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https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.inCharts | [183, 1] | [196, 77] | rw [PartialEquiv.left_inv _ m, PartialEquiv.left_inv _ (mem_extChartAt_source I z)] at fn | 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 : S β T
z : S
n : NontrivialHolomorphicAt f z
c :
βαΆ (a : S) in π z,
β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) a))) =
β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) z)))
w : S
fm : f w β (extChartAt π(β, β) (f z)).source
m : w β (extChartAt π(β, β) z).source
fn :
β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) w))) =
β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) z)))
β’ f w = 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 : S β T
z : S
n : NontrivialHolomorphicAt f z
c :
βαΆ (a : S) in π z,
β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) a))) =
β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) z)))
w : S
fm : f w β (extChartAt π(β, β) (f z)).source
m : w β (extChartAt π(β, β) z).source
fn : β(extChartAt π(β, β) (f z)) (f w) = β(extChartAt π(β, β) (f z)) (f z)
β’ f w = f z | 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 : S β T
z : S
n : NontrivialHolomorphicAt f z
c :
βαΆ (a : S) in π z,
β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) a))) =
β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) z)))
w : S
fm : f w β (extChartAt π(β, β) (f z)).source
m : w β (extChartAt π(β, β) z).source
fn :
β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) w))) =
β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) z)))
β’ f w = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.inCharts | [183, 1] | [196, 77] | exact ((PartialEquiv.injOn _).eq_iff fm (mem_extChartAt_source _ _)).mp fn | 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 : S β T
z : S
n : NontrivialHolomorphicAt f z
c :
βαΆ (a : S) in π z,
β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) a))) =
β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) z)))
w : S
fm : f w β (extChartAt π(β, β) (f z)).source
m : w β (extChartAt π(β, β) z).source
fn : β(extChartAt π(β, β) (f z)) (f w) = β(extChartAt π(β, β) (f z)) (f z)
β’ f w = f z | no goals | 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 : S β T
z : S
n : NontrivialHolomorphicAt f z
c :
βαΆ (a : S) in π z,
β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) a))) =
β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) z)))
w : S
fm : f w β (extChartAt π(β, β) (f z)).source
m : w β (extChartAt π(β, β) z).source
fn : β(extChartAt π(β, β) (f z)) (f w) = β(extChartAt π(β, β) (f z)) (f z)
β’ f w = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | refine le_antisymm ?_ n.holomorphicAt.continuousAt | 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 : S β T
z : S
n : NontrivialHolomorphicAt f z
β’ π (f z) = Filter.map f (π 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 : S β T
z : S
n : NontrivialHolomorphicAt f z
β’ π (f z) β€ Filter.map f (π 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 : S β T
z : S
n : NontrivialHolomorphicAt f z
β’ π (f z) = Filter.map f (π z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | generalize hg : (fun x β¦ extChartAt I (f z) (f ((extChartAt I z).symm x))) = g | 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 : S β T
z : S
n : NontrivialHolomorphicAt f z
β’ π (f z) β€ Filter.map f (π 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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
β’ π (f z) β€ Filter.map f (π 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 : S β T
z : S
n : NontrivialHolomorphicAt f z
β’ π (f z) β€ Filter.map f (π z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | have ga : AnalyticAt β g (extChartAt I z z) := by rw [β hg]; exact n.holomorphicAt.2 | 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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
β’ π (f z) β€ Filter.map f (π 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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
β’ π (f z) β€ Filter.map f (π 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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
β’ π (f z) β€ Filter.map f (π z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | cases' ga.eventually_constant_or_nhds_le_map_nhds with h 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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
β’ π (f z) β€ Filter.map f (π z) | case inl
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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
