartAt I ∞).symm) (↑(extChartAt I ∞) ∞) | case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ AnalyticAt ℂ (fun x => ↑(chartAt ℂ y) (OnePoint.rec y f (↑x)⁻¹)) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ AnalyticAt ℂ (↑(extChartAt I (fill f y ∞)) ∘ fill f y ∘ ↑(extChartAt I ∞).symm) (↑(extChartAt I ∞) ∞)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_inf | [454, 1] | [492, 53] | rw [e] | case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
e : (fun z => ↑(chartAt ℂ y) (OnePoint.rec y f (↑z)⁻¹)) = fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)
⊢ AnalyticAt ℂ (fun x => ↑(chartAt ℂ y) (OnePoint.rec y f (↑x)⁻¹)) 0 | case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
e : (fun z => ↑(chartAt ℂ y) (OnePoint.rec y f (↑z)⁻¹)) = fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)
⊢ AnalyticAt ℂ (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
e : (fun z => ↑(chartAt ℂ y) (OnePoint.rec y f (↑z)⁻¹)) = fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)
⊢ AnalyticAt ℂ (fun x => ↑(chartAt ℂ y) (OnePoint.rec y f (↑x)⁻¹)) 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_inf | [454, 1] | [492, 53] | clear e | case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
e : (fun z => ↑(chartAt ℂ y) (OnePoint.rec y f (↑z)⁻¹)) = fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)
⊢ AnalyticAt ℂ (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) 0 | case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ AnalyticAt ℂ (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
e : (fun z => ↑(chartAt ℂ y) (OnePoint.rec y f (↑z)⁻¹)) = fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)
⊢ AnalyticAt ℂ (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_inf | [454, 1] | [492, 53] | apply Complex.analyticAt_of_differentiable_on_punctured_nhds_of_continuousAt | case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ AnalyticAt ℂ (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) 0 | case right.hd
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ ∀ᶠ (z : ℂ) in 𝓝[≠] 0, DifferentiableAt ℂ (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) z
case right.hc
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ ContinuousAt (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ AnalyticAt ℂ (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_inf | [454, 1] | [492, 53] | funext z | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ (fun z => ↑(chartAt ℂ y) (OnePoint.rec y f (↑z)⁻¹)) = fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹) | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
⊢ ↑(chartAt ℂ y) (OnePoint.rec y f (↑z)⁻¹) = ↑(extChartAt I y) (if z = 0 then y else f z⁻¹) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ (fun z => ↑(chartAt ℂ y) (OnePoint.rec y f (↑z)⁻¹)) = fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_inf | [454, 1] | [492, 53] | by_cases z0 : z = 0 | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
⊢ ↑(chartAt ℂ y) (OnePoint.rec y f (↑z)⁻¹) = ↑(extChartAt I y) (if z = 0 then y else f z⁻¹) | case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
z0 : z = 0
⊢ ↑(chartAt ℂ y) (OnePoint.rec y f (↑z)⁻¹) = ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
z0 : ¬z = 0
⊢ ↑(chartAt ℂ y) (OnePoint.rec y f (↑z)⁻¹) = ↑(extChartAt I y) (if z = 0 then y else f z⁻¹) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
⊢ ↑(chartAt ℂ y) (OnePoint.rec y f (↑z)⁻¹) = ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_inf | [454, 1] | [492, 53] | simp only [if_pos z0, z0, coe_zero, inv_zero', rec_inf, extChartAt, PartialHomeomorph.extend,
modelWithCornersSelf_partialEquiv, PartialEquiv.trans_refl, PartialHomeomorph.toFun_eq_coe,
if_true] | case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
z0 : z = 0
⊢ ↑(chartAt ℂ y) (OnePoint.rec y f (↑z)⁻¹) = ↑(extChartAt I y) (if z = 0 then y else f z⁻¹) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
z0 : z = 0
⊢ ↑(chartAt ℂ y) (OnePoint.rec y f (↑z)⁻¹) = ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_inf | [454, 1] | [492, 53] | simp only [inv_coe z0, rec_coe, extChartAt, PartialHomeomorph.extend,
modelWithCornersSelf_partialEquiv, PartialEquiv.trans_refl, z0, ite_false,
PartialHomeomorph.toFun_eq_coe] | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
z0 : ¬z = 0
⊢ ↑(chartAt ℂ y) (OnePoint.rec y f (↑z)⁻¹) = ↑(extChartAt I y) (if z = 0 then y else f z⁻¹) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
z0 : ¬z = 0
⊢ ↑(chartAt ℂ y) (OnePoint.rec y f (↑z)⁻¹) = ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_inf | [454, 1] | [492, 53] | apply (inv_tendsto_atInf.eventually fa).mp | case right.hd
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ ∀ᶠ (z : ℂ) in 𝓝[≠] 0, DifferentiableAt ℂ (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) z | case right.hd
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ ∀ᶠ (x : ℂ) in 𝓝[≠] 0,
HolomorphicAt I I f x⁻¹ → DifferentiableAt ℂ (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) x | Please generate a tactic in lean4 to solve the state.
STATE:
case right.hd
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ ∀ᶠ (z : ℂ) in 𝓝[≠] 0, DifferentiableAt ℂ (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_inf | [454, 1] | [492, 53] | apply (inv_tendsto_atInf.eventually (fi.eventually
((isOpen_extChartAt_source I y).eventually_mem (mem_extChartAt_source I y)))).mp | case right.hd
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ ∀ᶠ (x : ℂ) in 𝓝[≠] 0,
HolomorphicAt I I f x⁻¹ → DifferentiableAt ℂ (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) x | case right.hd
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ ∀ᶠ (x : ℂ) in 𝓝[≠] 0,
f x⁻¹ ∈ (extChartAt I y).source →
HolomorphicAt I I f x⁻¹ → DifferentiableAt ℂ (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) x | Please generate a tactic in lean4 to solve the state.
STATE:
case right.hd
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ ∀ᶠ (x : ℂ) in 𝓝[≠] 0,
HolomorphicAt I I f x⁻¹ → DifferentiableAt ℂ (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_inf | [454, 1] | [492, 53] | apply eventually_nhdsWithin_of_forall | case right.hd
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ ∀ᶠ (x : ℂ) in 𝓝[≠] 0,
f x⁻¹ ∈ (extChartAt I y).source →
HolomorphicAt I I f x⁻¹ → DifferentiableAt ℂ (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) x | case right.hd.h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ ∀ x ∈ {0}ᶜ,
f x⁻¹ ∈ (extChartAt I y).source →
HolomorphicAt I I f x⁻¹ → DifferentiableAt ℂ (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) x | Please generate a tactic in lean4 to solve the state.
