url stringclasses 147 values | commit stringclasses 147 values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
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:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_lift' | [529, 1] | [535, 49] | apply ContinuousAt.continuousWithinAt | X : Type
inst✝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) | 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
gc : Continuous (uncurry g)
gi : ∀ (x : X), Tendsto (uncurry g) ((𝓝 x).prod atInf) atInf
x : X
z : 𝕊
a✝ : (x, z) ∈ univ
⊢ ContinuousAt (uncurry (lift' 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 : 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)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_lift' | [529, 1] | [535, 49] | 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
gc : Continuous (uncurry g)
gi : ∀ (x : X), Tendsto (uncurry g) ((𝓝 x).prod atInf) atInf
x : X
z : 𝕊
a✝ : (x, z) ∈ univ
⊢ ContinuousAt (uncurry (lift' g ∞)) (x, z) | 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
gc : Continuous (uncurry g)
gi : ∀ (x : X), Tendsto (uncurry g) ((𝓝 x).prod atInf) atInf
x : X
a✝ : (x, ∞) ∈ univ
⊢ ContinuousAt (uncurry (lift' g ∞)) (x, ∞)
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
gc : Continuous (uncurry g)
gi : ∀ (x : X), Tendsto (uncurry g) ((𝓝 x).prod atInf) atInf
x : X
x✝ : ℂ
a✝ : (x, ↑x✝) ∈ univ
⊢ ContinuousAt (uncurry (lift' g ∞)) (x, ↑x✝) | 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
gc : Continuous (uncurry g)
gi : ∀ (x : X), Tendsto (uncurry g) ((𝓝 x).prod atInf) atInf
x : X
z : 𝕊
a✝ : (x, z) ∈ univ
⊢ ContinuousAt (uncurry (lift' g ∞)) (x, z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_lift' | [529, 1] | [535, 49] | exact continuousAt_lift_inf' (gi x) | 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
gc : Continuous (uncurry g)
gi : ∀ (x : X), Tendsto (uncurry g) ((𝓝 x).prod atInf) atInf
x : X
a✝ : (x, ∞) ∈ univ
⊢ ContinuousAt (uncurry (lift' g ∞)) (x, ∞) | 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
gc : Continuous (uncurry g)
gi : ∀ (x : X), Tendsto (uncurry g) ((𝓝 x).prod atInf) atInf
x : X
a✝ : (x, ∞) ∈ univ
⊢ ContinuousAt (uncurry (lift' g ∞)) (x, ∞)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_lift' | [529, 1] | [535, 49] | exact continuousAt_lift_coe' gc.continuousAt | 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
gc : Continuous (uncurry g)
gi : ∀ (x : X), Tendsto (uncurry g) ((𝓝 x).prod atInf) atInf
x : X
x✝ : ℂ
a✝ : (x, ↑x✝) ∈ univ
⊢ ContinuousAt (uncurry (lift' g ∞)) (x, ↑x✝) | 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
gc : Continuous (uncurry g)
gi : ∀ (x : X), Tendsto (uncurry g) ((𝓝 x).prod atInf) atInf
x : X
x✝ : ℂ
a✝ : (x, ↑x✝) ∈ univ
⊢ ContinuousAt (uncurry (lift' g ∞)) (x, ↑x✝)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuousAt_lift_coe | [538, 1] | [541, 90] | refine ContinuousAt.comp 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
fc : ContinuousAt f z
⊢ ContinuousAt (uncurry fun x => f) ((), 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
fc : ContinuousAt f z
⊢ ContinuousAt (fun a => a.2) ((), 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
fc : ContinuousAt f z
⊢ ContinuousAt (uncurry fun x => f) ((), z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuousAt_lift_coe | [538, 1] | [541, 90] | exact continuousAt_snd | X : Type
inst✝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
fc : ContinuousAt f z
⊢ ContinuousAt (fun a => a.2) ((), 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
fc : ContinuousAt f z
⊢ ContinuousAt (fun a => a.2) ((), z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_lift | [550, 1] | [554, 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
fc : Continuous f
fi : Tendsto f atInf atInf
⊢ Continuous (lift f ∞) | X : Type
inst✝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
fc : Continuous f
fi : Tendsto f atInf atInf
⊢ ∀ (x : 𝕊), ContinuousAt (lift f ∞) 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
fc : Continuous f
fi : Tendsto f atInf atInf
⊢ Continuous (lift f ∞)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_lift | [550, 1] | [554, 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
fc : Continuous f
fi : Tendsto f atInf atInf
⊢ ∀ (x : 𝕊), ContinuousAt (lift f ∞) 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
fc : Continuous f
fi : Tendsto f atInf atInf
z : 𝕊
⊢ ContinuousAt (lift f ∞) 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
fc : Continuous f
fi : Tendsto f atInf atInf