h : βαΆ (z_1 : β) in π (β(extChartAt π(β, β) z) z), g z_1 = g (β(extChartAt π(β, β) z) z)
β’ π (f z) β€ Filter.map f (π z)
case inr
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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
h : π (g (β(extChartAt π(β, β) z) z)) β€ Filter.map g (π (β(extChartAt π(β, β) z) z))
β’ π (f z) β€ Filter.map f (π 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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
β’ π (f z) β€ Filter.map f (π z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | rw [β hg] | 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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
β’ AnalyticAt β g (β(extChartAt π(β, β) z) 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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
β’ AnalyticAt β (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) (β(extChartAt π(β, β) z) 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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
β’ AnalyticAt β g (β(extChartAt π(β, β) z) z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | exact n.holomorphicAt.2 | 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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
β’ AnalyticAt β (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) (β(extChartAt π(β, β) z) z) | 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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
β’ AnalyticAt β (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) (β(extChartAt π(β, β) z) z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | contrapose h | case inl
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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
h : βαΆ (z_1 : β) in π (β(extChartAt π(β, β) z) z), g z_1 = g (β(extChartAt π(β, β) z) z)
β’ π (f z) β€ Filter.map f (π z) | case inl
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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
h : Β¬π (f z) β€ Filter.map f (π z)
β’ Β¬βαΆ (z_1 : β) in π (β(extChartAt π(β, β) z) z), g z_1 = g (β(extChartAt π(β, β) z) z) | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
h : βαΆ (z_1 : β) in π (β(extChartAt π(β, β) z) z), g z_1 = g (β(extChartAt π(β, β) z) z)
β’ π (f z) β€ Filter.map f (π z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | clear h | case inl
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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
h : Β¬π (f z) β€ Filter.map f (π z)
β’ Β¬βαΆ (z_1 : β) in π (β(extChartAt π(β, β) z) z), g z_1 = g (β(extChartAt π(β, β) z) z) | case inl
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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
β’ Β¬βαΆ (z_1 : β) in π (β(extChartAt π(β, β) z) z), g z_1 = g (β(extChartAt π(β, β) z) z) | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
h : Β¬π (f z) β€ Filter.map f (π z)
β’ Β¬βαΆ (z_1 : β) in π (β(extChartAt π(β, β) z) z), g z_1 = g (β(extChartAt π(β, β) z) z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | simp only [Filter.not_eventually] | case inl
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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
β’ Β¬βαΆ (z_1 : β) in π (β(extChartAt π(β, β) z) z), g z_1 = g (β(extChartAt π(β, β) z) z) | case inl
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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
β’ βαΆ (x : β) in π (β(extChartAt π(β, β) z) z), Β¬g x = g (β(extChartAt π(β, β) z) z) | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
β’ Β¬βαΆ (z_1 : β) in π (β(extChartAt π(β, β) z) z), g z_1 = g (β(extChartAt π(β, β) z) z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | apply n.inCharts.nonconst.mp | case inl
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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
β’ βαΆ (x : β) in π (β(extChartAt π(β, β) z) z), Β¬g x = g (β(extChartAt π(β, β) z) z) | case inl
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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
β’ βαΆ (x : β) in π (β(extChartAt π(β, β) z) z),
β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x)) β
β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) z))) β
Β¬g x = g (β(extChartAt π(β, β) z) z) | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
β’ βαΆ (x : β) in π (β(extChartAt π(β, β) z) z), Β¬g x = g (β(extChartAt π(β, β) z) z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | simp only [β hg, Ne, imp_self, Filter.eventually_true] | case inl