STATE:
case right.hd
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ ∀ᶠ (x : ℂ) in 𝓝[≠] 0,
f x⁻¹ ∈ (extChartAt I y).source →
HolomorphicAt I I f x⁻¹ → DifferentiableAt ℂ (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_inf | [454, 1] | [492, 53] | intro z z0 m fa | case right.hd.h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ ∀ x ∈ {0}ᶜ,
f x⁻¹ ∈ (extChartAt I y).source →
HolomorphicAt I I f x⁻¹ → DifferentiableAt ℂ (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) x | case right.hd.h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa✝ : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
z0 : z ∈ {0}ᶜ
m : f z⁻¹ ∈ (extChartAt I y).source
fa : HolomorphicAt I I f z⁻¹
⊢ DifferentiableAt ℂ (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) z | Please generate a tactic in lean4 to solve the state.
STATE:
case right.hd.h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ ∀ x ∈ {0}ᶜ,
f x⁻¹ ∈ (extChartAt I y).source →
HolomorphicAt I I f x⁻¹ → DifferentiableAt ℂ (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_inf | [454, 1] | [492, 53] | simp only [Set.mem_compl_iff, Set.mem_singleton_iff] at z0 | case right.hd.h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa✝ : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
z0 : z ∈ {0}ᶜ
m : f z⁻¹ ∈ (extChartAt I y).source
fa : HolomorphicAt I I f z⁻¹
⊢ DifferentiableAt ℂ (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) z | case right.hd.h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa✝ : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
m : f z⁻¹ ∈ (extChartAt I y).source
fa : HolomorphicAt I I f z⁻¹
z0 : ¬z = 0
⊢ DifferentiableAt ℂ (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) z | Please generate a tactic in lean4 to solve the state.
STATE:
case right.hd.h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa✝ : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
z0 : z ∈ {0}ᶜ
m : f z⁻¹ ∈ (extChartAt I y).source
fa : HolomorphicAt I I f z⁻¹
⊢ DifferentiableAt ℂ (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_inf | [454, 1] | [492, 53] | have e : (fun z ↦ extChartAt I y (if z = 0 then y else f z⁻¹)) =ᶠ[𝓝 z]
fun z ↦ extChartAt I y (f z⁻¹) := by
refine (continuousAt_id.eventually_ne z0).mp (eventually_of_forall fun w w0 ↦ ?_)
simp only [Ne, id_eq] at w0; simp only [w0, if_false] | case right.hd.h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa✝ : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
m : f z⁻¹ ∈ (extChartAt I y).source
fa : HolomorphicAt I I f z⁻¹
z0 : ¬z = 0
⊢ DifferentiableAt ℂ (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) z | case right.hd.h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa✝ : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
m : f z⁻¹ ∈ (extChartAt I y).source
fa : HolomorphicAt I I f z⁻¹
z0 : ¬z = 0
e : (𝓝 z).EventuallyEq (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) fun z => ↑(extChartAt I y) (f z⁻¹)
⊢ DifferentiableAt ℂ (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) z | Please generate a tactic in lean4 to solve the state.
STATE:
case right.hd.h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa✝ : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
m : f z⁻¹ ∈ (extChartAt I y).source
fa : HolomorphicAt I I f z⁻¹
z0 : ¬z = 0
⊢ DifferentiableAt ℂ (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_inf | [454, 1] | [492, 53] | refine DifferentiableAt.congr_of_eventuallyEq ?_ e | case right.hd.h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa✝ : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
m : f z⁻¹ ∈ (extChartAt I y).source
fa : HolomorphicAt I I f z⁻¹
z0 : ¬z = 0
e : (𝓝 z).EventuallyEq (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) fun z => ↑(extChartAt I y) (f z⁻¹)
⊢ DifferentiableAt ℂ (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) z | case right.hd.h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa✝ : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
m : f z⁻¹ ∈ (extChartAt I y).source
fa : HolomorphicAt I I f z⁻¹
z0 : ¬z = 0
e : (𝓝 z).EventuallyEq (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) fun z => ↑(extChartAt I y) (f z⁻¹)
⊢ DifferentiableAt ℂ (fun z => ↑(extChartAt I y) (f z⁻¹)) z | Please generate a tactic in lean4 to solve the state.
STATE:
case right.hd.h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa✝ : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
m : f z⁻¹ ∈ (extChartAt I y).source
fa : HolomorphicAt I I f z⁻¹
z0 : ¬z = 0
e : (𝓝 z).EventuallyEq (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) fun z => ↑(extChartAt I y) (f z⁻¹)
⊢ DifferentiableAt ℂ (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_inf | [454, 1] | [492, 53] | apply AnalyticAt.differentiableAt | case right.hd.h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa✝ : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
m : f z⁻¹ ∈ (extChartAt I y).source
fa : HolomorphicAt I I f z⁻¹
z0 : ¬z = 0
e : (𝓝 z).EventuallyEq (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) fun z => ↑(extChartAt I y) (f z⁻¹)
⊢ DifferentiableAt ℂ (fun z => ↑(extChartAt I y) (f z⁻¹)) z | case right.hd.h.a
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa✝ : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
m : f z⁻¹ ∈ (extChartAt I y).source
fa : HolomorphicAt I I f z⁻¹
z0 : ¬z = 0
e : (𝓝 z).EventuallyEq (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) fun z => ↑(extChartAt I y) (f z⁻¹)
⊢ AnalyticAt ℂ (fun z => ↑(extChartAt I y) (f z⁻¹)) z | Please generate a tactic in lean4 to solve the state.
STATE:
case right.hd.h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa✝ : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
m : f z⁻¹ ∈ (extChartAt I y).source
fa : HolomorphicAt I I f z⁻¹
z0 : ¬z = 0
e : (𝓝 z).EventuallyEq (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) fun z => ↑(extChartAt I y) (f z⁻¹)
⊢ DifferentiableAt ℂ (fun z => ↑(extChartAt I y) (f z⁻¹)) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_inf | [454, 1] | [492, 53] | apply HolomorphicAt.analyticAt I I | case right.hd.h.a
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa✝ : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
m : f z⁻¹ ∈ (extChartAt I y).source
fa : HolomorphicAt I I f z⁻¹
z0 : ¬z = 0
e : (𝓝 z).EventuallyEq (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) fun z => ↑(extChartAt I y) (f z⁻¹)
⊢ AnalyticAt ℂ (fun z => ↑(extChartAt I y) (f z⁻¹)) z | case right.hd.h.a
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa✝ : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
m : f z⁻¹ ∈ (extChartAt I y).source
fa : HolomorphicAt I I f z⁻¹
z0 : ¬z = 0
e : (𝓝 z).EventuallyEq (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) fun z => ↑(extChartAt I y) (f z⁻¹)
⊢ HolomorphicAt I I (fun z => ↑(extChartAt I y) (f z⁻¹)) z | Please generate a tactic in lean4 to solve the state.