⊢ ∀ (x : 𝕊), ContinuousAt (lift f ∞) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_lift | [550, 1] | [554, 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
fc : Continuous f
fi : Tendsto f atInf atInf
z : 𝕊
⊢ ContinuousAt (lift f ∞) 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
fc : Continuous f
fi : Tendsto f atInf atInf
⊢ ContinuousAt (lift f ∞) ∞
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
fc : Continuous f
fi : Tendsto f atInf atInf
x✝ : ℂ
⊢ ContinuousAt (lift f ∞) ↑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
fc : Continuous f
fi : Tendsto f atInf atInf
z : 𝕊
⊢ ContinuousAt (lift f ∞) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_lift | [550, 1] | [554, 48] | exact continuousAt_lift_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
fc : Continuous f
fi : Tendsto f atInf atInf
⊢ ContinuousAt (lift f ∞) ∞ | 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
fc : Continuous f
fi : Tendsto f atInf atInf
⊢ ContinuousAt (lift f ∞) ∞
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_lift | [550, 1] | [554, 48] | exact continuousAt_lift_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
fc : Continuous f
fi : Tendsto f atInf atInf
x✝ : ℂ
⊢ ContinuousAt (lift f ∞) ↑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
fc : Continuous f
fi : Tendsto f atInf atInf
x✝ : ℂ
⊢ ContinuousAt (lift f ∞) ↑x✝
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_lift_coe | [557, 1] | [558, 100] | rw [lift_eq_fill] | X : Type
inst✝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
fa : AnalyticAt ℂ f z
⊢ HolomorphicAt I I (lift 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
fa : AnalyticAt ℂ f z
⊢ HolomorphicAt I I (fill (fun z => ↑(f z)) 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
fa : AnalyticAt ℂ f z
⊢ HolomorphicAt I I (lift f y) ↑z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_lift_coe | [557, 1] | [558, 100] | exact holomorphicAt_fill_coe ((holomorphic_coe _).comp (fa.holomorphicAt I I)) | X : Type
inst✝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
fa : AnalyticAt ℂ f z
⊢ HolomorphicAt I I (fill (fun z => ↑(f z)) 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
fa : AnalyticAt ℂ f z
⊢ HolomorphicAt I I (fill (fun z => ↑(f z)) y) ↑z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_lift_inf | [561, 1] | [565, 32] | rw [lift_eq_fill] | X : Type
inst✝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
fa : ∀ᶠ (z : ℂ) in atInf, AnalyticAt ℂ f z
fi : Tendsto f atInf atInf
⊢ HolomorphicAt I I (lift f ∞) ∞ | X : Type
inst✝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
fa : ∀ᶠ (z : ℂ) in atInf, AnalyticAt ℂ f z
fi : Tendsto f atInf atInf
⊢ HolomorphicAt I I (fill (fun z => ↑(f 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
fa : ∀ᶠ (z : ℂ) in atInf, AnalyticAt ℂ f z
fi : Tendsto f atInf atInf
⊢ HolomorphicAt I I (lift f ∞) ∞
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_lift_inf | [561, 1] | [565, 32] | apply holomorphicAt_fill_inf | X : Type
inst✝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
fa : ∀ᶠ (z : ℂ) in atInf, AnalyticAt ℂ f z
fi : Tendsto f atInf atInf
⊢ HolomorphicAt I I (fill (fun z => ↑(f z)) ∞) ∞ | case fa
X : Type
inst✝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
fa : ∀ᶠ (z : ℂ) in atInf, AnalyticAt ℂ f z
fi : Tendsto f atInf atInf
⊢ ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I (fun z => ↑(f z)) z
case 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
fa : ∀ᶠ (z : ℂ) in atInf, AnalyticAt ℂ f z
fi : Tendsto f atInf atInf
⊢ Tendsto (fun z => ↑(f z)) atInf (𝓝 ∞) | 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
fa : ∀ᶠ (z : ℂ) in atInf, AnalyticAt ℂ f z
fi : Tendsto f atInf atInf
⊢ HolomorphicAt I I (fill (fun z => ↑(f z)) ∞) ∞
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_lift_inf | [561, 1] | [565, 32] | exact fa.mp (eventually_of_forall fun z fa ↦ (holomorphic_coe _).comp (fa.holomorphicAt I I)) | case fa
X : Type
inst✝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
fa : ∀ᶠ (z : ℂ) in atInf, AnalyticAt ℂ f z
fi : Tendsto f atInf atInf
⊢ ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I (fun z => ↑(f z)) z
case 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
fa : ∀ᶠ (z : ℂ) in atInf, AnalyticAt ℂ f z
fi : Tendsto f atInf atInf
⊢ Tendsto (fun z => ↑(f z)) atInf (𝓝 ∞) | case 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
fa : ∀ᶠ (z : ℂ) in atInf, AnalyticAt ℂ f z
fi : Tendsto f atInf atInf
⊢ Tendsto (fun z => ↑(f z)) atInf (𝓝 ∞) | Please generate a tactic in lean4 to solve the state.