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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
β’ βαΆ (x : β) in π (β(extChartAt π(β, β) z) z),
β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x)) β
β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) z))) β
Β¬g x = g (β(extChartAt π(β, β) z) z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
β’ βαΆ (x : β) in π (β(extChartAt π(β, β) z) z),
β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x)) β
β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) z))) β
Β¬g x = g (β(extChartAt π(β, β) z) z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | simp only [β extChartAt_map_nhds' I z, Filter.map_map] at h | case inr
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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
h : π (g (β(extChartAt π(β, β) z) z)) β€ Filter.map g (π (β(extChartAt π(β, β) z) z))
β’ π (f z) β€ Filter.map f (π z) | case inr
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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
h : π (g (β(extChartAt π(β, β) z) z)) β€ Filter.map (g β β(extChartAt π(β, β) z)) (π z)
β’ π (f z) β€ Filter.map f (π z) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
h : π (g (β(extChartAt π(β, β) z) z)) β€ Filter.map g (π (β(extChartAt π(β, β) z) z))
β’ π (f z) β€ Filter.map f (π z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | replace h := @Filter.map_mono _ _ (extChartAt I (f z)).symm _ _ h | case inr
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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
h : π (g (β(extChartAt π(β, β) z) z)) β€ Filter.map (g β β(extChartAt π(β, β) z)) (π z)
β’ π (f z) β€ Filter.map f (π z) | case inr
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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
h :
Filter.map (β(extChartAt π(β, β) (f z)).symm) (π (g (β(extChartAt π(β, β) z) z))) β€
Filter.map (β(extChartAt π(β, β) (f z)).symm) (Filter.map (g β β(extChartAt π(β, β) z)) (π z))
β’ π (f z) β€ Filter.map f (π z) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
h : π (g (β(extChartAt π(β, β) z) z)) β€ Filter.map (g β β(extChartAt π(β, β) z)) (π z)
β’ π (f z) β€ Filter.map f (π z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | simp only [β hg] at h | case inr
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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
h :
Filter.map (β(extChartAt π(β, β) (f z)).symm) (π (g (β(extChartAt π(β, β) z) z))) β€
Filter.map (β(extChartAt π(β, β) (f z)).symm) (Filter.map (g β β(extChartAt π(β, β) z)) (π z))
β’ π (f z) β€ Filter.map f (π z) | case inr
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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
h :
Filter.map (β(extChartAt π(β, β) (f z)).symm)
(π (β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) z))))) β€
Filter.map (β(extChartAt π(β, β) (f z)).symm)
(Filter.map
((fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) β β(extChartAt π(β, β) z)) (π z))
β’ π (f z) β€ Filter.map f (π z) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
h :
Filter.map (β(extChartAt π(β, β) (f z)).symm) (π (g (β(extChartAt π(β, β) z) z))) β€
Filter.map (β(extChartAt π(β, β) (f z)).symm) (Filter.map (g β β(extChartAt π(β, β) z)) (π z))
β’ π (f z) β€ Filter.map f (π z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | rw [PartialEquiv.left_inv _ (mem_extChartAt_source I z)] at h | case inr
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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
h :
Filter.map (β(extChartAt π(β, β) (f z)).symm)
(π (β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) z))))) β€
Filter.map (β(extChartAt π(β, β) (f z)).symm)
(Filter.map
((fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) β β(extChartAt π(β, β) z)) (π z))
β’ π (f z) β€ Filter.map f (π z) | case inr
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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
h :
Filter.map (β(extChartAt π(β, β) (f z)).symm) (π (β(extChartAt π(β, β) (f z)) (f z))) β€
Filter.map (β(extChartAt π(β, β) (f z)).symm)
(Filter.map
((fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) β β(extChartAt π(β, β) z)) (π z))
β’ π (f z) β€ Filter.map f (π z) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
h :
Filter.map (β(extChartAt π(β, β) (f z)).symm)
(π (β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) z))))) β€
Filter.map (β(extChartAt π(β, β) (f z)).symm)
(Filter.map
((fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) β β(extChartAt π(β, β) z)) (π z))