STATE:
case right.hd.h.a
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa✝ : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
m : f z⁻¹ ∈ (extChartAt I y).source
fa : HolomorphicAt I I f z⁻¹
z0 : ¬z = 0
e : (𝓝 z).EventuallyEq (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) fun z => ↑(extChartAt I y) (f z⁻¹)
⊢ AnalyticAt ℂ (fun z => ↑(extChartAt I y) (f z⁻¹)) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_inf | [454, 1] | [492, 53] | refine (HolomorphicAt.extChartAt ?_).comp ?_ | case right.hd.h.a
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa✝ : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
m : f z⁻¹ ∈ (extChartAt I y).source
fa : HolomorphicAt I I f z⁻¹
z0 : ¬z = 0
e : (𝓝 z).EventuallyEq (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) fun z => ↑(extChartAt I y) (f z⁻¹)
⊢ HolomorphicAt I I (fun z => ↑(extChartAt I y) (f z⁻¹)) z | case right.hd.h.a.refine_1
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa✝ : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
m : f z⁻¹ ∈ (extChartAt I y).source
fa : HolomorphicAt I I f z⁻¹
z0 : ¬z = 0
e : (𝓝 z).EventuallyEq (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) fun z => ↑(extChartAt I y) (f z⁻¹)
⊢ f z⁻¹ ∈ (extChartAt I y).source
case right.hd.h.a.refine_2
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa✝ : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
m : f z⁻¹ ∈ (extChartAt I y).source
fa : HolomorphicAt I I f z⁻¹
z0 : ¬z = 0
e : (𝓝 z).EventuallyEq (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) fun z => ↑(extChartAt I y) (f z⁻¹)
⊢ HolomorphicAt I I (fun z => f z⁻¹) z | Please generate a tactic in lean4 to solve the state.
STATE:
case right.hd.h.a
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa✝ : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
m : f z⁻¹ ∈ (extChartAt I y).source
fa : HolomorphicAt I I f z⁻¹
z0 : ¬z = 0
e : (𝓝 z).EventuallyEq (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) fun z => ↑(extChartAt I y) (f z⁻¹)
⊢ HolomorphicAt I I (fun z => ↑(extChartAt I y) (f z⁻¹)) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_inf | [454, 1] | [492, 53] | exact m | case right.hd.h.a.refine_1
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa✝ : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
m : f z⁻¹ ∈ (extChartAt I y).source
fa : HolomorphicAt I I f z⁻¹
z0 : ¬z = 0
e : (𝓝 z).EventuallyEq (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) fun z => ↑(extChartAt I y) (f z⁻¹)
⊢ f z⁻¹ ∈ (extChartAt I y).source
case right.hd.h.a.refine_2
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa✝ : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
m : f z⁻¹ ∈ (extChartAt I y).source
fa : HolomorphicAt I I f z⁻¹
z0 : ¬z = 0
e : (𝓝 z).EventuallyEq (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) fun z => ↑(extChartAt I y) (f z⁻¹)
⊢ HolomorphicAt I I (fun z => f z⁻¹) z | case right.hd.h.a.refine_2
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa✝ : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
m : f z⁻¹ ∈ (extChartAt I y).source
fa : HolomorphicAt I I f z⁻¹
z0 : ¬z = 0
e : (𝓝 z).EventuallyEq (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) fun z => ↑(extChartAt I y) (f z⁻¹)
⊢ HolomorphicAt I I (fun z => f z⁻¹) z | Please generate a tactic in lean4 to solve the state.
STATE:
case right.hd.h.a.refine_1
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa✝ : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
m : f z⁻¹ ∈ (extChartAt I y).source
fa : HolomorphicAt I I f z⁻¹
z0 : ¬z = 0
e : (𝓝 z).EventuallyEq (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) fun z => ↑(extChartAt I y) (f z⁻¹)
⊢ f z⁻¹ ∈ (extChartAt I y).source
case right.hd.h.a.refine_2
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa✝ : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
m : f z⁻¹ ∈ (extChartAt I y).source
fa : HolomorphicAt I I f z⁻¹
z0 : ¬z = 0
e : (𝓝 z).EventuallyEq (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) fun z => ↑(extChartAt I y) (f z⁻¹)
⊢ HolomorphicAt I I (fun z => f z⁻¹) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_inf | [454, 1] | [492, 53] | exact fa.comp (holomorphicAt_id.inv z0) | case right.hd.h.a.refine_2
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa✝ : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
m : f z⁻¹ ∈ (extChartAt I y).source
fa : HolomorphicAt I I f z⁻¹
z0 : ¬z = 0
e : (𝓝 z).EventuallyEq (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) fun z => ↑(extChartAt I y) (f z⁻¹)
⊢ HolomorphicAt I I (fun z => f z⁻¹) z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right.hd.h.a.refine_2
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa✝ : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
m : f z⁻¹ ∈ (extChartAt I y).source
fa : HolomorphicAt I I f z⁻¹
z0 : ¬z = 0
e : (𝓝 z).EventuallyEq (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) fun z => ↑(extChartAt I y) (f z⁻¹)
⊢ HolomorphicAt I I (fun z => f z⁻¹) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_inf | [454, 1] | [492, 53] | refine (continuousAt_id.eventually_ne z0).mp (eventually_of_forall fun w w0 ↦ ?_) | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa✝ : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
m : f z⁻¹ ∈ (extChartAt I y).source
fa : HolomorphicAt I I f z⁻¹
z0 : ¬z = 0
⊢ (𝓝 z).EventuallyEq (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) fun z => ↑(extChartAt I y) (f z⁻¹) | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa✝ : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
m : f z⁻¹ ∈ (extChartAt I y).source
fa : HolomorphicAt I I f z⁻¹
z0 : ¬z = 0
w : ℂ
w0 : id w ≠ 0
⊢ (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) w = (fun z => ↑(extChartAt I y) (f z⁻¹)) w | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa✝ : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
m : f z⁻¹ ∈ (extChartAt I y).source
fa : HolomorphicAt I I f z⁻¹
z0 : ¬z = 0