STATE:
case fa
X : Type
inst✝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
fa : ∀ᶠ (z : ℂ) in atInf, AnalyticAt ℂ f z
fi : Tendsto f atInf atInf
⊢ ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I (fun z => ↑(f z)) z
case 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
fa : ∀ᶠ (z : ℂ) in atInf, AnalyticAt ℂ f z
fi : Tendsto f atInf atInf
⊢ Tendsto (fun z => ↑(f z)) atInf (𝓝 ∞)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicAt_lift_inf | [561, 1] | [565, 32] | exact coe_tendsto_inf.comp fi | case 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
fa : ∀ᶠ (z : ℂ) in atInf, AnalyticAt ℂ f z
fi : Tendsto f atInf atInf
⊢ Tendsto (fun z => ↑(f z)) atInf (𝓝 ∞) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case 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
fa : ∀ᶠ (z : ℂ) in atInf, AnalyticAt ℂ f z
fi : Tendsto f atInf atInf
⊢ Tendsto (fun z => ↑(f z)) atInf (𝓝 ∞)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_lift | [568, 1] | [572, 53] | 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
fa : AnalyticOn ℂ f univ
fi : Tendsto f atInf atInf
⊢ Holomorphic I I (lift f ∞) | X : Type
inst✝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
fa : AnalyticOn ℂ f univ
fi : Tendsto f atInf atInf
z : 𝕊
⊢ HolomorphicAt I I (lift f ∞) 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
fa : AnalyticOn ℂ f univ
fi : Tendsto f atInf atInf
⊢ Holomorphic I I (lift f ∞)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_lift | [568, 1] | [572, 53] | 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
fa : AnalyticOn ℂ f univ
fi : Tendsto f atInf atInf
z : 𝕊
⊢ HolomorphicAt I I (lift f ∞) 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
fa : AnalyticOn ℂ f univ
fi : Tendsto f atInf atInf
⊢ HolomorphicAt I I (lift f ∞) ∞
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
fa : AnalyticOn ℂ f univ
fi : Tendsto f atInf atInf
x✝ : ℂ
⊢ HolomorphicAt I I (lift f ∞) ↑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
fa : AnalyticOn ℂ f univ
fi : Tendsto f atInf atInf
z : 𝕊
⊢ HolomorphicAt I I (lift f ∞) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_lift | [568, 1] | [572, 53] | exact holomorphicAt_lift_inf (eventually_of_forall fun z ↦ fa z (mem_univ _)) 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
fa : AnalyticOn ℂ f univ
fi : Tendsto f atInf atInf
⊢ HolomorphicAt I I (lift f ∞) ∞ | 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
fa : AnalyticOn ℂ f univ
fi : Tendsto f atInf atInf
⊢ HolomorphicAt I I (lift f ∞) ∞
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_lift | [568, 1] | [572, 53] | exact holomorphicAt_lift_coe (fa _ (mem_univ _)) | 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
fa : AnalyticOn ℂ f univ
fi : Tendsto f atInf atInf
x✝ : ℂ
⊢ HolomorphicAt I I (lift f ∞) ↑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
fa : AnalyticOn ℂ f univ
fi : Tendsto f atInf atInf
x✝ : ℂ
⊢ HolomorphicAt I I (lift f ∞) ↑x✝
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicLift' | [575, 1] | [585, 69] | apply osgoodManifold (continuous_lift' fa.continuous 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 → ℂ → ℂ
fa : AnalyticOn ℂ (uncurry f) univ
fi : ∀ (x : ℂ), Tendsto (uncurry f) ((𝓝 x).prod atInf) atInf
⊢ Holomorphic (I.prod I) I (uncurry (lift' f ∞)) | case f0
X : Type
inst✝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 → ℂ → ℂ
fa : AnalyticOn ℂ (uncurry f) univ
fi : ∀ (x : ℂ), Tendsto (uncurry f) ((𝓝 x).prod atInf) atInf
⊢ ∀ (x : ℂ) (y : 𝕊), HolomorphicAt I I (fun x => uncurry (lift' f ∞) (x, y)) x
case f1
X : Type
inst✝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 → ℂ → ℂ
fa : AnalyticOn ℂ (uncurry f) univ
fi : ∀ (x : ℂ), Tendsto (uncurry f) ((𝓝 x).prod atInf) atInf
⊢ ∀ (x : ℂ) (y : 𝕊), HolomorphicAt I I (fun y => uncurry (lift' f ∞) (x, y)) y | 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 → ℂ → ℂ
fa : AnalyticOn ℂ (uncurry f) univ
fi : ∀ (x : ℂ), Tendsto (uncurry f) ((𝓝 x).prod atInf) atInf
⊢ Holomorphic (I.prod I) I (uncurry (lift' f ∞))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicLift' | [575, 1] | [585, 69] | intro x z | case f0
X : Type
inst✝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 → ℂ → ℂ
fa : AnalyticOn ℂ (uncurry f) univ
fi : ∀ (x : ℂ), Tendsto (uncurry f) ((𝓝 x).prod atInf) atInf
⊢ ∀ (x : ℂ) (y : 𝕊), HolomorphicAt I I (fun x => uncurry (lift' f ∞) (x, y)) x | case f0
X : Type
inst✝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 → ℂ → ℂ
fa : AnalyticOn ℂ (uncurry f) univ
fi : ∀ (x : ℂ), Tendsto (uncurry f) ((𝓝 x).prod atInf) atInf
x : ℂ
z : 𝕊
⊢ HolomorphicAt I I (fun x => uncurry (lift' f ∞) (x, z)) x | Please generate a tactic in lean4 to solve the state.
