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|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_lift' | [529, 1] | [535, 49] | apply ContinuousAt.continuousWithinAt | X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
xβ : X
zβ : β
gc : Continuous (uncurry g)
gi : β (x : X), Tendsto (uncurry g) ((π x).prod atInf) atInf
x : X
z : π
aβ : (x, z) β univ
β’ ContinuousWithinAt (uncurry (lift' g β)) univ (x, z) | case h
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
xβ : X
zβ : β
gc : Continuous (uncurry g)
gi : β (x : X), Tendsto (uncurry g) ((π x).prod atInf) atInf
x : X
z : π
aβ : (x, z) β univ
β’ ContinuousAt (uncurry (lift' g β)) (x, z) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
xβ : X
zβ : β
gc : Continuous (uncurry g)
gi : β (x : X), Tendsto (uncurry g) ((π x).prod atInf) atInf
x : X
z : π
aβ : (x, z) β univ
β’ ContinuousWithinAt (uncurry (lift' g β)) 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
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
xβ : X
zβ : β
gc : Continuous (uncurry g)
gi : β (x : X), Tendsto (uncurry g) ((π x).prod atInf) atInf
x : X
z : π
aβ : (x, z) β univ
β’ ContinuousAt (uncurry (lift' g β)) (x, z) | case h.hβ
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
xβ : X
z : β
gc : Continuous (uncurry g)
gi : β (x : X), Tendsto (uncurry g) ((π x).prod atInf) atInf
x : X
aβ : (x, β) β univ
β’ ContinuousAt (uncurry (lift' g β)) (x, β)
case h.hβ
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
xβΒΉ : X
z : β
gc : Continuous (uncurry g)
gi : β (x : X), Tendsto (uncurry g) ((π x).prod atInf) atInf
x : X
xβ : β
aβ : (x, βxβ) β univ
β’ ContinuousAt (uncurry (lift' g β)) (x, βxβ) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
xβ : X
zβ : β
gc : Continuous (uncurry g)
gi : β (x : X), Tendsto (uncurry g) ((π x).prod atInf) atInf
x : X
z : π
aβ : (x, z) β univ
β’ ContinuousAt (uncurry (lift' g β)) (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.hβ
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
xβ : X
z : β
gc : Continuous (uncurry g)
gi : β (x : X), Tendsto (uncurry g) ((π x).prod atInf) atInf
x : X
aβ : (x, β) β univ
β’ ContinuousAt (uncurry (lift' g β)) (x, β) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hβ
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
xβ : X
z : β
gc : Continuous (uncurry g)
gi : β (x : X), Tendsto (uncurry g) ((π x).prod atInf) atInf
x : X
aβ : (x, β) β univ
β’ ContinuousAt (uncurry (lift' g β)) (x, β)
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.hβ
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
xβΒΉ : X
z : β
gc : Continuous (uncurry g)
gi : β (x : X), Tendsto (uncurry g) ((π x).prod atInf) atInf
x : X
xβ : β
aβ : (x, βxβ) β univ
β’ ContinuousAt (uncurry (lift' g β)) (x, βxβ) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hβ
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
xβΒΉ : X
z : β
gc : Continuous (uncurry g)
gi : β (x : X), Tendsto (uncurry g) ((π x).prod atInf) atInf
x : X
xβ : β
aβ : (x, βxβ) β univ
β’ ContinuousAt (uncurry (lift' g β)) (x, βxβ)
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
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
z : β
fc : ContinuousAt f z
β’ ContinuousAt (uncurry fun x => f) ((), z) | X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
z : β
fc : ContinuousAt f z
β’ ContinuousAt (fun a => a.2) ((), z) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
z : β
fc : ContinuousAt f z
β’ 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
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
z : β
fc : ContinuousAt f z
β’ ContinuousAt (fun a => a.2) ((), z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
z : β
fc : ContinuousAt f z
β’ 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
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
z : β
fc : Continuous f
fi : Tendsto f atInf atInf
β’ Continuous (lift f β) | X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
z : β
fc : Continuous f
fi : Tendsto f atInf atInf
β’ β (x : π), ContinuousAt (lift f β) x | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
z : β
fc : Continuous f
fi : Tendsto f atInf atInf
β’ Continuous (lift f β)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_lift | [550, 1] | [554, 48] | intro z | X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
z : β
fc : Continuous f
fi : Tendsto f atInf atInf
β’ β (x : π), ContinuousAt (lift f β) x | X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
zβ : β
fc : Continuous f
fi : Tendsto f atInf atInf
z : π
β’ ContinuousAt (lift f β) z | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
z : β
fc : Continuous f
fi : Tendsto f atInf atInf
β’ β (x : π), ContinuousAt (lift f β) 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
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
zβ : β
fc : Continuous f
fi : Tendsto f atInf atInf
z : π
β’ ContinuousAt (lift f β) z | case hβ
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
z : β
fc : Continuous f
fi : Tendsto f atInf atInf
β’ ContinuousAt (lift f β) β
case hβ
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
z : β
fc : Continuous f
fi : Tendsto f atInf atInf
xβ : β
β’ ContinuousAt (lift f β) βxβ | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
zβ : β
fc : Continuous f
fi : Tendsto f atInf atInf
z : π
β’ ContinuousAt (lift f β) 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 hβ
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
z : β
fc : Continuous f
fi : Tendsto f atInf atInf
β’ ContinuousAt (lift f β) β | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hβ
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
z : β
fc : Continuous f
fi : Tendsto f atInf atInf
β’ ContinuousAt (lift f β) β
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 hβ
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
z : β
fc : Continuous f
fi : Tendsto f atInf atInf
xβ : β
β’ ContinuousAt (lift f β) βxβ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hβ
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
z : β
fc : Continuous f
fi : Tendsto f atInf atInf
xβ : β
β’ ContinuousAt (lift f β) βxβ
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
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
z : β
fa : AnalyticAt β f z
β’ HolomorphicAt I I (lift f y) βz | X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
z : β
fa : AnalyticAt β f z
β’ HolomorphicAt I I (fill (fun z => β(f z)) y) βz | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
z : β
fa : AnalyticAt β f z
β’ HolomorphicAt I I (lift f y) βz
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
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
z : β
fa : AnalyticAt β f z
β’ HolomorphicAt I I (fill (fun z => β(f z)) y) βz | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
z : β
fa : AnalyticAt β f z
β’ HolomorphicAt I I (fill (fun z => β(f z)) y) βz
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
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
z : β
fa : βαΆ (z : β) in atInf, AnalyticAt β f z
fi : Tendsto f atInf atInf
β’ HolomorphicAt I I (lift f β) β | X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
z : β
fa : βαΆ (z : β) in atInf, AnalyticAt β f z
fi : Tendsto f atInf atInf
β’ HolomorphicAt I I (fill (fun z => β(f z)) β) β | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
z : β
fa : βαΆ (z : β) in atInf, AnalyticAt β f z
fi : Tendsto f atInf atInf
β’ HolomorphicAt I I (lift f β) β
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
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
z : β
fa : βαΆ (z : β) in atInf, AnalyticAt β f z
fi : Tendsto f atInf atInf
β’ HolomorphicAt I I (fill (fun z => β(f z)) β) β | case fa
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
z : β
fa : βαΆ (z : β) in atInf, AnalyticAt β f z
fi : Tendsto f atInf atInf
β’ βαΆ (z : β) in atInf, HolomorphicAt I I (fun z => β(f z)) z
case fi
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
z : β
fa : βαΆ (z : β) in atInf, AnalyticAt β f z
fi : Tendsto f atInf atInf
β’ Tendsto (fun z => β(f z)) atInf (π β) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
z : β
fa : βαΆ (z : β) in atInf, AnalyticAt β f z
fi : Tendsto f atInf atInf
β’ HolomorphicAt I I (fill (fun z => β(f z)) β) β
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 β¦ (holomorphic_coe _).comp (fa.holomorphicAt I I)) | case fa
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
z : β
fa : βαΆ (z : β) in atInf, AnalyticAt β f z
fi : Tendsto f atInf atInf
β’ βαΆ (z : β) in atInf, HolomorphicAt I I (fun z => β(f z)) z
case fi
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
z : β
fa : βαΆ (z : β) in atInf, AnalyticAt β f z
fi : Tendsto f atInf atInf
β’ Tendsto (fun z => β(f z)) atInf (π β) | case fi
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
z : β
fa : βαΆ (z : β) in atInf, AnalyticAt β f z
fi : Tendsto f atInf atInf
β’ Tendsto (fun z => β(f z)) atInf (π β) | Please generate a tactic in lean4 to solve the state.