⊢ (𝓝 z).EventuallyEq (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) fun z => ↑(extChartAt I y) (f z⁻¹)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_inf | [454, 1] | [492, 53] | simp only [Ne, id_eq] at w0 | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa✝ : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
m : f z⁻¹ ∈ (extChartAt I y).source
fa : HolomorphicAt I I f z⁻¹
z0 : ¬z = 0
w : ℂ
w0 : id w ≠ 0
⊢ (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) w = (fun z => ↑(extChartAt I y) (f z⁻¹)) w | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa✝ : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
m : f z⁻¹ ∈ (extChartAt I y).source
fa : HolomorphicAt I I f z⁻¹
z0 : ¬z = 0
w : ℂ
w0 : ¬w = 0
⊢ (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) w = (fun z => ↑(extChartAt I y) (f z⁻¹)) w | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa✝ : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
m : f z⁻¹ ∈ (extChartAt I y).source
fa : HolomorphicAt I I f z⁻¹
z0 : ¬z = 0
w : ℂ
w0 : id w ≠ 0
⊢ (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) w = (fun z => ↑(extChartAt I y) (f z⁻¹)) w
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_inf | [454, 1] | [492, 53] | simp only [w0, if_false] | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa✝ : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
m : f z⁻¹ ∈ (extChartAt I y).source
fa : HolomorphicAt I I f z⁻¹
z0 : ¬z = 0
w : ℂ
w0 : ¬w = 0
⊢ (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) w = (fun z => ↑(extChartAt I y) (f z⁻¹)) w | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa✝ : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
m : f z⁻¹ ∈ (extChartAt I y).source
fa : HolomorphicAt I I f z⁻¹
z0 : ¬z = 0
w : ℂ
w0 : ¬w = 0
⊢ (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) w = (fun z => ↑(extChartAt I y) (f z⁻¹)) w
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_inf | [454, 1] | [492, 53] | refine (continuousAt_extChartAt' I ?_).comp ?_ | case right.hc
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ ContinuousAt (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) 0 | case right.hc.refine_1
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ (if 0 = 0 then y else f 0⁻¹) ∈ (extChartAt I y).source
case right.hc.refine_2
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ ContinuousAt (fun z => if z = 0 then y else f z⁻¹) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case right.hc
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ ContinuousAt (fun z => ↑(extChartAt I y) (if z = 0 then y else f z⁻¹)) 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_inf | [454, 1] | [492, 53] | simp only [eq_self_iff_true, if_pos, mem_extChartAt_source] | case right.hc.refine_1
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ (if 0 = 0 then y else f 0⁻¹) ∈ (extChartAt I y).source | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right.hc.refine_1
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ (if 0 = 0 then y else f 0⁻¹) ∈ (extChartAt I y).source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_inf | [454, 1] | [492, 53] | simp only [← continuousWithinAt_compl_self, ContinuousWithinAt] | case right.hc.refine_2
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ ContinuousAt (fun z => if z = 0 then y else f z⁻¹) 0 | case right.hc.refine_2
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ Tendsto (fun z => if z = 0 then y else f z⁻¹) (𝓝[≠] 0) (𝓝 (if True then y else f 0⁻¹)) | Please generate a tactic in lean4 to solve the state.
STATE:
case right.hc.refine_2
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ ContinuousAt (fun z => if z = 0 then y else f z⁻¹) 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_inf | [454, 1] | [492, 53] | apply tendsto_nhdsWithin_congr (f := fun z ↦ f z⁻¹) | case right.hc.refine_2
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ Tendsto (fun z => if z = 0 then y else f z⁻¹) (𝓝[≠] 0) (𝓝 (if True then y else f 0⁻¹)) | case right.hc.refine_2.hfg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ ∀ x ∈ {0}ᶜ, f x⁻¹ = if x = 0 then y else f x⁻¹
case right.hc.refine_2.hf
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ Tendsto (fun z => f z⁻¹) (𝓝[≠] 0) (𝓝 (if True then y else f 0⁻¹)) | Please generate a tactic in lean4 to solve the state.
STATE:
case right.hc.refine_2
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ Tendsto (fun z => if z = 0 then y else f z⁻¹) (𝓝[≠] 0) (𝓝 (if True then y else f 0⁻¹))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_inf | [454, 1] | [492, 53] | intro z z0 | case right.hc.refine_2.hfg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ ∀ x ∈ {0}ᶜ, f x⁻¹ = if x = 0 then y else f x⁻¹
case right.hc.refine_2.hf
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ Tendsto (fun z => f z⁻¹) (𝓝[≠] 0) (𝓝 (if True then y else f 0⁻¹)) | case right.hc.refine_2.hfg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
z0 : z ∈ {0}ᶜ
⊢ f z⁻¹ = if z = 0 then y else f z⁻¹
case right.hc.refine_2.hf
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ Tendsto (fun z => f z⁻¹) (𝓝[≠] 0) (𝓝 (if True then y else f 0⁻¹)) | Please generate a tactic in lean4 to solve the state.
STATE:
case right.hc.refine_2.hfg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ ∀ x ∈ {0}ᶜ, f x⁻¹ = if x = 0 then y else f x⁻¹
case right.hc.refine_2.hf
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ Tendsto (fun z => f z⁻¹) (𝓝[≠] 0) (𝓝 (if True then y else f 0⁻¹))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_inf | [454, 1] | [492, 53] | simp only [Set.mem_compl_iff, Set.mem_singleton_iff] at z0 | case right.hc.refine_2.hfg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
z0 : z ∈ {0}ᶜ
⊢ f z⁻¹ = if z = 0 then y else f z⁻¹
case right.hc.refine_2.hf
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ Tendsto (fun z => f z⁻¹) (𝓝[≠] 0) (𝓝 (if True then y else f 0⁻¹)) | case right.hc.refine_2.hfg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
z0 : ¬z = 0
⊢ f z⁻¹ = if z = 0 then y else f z⁻¹
case right.hc.refine_2.hf
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ Tendsto (fun z => f z⁻¹) (𝓝[≠] 0) (𝓝 (if True then y else f 0⁻¹)) | Please generate a tactic in lean4 to solve the state.