STATE:
case f0
X : Type
inst✝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 → ℂ → ℂ
fa : AnalyticOn ℂ (uncurry f) univ
fi : ∀ (x : ℂ), Tendsto (uncurry f) ((𝓝 x).prod atInf) atInf
⊢ ∀ (x : ℂ) (y : 𝕊), HolomorphicAt I I (fun x => uncurry (lift' f ∞) (x, y)) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicLift' | [575, 1] | [585, 69] | induction z using OnePoint.rec | case f0
X : Type
inst✝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 → ℂ → ℂ
fa : AnalyticOn ℂ (uncurry f) univ
fi : ∀ (x : ℂ), Tendsto (uncurry f) ((𝓝 x).prod atInf) atInf
x : ℂ
z : 𝕊
⊢ HolomorphicAt I I (fun x => uncurry (lift' f ∞) (x, z)) x | case f0.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 → ℂ → ℂ
fa : AnalyticOn ℂ (uncurry f) univ
fi : ∀ (x : ℂ), Tendsto (uncurry f) ((𝓝 x).prod atInf) atInf
x : ℂ
⊢ HolomorphicAt I I (fun x => uncurry (lift' f ∞) (x, ∞)) x
case f0.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
f : ℂ → ℂ → ℂ
fa : AnalyticOn ℂ (uncurry f) univ
fi : ∀ (x : ℂ), Tendsto (uncurry f) ((𝓝 x).prod atInf) atInf
x x✝ : ℂ
⊢ HolomorphicAt I I (fun x => uncurry (lift' f ∞) (x, ↑x✝)) x | Please generate a tactic in lean4 to solve the state.
STATE:
case f0
X : Type
inst✝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 → ℂ → ℂ
fa : AnalyticOn ℂ (uncurry f) univ
fi : ∀ (x : ℂ), Tendsto (uncurry f) ((𝓝 x).prod atInf) atInf
x : ℂ
z : 𝕊
⊢ HolomorphicAt I I (fun x => uncurry (lift' f ∞) (x, z)) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicLift' | [575, 1] | [585, 69] | simp only [uncurry, lift_inf'] | case f0.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 → ℂ → ℂ
fa : AnalyticOn ℂ (uncurry f) univ
fi : ∀ (x : ℂ), Tendsto (uncurry f) ((𝓝 x).prod atInf) atInf
x : ℂ
⊢ HolomorphicAt I I (fun x => uncurry (lift' f ∞) (x, ∞)) x | case f0.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 → ℂ → ℂ
fa : AnalyticOn ℂ (uncurry f) univ
fi : ∀ (x : ℂ), Tendsto (uncurry f) ((𝓝 x).prod atInf) atInf
x : ℂ
⊢ HolomorphicAt I I (fun x => ∞) x | Please generate a tactic in lean4 to solve the state.
STATE:
case f0.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 → ℂ → ℂ
fa : AnalyticOn ℂ (uncurry f) univ
fi : ∀ (x : ℂ), Tendsto (uncurry f) ((𝓝 x).prod atInf) atInf
x : ℂ
⊢ HolomorphicAt I I (fun x => uncurry (lift' f ∞) (x, ∞)) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicLift' | [575, 1] | [585, 69] | exact holomorphicAt_const | case f0.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 → ℂ → ℂ
fa : AnalyticOn ℂ (uncurry f) univ
fi : ∀ (x : ℂ), Tendsto (uncurry f) ((𝓝 x).prod atInf) atInf
x : ℂ
⊢ HolomorphicAt I I (fun x => ∞) x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case f0.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 → ℂ → ℂ
fa : AnalyticOn ℂ (uncurry f) univ
fi : ∀ (x : ℂ), Tendsto (uncurry f) ((𝓝 x).prod atInf) atInf
x : ℂ
⊢ HolomorphicAt I I (fun x => ∞) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicLift' | [575, 1] | [585, 69] | exact (holomorphic_coe _).comp ((fa _ (mem_univ ⟨_,_⟩)).along_fst.holomorphicAt _ _) | case f0.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
f : ℂ → ℂ → ℂ
fa : AnalyticOn ℂ (uncurry f) univ
fi : ∀ (x : ℂ), Tendsto (uncurry f) ((𝓝 x).prod atInf) atInf
x x✝ : ℂ
⊢ HolomorphicAt I I (fun x => uncurry (lift' f ∞) (x, ↑x✝)) x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case f0.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
f : ℂ → ℂ → ℂ
fa : AnalyticOn ℂ (uncurry f) univ
fi : ∀ (x : ℂ), Tendsto (uncurry f) ((𝓝 x).prod atInf) atInf
x x✝ : ℂ
⊢ HolomorphicAt I I (fun x => uncurry (lift' f ∞) (x, ↑x✝)) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicLift' | [575, 1] | [585, 69] | intro x z | case f1
X : Type
inst✝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 → ℂ → ℂ
fa : AnalyticOn ℂ (uncurry f) univ
fi : ∀ (x : ℂ), Tendsto (uncurry f) ((𝓝 x).prod atInf) atInf
⊢ ∀ (x : ℂ) (y : 𝕊), HolomorphicAt I I (fun y => uncurry (lift' f ∞) (x, y)) y | case f1
X : Type
inst✝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 → ℂ → ℂ
fa : AnalyticOn ℂ (uncurry f) univ
fi : ∀ (x : ℂ), Tendsto (uncurry f) ((𝓝 x).prod atInf) atInf
x : ℂ
z : 𝕊
⊢ HolomorphicAt I I (fun y => uncurry (lift' f ∞) (x, y)) z | Please generate a tactic in lean4 to solve the state.