β’ π (f z) β€ Filter.map f (π z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | simp only [extChartAt_symm_map_nhds' I (f z), Filter.map_map, Function.comp] at h | case inr
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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
h :
Filter.map (β(extChartAt π(β, β) (f z)).symm) (π (β(extChartAt π(β, β) (f z)) (f z))) β€
Filter.map (β(extChartAt π(β, β) (f z)).symm)
(Filter.map
((fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) β β(extChartAt π(β, β) z)) (π z))
β’ π (f z) β€ Filter.map f (π z) | case inr
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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
h :
π (f z) β€
Filter.map
(fun x =>
β(extChartAt π(β, β) (f z)).symm
(β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x)))))
(π z)
β’ π (f z) β€ Filter.map f (π z) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
h :
Filter.map (β(extChartAt π(β, β) (f z)).symm) (π (β(extChartAt π(β, β) (f z)) (f z))) β€
Filter.map (β(extChartAt π(β, β) (f z)).symm)
(Filter.map
((fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) β β(extChartAt π(β, β) z)) (π z))
β’ π (f z) β€ Filter.map f (π z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | have e : (fun w β¦ (extChartAt I (f z)).symm
(extChartAt I (f z) (f ((extChartAt I z).symm (extChartAt I z w))))) =αΆ [π z] f := by
apply ((isOpen_extChartAt_source I z).eventually_mem (mem_extChartAt_source I z)).mp
apply (n.holomorphicAt.continuousAt.eventually_mem (extChartAt_source_mem_nhds I (f z))).mp
refine eventually_of_forall fun w fm m β¦ ?_
simp only [PartialEquiv.left_inv _ m, PartialEquiv.left_inv _ fm] | case inr
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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
h :
π (f z) β€
Filter.map
(fun x =>
β(extChartAt π(β, β) (f z)).symm
(β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x)))))
(π z)
β’ π (f z) β€ Filter.map f (π z) | case inr
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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
h :
π (f z) β€
Filter.map
(fun x =>
β(extChartAt π(β, β) (f z)).symm
(β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x)))))
(π z)
e :
(π z).EventuallyEq
(fun w =>
β(extChartAt π(β, β) (f z)).symm
(β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) w)))))
f
β’ π (f z) β€ Filter.map f (π z) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
h :
π (f z) β€
Filter.map
(fun x =>
β(extChartAt π(β, β) (f z)).symm
(β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x)))))
(π z)
β’ π (f z) β€ Filter.map f (π z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | rw [Filter.map_congr e] at h | case inr
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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
h :
π (f z) β€
Filter.map
(fun x =>
β(extChartAt π(β, β) (f z)).symm
(β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x)))))
(π z)
e :
(π z).EventuallyEq
(fun w =>
β(extChartAt π(β, β) (f z)).symm
(β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) w)))))
f
β’ π (f z) β€ Filter.map f (π z) | case inr
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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
h : π (f z) β€ Filter.map f (π z)
e :
(π z).EventuallyEq
(fun w =>
β(extChartAt π(β, β) (f z)).symm
(β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) w)))))
f
β’ π (f z) β€ Filter.map f (π z) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
h :
π (f z) β€
Filter.map
(fun x =>
β(extChartAt π(β, β) (f z)).symm
(β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x)))))
(π z)
e :
(π z).EventuallyEq
(fun w =>
β(extChartAt π(β, β) (f z)).symm
(β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) w)))))
f
β’ π (f z) β€ Filter.map f (π z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | exact h | case inr
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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
h : π (f z) β€ Filter.map f (π z)
e :
(π z).EventuallyEq
(fun w =>
β(extChartAt π(β, β) (f z)).symm
(β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) w)))))
f
β’ π (f z) β€ Filter.map f (π z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
h : π (f z) β€ Filter.map f (π z)
e :
(π z).EventuallyEq
(fun w =>
β(extChartAt π(β, β) (f z)).symm
(β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) w)))))
f
β’ π (f z) β€ Filter.map f (π z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | apply ((isOpen_extChartAt_source I z).eventually_mem (mem_extChartAt_source I z)).mp | 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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