STATE:
case fa
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
z : β
fa : βαΆ (z : β) in atInf, AnalyticAt β f z
fi : Tendsto f atInf atInf
β’ βαΆ (z : β) in atInf, HolomorphicAt I I (fun z => β(f z)) z
case fi
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
z : β
fa : βαΆ (z : β) in atInf, AnalyticAt β f z
fi : Tendsto f atInf atInf
β’ Tendsto (fun z => β(f z)) atInf (π β)
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
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
z : β
fa : βαΆ (z : β) in atInf, AnalyticAt β f z
fi : Tendsto f atInf atInf
β’ Tendsto (fun z => β(f z)) atInf (π β) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case fi
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
z : β
fa : βαΆ (z : β) in atInf, AnalyticAt β f z
fi : Tendsto f atInf atInf
β’ Tendsto (fun z => β(f z)) atInf (π β)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphic_lift | [568, 1] | [572, 53] | intro z | X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
z : β
fa : AnalyticOn β f univ
fi : Tendsto f atInf atInf
β’ Holomorphic I I (lift f β) | X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
zβ : β
fa : AnalyticOn β f univ
fi : Tendsto f atInf atInf
z : π
β’ HolomorphicAt I I (lift f β) z | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
z : β
fa : AnalyticOn β f univ
fi : Tendsto f atInf atInf
β’ Holomorphic I I (lift f β)
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
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
zβ : β
fa : AnalyticOn β f univ
fi : Tendsto f atInf atInf
z : π
β’ HolomorphicAt I I (lift f β) z | case hβ
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
z : β
fa : AnalyticOn β f univ
fi : Tendsto f atInf atInf
β’ HolomorphicAt I I (lift f β) β
case hβ
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
z : β
fa : AnalyticOn β f univ
fi : Tendsto f atInf atInf
xβ : β
β’ HolomorphicAt I I (lift f β) βxβ | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
zβ : β
fa : AnalyticOn β f univ
fi : Tendsto f atInf atInf
z : π
β’ HolomorphicAt I I (lift f β) 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 β¦ fa z (mem_univ _)) fi | case hβ
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
z : β
fa : AnalyticOn β f univ
fi : Tendsto f atInf atInf
β’ HolomorphicAt I I (lift f β) β | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hβ
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
z : β
fa : AnalyticOn β f univ
fi : Tendsto f atInf atInf
β’ HolomorphicAt I I (lift f β) β
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 hβ
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
z : β
fa : AnalyticOn β f univ
fi : Tendsto f atInf atInf
xβ : β
β’ HolomorphicAt I I (lift f β) βxβ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hβ
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
f : β β β
g : X β β β β
y : π
x : X
z : β
fa : AnalyticOn β f univ
fi : Tendsto f atInf atInf
xβ : β
β’ HolomorphicAt I I (lift f β) βxβ
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
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
fβ : β β β
g : X β β β β
y : π
x : X
z : β
f : β β β β β
fa : AnalyticOn β (uncurry f) univ
fi : β (x : β), Tendsto (uncurry f) ((π x).prod atInf) atInf
β’ Holomorphic (I.prod I) I (uncurry (lift' f β)) | case f0
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
fβ : β β β
g : X β β β β
y : π
x : X
z : β
f : β β β β β
fa : AnalyticOn β (uncurry f) univ
fi : β (x : β), Tendsto (uncurry f) ((π x).prod atInf) atInf
β’ β (x : β) (y : π), HolomorphicAt I I (fun x => uncurry (lift' f β) (x, y)) x
case f1
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
fβ : β β β
g : X β β β β
y : π
x : X
z : β
f : β β β β β
fa : AnalyticOn β (uncurry f) univ
fi : β (x : β), Tendsto (uncurry f) ((π x).prod atInf) atInf
β’ β (x : β) (y : π), HolomorphicAt I I (fun y => uncurry (lift' f β) (x, y)) y | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
fβ : β β β
g : X β β β β
y : π
x : X
z : β
f : β β β β β
fa : AnalyticOn β (uncurry f) univ
fi : β (x : β), Tendsto (uncurry f) ((π x).prod atInf) atInf
β’ Holomorphic (I.prod I) I (uncurry (lift' f β))
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
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
fβ : β β β
g : X β β β β
y : π
x : X
z : β
f : β β β β β
fa : AnalyticOn β (uncurry f) univ
fi : β (x : β), Tendsto (uncurry f) ((π x).prod atInf) atInf
β’ β (x : β) (y : π), HolomorphicAt I I (fun x => uncurry (lift' f β) (x, y)) x | case f0
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
fβ : β β β
g : X β β β β
y : π
xβ : X
zβ : β
f : β β β β β
fa : AnalyticOn β (uncurry f) univ
fi : β (x : β), Tendsto (uncurry f) ((π x).prod atInf) atInf
x : β
z : π
β’ HolomorphicAt I I (fun x => uncurry (lift' f β) (x, z)) x | Please generate a tactic in lean4 to solve the state.
STATE:
case f0
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
fβ : β β β
g : X β β β β
y : π
x : X
z : β
f : β β β β β
fa : AnalyticOn β (uncurry f) univ
fi : β (x : β), Tendsto (uncurry f) ((π x).prod atInf) atInf
β’ β (x : β) (y : π), HolomorphicAt I I (fun x => uncurry (lift' f β) (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
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
fβ : β β β
g : X β β β β
y : π
xβ : X
zβ : β
f : β β β β β
fa : AnalyticOn β (uncurry f) univ
fi : β (x : β), Tendsto (uncurry f) ((π x).prod atInf) atInf
x : β
z : π
β’ HolomorphicAt I I (fun x => uncurry (lift' f β) (x, z)) x | case f0.hβ
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
fβ : β β β
g : X β β β β
y : π
xβ : X
z : β
f : β β β β β
fa : AnalyticOn β (uncurry f) univ
fi : β (x : β), Tendsto (uncurry f) ((π x).prod atInf) atInf
x : β
β’ HolomorphicAt I I (fun x => uncurry (lift' f β) (x, β)) x
case f0.hβ
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
fβ : β β β
g : X β β β β
y : π
xβΒΉ : X
z : β
f : β β β β β
fa : AnalyticOn β (uncurry f) univ
fi : β (x : β), Tendsto (uncurry f) ((π x).prod atInf) atInf
x xβ : β
β’ HolomorphicAt I I (fun x => uncurry (lift' f β) (x, βxβ)) x | Please generate a tactic in lean4 to solve the state.
STATE:
case f0
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
fβ : β β β
g : X β β β β
y : π
xβ : X
zβ : β
f : β β β β β
fa : AnalyticOn β (uncurry f) univ
fi : β (x : β), Tendsto (uncurry f) ((π x).prod atInf) atInf
x : β
z : π
β’ HolomorphicAt I I (fun x => uncurry (lift' f β) (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.hβ
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
fβ : β β β
g : X β β β β
y : π
xβ : X
z : β
f : β β β β β
fa : AnalyticOn β (uncurry f) univ
fi : β (x : β), Tendsto (uncurry f) ((π x).prod atInf) atInf
x : β
β’ HolomorphicAt I I (fun x => uncurry (lift' f β) (x, β)) x | case f0.hβ
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
fβ : β β β
g : X β β β β
y : π
xβ : X
z : β
f : β β β β β
fa : AnalyticOn β (uncurry f) univ
fi : β (x : β), Tendsto (uncurry f) ((π x).prod atInf) atInf
x : β
β’ HolomorphicAt I I (fun x => β) x | Please generate a tactic in lean4 to solve the state.
STATE:
case f0.hβ
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
fβ : β β β
g : X β β β β
y : π
xβ : X
z : β
f : β β β β β
fa : AnalyticOn β (uncurry f) univ
fi : β (x : β), Tendsto (uncurry f) ((π x).prod atInf) atInf
x : β
β’ HolomorphicAt I I (fun x => uncurry (lift' f β) (x, β)) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.holomorphicLift' | [575, 1] | [585, 69] | exact holomorphicAt_const | case f0.hβ
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
fβ : β β β
g : X β β β β
y : π
xβ : X
z : β
f : β β β β β
fa : AnalyticOn β (uncurry f) univ
fi : β (x : β), Tendsto (uncurry f) ((π x).prod atInf) atInf
x : β
β’ HolomorphicAt I I (fun x => β) x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case f0.hβ
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
fβ : β β β
g : X β β β β
y : π
xβ : X
z : β
f : β β β β β
fa : AnalyticOn β (uncurry f) univ
fi : β (x : β), Tendsto (uncurry f) ((π x).prod atInf) atInf
x : β
β’ HolomorphicAt I I (fun x => β) 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 β¨_,_β©)).along_fst.holomorphicAt _ _) | case f0.hβ
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
fβ : β β β
g : X β β β β
y : π
xβΒΉ : X
z : β
f : β β β β β
fa : AnalyticOn β (uncurry f) univ
fi : β (x : β), Tendsto (uncurry f) ((π x).prod atInf) atInf
x xβ : β
β’ HolomorphicAt I I (fun x => uncurry (lift' f β) (x, βxβ)) x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case f0.hβ
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
fβ : β β β
g : X β β β β
y : π
xβΒΉ : X
z : β
f : β β β β β
fa : AnalyticOn β (uncurry f) univ
fi : β (x : β), Tendsto (uncurry f) ((π x).prod atInf) atInf
x xβ : β
β’ HolomorphicAt I I (fun x => uncurry (lift' f β) (x, βxβ)) 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
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
fβ : β β β
g : X β β β β
y : π
x : X
z : β
f : β β β β β
fa : AnalyticOn β (uncurry f) univ
fi : β (x : β), Tendsto (uncurry f) ((π x).prod atInf) atInf
β’ β (x : β) (y : π), HolomorphicAt I I (fun y => uncurry (lift' f β) (x, y)) y | case f1
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
fβ : β β β
g : X β β β β
y : π
xβ : X
zβ : β
f : β β β β β
fa : AnalyticOn β (uncurry f) univ
fi : β (x : β), Tendsto (uncurry f) ((π x).prod atInf) atInf
x : β
z : π
β’ HolomorphicAt I I (fun y => uncurry (lift' f β) (x, y)) z | Please generate a tactic in lean4 to solve the state.