STATE:
case f1
X : Type
inst✝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 → ℂ → ℂ
fa : AnalyticOn ℂ (uncurry f) univ
fi : ∀ (x : ℂ), Tendsto (uncurry f) ((𝓝 x).prod atInf) atInf
⊢ ∀ (x : ℂ) (y : 𝕊), HolomorphicAt I I (fun y => uncurry (lift' f ∞) (x, y)) y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicLift' | [575, 1] | [585, 69] | exact holomorphic_lift (fun _ _ ↦ (fa _ (mem_univ ⟨_,_⟩)).along_snd)
((fi x).comp (tendsto_const_nhds.prod_mk Filter.tendsto_id)) z | case f1
X : Type
inst✝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 → ℂ → ℂ
fa : AnalyticOn ℂ (uncurry f) univ
fi : ∀ (x : ℂ), Tendsto (uncurry f) ((𝓝 x).prod atInf) atInf
x : ℂ
z : 𝕊
⊢ HolomorphicAt I I (fun y => uncurry (lift' f ∞) (x, y)) z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case f1
X : Type
inst✝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 → ℂ → ℂ
fa : AnalyticOn ℂ (uncurry f) univ
fi : ∀ (x : ℂ), Tendsto (uncurry f) ((𝓝 x).prod atInf) atInf
x : ℂ
z : 𝕊
⊢ HolomorphicAt I I (fun y => uncurry (lift' f ∞) (x, y)) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Interval.mem_approx_iter_sqrt | [100, 1] | [107, 9] | rw [iter_sqrt] | a : ℝ
x : Interval
n : ℕ
ax : a ∈ approx x
⊢ a ^ 2 ^ (-↑n) ∈ approx (x.iter_sqrt n) | a : ℝ
x : Interval
n : ℕ
ax : a ∈ approx x
⊢ a ^ 2 ^ (-↑n) ∈ approx (x.log.scaleB' (-Fixed.ofNat0 n)).exp | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℝ
x : Interval
n : ℕ
ax : a ∈ approx x
⊢ a ^ 2 ^ (-↑n) ∈ approx (x.iter_sqrt n)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Interval.mem_approx_iter_sqrt | [100, 1] | [107, 9] | by_cases a0 : a ≤ 0 | a : ℝ
x : Interval
n : ℕ
ax : a ∈ approx x
⊢ a ^ 2 ^ (-↑n) ∈ approx (x.log.scaleB' (-Fixed.ofNat0 n)).exp | case pos
a : ℝ
x : Interval
n : ℕ
ax : a ∈ approx x
a0 : a ≤ 0
⊢ a ^ 2 ^ (-↑n) ∈ approx (x.log.scaleB' (-Fixed.ofNat0 n)).exp
case neg
a : ℝ
x : Interval
n : ℕ
ax : a ∈ approx x
a0 : ¬a ≤ 0
⊢ a ^ 2 ^ (-↑n) ∈ approx (x.log.scaleB' (-Fixed.ofNat0 n)).exp | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℝ
x : Interval
n : ℕ
ax : a ∈ approx x
⊢ a ^ 2 ^ (-↑n) ∈ approx (x.log.scaleB' (-Fixed.ofNat0 n)).exp
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Interval.mem_approx_iter_sqrt | [100, 1] | [107, 9] | simp only [log_nonpos a0 ax, nan_scaleB', exp_nan, approx_nan, mem_univ] | case pos
a : ℝ
x : Interval
n : ℕ
ax : a ∈ approx x
a0 : a ≤ 0
⊢ a ^ 2 ^ (-↑n) ∈ approx (x.log.scaleB' (-Fixed.ofNat0 n)).exp | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
a : ℝ
x : Interval
n : ℕ
ax : a ∈ approx x
a0 : a ≤ 0
⊢ a ^ 2 ^ (-↑n) ∈ approx (x.log.scaleB' (-Fixed.ofNat0 n)).exp
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Interval.mem_approx_iter_sqrt | [100, 1] | [107, 9] | rw [Real.rpow_def_of_pos (not_le.mp a0)] | case neg
a : ℝ
x : Interval
n : ℕ
ax : a ∈ approx x
a0 : ¬a ≤ 0
⊢ a ^ 2 ^ (-↑n) ∈ approx (x.log.scaleB' (-Fixed.ofNat0 n)).exp | case neg
a : ℝ
x : Interval
n : ℕ
ax : a ∈ approx x
a0 : ¬a ≤ 0
⊢ (a.log * 2 ^ (-↑n)).exp ∈ approx (x.log.scaleB' (-Fixed.ofNat0 n)).exp | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