STATE:
case right.hc.refine_2.hfg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
z0 : z ∈ {0}ᶜ
⊢ f z⁻¹ = if z = 0 then y else f z⁻¹
case right.hc.refine_2.hf
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ Tendsto (fun z => f z⁻¹) (𝓝[≠] 0) (𝓝 (if True then y else f 0⁻¹))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_inf | [454, 1] | [492, 53] | simp only [z0, if_false] | case right.hc.refine_2.hfg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
z0 : ¬z = 0
⊢ f z⁻¹ = if z = 0 then y else f z⁻¹
case right.hc.refine_2.hf
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ Tendsto (fun z => f z⁻¹) (𝓝[≠] 0) (𝓝 (if True then y else f 0⁻¹)) | case right.hc.refine_2.hf
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ Tendsto (fun z => f z⁻¹) (𝓝[≠] 0) (𝓝 (if True then y else f 0⁻¹)) | Please generate a tactic in lean4 to solve the state.
STATE:
case right.hc.refine_2.hfg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
z : ℂ
z0 : ¬z = 0
⊢ f z⁻¹ = if z = 0 then y else f z⁻¹
case right.hc.refine_2.hf
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ Tendsto (fun z => f z⁻¹) (𝓝[≠] 0) (𝓝 (if True then y else f 0⁻¹))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_fill_inf | [454, 1] | [492, 53] | exact Filter.Tendsto.comp fi inv_tendsto_atInf | case right.hc.refine_2.hf
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ Tendsto (fun z => f z⁻¹) (𝓝[≠] 0) (𝓝 (if True then y else f 0⁻¹)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right.hc.refine_2.hf
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I f z
fi : Tendsto f atInf (𝓝 y)
⊢ Tendsto (fun z => f z⁻¹) (𝓝[≠] 0) (𝓝 (if True then y else f 0⁻¹))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_fill | [495, 1] | [499, 40] | intro z | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : Holomorphic I I f
fi : Tendsto f atInf (𝓝 y)
⊢ Holomorphic I I (fill f y) | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa : Holomorphic I I f
fi : Tendsto f atInf (𝓝 y)
z : 𝕊
⊢ HolomorphicAt I I (fill f y) z | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : Holomorphic I I f
fi : Tendsto f atInf (𝓝 y)
⊢ Holomorphic I I (fill f y)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_fill | [495, 1] | [499, 40] | induction z using OnePoint.rec | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa : Holomorphic I I f
fi : Tendsto f atInf (𝓝 y)
z : 𝕊
⊢ HolomorphicAt I I (fill f y) z | case h₁
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : Holomorphic I I f
fi : Tendsto f atInf (𝓝 y)
⊢ HolomorphicAt I I (fill f y) ∞
case h₂
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : Holomorphic I I f
fi : Tendsto f atInf (𝓝 y)
x✝ : ℂ
⊢ HolomorphicAt I I (fill f y) ↑x✝ | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z✝ : ℂ
f : ℂ → T
y : T
fa : Holomorphic I I f
fi : Tendsto f atInf (𝓝 y)
z : 𝕊
⊢ HolomorphicAt I I (fill f y) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_fill | [495, 1] | [499, 40] | exact holomorphicAt_fill_inf (eventually_of_forall fa) fi | case h₁
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : Holomorphic I I f
fi : Tendsto f atInf (𝓝 y)
⊢ HolomorphicAt I I (fill f y) ∞ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : Holomorphic I I f
fi : Tendsto f atInf (𝓝 y)
⊢ HolomorphicAt I I (fill f y) ∞
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_fill | [495, 1] | [499, 40] | exact holomorphicAt_fill_coe (fa _) | case h₂
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : Holomorphic I I f
fi : Tendsto f atInf (𝓝 y)
x✝ : ℂ
⊢ HolomorphicAt I I (fill f y) ↑x✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f✝ : ℂ → ℂ
g : X → ℂ → ℂ
y✝ : 𝕊
x : X
z : ℂ
f : ℂ → T
y : T
fa : Holomorphic I I f
fi : Tendsto f atInf (𝓝 y)
x✝ : ℂ
⊢ HolomorphicAt I I (fill f y) ↑x✝
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuousAt_lift_coe' | [502, 1] | [506, 50] | simp only [lift', ContinuousAt, uncurry, rec_coe, OnePoint.nhds_coe_eq, prod_nhds_eq,
Filter.tendsto_map'_iff, Function.comp] | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
gc : ContinuousAt (uncurry g) (x, z)
⊢ ContinuousAt (uncurry (lift' g y)) (x, ↑z) | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
gc : ContinuousAt (uncurry g) (x, z)
⊢ Tendsto (fun x => ↑(g x.1 x.2)) (𝓝 (x, z)) (Filter.map OnePoint.some (𝓝 (g x z))) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
gc : ContinuousAt (uncurry g) (x, z)