STATE:
case f1
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
fβ : β β β
g : X β β β β
y : π
x : X
z : β
f : β β β β β
fa : AnalyticOn β (uncurry f) univ
fi : β (x : β), Tendsto (uncurry f) ((π x).prod atInf) atInf
β’ β (x : β) (y : π), HolomorphicAt I I (fun y => uncurry (lift' f β) (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 _ _ β¦ (fa _ (mem_univ β¨_,_β©)).along_snd)
((fi x).comp (tendsto_const_nhds.prod_mk Filter.tendsto_id)) z | case f1
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
fβ : β β β
g : X β β β β
y : π
xβ : X
zβ : β
f : β β β β β
fa : AnalyticOn β (uncurry f) univ
fi : β (x : β), Tendsto (uncurry f) ((π x).prod atInf) atInf
x : β
z : π
β’ HolomorphicAt I I (fun y => uncurry (lift' f β) (x, y)) z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case f1
X : Type
instββ΄ : TopologicalSpace X
Y : Type
instβΒ³ : TopologicalSpace Y
T : Type
instβΒ² : TopologicalSpace T
instβΒΉ : ChartedSpace β T
instβ : AnalyticManifold I T
fβ : β β β
g : X β β β β
y : π
xβ : X
zβ : β
f : β β β β β
fa : AnalyticOn β (uncurry f) univ
fi : β (x : β), Tendsto (uncurry f) ((π x).prod atInf) atInf
x : β
z : π
β’ HolomorphicAt I I (fun y => uncurry (lift' f β) (x, y)) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj | [24, 1] | [35, 61] | rcases complex_inverse_fun' fa nc with β¨g, ga, gf, fgβ© | S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : S β T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z β 0
β’ βαΆ (p : S Γ S) in π (z, z), f p.1 = f p.2 β p.1 = p.2 | case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : S β T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z β 0
g : T β S
ga : HolomorphicAt I I g (f z)
gf : βαΆ (x : S) in π z, g (f x) = x
fg : βαΆ (x : T) in π (f z), f (g x) = x
β’ βαΆ (p : S Γ S) in π (z, z), f p.1 = f p.2 β p.1 = p.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : S β T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z β 0
β’ βαΆ (p : S Γ S) in π (z, z), f p.1 = f p.2 β p.1 = p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj | [24, 1] | [35, 61] | have n : NontrivialHolomorphicAt g (f z) := by
rw [β gf.self_of_nhds] at fa
refine (NontrivialHolomorphicAt.anti ?_ fa ga).2
exact (nontrivialHolomorphicAt_id _).congr (Filter.EventuallyEq.symm fg) | case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : S β T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z β 0
g : T β S
ga : HolomorphicAt I I g (f z)
gf : βαΆ (x : S) in π z, g (f x) = x
fg : βαΆ (x : T) in π (f z), f (g x) = x
β’ βαΆ (p : S Γ S) in π (z, z), f p.1 = f p.2 β p.1 = p.2 | case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : S β T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z β 0
g : T β S
ga : HolomorphicAt I I g (f z)
gf : βαΆ (x : S) in π z, g (f x) = x
fg : βαΆ (x : T) in π (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
β’ βαΆ (p : S Γ S) in π (z, z), f p.1 = f p.2 β p.1 = p.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : S β T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z β 0
g : T β S
ga : HolomorphicAt I I g (f z)
gf : βαΆ (x : S) in π z, g (f x) = x
fg : βαΆ (x : T) in π (f z), f (g x) = x
β’ βαΆ (p : S Γ S) in π (z, z), f p.1 = f p.2 β p.1 = p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj | [24, 1] | [35, 61] | have o := n.nhds_eq_map_nhds | case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : S β T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z β 0
g : T β S
ga : HolomorphicAt I I g (f z)
gf : βαΆ (x : S) in π z, g (f x) = x
fg : βαΆ (x : T) in π (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
β’ βαΆ (p : S Γ S) in π (z, z), f p.1 = f p.2 β p.1 = p.2 | case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : S β T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z β 0
g : T β S
ga : HolomorphicAt I I g (f z)
gf : βαΆ (x : S) in π z, g (f x) = x
fg : βαΆ (x : T) in π (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : π (g (f z)) = Filter.map g (π (f z))
β’ βαΆ (p : S Γ S) in π (z, z), f p.1 = f p.2 β p.1 = p.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : S β T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z β 0
g : T β S
ga : HolomorphicAt I I g (f z)
gf : βαΆ (x : S) in π z, g (f x) = x
fg : βαΆ (x : T) in π (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
β’ βαΆ (p : S Γ S) in π (z, z), f p.1 = f p.2 β p.1 = p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj | [24, 1] | [35, 61] | rw [gf.self_of_nhds] at o | case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : S β T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z β 0
g : T β S
ga : HolomorphicAt I I g (f z)
gf : βαΆ (x : S) in π z, g (f x) = x
fg : βαΆ (x : T) in π (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : π (g (f z)) = Filter.map g (π (f z))
β’ βαΆ (p : S Γ S) in π (z, z), f p.1 = f p.2 β p.1 = p.2 | case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : S β T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z β 0
g : T β S
ga : HolomorphicAt I I g (f z)
gf : βαΆ (x : S) in π z, g (f x) = x
fg : βαΆ (x : T) in π (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : π z = Filter.map g (π (f z))
β’ βαΆ (p : S Γ S) in π (z, z), f p.1 = f p.2 β p.1 = p.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : S β T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z β 0
g : T β S
ga : HolomorphicAt I I g (f z)
gf : βαΆ (x : S) in π z, g (f x) = x
fg : βαΆ (x : T) in π (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : π (g (f z)) = Filter.map g (π (f z))
β’ βαΆ (p : S Γ S) in π (z, z), f p.1 = f p.2 β p.1 = p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj | [24, 1] | [35, 61] | simp only [nhds_prod_eq, o, Filter.prod_map_map_eq, Filter.eventually_map] | case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : S β T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z β 0
g : T β S
ga : HolomorphicAt I I g (f z)
gf : βαΆ (x : S) in π z, g (f x) = x
fg : βαΆ (x : T) in π (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : π z = Filter.map g (π (f z))
β’ βαΆ (p : S Γ S) in π (z, z), f p.1 = f p.2 β p.1 = p.2 | case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : S β T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z β 0
g : T β S
ga : HolomorphicAt I I g (f z)
gf : βαΆ (x : S) in π z, g (f x) = x
fg : βαΆ (x : T) in π (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : π z = Filter.map g (π (f z))
β’ βαΆ (a : T Γ T) in π (f z) ΓΛ’ π (f z), f (g a.1) = f (g a.2) β g a.1 = g a.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : S β T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z β 0
g : T β S
ga : HolomorphicAt I I g (f z)
gf : βαΆ (x : S) in π z, g (f x) = x
fg : βαΆ (x : T) in π (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : π z = Filter.map g (π (f z))
β’ βαΆ (p : S Γ S) in π (z, z), f p.1 = f p.2 β p.1 = p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj | [24, 1] | [35, 61] | refine (fg.prod_mk fg).mp (eventually_of_forall ?_) | case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : S β T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z β 0
g : T β S
ga : HolomorphicAt I I g (f z)
gf : βαΆ (x : S) in π z, g (f x) = x
fg : βαΆ (x : T) in π (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : π z = Filter.map g (π (f z))
β’ βαΆ (a : T Γ T) in π (f z) ΓΛ’ π (f z), f (g a.1) = f (g a.2) β g a.1 = g a.2 | case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : S β T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z β 0
g : T β S
ga : HolomorphicAt I I g (f z)
gf : βαΆ (x : S) in π z, g (f x) = x
fg : βαΆ (x : T) in π (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : π z = Filter.map g (π (f z))
β’ β (x : T Γ T), f (g x.1) = x.1 β§ f (g x.2) = x.2 β f (g x.1) = f (g x.2) β g x.1 = g x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : S β T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z β 0
g : T β S
ga : HolomorphicAt I I g (f z)
gf : βαΆ (x : S) in π z, g (f x) = x
fg : βαΆ (x : T) in π (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : π z = Filter.map g (π (f z))
β’ βαΆ (a : T Γ T) in π (f z) ΓΛ’ π (f z), f (g a.1) = f (g a.2) β g a.1 = g a.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj | [24, 1] | [35, 61] | intro β¨x, yβ© β¨ex, eyβ© h | case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : S β T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z β 0
g : T β S
ga : HolomorphicAt I I g (f z)
gf : βαΆ (x : S) in π z, g (f x) = x
fg : βαΆ (x : T) in π (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : π z = Filter.map g (π (f z))
β’ β (x : T Γ T), f (g x.1) = x.1 β§ f (g x.2) = x.2 β f (g x.1) = f (g x.2) β g x.1 = g x.2 | case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : S β T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z β 0
g : T β S
ga : HolomorphicAt I I g (f z)
gf : βαΆ (x : S) in π z, g (f x) = x
fg : βαΆ (x : T) in π (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : π z = Filter.map g (π (f z))
x y : T
ex : f (g (x, y).1) = (x, y).1
ey : f (g (x, y).2) = (x, y).2
h : f (g (x, y).1) = f (g (x, y).2)
β’ g (x, y).1 = g (x, y).2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : S β T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z β 0
g : T β S
ga : HolomorphicAt I I g (f z)
gf : βαΆ (x : S) in π z, g (f x) = x
fg : βαΆ (x : T) in π (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : π z = Filter.map g (π (f z))
β’ β (x : T Γ T), f (g x.1) = x.1 β§ f (g x.2) = x.2 β f (g x.1) = f (g x.2) β g x.1 = g x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj | [24, 1] | [35, 61] | simp only at ex ey | case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : S β T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z β 0
g : T β S
ga : HolomorphicAt I I g (f z)
gf : βαΆ (x : S) in π z, g (f x) = x
fg : βαΆ (x : T) in π (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : π z = Filter.map g (π (f z))
x y : T
ex : f (g (x, y).1) = (x, y).1
ey : f (g (x, y).2) = (x, y).2
h : f (g (x, y).1) = f (g (x, y).2)
β’ g (x, y).1 = g (x, y).2 | case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : S β T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z β 0
g : T β S
ga : HolomorphicAt I I g (f z)
gf : βαΆ (x : S) in π z, g (f x) = x
fg : βαΆ (x : T) in π (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : π z = Filter.map g (π (f z))
x y : T
ex : f (g x) = x
ey : f (g y) = y
h : f (g (x, y).1) = f (g (x, y).2)