a : ℝ
x : Interval
n : ℕ
ax : a ∈ approx x
a0 : ¬a ≤ 0
⊢ a ^ 2 ^ (-↑n) ∈ approx (x.log.scaleB' (-Fixed.ofNat0 n)).exp
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Interval.mem_approx_iter_sqrt | [100, 1] | [107, 9] | mono | case neg
a : ℝ
x : Interval
n : ℕ
ax : a ∈ approx x
a0 : ¬a ≤ 0
⊢ (a.log * 2 ^ (-↑n)).exp ∈ approx (x.log.scaleB' (-Fixed.ofNat0 n)).exp | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
a : ℝ
x : Interval
n : ℕ
ax : a ∈ approx x
a0 : ¬a ≤ 0
⊢ (a.log * 2 ^ (-↑n)).exp ∈ approx (x.log.scaleB' (-Fixed.ofNat0 n)).exp
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Interval.mem_approx_iter_sqrt' | [109, 1] | [118, 32] | generalize hb : a^(2^n) = b at ax | a : ℝ
x : Interval
n : ℕ
a0 : 0 ≤ a
ax : a ^ 2 ^ n ∈ approx x
⊢ a ∈ approx (x.iter_sqrt n) | a : ℝ
x : Interval
n : ℕ
a0 : 0 ≤ a
b : ℝ
hb : a ^ 2 ^ n = b
ax : b ∈ approx x
⊢ a ∈ approx (x.iter_sqrt n) | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℝ
x : Interval
n : ℕ
a0 : 0 ≤ a
ax : a ^ 2 ^ n ∈ approx x
⊢ a ∈ approx (x.iter_sqrt n)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Interval.mem_approx_iter_sqrt' | [109, 1] | [118, 32] | have ab : a = b ^ (2 ^ (-n : ℝ) : ℝ) := by
have e : (2:ℝ)^n = (2^n : ℕ) := by norm_num
rw [←hb, Real.rpow_neg (by norm_num), Real.rpow_natCast, e,
Real.pow_rpow_inv_natCast a0 (pow_ne_zero _ (by norm_num))] | a : ℝ
x : Interval
n : ℕ
a0 : 0 ≤ a
b : ℝ
hb : a ^ 2 ^ n = b
ax : b ∈ approx x
⊢ a ∈ approx (x.iter_sqrt n) | a : ℝ
x : Interval
n : ℕ
a0 : 0 ≤ a
b : ℝ
hb : a ^ 2 ^ n = b
ax : b ∈ approx x
ab : a = b ^ 2 ^ (-↑n)
⊢ a ∈ approx (x.iter_sqrt n) | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℝ
x : Interval
n : ℕ
a0 : 0 ≤ a
b : ℝ
hb : a ^ 2 ^ n = b
ax : b ∈ approx x
⊢ a ∈ approx (x.iter_sqrt n)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Interval.mem_approx_iter_sqrt' | [109, 1] | [118, 32] | rw [ab] | a : ℝ
x : Interval
n : ℕ
a0 : 0 ≤ a
b : ℝ
hb : a ^ 2 ^ n = b
ax : b ∈ approx x
ab : a = b ^ 2 ^ (-↑n)
⊢ a ∈ approx (x.iter_sqrt n) | a : ℝ
x : Interval
n : ℕ
a0 : 0 ≤ a
b : ℝ
hb : a ^ 2 ^ n = b
ax : b ∈ approx x
ab : a = b ^ 2 ^ (-↑n)
⊢ b ^ 2 ^ (-↑n) ∈ approx (x.iter_sqrt n) | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℝ
x : Interval
n : ℕ
a0 : 0 ≤ a
b : ℝ
hb : a ^ 2 ^ n = b
ax : b ∈ approx x
ab : a = b ^ 2 ^ (-↑n)
⊢ a ∈ approx (x.iter_sqrt n)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Interval.mem_approx_iter_sqrt' | [109, 1] | [118, 32] | exact mem_approx_iter_sqrt ax | a : ℝ
x : Interval
n : ℕ
a0 : 0 ≤ a
b : ℝ
hb : a ^ 2 ^ n = b
ax : b ∈ approx x
ab : a = b ^ 2 ^ (-↑n)
⊢ b ^ 2 ^ (-↑n) ∈ approx (x.iter_sqrt n) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℝ
x : Interval
n : ℕ
a0 : 0 ≤ a
b : ℝ
hb : a ^ 2 ^ n = b
ax : b ∈ approx x
ab : a = b ^ 2 ^ (-↑n)
⊢ b ^ 2 ^ (-↑n) ∈ approx (x.iter_sqrt n)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Interval.mem_approx_iter_sqrt' | [109, 1] | [118, 32] | have e : (2:ℝ)^n = (2^n : ℕ) := by norm_num | a : ℝ
x : Interval
n : ℕ
a0 : 0 ≤ a
b : ℝ