⊢ ContinuousAt (uncurry (lift' g y)) (x, ↑z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuousAt_lift_coe' | [502, 1] | [506, 50] | exact Filter.Tendsto.comp Filter.tendsto_map gc | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
gc : ContinuousAt (uncurry g) (x, z)
⊢ Tendsto (fun x => ↑(g x.1 x.2)) (𝓝 (x, z)) (Filter.map OnePoint.some (𝓝 (g x z))) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
gc : ContinuousAt (uncurry g) (x, z)
⊢ Tendsto (fun x => ↑(g x.1 x.2)) (𝓝 (x, z)) (Filter.map OnePoint.some (𝓝 (g x z)))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuousAt_lift_inf' | [509, 1] | [526, 45] | simp only [ContinuousAt, Filter.Tendsto, Filter.le_def, Filter.mem_map] | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
gi : Tendsto (uncurry g) ((𝓝 x).prod atInf) atInf
⊢ ContinuousAt (uncurry (lift' g ∞)) (x, ∞) | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
gi : Tendsto (uncurry g) ((𝓝 x).prod atInf) atInf
⊢ ∀ x_1 ∈ 𝓝 (uncurry (lift' g ∞) (x, ∞)), uncurry (lift' g ∞) ⁻¹' x_1 ∈ 𝓝 (x, ∞) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
gi : Tendsto (uncurry g) ((𝓝 x).prod atInf) atInf
⊢ ContinuousAt (uncurry (lift' g ∞)) (x, ∞)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuousAt_lift_inf' | [509, 1] | [526, 45] | intro s m | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
gi : Tendsto (uncurry g) ((𝓝 x).prod atInf) atInf
⊢ ∀ x_1 ∈ 𝓝 (uncurry (lift' g ∞) (x, ∞)), uncurry (lift' g ∞) ⁻¹' x_1 ∈ 𝓝 (x, ∞) | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
gi : Tendsto (uncurry g) ((𝓝 x).prod atInf) atInf
s : Set 𝕊
m : s ∈ 𝓝 (uncurry (lift' g ∞) (x, ∞))
⊢ uncurry (lift' g ∞) ⁻¹' s ∈ 𝓝 (x, ∞) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
gi : Tendsto (uncurry g) ((𝓝 x).prod atInf) atInf
⊢ ∀ x_1 ∈ 𝓝 (uncurry (lift' g ∞) (x, ∞)), uncurry (lift' g ∞) ⁻¹' x_1 ∈ 𝓝 (x, ∞)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuousAt_lift_inf' | [509, 1] | [526, 45] | simp only [OnePoint.nhds_infty_eq, Filter.coclosedCompact_eq_cocompact, Filter.mem_sup,
Filter.mem_map, Filter.mem_pure, ← atInf_eq_cocompact, lift', rec_inf, uncurry] at m | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
gi : Tendsto (uncurry g) ((𝓝 x).prod atInf) atInf
s : Set 𝕊
m : s ∈ 𝓝 (uncurry (lift' g ∞) (x, ∞))
⊢ uncurry (lift' g ∞) ⁻¹' s ∈ 𝓝 (x, ∞) | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
gi : Tendsto (uncurry g) ((𝓝 x).prod atInf) atInf
s : Set 𝕊
m : OnePoint.some ⁻¹' s ∈ atInf ∧ ∞ ∈ s
⊢ uncurry (lift' g ∞) ⁻¹' s ∈ 𝓝 (x, ∞) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
gi : Tendsto (uncurry g) ((𝓝 x).prod atInf) atInf
s : Set 𝕊
m : s ∈ 𝓝 (uncurry (lift' g ∞) (x, ∞))
⊢ uncurry (lift' g ∞) ⁻¹' s ∈ 𝓝 (x, ∞)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuousAt_lift_inf' | [509, 1] | [526, 45] | simp only [true_imp_iff, mem_setOf, uncurry, Tendsto] at gi | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
gi : Tendsto (uncurry g) ((𝓝 x).prod atInf) atInf
s : Set 𝕊
m : OnePoint.some ⁻¹' s ∈ atInf ∧ ∞ ∈ s
⊢ uncurry (lift' g ∞) ⁻¹' s ∈ 𝓝 (x, ∞) | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
gi : Filter.map (uncurry g) ((𝓝 x).prod atInf) ≤ atInf
s : Set 𝕊
m : OnePoint.some ⁻¹' s ∈ atInf ∧ ∞ ∈ s
⊢ uncurry (lift' g ∞) ⁻¹' s ∈ 𝓝 (x, ∞) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
gi : Tendsto (uncurry g) ((𝓝 x).prod atInf) atInf
s : Set 𝕊
m : OnePoint.some ⁻¹' s ∈ atInf ∧ ∞ ∈ s
⊢ uncurry (lift' g ∞) ⁻¹' s ∈ 𝓝 (x, ∞)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuousAt_lift_inf' | [509, 1] | [526, 45] | specialize gi m.1 | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
gi : Filter.map (uncurry g) ((𝓝 x).prod atInf) ≤ atInf
s : Set 𝕊
m : OnePoint.some ⁻¹' s ∈ atInf ∧ ∞ ∈ s
⊢ uncurry (lift' g ∞) ⁻¹' s ∈ 𝓝 (x, ∞) | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
s : Set 𝕊
m : OnePoint.some ⁻¹' s ∈ atInf ∧ ∞ ∈ s
gi : OnePoint.some ⁻¹' s ∈ Filter.map (uncurry g) ((𝓝 x).prod atInf)
⊢ uncurry (lift' g ∞) ⁻¹' s ∈ 𝓝 (x, ∞) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
gi : Filter.map (uncurry g) ((𝓝 x).prod atInf) ≤ atInf
s : Set 𝕊
m : OnePoint.some ⁻¹' s ∈ atInf ∧ ∞ ∈ s
⊢ uncurry (lift' g ∞) ⁻¹' s ∈ 𝓝 (x, ∞)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuousAt_lift_inf' | [509, 1] | [526, 45] | simp only [Filter.mem_map, preimage_preimage] at gi | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
s : Set 𝕊
m : OnePoint.some ⁻¹' s ∈ atInf ∧ ∞ ∈ s
gi : OnePoint.some ⁻¹' s ∈ Filter.map (uncurry g) ((𝓝 x).prod atInf)
⊢ uncurry (lift' g ∞) ⁻¹' s ∈ 𝓝 (x, ∞) | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