β’ g (x, y).1 = g (x, y).2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : S β T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z β 0
g : T β S
ga : HolomorphicAt I I g (f z)
gf : βαΆ (x : S) in π z, g (f x) = x
fg : βαΆ (x : T) in π (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : π z = Filter.map g (π (f z))
x y : T
ex : f (g (x, y).1) = (x, y).1
ey : f (g (x, y).2) = (x, y).2
h : f (g (x, y).1) = f (g (x, y).2)
β’ g (x, y).1 = g (x, y).2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj | [24, 1] | [35, 61] | simp only [ex, ey] at h | case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : S β T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z β 0
g : T β S
ga : HolomorphicAt I I g (f z)
gf : βαΆ (x : S) in π z, g (f x) = x
fg : βαΆ (x : T) in π (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : π z = Filter.map g (π (f z))
x y : T
ex : f (g x) = x
ey : f (g y) = y
h : f (g (x, y).1) = f (g (x, y).2)
β’ g (x, y).1 = g (x, y).2 | case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : S β T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z β 0
g : T β S
ga : HolomorphicAt I I g (f z)
gf : βαΆ (x : S) in π z, g (f x) = x
fg : βαΆ (x : T) in π (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : π z = Filter.map g (π (f z))
x y : T
ex : f (g x) = x
ey : f (g y) = y
h : x = y
β’ g (x, y).1 = g (x, y).2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : S β T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z β 0
g : T β S
ga : HolomorphicAt I I g (f z)
gf : βαΆ (x : S) in π z, g (f x) = x
fg : βαΆ (x : T) in π (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : π z = Filter.map g (π (f z))
x y : T
ex : f (g x) = x
ey : f (g y) = y
h : f (g (x, y).1) = f (g (x, y).2)
β’ g (x, y).1 = g (x, y).2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj | [24, 1] | [35, 61] | simp only [h] | case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : S β T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z β 0
g : T β S
ga : HolomorphicAt I I g (f z)
gf : βαΆ (x : S) in π z, g (f x) = x
fg : βαΆ (x : T) in π (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : π z = Filter.map g (π (f z))
x y : T
ex : f (g x) = x
ey : f (g y) = y
h : x = y
β’ g (x, y).1 = g (x, y).2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : S β T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z β 0
g : T β S
ga : HolomorphicAt I I g (f z)
gf : βαΆ (x : S) in π z, g (f x) = x
fg : βαΆ (x : T) in π (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : π z = Filter.map g (π (f z))
x y : T
ex : f (g x) = x
ey : f (g y) = y
h : x = y
β’ g (x, y).1 = g (x, y).2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj | [24, 1] | [35, 61] | rw [β gf.self_of_nhds] at fa | S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : S β T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z β 0
g : T β S
ga : HolomorphicAt I I g (f z)
gf : βαΆ (x : S) in π z, g (f x) = x
fg : βαΆ (x : T) in π (f z), f (g x) = x
β’ NontrivialHolomorphicAt g (f z) | S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : S β T
z : S
nc : mfderiv I I f z β 0
g : T β S
fa : HolomorphicAt I I f (g (f z))
ga : HolomorphicAt I I g (f z)
gf : βαΆ (x : S) in π z, g (f x) = x
fg : βαΆ (x : T) in π (f z), f (g x) = x
β’ NontrivialHolomorphicAt g (f z) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : S β T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z β 0
g : T β S
ga : HolomorphicAt I I g (f z)
gf : βαΆ (x : S) in π z, g (f x) = x
fg : βαΆ (x : T) in π (f z), f (g x) = x
β’ NontrivialHolomorphicAt g (f z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj | [24, 1] | [35, 61] | refine (NontrivialHolomorphicAt.anti ?_ fa ga).2 | S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : S β T
z : S
nc : mfderiv I I f z β 0
g : T β S
fa : HolomorphicAt I I f (g (f z))
ga : HolomorphicAt I I g (f z)
gf : βαΆ (x : S) in π z, g (f x) = x
fg : βαΆ (x : T) in π (f z), f (g x) = x
β’ NontrivialHolomorphicAt g (f z) | S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : S β T
z : S
nc : mfderiv I I f z β 0
g : T β S
fa : HolomorphicAt I I f (g (f z))
ga : HolomorphicAt I I g (f z)
gf : βαΆ (x : S) in π z, g (f x) = x
fg : βαΆ (x : T) in π (f z), f (g x) = x
β’ NontrivialHolomorphicAt (fun z => f (g z)) (f z) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : S β T
z : S
nc : mfderiv I I f z β 0
g : T β S
fa : HolomorphicAt I I f (g (f z))
ga : HolomorphicAt I I g (f z)
gf : βαΆ (x : S) in π z, g (f x) = x
fg : βαΆ (x : T) in π (f z), f (g x) = x
β’ NontrivialHolomorphicAt g (f z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj | [24, 1] | [35, 61] | exact (nontrivialHolomorphicAt_id _).congr (Filter.EventuallyEq.symm fg) | S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : S β T
z : S
nc : mfderiv I I f z β 0
g : T β S
fa : HolomorphicAt I I f (g (f z))
ga : HolomorphicAt I I g (f z)
gf : βαΆ (x : S) in π z, g (f x) = x
fg : βαΆ (x : T) in π (f z), f (g x) = x
β’ NontrivialHolomorphicAt (fun z => f (g z)) (f z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : S β T
z : S
nc : mfderiv I I f z β 0
g : T β S
fa : HolomorphicAt I I f (g (f z))
ga : HolomorphicAt I I g (f z)
gf : βαΆ (x : S) in π z, g (f x) = x
fg : βαΆ (x : T) in π (f z), f (g x) = x
β’ NontrivialHolomorphicAt (fun z => f (g z)) (f z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | rcases complex_inverse_fun fa nc with β¨g, ga, gf, fgβ© | S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
β’ βαΆ (p : (β Γ S) Γ β Γ S) in π ((c, z), c, z), p.1.1 = p.2.1 β f p.1.1 p.1.2 = f p.2.1 p.2.2 β p.1 = p.2 | case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β β T β S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
β’ βαΆ (p : (β Γ S) Γ β Γ S) in π ((c, z), c, z), p.1.1 = p.2.1 β f p.1.1 p.1.2 = f p.2.1 p.2.2 β p.1 = p.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
β’ βαΆ (p : (β Γ S) Γ β Γ S) in π ((c, z), c, z), p.1.1 = p.2.1 β f p.1.1 p.1.2 = f p.2.1 p.2.2 β p.1 = p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | have n : NontrivialHolomorphicAt (g c) (f c z) := by
have e : (c, z) = (c, g c (f c z)) := by rw [gf.self_of_nhds]
rw [e] at fa
refine (NontrivialHolomorphicAt.anti ?_ fa.along_snd ga.along_snd).2
refine (nontrivialHolomorphicAt_id _).congr ?_
refine ((continuousAt_const.prod continuousAt_id).eventually fg).mp (eventually_of_forall ?_)
exact fun _ e β¦ e.symm | case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β β T β S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
β’ βαΆ (p : (β Γ S) Γ β Γ S) in π ((c, z), c, z), p.1.1 = p.2.1 β f p.1.1 p.1.2 = f p.2.1 p.2.2 β p.1 = p.2 | case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β β T β S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
β’ βαΆ (p : (β Γ S) Γ β Γ S) in π ((c, z), c, z), p.1.1 = p.2.1 β f p.1.1 p.1.2 = f p.2.1 p.2.2 β p.1 = p.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β β T β S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
β’ βαΆ (p : (β Γ S) Γ β Γ S) in π ((c, z), c, z), p.1.1 = p.2.1 β f p.1.1 p.1.2 = f p.2.1 p.2.2 β p.1 = p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | have o := n.nhds_eq_map_nhds_param ga | case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β β T β S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
β’ βαΆ (p : (β Γ S) Γ β Γ S) in π ((c, z), c, z), p.1.1 = p.2.1 β f p.1.1 p.1.2 = f p.2.1 p.2.2 β p.1 = p.2 | case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β β T β S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : π (c, g c (f c z)) = Filter.map (fun p => (p.1, g p.1 p.2)) (π (c, f c z))
β’ βαΆ (p : (β Γ S) Γ β Γ S) in π ((c, z), c, z), p.1.1 = p.2.1 β f p.1.1 p.1.2 = f p.2.1 p.2.2 β p.1 = p.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β β T β S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
β’ βαΆ (p : (β Γ S) Γ β Γ S) in π ((c, z), c, z), p.1.1 = p.2.1 β f p.1.1 p.1.2 = f p.2.1 p.2.2 β p.1 = p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | rw [gf.self_of_nhds] at o | case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β β T β S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : π (c, g c (f c z)) = Filter.map (fun p => (p.1, g p.1 p.2)) (π (c, f c z))
β’ βαΆ (p : (β Γ S) Γ β Γ S) in π ((c, z), c, z), p.1.1 = p.2.1 β f p.1.1 p.1.2 = f p.2.1 p.2.2 β p.1 = p.2 | case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β β T β S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : π (c, (c, z).2) = Filter.map (fun p => (p.1, g p.1 p.2)) (π (c, f c z))
β’ βαΆ (p : (β Γ S) Γ β Γ S) in π ((c, z), c, z), p.1.1 = p.2.1 β f p.1.1 p.1.2 = f p.2.1 p.2.2 β p.1 = p.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β β T β S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : π (c, g c (f c z)) = Filter.map (fun p => (p.1, g p.1 p.2)) (π (c, f c z))
β’ βαΆ (p : (β Γ S) Γ β Γ S) in π ((c, z), c, z), p.1.1 = p.2.1 β f p.1.1 p.1.2 = f p.2.1 p.2.2 β p.1 = p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | simp only at o | case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β β T β S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : π (c, (c, z).2) = Filter.map (fun p => (p.1, g p.1 p.2)) (π (c, f c z))
β’ βαΆ (p : (β Γ S) Γ β Γ S) in π ((c, z), c, z), p.1.1 = p.2.1 β f p.1.1 p.1.2 = f p.2.1 p.2.2 β p.1 = p.2 | case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β β T β S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : π (c, z) = Filter.map (fun p => (p.1, g p.1 p.2)) (π (c, f c z))
β’ βαΆ (p : (β Γ S) Γ β Γ S) in π ((c, z), c, z), p.1.1 = p.2.1 β f p.1.1 p.1.2 = f p.2.1 p.2.2 β p.1 = p.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β β T β S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : π (c, (c, z).2) = Filter.map (fun p => (p.1, g p.1 p.2)) (π (c, f c z))
β’ βαΆ (p : (β Γ S) Γ β Γ S) in π ((c, z), c, z), p.1.1 = p.2.1 β f p.1.1 p.1.2 = f p.2.1 p.2.2 β p.1 = p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | rw [nhds_prod_eq, o] | case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β β T β S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : π (c, z) = Filter.map (fun p => (p.1, g p.1 p.2)) (π (c, f c z))
β’ βαΆ (p : (β Γ S) Γ β Γ S) in π ((c, z), c, z), p.1.1 = p.2.1 β f p.1.1 p.1.2 = f p.2.1 p.2.2 β p.1 = p.2 | case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β β T β S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : π (c, z) = Filter.map (fun p => (p.1, g p.1 p.2)) (π (c, f c z))