hb : a ^ 2 ^ n = b
ax : b ∈ approx x
⊢ a = b ^ 2 ^ (-↑n) | a : ℝ
x : Interval
n : ℕ
a0 : 0 ≤ a
b : ℝ
hb : a ^ 2 ^ n = b
ax : b ∈ approx x
e : 2 ^ n = ↑(2 ^ n)
⊢ a = b ^ 2 ^ (-↑n) | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℝ
x : Interval
n : ℕ
a0 : 0 ≤ a
b : ℝ
hb : a ^ 2 ^ n = b
ax : b ∈ approx x
⊢ a = b ^ 2 ^ (-↑n)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Interval.mem_approx_iter_sqrt' | [109, 1] | [118, 32] | rw [←hb, Real.rpow_neg (by norm_num), Real.rpow_natCast, e,
Real.pow_rpow_inv_natCast a0 (pow_ne_zero _ (by norm_num))] | a : ℝ
x : Interval
n : ℕ
a0 : 0 ≤ a
b : ℝ
hb : a ^ 2 ^ n = b
ax : b ∈ approx x
e : 2 ^ n = ↑(2 ^ n)
⊢ a = b ^ 2 ^ (-↑n) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℝ
x : Interval
n : ℕ
a0 : 0 ≤ a
b : ℝ
hb : a ^ 2 ^ n = b
ax : b ∈ approx x
e : 2 ^ n = ↑(2 ^ n)
⊢ a = b ^ 2 ^ (-↑n)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Interval.mem_approx_iter_sqrt' | [109, 1] | [118, 32] | norm_num | a : ℝ
x : Interval
n : ℕ
a0 : 0 ≤ a
b : ℝ
hb : a ^ 2 ^ n = b
ax : b ∈ approx x
⊢ 2 ^ n = ↑(2 ^ n) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℝ
x : Interval
n : ℕ
a0 : 0 ≤ a
b : ℝ
hb : a ^ 2 ^ n = b
ax : b ∈ approx x
⊢ 2 ^ n = ↑(2 ^ n)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Interval.mem_approx_iter_sqrt' | [109, 1] | [118, 32] | norm_num | a : ℝ
x : Interval
n : ℕ
a0 : 0 ≤ a
b : ℝ
hb : a ^ 2 ^ n = b
ax : b ∈ approx x
e : 2 ^ n = ↑(2 ^ n)
⊢ 0 ≤ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℝ
x : Interval
n : ℕ
a0 : 0 ≤ a
b : ℝ
hb : a ^ 2 ^ n = b
ax : b ∈ approx x
e : 2 ^ n = ↑(2 ^ n)
⊢ 0 ≤ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Interval.mem_approx_iter_sqrt' | [109, 1] | [118, 32] | norm_num | a : ℝ
x : Interval
n : ℕ
a0 : 0 ≤ a
b : ℝ
hb : a ^ 2 ^ n = b
ax : b ∈ approx x
e : 2 ^ n = ↑(2 ^ n)
⊢ 2 ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℝ
x : Interval
n : ℕ
a0 : 0 ≤ a
b : ℝ
hb : a ^ 2 ^ n = b
ax : b ∈ approx x
e : 2 ^ n = ↑(2 ^ n)
⊢ 2 ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_small | [120, 1] | [130, 33] | refine ss ⟨?_, ?_⟩ | c' z' : ℂ
c4 : Complex.abs c' ≤ 4
z4 : Complex.abs z' ≤ 4
ss : Icc 0.216 1 ⊆ approx potential_small
⊢ ⋯.potential c' ↑z' ∈ approx potential_small | case refine_1
c' z' : ℂ
c4 : Complex.abs c' ≤ 4
z4 : Complex.abs z' ≤ 4
ss : Icc 0.216 1 ⊆ approx potential_small
⊢ 0.216 ≤ ⋯.potential c' ↑z'
case refine_2
c' z' : ℂ
c4 : Complex.abs c' ≤ 4
z4 : Complex.abs z' ≤ 4
ss : Icc 0.216 1 ⊆ approx potential_small
⊢ ⋯.potential c' ↑z' ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
c' z' : ℂ
c4 : Complex.abs c' ≤ 4
z4 : Complex.abs z' ≤ 4
ss : Icc 0.216 1 ⊆ approx potential_small
⊢ ⋯.potential c' ↑z' ∈ approx potential_small
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_small | [120, 1] | [130, 33] | rw [potential_small] | c' z' : ℂ
c4 : Complex.abs c' ≤ 4
z4 : Complex.abs z' ≤ 4
⊢ Icc 0.216 1 ⊆ approx potential_small | c' z' : ℂ
c4 : Complex.abs c' ≤ 4
z4 : Complex.abs z' ≤ 4
⊢ Icc 0.216 1 ⊆ approx (0.216 ∪ 1) | Please generate a tactic in lean4 to solve the state.