s : Set 𝕊
m : OnePoint.some ⁻¹' s ∈ atInf ∧ ∞ ∈ s
gi : (fun x => ↑(uncurry g x)) ⁻¹' s ∈ (𝓝 x).prod atInf
⊢ uncurry (lift' g ∞) ⁻¹' s ∈ 𝓝 (x, ∞) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
s : Set 𝕊
m : OnePoint.some ⁻¹' s ∈ atInf ∧ ∞ ∈ s
gi : OnePoint.some ⁻¹' s ∈ Filter.map (uncurry g) ((𝓝 x).prod atInf)
⊢ uncurry (lift' g ∞) ⁻¹' s ∈ 𝓝 (x, ∞)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuousAt_lift_inf' | [509, 1] | [526, 45] | rw [e] | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
s : Set 𝕊
m : OnePoint.some ⁻¹' s ∈ atInf ∧ ∞ ∈ s
gi : (fun x => ↑(uncurry g x)) ⁻¹' s ∈ (𝓝 x).prod atInf
e : uncurry (lift' g ∞) ⁻¹' s = (fun x => (x.1, ↑x.2)) '' ((fun x => ↑(g x.1 x.2)) ⁻¹' s) ∪ univ ×ˢ {∞}
⊢ uncurry (lift' g ∞) ⁻¹' s ∈ 𝓝 (x, ∞) | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
s : Set 𝕊
m : OnePoint.some ⁻¹' s ∈ atInf ∧ ∞ ∈ s
gi : (fun x => ↑(uncurry g x)) ⁻¹' s ∈ (𝓝 x).prod atInf
e : uncurry (lift' g ∞) ⁻¹' s = (fun x => (x.1, ↑x.2)) '' ((fun x => ↑(g x.1 x.2)) ⁻¹' s) ∪ univ ×ˢ {∞}
⊢ (fun x => (x.1, ↑x.2)) '' ((fun x => ↑(g x.1 x.2)) ⁻¹' s) ∪ univ ×ˢ {∞} ∈ 𝓝 (x, ∞) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
s : Set 𝕊
m : OnePoint.some ⁻¹' s ∈ atInf ∧ ∞ ∈ s
gi : (fun x => ↑(uncurry g x)) ⁻¹' s ∈ (𝓝 x).prod atInf
e : uncurry (lift' g ∞) ⁻¹' s = (fun x => (x.1, ↑x.2)) '' ((fun x => ↑(g x.1 x.2)) ⁻¹' s) ∪ univ ×ˢ {∞}
⊢ uncurry (lift' g ∞) ⁻¹' s ∈ 𝓝 (x, ∞)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuousAt_lift_inf' | [509, 1] | [526, 45] | exact prod_mem_inf_of_mem_atInf gi | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
s : Set 𝕊
m : OnePoint.some ⁻¹' s ∈ atInf ∧ ∞ ∈ s
gi : (fun x => ↑(uncurry g x)) ⁻¹' s ∈ (𝓝 x).prod atInf
e : uncurry (lift' g ∞) ⁻¹' s = (fun x => (x.1, ↑x.2)) '' ((fun x => ↑(g x.1 x.2)) ⁻¹' s) ∪ univ ×ˢ {∞}
⊢ (fun x => (x.1, ↑x.2)) '' ((fun x => ↑(g x.1 x.2)) ⁻¹' s) ∪ univ ×ˢ {∞} ∈ 𝓝 (x, ∞) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
s : Set 𝕊
m : OnePoint.some ⁻¹' s ∈ atInf ∧ ∞ ∈ s
gi : (fun x => ↑(uncurry g x)) ⁻¹' s ∈ (𝓝 x).prod atInf
e : uncurry (lift' g ∞) ⁻¹' s = (fun x => (x.1, ↑x.2)) '' ((fun x => ↑(g x.1 x.2)) ⁻¹' s) ∪ univ ×ˢ {∞}
⊢ (fun x => (x.1, ↑x.2)) '' ((fun x => ↑(g x.1 x.2)) ⁻¹' s) ∪ univ ×ˢ {∞} ∈ 𝓝 (x, ∞)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuousAt_lift_inf' | [509, 1] | [526, 45] | apply Set.ext | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
s : Set 𝕊
m : OnePoint.some ⁻¹' s ∈ atInf ∧ ∞ ∈ s
gi : (fun x => ↑(uncurry g x)) ⁻¹' s ∈ (𝓝 x).prod atInf
⊢ uncurry (lift' g ∞) ⁻¹' s = (fun x => (x.1, ↑x.2)) '' ((fun x => ↑(g x.1 x.2)) ⁻¹' s) ∪ univ ×ˢ {∞} | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
s : Set 𝕊
m : OnePoint.some ⁻¹' s ∈ atInf ∧ ∞ ∈ s
gi : (fun x => ↑(uncurry g x)) ⁻¹' s ∈ (𝓝 x).prod atInf
⊢ ∀ (x : X × 𝕊),
x ∈ uncurry (lift' g ∞) ⁻¹' s ↔ x ∈ (fun x => (x.1, ↑x.2)) '' ((fun x => ↑(g x.1 x.2)) ⁻¹' s) ∪ univ ×ˢ {∞} | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
s : Set 𝕊
m : OnePoint.some ⁻¹' s ∈ atInf ∧ ∞ ∈ s
gi : (fun x => ↑(uncurry g x)) ⁻¹' s ∈ (𝓝 x).prod atInf
⊢ uncurry (lift' g ∞) ⁻¹' s = (fun x => (x.1, ↑x.2)) '' ((fun x => ↑(g x.1 x.2)) ⁻¹' s) ∪ univ ×ˢ {∞}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuousAt_lift_inf' | [509, 1] | [526, 45] | intro ⟨x, z⟩ | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
s : Set 𝕊
m : OnePoint.some ⁻¹' s ∈ atInf ∧ ∞ ∈ s
gi : (fun x => ↑(uncurry g x)) ⁻¹' s ∈ (𝓝 x).prod atInf
⊢ ∀ (x : X × 𝕊),
x ∈ uncurry (lift' g ∞) ⁻¹' s ↔ x ∈ (fun x => (x.1, ↑x.2)) '' ((fun x => ↑(g x.1 x.2)) ⁻¹' s) ∪ univ ×ˢ {∞} | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x✝ : X
z✝ : ℂ
s : Set 𝕊
m : OnePoint.some ⁻¹' s ∈ atInf ∧ ∞ ∈ s
gi : (fun x => ↑(uncurry g x)) ⁻¹' s ∈ (𝓝 x✝).prod atInf
x : X
z : 𝕊
⊢ (x, z) ∈ uncurry (lift' g ∞) ⁻¹' s ↔ (x, z) ∈ (fun x => (x.1, ↑x.2)) '' ((fun x => ↑(g x.1 x.2)) ⁻¹' s) ∪ univ ×ˢ {∞} | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
s : Set 𝕊
m : OnePoint.some ⁻¹' s ∈ atInf ∧ ∞ ∈ s
gi : (fun x => ↑(uncurry g x)) ⁻¹' s ∈ (𝓝 x).prod atInf
⊢ ∀ (x : X × 𝕊),
x ∈ uncurry (lift' g ∞) ⁻¹' s ↔ x ∈ (fun x => (x.1, ↑x.2)) '' ((fun x => ↑(g x.1 x.2)) ⁻¹' s) ∪ univ ×ˢ {∞}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuousAt_lift_inf' | [509, 1] | [526, 45] | induction z using OnePoint.rec | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x✝ : X
z✝ : ℂ
s : Set 𝕊
m : OnePoint.some ⁻¹' s ∈ atInf ∧ ∞ ∈ s
gi : (fun x => ↑(uncurry g x)) ⁻¹' s ∈ (𝓝 x✝).prod atInf
x : X
z : 𝕊