β’ βαΆ (p : (β Γ S) Γ β Γ S) in
Filter.map (fun p => (p.1, g p.1 p.2)) (π (c, f c z)) ΓΛ’ Filter.map (fun p => (p.1, g p.1 p.2)) (π (c, f c z)),
p.1.1 = p.2.1 β f p.1.1 p.1.2 = f p.2.1 p.2.2 β p.1 = p.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β β T β S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : π (c, z) = Filter.map (fun p => (p.1, g p.1 p.2)) (π (c, f c z))
β’ βαΆ (p : (β Γ S) Γ β Γ S) in π ((c, z), c, z), p.1.1 = p.2.1 β f p.1.1 p.1.2 = f p.2.1 p.2.2 β p.1 = p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | simp only [Filter.prod_map_map_eq, Filter.eventually_map] | case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β β T β S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : π (c, z) = Filter.map (fun p => (p.1, g p.1 p.2)) (π (c, f c z))
β’ βαΆ (p : (β Γ S) Γ β Γ S) in
Filter.map (fun p => (p.1, g p.1 p.2)) (π (c, f c z)) ΓΛ’ Filter.map (fun p => (p.1, g p.1 p.2)) (π (c, f c z)),
p.1.1 = p.2.1 β f p.1.1 p.1.2 = f p.2.1 p.2.2 β p.1 = p.2 | case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β β T β S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : π (c, z) = Filter.map (fun p => (p.1, g p.1 p.2)) (π (c, f c z))
β’ βαΆ (a : (β Γ T) Γ β Γ T) in π (c, f c z) ΓΛ’ π (c, f c z),
a.1.1 = a.2.1 β f a.1.1 (g a.1.1 a.1.2) = f a.2.1 (g a.2.1 a.2.2) β (a.1.1, g a.1.1 a.1.2) = (a.2.1, g a.2.1 a.2.2) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β β T β S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : π (c, z) = Filter.map (fun p => (p.1, g p.1 p.2)) (π (c, f c z))
β’ βαΆ (p : (β Γ S) Γ β Γ S) in
Filter.map (fun p => (p.1, g p.1 p.2)) (π (c, f c z)) ΓΛ’ Filter.map (fun p => (p.1, g p.1 p.2)) (π (c, f c z)),
p.1.1 = p.2.1 β f p.1.1 p.1.2 = f p.2.1 p.2.2 β p.1 = p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | refine (fg.prod_mk fg).mp (eventually_of_forall ?_) | case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β β T β S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : π (c, z) = Filter.map (fun p => (p.1, g p.1 p.2)) (π (c, f c z))
β’ βαΆ (a : (β Γ T) Γ β Γ T) in π (c, f c z) ΓΛ’ π (c, f c z),
a.1.1 = a.2.1 β f a.1.1 (g a.1.1 a.1.2) = f a.2.1 (g a.2.1 a.2.2) β (a.1.1, g a.1.1 a.1.2) = (a.2.1, g a.2.1 a.2.2) | case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β β T β S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : π (c, z) = Filter.map (fun p => (p.1, g p.1 p.2)) (π (c, f c z))
β’ β (x : (β Γ T) Γ β Γ T),
f x.1.1 (g x.1.1 x.1.2) = x.1.2 β§ f x.2.1 (g x.2.1 x.2.2) = x.2.2 β
x.1.1 = x.2.1 β
f x.1.1 (g x.1.1 x.1.2) = f x.2.1 (g x.2.1 x.2.2) β (x.1.1, g x.1.1 x.1.2) = (x.2.1, g x.2.1 x.2.2) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β β T β S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : π (c, z) = Filter.map (fun p => (p.1, g p.1 p.2)) (π (c, f c z))
β’ βαΆ (a : (β Γ T) Γ β Γ T) in π (c, f c z) ΓΛ’ π (c, f c z),
a.1.1 = a.2.1 β f a.1.1 (g a.1.1 a.1.2) = f a.2.1 (g a.2.1 a.2.2) β (a.1.1, g a.1.1 a.1.2) = (a.2.1, g a.2.1 a.2.2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | intro β¨x, yβ© β¨ex, eyβ© h1 h2 | case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β β T β S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : π (c, z) = Filter.map (fun p => (p.1, g p.1 p.2)) (π (c, f c z))
β’ β (x : (β Γ T) Γ β Γ T),
f x.1.1 (g x.1.1 x.1.2) = x.1.2 β§ f x.2.1 (g x.2.1 x.2.2) = x.2.2 β
x.1.1 = x.2.1 β
f x.1.1 (g x.1.1 x.1.2) = f x.2.1 (g x.2.1 x.2.2) β (x.1.1, g x.1.1 x.1.2) = (x.2.1, g x.2.1 x.2.2) | case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β β T β S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : π (c, z) = Filter.map (fun p => (p.1, g p.1 p.2)) (π (c, f c z))
x y : β Γ T
ex : f (x, y).1.1 (g (x, y).1.1 (x, y).1.2) = (x, y).1.2
ey : f (x, y).2.1 (g (x, y).2.1 (x, y).2.2) = (x, y).2.2
h1 : (x, y).1.1 = (x, y).2.1
h2 : f (x, y).1.1 (g (x, y).1.1 (x, y).1.2) = f (x, y).2.1 (g (x, y).2.1 (x, y).2.2)
β’ ((x, y).1.1, g (x, y).1.1 (x, y).1.2) = ((x, y).2.1, g (x, y).2.1 (x, y).2.2) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β β T β S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : π (c, z) = Filter.map (fun p => (p.1, g p.1 p.2)) (π (c, f c z))
β’ β (x : (β Γ T) Γ β Γ T),
f x.1.1 (g x.1.1 x.1.2) = x.1.2 β§ f x.2.1 (g x.2.1 x.2.2) = x.2.2 β
x.1.1 = x.2.1 β
f x.1.1 (g x.1.1 x.1.2) = f x.2.1 (g x.2.1 x.2.2) β (x.1.1, g x.1.1 x.1.2) = (x.2.1, g x.2.1 x.2.2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | simp only at h1 | case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β β T β S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : π (c, z) = Filter.map (fun p => (p.1, g p.1 p.2)) (π (c, f c z))
x y : β Γ T
ex : f (x, y).1.1 (g (x, y).1.1 (x, y).1.2) = (x, y).1.2
ey : f (x, y).2.1 (g (x, y).2.1 (x, y).2.2) = (x, y).2.2
h1 : (x, y).1.1 = (x, y).2.1
h2 : f (x, y).1.1 (g (x, y).1.1 (x, y).1.2) = f (x, y).2.1 (g (x, y).2.1 (x, y).2.2)
β’ ((x, y).1.1, g (x, y).1.1 (x, y).1.2) = ((x, y).2.1, g (x, y).2.1 (x, y).2.2) | case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β β T β S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : π (c, z) = Filter.map (fun p => (p.1, g p.1 p.2)) (π (c, f c z))
x y : β Γ T
ex : f (x, y).1.1 (g (x, y).1.1 (x, y).1.2) = (x, y).1.2
ey : f (x, y).2.1 (g (x, y).2.1 (x, y).2.2) = (x, y).2.2
h1 : x.1 = y.1
h2 : f (x, y).1.1 (g (x, y).1.1 (x, y).1.2) = f (x, y).2.1 (g (x, y).2.1 (x, y).2.2)
β’ ((x, y).1.1, g (x, y).1.1 (x, y).1.2) = ((x, y).2.1, g (x, y).2.1 (x, y).2.2) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β β T β S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : π (c, z) = Filter.map (fun p => (p.1, g p.1 p.2)) (π (c, f c z))
x y : β Γ T
ex : f (x, y).1.1 (g (x, y).1.1 (x, y).1.2) = (x, y).1.2
ey : f (x, y).2.1 (g (x, y).2.1 (x, y).2.2) = (x, y).2.2
h1 : (x, y).1.1 = (x, y).2.1
h2 : f (x, y).1.1 (g (x, y).1.1 (x, y).1.2) = f (x, y).2.1 (g (x, y).2.1 (x, y).2.2)
β’ ((x, y).1.1, g (x, y).1.1 (x, y).1.2) = ((x, y).2.1, g (x, y).2.1 (x, y).2.2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | simp only [h1] at ex ey h2 β’ | case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β β T β S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : π (c, z) = Filter.map (fun p => (p.1, g p.1 p.2)) (π (c, f c z))
x y : β Γ T
ex : f (x, y).1.1 (g (x, y).1.1 (x, y).1.2) = (x, y).1.2
ey : f (x, y).2.1 (g (x, y).2.1 (x, y).2.2) = (x, y).2.2
h1 : x.1 = y.1
h2 : f (x, y).1.1 (g (x, y).1.1 (x, y).1.2) = f (x, y).2.1 (g (x, y).2.1 (x, y).2.2)
β’ ((x, y).1.1, g (x, y).1.1 (x, y).1.2) = ((x, y).2.1, g (x, y).2.1 (x, y).2.2) | case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β β T β S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : π (c, z) = Filter.map (fun p => (p.1, g p.1 p.2)) (π (c, f c z))
x y : β Γ T
ey : f y.1 (g y.1 y.2) = y.2
h1 : x.1 = y.1
ex : f y.1 (g y.1 x.2) = x.2
h2 : f y.1 (g y.1 x.2) = f y.1 (g y.1 y.2)
β’ (y.1, g y.1 x.2) = (y.1, g y.1 y.2) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β β T β S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : π (c, z) = Filter.map (fun p => (p.1, g p.1 p.2)) (π (c, f c z))
x y : β Γ T
ex : f (x, y).1.1 (g (x, y).1.1 (x, y).1.2) = (x, y).1.2
ey : f (x, y).2.1 (g (x, y).2.1 (x, y).2.2) = (x, y).2.2
h1 : x.1 = y.1
h2 : f (x, y).1.1 (g (x, y).1.1 (x, y).1.2) = f (x, y).2.1 (g (x, y).2.1 (x, y).2.2)
β’ ((x, y).1.1, g (x, y).1.1 (x, y).1.2) = ((x, y).2.1, g (x, y).2.1 (x, y).2.2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | simp only [ex, ey] at h2 | case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β β T β S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : π (c, z) = Filter.map (fun p => (p.1, g p.1 p.2)) (π (c, f c z))
x y : β Γ T
ey : f y.1 (g y.1 y.2) = y.2
h1 : x.1 = y.1
ex : f y.1 (g y.1 x.2) = x.2
h2 : f y.1 (g y.1 x.2) = f y.1 (g y.1 y.2)
β’ (y.1, g y.1 x.2) = (y.1, g y.1 y.2) | case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β β T β S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : π (c, z) = Filter.map (fun p => (p.1, g p.1 p.2)) (π (c, f c z))
x y : β Γ T
ey : f y.1 (g y.1 y.2) = y.2
h1 : x.1 = y.1
ex : f y.1 (g y.1 x.2) = x.2
h2 : x.2 = y.2
β’ (y.1, g y.1 x.2) = (y.1, g y.1 y.2) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β β T β S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : π (c, z) = Filter.map (fun p => (p.1, g p.1 p.2)) (π (c, f c z))
x y : β Γ T
ey : f y.1 (g y.1 y.2) = y.2
h1 : x.1 = y.1
ex : f y.1 (g y.1 x.2) = x.2
h2 : f y.1 (g y.1 x.2) = f y.1 (g y.1 y.2)
β’ (y.1, g y.1 x.2) = (y.1, g y.1 y.2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | simp only [h2] | case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β β T β S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : π (c, z) = Filter.map (fun p => (p.1, g p.1 p.2)) (π (c, f c z))
x y : β Γ T
ey : f y.1 (g y.1 y.2) = y.2
h1 : x.1 = y.1
ex : f y.1 (g y.1 x.2) = x.2
h2 : x.2 = y.2
β’ (y.1, g y.1 x.2) = (y.1, g y.1 y.2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β β T β S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : π (c, z) = Filter.map (fun p => (p.1, g p.1 p.2)) (π (c, f c z))
x y : β Γ T
ey : f y.1 (g y.1 y.2) = y.2
h1 : x.1 = y.1
ex : f y.1 (g y.1 x.2) = x.2
h2 : x.2 = y.2
β’ (y.1, g y.1 x.2) = (y.1, g y.1 y.2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | have e : (c, z) = (c, g c (f c z)) := by rw [gf.self_of_nhds] | S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β β T β S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
β’ NontrivialHolomorphicAt (g c) (f c z) | S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β β T β S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
e : (c, z) = (c, g c (f c z))
β’ NontrivialHolomorphicAt (g c) (f c z) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β β T β S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
β’ NontrivialHolomorphicAt (g c) (f c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | rw [e] at fa | S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β β T β S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
e : (c, z) = (c, g c (f c z))
β’ NontrivialHolomorphicAt (g c) (f c z) | S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
nc : mfderiv I I (f c) z β 0
g : β β T β S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, g c (f c z))
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
e : (c, z) = (c, g c (f c z))
β’ NontrivialHolomorphicAt (g c) (f c z) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β β T β S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