STATE:
c' z' : ℂ
c4 : Complex.abs c' ≤ 4
z4 : Complex.abs z' ≤ 4
⊢ Icc 0.216 1 ⊆ approx potential_small
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_small | [120, 1] | [130, 33] | apply Icc_subset_approx | c' z' : ℂ
c4 : Complex.abs c' ≤ 4
z4 : Complex.abs z' ≤ 4
⊢ Icc 0.216 1 ⊆ approx (0.216 ∪ 1) | case ax
c' z' : ℂ
c4 : Complex.abs c' ≤ 4
z4 : Complex.abs z' ≤ 4
⊢ 0.216 ∈ approx (0.216 ∪ 1)
case bx
c' z' : ℂ
c4 : Complex.abs c' ≤ 4
z4 : Complex.abs z' ≤ 4
⊢ 1 ∈ approx (0.216 ∪ 1) | Please generate a tactic in lean4 to solve the state.
STATE:
c' z' : ℂ
c4 : Complex.abs c' ≤ 4
z4 : Complex.abs z' ≤ 4
⊢ Icc 0.216 1 ⊆ approx (0.216 ∪ 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_small | [120, 1] | [130, 33] | exact Interval.approx_union_left (Interval.approx_ofScientific _ _ _) | case ax
c' z' : ℂ
c4 : Complex.abs c' ≤ 4
z4 : Complex.abs z' ≤ 4
⊢ 0.216 ∈ approx (0.216 ∪ 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case ax
c' z' : ℂ
c4 : Complex.abs c' ≤ 4
z4 : Complex.abs z' ≤ 4
⊢ 0.216 ∈ approx (0.216 ∪ 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_small | [120, 1] | [130, 33] | exact Interval.approx_union_right Interval.mem_approx_one | case bx
c' z' : ℂ
c4 : Complex.abs c' ≤ 4
z4 : Complex.abs z' ≤ 4
⊢ 1 ∈ approx (0.216 ∪ 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case bx
c' z' : ℂ
c4 : Complex.abs c' ≤ 4
z4 : Complex.abs z' ≤ 4
⊢ 1 ∈ approx (0.216 ∪ 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_small | [120, 1] | [130, 33] | exact le_potential c4 z4 | case refine_1
c' z' : ℂ
c4 : Complex.abs c' ≤ 4
z4 : Complex.abs z' ≤ 4
ss : Icc 0.216 1 ⊆ approx potential_small
⊢ 0.216 ≤ ⋯.potential c' ↑z' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
c' z' : ℂ
c4 : Complex.abs c' ≤ 4
z4 : Complex.abs z' ≤ 4
ss : Icc 0.216 1 ⊆ approx potential_small
⊢ 0.216 ≤ ⋯.potential c' ↑z'
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_small | [120, 1] | [130, 33] | apply Super.potential_le_one | case refine_2
c' z' : ℂ
c4 : Complex.abs c' ≤ 4
z4 : Complex.abs z' ≤ 4
ss : Icc 0.216 1 ⊆ approx potential_small
⊢ ⋯.potential c' ↑z' ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
c' z' : ℂ
c4 : Complex.abs c' ≤ 4
z4 : Complex.abs z' ≤ 4
ss : Icc 0.216 1 ⊆ approx potential_small
⊢ ⋯.potential c' ↑z' ≤ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | rw [potential_large] | c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
⊢ ⋯.potential c' ↑z' ∈ approx z.potential_large | c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
⊢ ⋯.potential c' ↑z' ∈
approx
(let s := z.normSq;
let log_s := s.log;
(-log_s.div2).exp.grow (0.8095 * (log_s * -0.9635).exp).hi) | Please generate a tactic in lean4 to solve the state.
STATE:
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
⊢ ⋯.potential c' ↑z' ∈ approx z.potential_large
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | apply Interval.approx_grow (potential_approx 2 (le_trans (by norm_num) z6) cz) | c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
⊢ ⋯.potential c' ↑z' ∈
approx
(let s := z.normSq;
let log_s := s.log;
(-log_s.div2).exp.grow (0.8095 * (log_s * -0.9635).exp).hi) | case be
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
⊢ (0.8095 * (z.normSq.log * -0.9635).exp).hi ≠ nan →
potential_error 2 c' z' ≤ (0.8095 * (z.normSq.log * -0.9635).exp).hi.val
case zx
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
⊢ 1 / Complex.abs z' ∈ approx (-z.normSq.log.div2).exp | Please generate a tactic in lean4 to solve the state.
STATE:
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
⊢ ⋯.potential c' ↑z' ∈
approx
(let s := z.normSq;
let log_s := s.log;
(-log_s.div2).exp.grow (0.8095 * (log_s * -0.9635).exp).hi)
TACTIC:
|
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