⊢ (x, z) ∈ uncurry (lift' g ∞) ⁻¹' s ↔ (x, z) ∈ (fun x => (x.1, ↑x.2)) '' ((fun x => ↑(g x.1 x.2)) ⁻¹' s) ∪ univ ×ˢ {∞} | case h.h₁
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x✝ : X
z : ℂ
s : Set 𝕊
m : OnePoint.some ⁻¹' s ∈ atInf ∧ ∞ ∈ s
gi : (fun x => ↑(uncurry g x)) ⁻¹' s ∈ (𝓝 x✝).prod atInf
x : X
⊢ (x, ∞) ∈ uncurry (lift' g ∞) ⁻¹' s ↔ (x, ∞) ∈ (fun x => (x.1, ↑x.2)) '' ((fun x => ↑(g x.1 x.2)) ⁻¹' s) ∪ univ ×ˢ {∞}
case h.h₂
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x✝¹ : X
z : ℂ
s : Set 𝕊
m : OnePoint.some ⁻¹' s ∈ atInf ∧ ∞ ∈ s
gi : (fun x => ↑(uncurry g x)) ⁻¹' s ∈ (𝓝 x✝¹).prod atInf
x : X
x✝ : ℂ
⊢ (x, ↑x✝) ∈ uncurry (lift' g ∞) ⁻¹' s ↔
(x, ↑x✝) ∈ (fun x => (x.1, ↑x.2)) '' ((fun x => ↑(g x.1 x.2)) ⁻¹' s) ∪ univ ×ˢ {∞} | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x✝ : X
z✝ : ℂ
s : Set 𝕊
m : OnePoint.some ⁻¹' s ∈ atInf ∧ ∞ ∈ s
gi : (fun x => ↑(uncurry g x)) ⁻¹' s ∈ (𝓝 x✝).prod atInf
x : X
z : 𝕊
⊢ (x, z) ∈ uncurry (lift' g ∞) ⁻¹' s ↔ (x, z) ∈ (fun x => (x.1, ↑x.2)) '' ((fun x => ↑(g x.1 x.2)) ⁻¹' s) ∪ univ ×ˢ {∞}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuousAt_lift_inf' | [509, 1] | [526, 45] | simp only [mem_preimage, mem_image, mem_union, mem_prod_eq, mem_univ, true_and_iff,
mem_singleton_iff, eq_self_iff_true, or_true_iff, iff_true_iff, uncurry, lift', rec_inf,
m.2] | case h.h₁
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x✝ : X
z : ℂ
s : Set 𝕊
m : OnePoint.some ⁻¹' s ∈ atInf ∧ ∞ ∈ s
gi : (fun x => ↑(uncurry g x)) ⁻¹' s ∈ (𝓝 x✝).prod atInf
x : X
⊢ (x, ∞) ∈ uncurry (lift' g ∞) ⁻¹' s ↔ (x, ∞) ∈ (fun x => (x.1, ↑x.2)) '' ((fun x => ↑(g x.1 x.2)) ⁻¹' s) ∪ univ ×ˢ {∞} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h₁
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x✝ : X
z : ℂ
s : Set 𝕊
m : OnePoint.some ⁻¹' s ∈ atInf ∧ ∞ ∈ s
gi : (fun x => ↑(uncurry g x)) ⁻¹' s ∈ (𝓝 x✝).prod atInf
x : X
⊢ (x, ∞) ∈ uncurry (lift' g ∞) ⁻¹' s ↔ (x, ∞) ∈ (fun x => (x.1, ↑x.2)) '' ((fun x => ↑(g x.1 x.2)) ⁻¹' s) ∪ univ ×ˢ {∞}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuousAt_lift_inf' | [509, 1] | [526, 45] | simp only [uncurry, lift', mem_preimage, rec_coe, prod_singleton, image_univ, mem_union,
mem_image, Prod.ext_iff, coe_eq_coe, Prod.exists, exists_eq_right_right, exists_eq_right,
mem_range, OnePoint.infty_ne_coe, and_false, exists_false, or_false] | case h.h₂
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x✝¹ : X
z : ℂ
s : Set 𝕊
m : OnePoint.some ⁻¹' s ∈ atInf ∧ ∞ ∈ s
gi : (fun x => ↑(uncurry g x)) ⁻¹' s ∈ (𝓝 x✝¹).prod atInf
x : X
x✝ : ℂ
⊢ (x, ↑x✝) ∈ uncurry (lift' g ∞) ⁻¹' s ↔
(x, ↑x✝) ∈ (fun x => (x.1, ↑x.2)) '' ((fun x => ↑(g x.1 x.2)) ⁻¹' s) ∪ univ ×ˢ {∞} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h₂
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x✝¹ : X
z : ℂ
s : Set 𝕊
m : OnePoint.some ⁻¹' s ∈ atInf ∧ ∞ ∈ s
gi : (fun x => ↑(uncurry g x)) ⁻¹' s ∈ (𝓝 x✝¹).prod atInf
x : X
x✝ : ℂ
⊢ (x, ↑x✝) ∈ uncurry (lift' g ∞) ⁻¹' s ↔
(x, ↑x✝) ∈ (fun x => (x.1, ↑x.2)) '' ((fun x => ↑(g x.1 x.2)) ⁻¹' s) ∪ univ ×ˢ {∞}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_lift' | [529, 1] | [535, 49] | rw [continuous_iff_continuousOn_univ] | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
gc : Continuous (uncurry g)
gi : ∀ (x : X), Tendsto (uncurry g) ((𝓝 x).prod atInf) atInf
⊢ Continuous (uncurry (lift' g ∞)) | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
gc : Continuous (uncurry g)
gi : ∀ (x : X), Tendsto (uncurry g) ((𝓝 x).prod atInf) atInf
⊢ ContinuousOn (uncurry (lift' g ∞)) univ | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
gc : Continuous (uncurry g)
gi : ∀ (x : X), Tendsto (uncurry g) ((𝓝 x).prod atInf) atInf
⊢ Continuous (uncurry (lift' g ∞))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_lift' | [529, 1] | [535, 49] | intro ⟨x, z⟩ _ | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
gc : Continuous (uncurry g)
gi : ∀ (x : X), Tendsto (uncurry g) ((𝓝 x).prod atInf) atInf
⊢ ContinuousOn (uncurry (lift' g ∞)) univ | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x✝ : X
z✝ : ℂ
gc : Continuous (uncurry g)
gi : ∀ (x : X), Tendsto (uncurry g) ((𝓝 x).prod atInf) atInf
x : X
z : 𝕊
a✝ : (x, z) ∈ univ
⊢ ContinuousWithinAt (uncurry (lift' g ∞)) univ (x, z) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
g : X → ℂ → ℂ
y : 𝕊
x : X
z : ℂ
gc : Continuous (uncurry g)
gi : ∀ (x : X), Tendsto (uncurry g) ((𝓝 x).prod atInf) atInf
⊢ ContinuousOn (uncurry (lift' g ∞)) univ
TACTIC:
|
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