e : (c, z) = (c, g c (f c z))
β’ NontrivialHolomorphicAt (g c) (f c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | refine (NontrivialHolomorphicAt.anti ?_ fa.along_snd ga.along_snd).2 | S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
nc : mfderiv I I (f c) z β 0
g : β β T β S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, g c (f c z))
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
e : (c, z) = (c, g c (f c z))
β’ NontrivialHolomorphicAt (g c) (f c z) | S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
nc : mfderiv I I (f c) z β 0
g : β β T β S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, g c (f c z))
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
e : (c, z) = (c, g c (f c z))
β’ NontrivialHolomorphicAt (fun z => f c (g c z)) (f c z) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
nc : mfderiv I I (f c) z β 0
g : β β T β S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, g c (f c z))
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
e : (c, z) = (c, g c (f c z))
β’ NontrivialHolomorphicAt (g c) (f c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | refine (nontrivialHolomorphicAt_id _).congr ?_ | S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
nc : mfderiv I I (f c) z β 0
g : β β T β S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, g c (f c z))
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
e : (c, z) = (c, g c (f c z))
β’ NontrivialHolomorphicAt (fun z => f c (g c z)) (f c z) | S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
nc : mfderiv I I (f c) z β 0
g : β β T β S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, g c (f c z))
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
e : (c, z) = (c, g c (f c z))
β’ (π (f c z)).EventuallyEq (fun w => w) fun z => f c (g c z) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
nc : mfderiv I I (f c) z β 0
g : β β T β S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, g c (f c z))
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
e : (c, z) = (c, g c (f c z))
β’ NontrivialHolomorphicAt (fun z => f c (g c z)) (f c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | refine ((continuousAt_const.prod continuousAt_id).eventually fg).mp (eventually_of_forall ?_) | S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
nc : mfderiv I I (f c) z β 0
g : β β T β S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, g c (f c z))
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
e : (c, z) = (c, g c (f c z))
β’ (π (f c z)).EventuallyEq (fun w => w) fun z => f c (g c z) | S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
nc : mfderiv I I (f c) z β 0
g : β β T β S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, g c (f c z))
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
e : (c, z) = (c, g c (f c z))
β’ β (x : T), f (c, id x).1 (g (c, id x).1 (c, id x).2) = (c, id x).2 β (fun w => w) x = (fun z => f c (g c z)) x | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
nc : mfderiv I I (f c) z β 0
g : β β T β S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, g c (f c z))
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
e : (c, z) = (c, g c (f c z))
β’ (π (f c z)).EventuallyEq (fun w => w) fun z => f c (g c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | exact fun _ e β¦ e.symm | S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
nc : mfderiv I I (f c) z β 0
g : β β T β S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, g c (f c z))
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
e : (c, z) = (c, g c (f c z))
β’ β (x : T), f (c, id x).1 (g (c, id x).1 (c, id x).2) = (c, id x).2 β (fun w => w) x = (fun z => f c (g c z)) x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
nc : mfderiv I I (f c) z β 0
g : β β T β S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, g c (f c z))
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
e : (c, z) = (c, g c (f c z))
β’ β (x : T), f (c, id x).1 (g (c, id x).1 (c, id x).2) = (c, id x).2 β (fun w => w) x = (fun z => f c (g c z)) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | rw [gf.self_of_nhds] | S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β β T β S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
β’ (c, z) = (c, g c (f c z)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β β T β S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : βαΆ (x : β Γ S) in π (c, z), g x.1 (f x.1 x.2) = x.2
fg : βαΆ (x : β Γ T) in π (c, f c z), f x.1 (g x.1 x.2) = x.2
β’ (c, z) = (c, g c (f c z))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj' | [58, 1] | [67, 65] | set g : β Γ S Γ S β (β Γ S) Γ β Γ S := fun p β¦ ((p.1, p.2.1), (p.1, p.2.2)) | S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
β’ βαΆ (p : β Γ S Γ S) in π (c, z, z), f p.1 p.2.1 = f p.1 p.2.2 β p.2.1 = p.2.2 | S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β Γ S Γ S β (β Γ S) Γ β Γ S := fun p => ((p.1, p.2.1), p.1, p.2.2)
β’ βαΆ (p : β Γ S Γ S) in π (c, z, z), f p.1 p.2.1 = f p.1 p.2.2 β p.2.1 = p.2.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
β’ βαΆ (p : β Γ S Γ S) in π (c, z, z), f p.1 p.2.1 = f p.1 p.2.2 β p.2.1 = p.2.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj' | [58, 1] | [67, 65] | refine (t.eventually (fa.local_inj'' nc)).mp (eventually_of_forall ?_) | S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β Γ S Γ S β (β Γ S) Γ β Γ S := fun p => ((p.1, p.2.1), p.1, p.2.2)
t : Tendsto g (π (c, z, z)) (π ((c, z), c, z))
β’ βαΆ (p : β Γ S Γ S) in π (c, z, z), f p.1 p.2.1 = f p.1 p.2.2 β p.2.1 = p.2.2 | S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β Γ S Γ S β (β Γ S) Γ β Γ S := fun p => ((p.1, p.2.1), p.1, p.2.2)
t : Tendsto g (π (c, z, z)) (π ((c, z), c, z))
β’ β (x : β Γ S Γ S),
((g x).1.1 = (g x).2.1 β f (g x).1.1 (g x).1.2 = f (g x).2.1 (g x).2.2 β (g x).1 = (g x).2) β
f x.1 x.2.1 = f x.1 x.2.2 β x.2.1 = x.2.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β Γ S Γ S β (β Γ S) Γ β Γ S := fun p => ((p.1, p.2.1), p.1, p.2.2)
t : Tendsto g (π (c, z, z)) (π ((c, z), c, z))
β’ βαΆ (p : β Γ S Γ S) in π (c, z, z), f p.1 p.2.1 = f p.1 p.2.2 β p.2.1 = p.2.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj' | [58, 1] | [67, 65] | intro β¨e, x, yβ© inj fe | S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β Γ S Γ S β (β Γ S) Γ β Γ S := fun p => ((p.1, p.2.1), p.1, p.2.2)
t : Tendsto g (π (c, z, z)) (π ((c, z), c, z))
β’ β (x : β Γ S Γ S),
((g x).1.1 = (g x).2.1 β f (g x).1.1 (g x).1.2 = f (g x).2.1 (g x).2.2 β (g x).1 = (g x).2) β
f x.1 x.2.1 = f x.1 x.2.2 β x.2.1 = x.2.2 | S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β Γ S Γ S β (β Γ S) Γ β Γ S := fun p => ((p.1, p.2.1), p.1, p.2.2)
t : Tendsto g (π (c, z, z)) (π ((c, z), c, z))
e : β
x y : S
inj :
(g (e, x, y)).1.1 = (g (e, x, y)).2.1 β
f (g (e, x, y)).1.1 (g (e, x, y)).1.2 = f (g (e, x, y)).2.1 (g (e, x, y)).2.2 β (g (e, x, y)).1 = (g (e, x, y)).2
fe : f (e, x, y).1 (e, x, y).2.1 = f (e, x, y).1 (e, x, y).2.2
β’ (e, x, y).2.1 = (e, x, y).2.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β Γ S Γ S β (β Γ S) Γ β Γ S := fun p => ((p.1, p.2.1), p.1, p.2.2)
t : Tendsto g (π (c, z, z)) (π ((c, z), c, z))
β’ β (x : β Γ S Γ S),
((g x).1.1 = (g x).2.1 β f (g x).1.1 (g x).1.2 = f (g x).2.1 (g x).2.2 β (g x).1 = (g x).2) β
f x.1 x.2.1 = f x.1 x.2.2 β x.2.1 = x.2.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj' | [58, 1] | [67, 65] | exact (Prod.ext_iff.mp (inj rfl fe)).2 | S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β Γ S Γ S β (β Γ S) Γ β Γ S := fun p => ((p.1, p.2.1), p.1, p.2.2)
t : Tendsto g (π (c, z, z)) (π ((c, z), c, z))
e : β
x y : S
inj :
(g (e, x, y)).1.1 = (g (e, x, y)).2.1 β
f (g (e, x, y)).1.1 (g (e, x, y)).1.2 = f (g (e, x, y)).2.1 (g (e, x, y)).2.2 β (g (e, x, y)).1 = (g (e, x, y)).2
fe : f (e, x, y).1 (e, x, y).2.1 = f (e, x, y).1 (e, x, y).2.2
β’ (e, x, y).2.1 = (e, x, y).2.2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β Γ S Γ S β (β Γ S) Γ β Γ S := fun p => ((p.1, p.2.1), p.1, p.2.2)
t : Tendsto g (π (c, z, z)) (π ((c, z), c, z))
e : β
x y : S
inj :
(g (e, x, y)).1.1 = (g (e, x, y)).2.1 β
f (g (e, x, y)).1.1 (g (e, x, y)).1.2 = f (g (e, x, y)).2.1 (g (e, x, y)).2.2 β (g (e, x, y)).1 = (g (e, x, y)).2
fe : f (e, x, y).1 (e, x, y).2.1 = f (e, x, y).1 (e, x, y).2.2
β’ (e, x, y).2.1 = (e, x, y).2.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj' | [58, 1] | [67, 65] | apply Continuous.continuousAt | S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β Γ S Γ S β (β Γ S) Γ β Γ S := fun p => ((p.1, p.2.1), p.1, p.2.2)
β’ Tendsto g (π (c, z, z)) (π ((c, z), c, z)) | case h
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β Γ S Γ S β (β Γ S) Γ β Γ S := fun p => ((p.1, p.2.1), p.1, p.2.2)
β’ Continuous g | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β Γ S Γ S β (β Γ S) Γ β Γ S := fun p => ((p.1, p.2.1), p.1, p.2.2)
β’ Tendsto g (π (c, z, z)) (π ((c, z), c, z))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj' | [58, 1] | [67, 65] | apply Continuous.prod_mk | case h
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β Γ S Γ S β (β Γ S) Γ β Γ S := fun p => ((p.1, p.2.1), p.1, p.2.2)
β’ Continuous g | case h.hf
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β Γ S Γ S β (β Γ S) Γ β Γ S := fun p => ((p.1, p.2.1), p.1, p.2.2)
β’ Continuous fun x => (x.1, x.2.1)
case h.hg
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β Γ S Γ S β (β Γ S) Γ β Γ S := fun p => ((p.1, p.2.1), p.1, p.2.2)
β’ Continuous fun x => (x.1, x.2.2) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β Γ S Γ S β (β Γ S) Γ β Γ S := fun p => ((p.1, p.2.1), p.1, p.2.2)
β’ Continuous g
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj' | [58, 1] | [67, 65] | exact continuous_fst.prod_mk (continuous_fst.comp continuous_snd) | case h.hf
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β Γ S Γ S β (β Γ S) Γ β Γ S := fun p => ((p.1, p.2.1), p.1, p.2.2)
β’ Continuous fun x => (x.1, x.2.1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hf
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β Γ S Γ S β (β Γ S) Γ β Γ S := fun p => ((p.1, p.2.1), p.1, p.2.2)
β’ Continuous fun x => (x.1, x.2.1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj' | [58, 1] | [67, 65] | exact continuous_fst.prod_mk (continuous_snd.comp continuous_snd) | case h.hg
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β Γ S Γ S β (β Γ S) Γ β Γ S := fun p => ((p.1, p.2.1), p.1, p.2.2)
β’ Continuous fun x => (x.1, x.2.2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hg
S : Type
instβΒ³ : TopologicalSpace S
instβΒ² : ChartedSpace β S
cms : AnalyticManifold I S
T : Type
instβΒΉ : TopologicalSpace T
instβ : ChartedSpace β T
cmt : AnalyticManifold I T
f : β β S β T
c : β
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z β 0
g : β Γ S Γ S β (β Γ S) Γ β Γ S := fun p => ((p.1, p.2.1), p.1, p.2.2)
β’ Continuous fun x => (x.1, x.2.2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | nontrivial_local_of_global | [46, 1] | [61, 29] | have fh : HolomorphicOn I I f (closedBall z r) := fun _ m β¦ (fa _ m).holomorphicAt I I | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
ef : β w β sphere z r, e β€ βf w - f zβ
β’ NontrivialHolomorphicAt f z | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
ef : β w β sphere z r, e β€ βf w - f zβ
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
β’ NontrivialHolomorphicAt f z | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
ef : β w β sphere z r, e β€ βf w - f zβ
β’ NontrivialHolomorphicAt f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | nontrivial_local_of_global | [46, 1] | [61, 29] | have zs : z β closedBall z r := mem_closedBall_self rp.le | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
ef : β w β sphere z r, e β€ βf w - f zβ
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
β’ NontrivialHolomorphicAt f z | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
ef : β w β sphere z r, e β€ βf w - f zβ
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
β’ NontrivialHolomorphicAt f z | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
ef : β w β sphere z r, e β€ βf w - f zβ
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
β’ NontrivialHolomorphicAt f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | nontrivial_local_of_global | [46, 1] | [61, 29] | use fh _ zs | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
ef : β w β sphere z r, e β€ βf w - f zβ
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
β’ NontrivialHolomorphicAt f z | case nonconst
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
ef : β w β sphere z r, e β€ βf w - f zβ
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
β’ βαΆ (w : β) in π z, f w β f z | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
ef : β w β sphere z r, e β€ βf w - f zβ
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
β’ NontrivialHolomorphicAt f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | nontrivial_local_of_global | [46, 1] | [61, 29] | contrapose ef | case nonconst
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
ef : β w β sphere z r, e β€ βf w - f zβ
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
β’ βαΆ (w : β) in π z, f w β f z | case nonconst
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
ef : Β¬βαΆ (w : β) in π z, f w β f z
β’ Β¬β w β sphere z r, e β€ βf w - f zβ | Please generate a tactic in lean4 to solve the state.
STATE:
case nonconst
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
ef : β w β sphere z r, e β€ βf w - f zβ
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
β’ βαΆ (w : β) in π z, f w β f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | nontrivial_local_of_global | [46, 1] | [61, 29] | simp only [Filter.not_frequently, not_not] at ef | case nonconst
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
ef : Β¬βαΆ (w : β) in π z, f w β f z
β’ Β¬β w β sphere z r, e β€ βf w - f zβ | case nonconst
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
ef : βαΆ (x : β) in π z, f x = f z
β’ Β¬β w β sphere z r, e β€ βf w - f zβ | Please generate a tactic in lean4 to solve the state.
STATE:
case nonconst
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
ef : Β¬βαΆ (w : β) in π z, f w β f z
β’ Β¬β w β sphere z r, e β€ βf w - f zβ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | nontrivial_local_of_global | [46, 1] | [61, 29] | simp only [not_forall, not_le] | case nonconst
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
ef : βαΆ (x : β) in π z, f x = f z
β’ Β¬β w β sphere z r, e β€ βf w - f zβ | case nonconst
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
ef : βαΆ (x : β) in π z, f x = f z
β’ β x, β (_ : x β sphere z r), βf x - f zβ < e | Please generate a tactic in lean4 to solve the state.
STATE:
case nonconst
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
ef : βαΆ (x : β) in π z, f x = f z
β’ Β¬β w β sphere z r, e β€ βf w - f zβ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | nontrivial_local_of_global | [46, 1] | [61, 29] | have zrs : z + r β sphere z r := by
simp only [mem_sphere, Complex.dist_eq, add_sub_cancel_left, Complex.abs_ofReal, abs_of_pos rp] | case nonconst
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
ef : βαΆ (x : β) in π z, f x = f z
β’ β x, β (_ : x β sphere z r), βf x - f zβ < e | case nonconst
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
ef : βαΆ (x : β) in π z, f x = f z
zrs : z + βr β sphere z r
β’ β x, β (_ : x β sphere z r), βf x - f zβ < e | Please generate a tactic in lean4 to solve the state.
STATE:
case nonconst
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
ef : βαΆ (x : β) in π z, f x = f z
β’ β x, β (_ : x β sphere z r), βf x - f zβ < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | nontrivial_local_of_global | [46, 1] | [61, 29] | use z + r, zrs | case nonconst
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
ef : βαΆ (x : β) in π z, f x = f z
zrs : z + βr β sphere z r
β’ β x, β (_ : x β sphere z r), βf x - f zβ < e | case h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
ef : βαΆ (x : β) in π z, f x = f z
zrs : z + βr β sphere z r
β’ βf (z + βr) - f zβ < e | Please generate a tactic in lean4 to solve the state.
STATE:
case nonconst
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
ef : βαΆ (x : β) in π z, f x = f z
zrs : z + βr β sphere z r
β’ β x, β (_ : x β sphere z r), βf x - f zβ < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | nontrivial_local_of_global | [46, 1] | [61, 29] | simp only [fh.const_of_locally_const' zs (convex_closedBall z r).isPreconnected ef (z + r)
(Metric.sphere_subset_closedBall zrs),
sub_self, norm_zero, ep] | case h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
ef : βαΆ (x : β) in π z, f x = f z
zrs : z + βr β sphere z r
β’ βf (z + βr) - f zβ < e | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
ef : βαΆ (x : β) in π z, f x = f z
zrs : z + βr β sphere z r
β’ βf (z + βr) - f zβ < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | nontrivial_local_of_global | [46, 1] | [61, 29] | simp only [mem_sphere, Complex.dist_eq, add_sub_cancel_left, Complex.abs_ofReal, abs_of_pos rp] | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
ef : βαΆ (x : β) in π z, f x = f z
β’ z + βr β sphere z r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
ef : βαΆ (x : β) in π z, f x = f z
β’ z + βr β sphere z r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | have fn : β d, d β u β βαΆ w in π z, f d w β f d z := by
refine fun d m β¦ (nontrivial_local_of_global (fa.along_snd.mono ?_) rp ep (ef d m)).nonconst
simp only [β closedBall_prod_same, mem_prod_eq, setOf_mem_eq, iff_true_iff.mpr m,
true_and_iff, subset_refl] | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
β’ (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r β π (c, f c z) | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
β’ (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r β π (c, f c z) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
β’ (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r β π (c, f c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | have op : β d, d β u β ball (f d z) (e / 2) β f d '' closedBall z r := by
intro d du; refine DiffContOnCl.ball_subset_image_closedBall ?_ rp (ef d du) (fn d du)
have e : f d = uncurry f β fun w β¦ (d, w) := rfl
rw [e]; apply DifferentiableOn.diffContOnCl; apply AnalyticOn.differentiableOn
refine fa.comp (analyticOn_const.prod (analyticOn_id _)) ?_
intro w wr; simp only [closure_ball _ rp.ne'] at wr
simp only [β closedBall_prod_same, mem_prod_eq, du, wr, true_and_iff, du] | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
β’ (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r β π (c, f c z) | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
β’ (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r β π (c, f c z) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
β’ (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r β π (c, f c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | rcases Metric.continuousAt_iff.mp
(fa (c, z) (mk_mem_prod (mem_of_mem_nhds un) (mem_closedBall_self rp.le))).continuousAt
(e / 4) (by linarith) with
β¨s, sp, shβ© | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
β’ (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r β π (c, f c z) | case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
β’ (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r β π (c, f c z) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
β’ (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r β π (c, f c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | rw [mem_nhds_prod_iff] | case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
β’ (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r β π (c, f c z) | case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
β’ β u_1 β π c, β v β π (f c z), u_1 ΓΛ’ v β (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
β’ (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r β π (c, f c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | refine β¨u β© ball c s, Filter.inter_mem un (Metric.ball_mem_nhds c (by linarith)), ?_β© | case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
β’ β u_1 β π c, β v β π (f c z), u_1 ΓΛ’ v β (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r | case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
β’ β v β π (f c z), (u β© ball c s) ΓΛ’ v β (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
β’ β u_1 β π c, β v β π (f c z), u_1 ΓΛ’ v β (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | use ball (f c z) (e / 4), Metric.ball_mem_nhds _ (by linarith) | case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
β’ β v β π (f c z), (u β© ball c s) ΓΛ’ v β (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r | case right
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
β’ (u β© ball c s) ΓΛ’ ball (f c z) (e / 4) β (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
β’ β v β π (f c z), (u β© ball c s) ΓΛ’ v β (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | intro β¨d, wβ© m | case right
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
β’ (u β© ball c s) ΓΛ’ ball (f c z) (e / 4) β (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r | case right
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : β
m : (d, w) β (u β© ball c s) ΓΛ’ ball (f c z) (e / 4)
β’ (d, w) β (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
β’ (u β© ball c s) ΓΛ’ ball (f c z) (e / 4) β (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | simp only [mem_inter_iff, mem_prod_eq, mem_image, @mem_ball _ _ c, lt_min_iff] at m op β’ | case right
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : β
m : (d, w) β (u β© ball c s) ΓΛ’ ball (f c z) (e / 4)
β’ (d, w) β (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r | case right
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : β
m : (d β u β§ dist d c < s) β§ w β ball (f c z) (e / 4)
β’ β x, (x.1 β u β§ x.2 β closedBall z r) β§ (x.1, f x.1 x.2) = (d, w) | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : β
m : (d, w) β (u β© ball c s) ΓΛ’ ball (f c z) (e / 4)
β’ (d, w) β (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | have wm : w β ball (f d z) (e / 2) := by
simp only [mem_ball] at m β’
specialize @sh β¨d, zβ©; simp only [Prod.dist_eq, dist_self, Function.uncurry] at sh
specialize sh (max_lt m.1.2 sp); rw [dist_comm] at sh
calc dist w (f d z)
_ β€ dist w (f c z) + dist (f c z) (f d z) := by bound
_ < e / 4 + dist (f c z) (f d z) := by linarith [m.2]
_ β€ e / 4 + e / 4 := by linarith [sh]
_ = e / 2 := by ring | case right
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : β
m : (d β u β§ dist d c < s) β§ w β ball (f c z) (e / 4)
β’ β x, (x.1 β u β§ x.2 β closedBall z r) β§ (x.1, f x.1 x.2) = (d, w) | case right
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : β
m : (d β u β§ dist d c < s) β§ w β ball (f c z) (e / 4)
wm : w β ball (f d z) (e / 2)
β’ β x, (x.1 β u β§ x.2 β closedBall z r) β§ (x.1, f x.1 x.2) = (d, w) | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : β
m : (d β u β§ dist d c < s) β§ w β ball (f c z) (e / 4)
β’ β x, (x.1 β u β§ x.2 β closedBall z r) β§ (x.1, f x.1 x.2) = (d, w)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | specialize op d m.1.1 wm | case right
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : β
m : (d β u β§ dist d c < s) β§ w β ball (f c z) (e / 4)
wm : w β ball (f d z) (e / 2)
β’ β x, (x.1 β u β§ x.2 β closedBall z r) β§ (x.1, f x.1 x.2) = (d, w) | case right
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : β
m : (d β u β§ dist d c < s) β§ w β ball (f c z) (e / 4)
wm : w β ball (f d z) (e / 2)
op : w β f d '' closedBall z r
β’ β x, (x.1 β u β§ x.2 β closedBall z r) β§ (x.1, f x.1 x.2) = (d, w) | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : β
m : (d β u β§ dist d c < s) β§ w β ball (f c z) (e / 4)
wm : w β ball (f d z) (e / 2)
β’ β x, (x.1 β u β§ x.2 β closedBall z r) β§ (x.1, f x.1 x.2) = (d, w)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | rcases (mem_image _ _ _).mp op with β¨y, yr, ywβ© | case right
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : β
m : (d β u β§ dist d c < s) β§ w β ball (f c z) (e / 4)
wm : w β ball (f d z) (e / 2)
op : w β f d '' closedBall z r
β’ β x, (x.1 β u β§ x.2 β closedBall z r) β§ (x.1, f x.1 x.2) = (d, w) | case right.intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : β
m : (d β u β§ dist d c < s) β§ w β ball (f c z) (e / 4)
wm : w β ball (f d z) (e / 2)
op : w β f d '' closedBall z r
y : β
yr : y β closedBall z r
yw : f d y = w
β’ β x, (x.1 β u β§ x.2 β closedBall z r) β§ (x.1, f x.1 x.2) = (d, w) | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : β
m : (d β u β§ dist d c < s) β§ w β ball (f c z) (e / 4)
wm : w β ball (f d z) (e / 2)
op : w β f d '' closedBall z r
β’ β x, (x.1 β u β§ x.2 β closedBall z r) β§ (x.1, f x.1 x.2) = (d, w)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | useβ¨d, yβ© | case right.intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : β
m : (d β u β§ dist d c < s) β§ w β ball (f c z) (e / 4)
wm : w β ball (f d z) (e / 2)
op : w β f d '' closedBall z r
y : β
yr : y β closedBall z r
yw : f d y = w
β’ β x, (x.1 β u β§ x.2 β closedBall z r) β§ (x.1, f x.1 x.2) = (d, w) | case h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : β
m : (d β u β§ dist d c < s) β§ w β ball (f c z) (e / 4)
wm : w β ball (f d z) (e / 2)
op : w β f d '' closedBall z r
y : β
yr : y β closedBall z r
yw : f d y = w
β’ ((d, y).1 β u β§ (d, y).2 β closedBall z r) β§ ((d, y).1, f (d, y).1 (d, y).2) = (d, w) | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : β
m : (d β u β§ dist d c < s) β§ w β ball (f c z) (e / 4)
wm : w β ball (f d z) (e / 2)
op : w β f d '' closedBall z r
y : β
yr : y β closedBall z r
yw : f d y = w
β’ β x, (x.1 β u β§ x.2 β closedBall z r) β§ (x.1, f x.1 x.2) = (d, w)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | simp only [mem_prod_eq, Prod.ext_iff, yw, and_true_iff, eq_self_iff_true, true_and_iff, yr, m.1.1] | case h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : β
m : (d β u β§ dist d c < s) β§ w β ball (f c z) (e / 4)
wm : w β ball (f d z) (e / 2)
op : w β f d '' closedBall z r
y : β
yr : y β closedBall z r
yw : f d y = w
β’ ((d, y).1 β u β§ (d, y).2 β closedBall z r) β§ ((d, y).1, f (d, y).1 (d, y).2) = (d, w) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : β
m : (d β u β§ dist d c < s) β§ w β ball (f c z) (e / 4)
wm : w β ball (f d z) (e / 2)
op : w β f d '' closedBall z r
y : β
yr : y β closedBall z r
yw : f d y = w
β’ ((d, y).1 β u β§ (d, y).2 β closedBall z r) β§ ((d, y).1, f (d, y).1 (d, y).2) = (d, w)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | refine fun d m β¦ (nontrivial_local_of_global (fa.along_snd.mono ?_) rp ep (ef d m)).nonconst | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
β’ β d β u, βαΆ (w : β) in π z, f d w β f d z | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
d : β
m : d β u
β’ closedBall z r β {y | (d, y) β u ΓΛ’ closedBall z r} | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
β’ β d β u, βαΆ (w : β) in π z, f d w β f d z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | simp only [β closedBall_prod_same, mem_prod_eq, setOf_mem_eq, iff_true_iff.mpr m,
true_and_iff, subset_refl] | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
d : β
m : d β u
β’ closedBall z r β {y | (d, y) β u ΓΛ’ closedBall z r} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
d : β
m : d β u
β’ closedBall z r β {y | (d, y) β u ΓΛ’ closedBall z r}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | intro d du | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
β’ β d β u, ball (f d z) (e / 2) β f d '' closedBall z r | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
d : β
du : d β u
β’ ball (f d z) (e / 2) β f d '' closedBall z r | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
β’ β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | refine DiffContOnCl.ball_subset_image_closedBall ?_ rp (ef d du) (fn d du) | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
d : β
du : d β u
β’ ball (f d z) (e / 2) β f d '' closedBall z r | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
d : β
du : d β u
β’ DiffContOnCl β (f d) (ball z r) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
d : β
du : d β u
β’ ball (f d z) (e / 2) β f d '' closedBall z r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | have e : f d = uncurry f β fun w β¦ (d, w) := rfl | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
d : β
du : d β u
β’ DiffContOnCl β (f d) (ball z r) | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
eβ r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < eβ
un : u β π c
ef : β d β u, β w β sphere z r, eβ β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
d : β
du : d β u
e : f d = uncurry f β fun w => (d, w)
β’ DiffContOnCl β (f d) (ball z r) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
d : β
du : d β u
β’ DiffContOnCl β (f d) (ball z r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | rw [e] | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
eβ r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < eβ
un : u β π c
ef : β d β u, β w β sphere z r, eβ β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
d : β
du : d β u
e : f d = uncurry f β fun w => (d, w)
β’ DiffContOnCl β (f d) (ball z r) | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
eβ r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < eβ
un : u β π c
ef : β d β u, β w β sphere z r, eβ β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
d : β
du : d β u
e : f d = uncurry f β fun w => (d, w)
β’ DiffContOnCl β (uncurry f β fun w => (d, w)) (ball z r) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
eβ r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < eβ
un : u β π c
ef : β d β u, β w β sphere z r, eβ β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
d : β
du : d β u
e : f d = uncurry f β fun w => (d, w)
β’ DiffContOnCl β (f d) (ball z r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | apply DifferentiableOn.diffContOnCl | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
eβ r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < eβ
un : u β π c
ef : β d β u, β w β sphere z r, eβ β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
d : β
du : d β u
e : f d = uncurry f β fun w => (d, w)
β’ DiffContOnCl β (uncurry f β fun w => (d, w)) (ball z r) | case h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
eβ r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < eβ
un : u β π c
ef : β d β u, β w β sphere z r, eβ β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
d : β
du : d β u
e : f d = uncurry f β fun w => (d, w)
β’ DifferentiableOn β (uncurry f β fun w => (d, w)) (closure (ball z r)) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
eβ r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < eβ
un : u β π c
ef : β d β u, β w β sphere z r, eβ β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
d : β
du : d β u
e : f d = uncurry f β fun w => (d, w)
β’ DiffContOnCl β (uncurry f β fun w => (d, w)) (ball z r)
TACTIC:
|
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