h :
π (f z) β€
Filter.map
(fun x =>
β(extChartAt π(β, β) (f z)).symm
(β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x)))))
(π z)
β’ (π z).EventuallyEq
(fun w =>
β(extChartAt π(β, β) (f z)).symm
(β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) w)))))
f | 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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
h :
π (f z) β€
Filter.map
(fun x =>
β(extChartAt π(β, β) (f z)).symm
(β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x)))))
(π z)
β’ βαΆ (x : S) in π z,
x β (extChartAt π(β, β) z).source β
(fun w =>
β(extChartAt π(β, β) (f z)).symm
(β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) w)))))
x =
f x | 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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
h :
π (f z) β€
Filter.map
(fun x =>
β(extChartAt π(β, β) (f z)).symm
(β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x)))))
(π z)
β’ (π z).EventuallyEq
(fun w =>
β(extChartAt π(β, β) (f z)).symm
(β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) w)))))
f
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | apply (n.holomorphicAt.continuousAt.eventually_mem (extChartAt_source_mem_nhds I (f z))).mp | 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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
h :
π (f z) β€
Filter.map
(fun x =>
β(extChartAt π(β, β) (f z)).symm
(β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x)))))
(π z)
β’ βαΆ (x : S) in π z,
x β (extChartAt π(β, β) z).source β
(fun w =>
β(extChartAt π(β, β) (f z)).symm
(β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) w)))))
x =
f x | 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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
h :
π (f z) β€
Filter.map
(fun x =>
β(extChartAt π(β, β) (f z)).symm
(β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x)))))
(π z)
β’ βαΆ (x : S) in π z,
f x β (extChartAt π(β, β) (f z)).source β
x β (extChartAt π(β, β) z).source β
(fun w =>
β(extChartAt π(β, β) (f z)).symm
(β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) w)))))
x =
f x | 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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
h :
π (f z) β€
Filter.map
(fun x =>
β(extChartAt π(β, β) (f z)).symm
(β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x)))))
(π z)
β’ βαΆ (x : S) in π z,
x β (extChartAt π(β, β) z).source β
(fun w =>
β(extChartAt π(β, β) (f z)).symm
(β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) w)))))
x =
f x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | refine eventually_of_forall fun w fm m β¦ ?_ | 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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
h :
π (f z) β€
Filter.map
(fun x =>
β(extChartAt π(β, β) (f z)).symm
(β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x)))))
(π z)
β’ βαΆ (x : S) in π z,
f x β (extChartAt π(β, β) (f z)).source β
x β (extChartAt π(β, β) z).source β
(fun w =>
β(extChartAt π(β, β) (f z)).symm
(β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) w)))))
x =
f x | 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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
h :
π (f z) β€
Filter.map
(fun x =>
β(extChartAt π(β, β) (f z)).symm
(β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x)))))
(π z)
w : S
fm : f w β (extChartAt π(β, β) (f z)).source
m : w β (extChartAt π(β, β) z).source
β’ (fun w =>
β(extChartAt π(β, β) (f z)).symm
(β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) w)))))
w =
f w | 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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
h :
π (f z) β€
Filter.map
(fun x =>
β(extChartAt π(β, β) (f z)).symm
(β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x)))))
(π z)
β’ βαΆ (x : S) in π z,
f x β (extChartAt π(β, β) (f z)).source β
x β (extChartAt π(β, β) z).source β
(fun w =>
β(extChartAt π(β, β) (f z)).symm
(β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) w)))))
x =
f x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds | [201, 1] | [225, 42] | simp only [PartialEquiv.left_inv _ m, PartialEquiv.left_inv _ fm] | 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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
h :
π (f z) β€
Filter.map
(fun x =>
β(extChartAt π(β, β) (f z)).symm
(β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x)))))
(π z)
w : S
fm : f w β (extChartAt π(β, β) (f z)).source
m : w β (extChartAt π(β, β) z).source
β’ (fun w =>
β(extChartAt π(β, β) (f z)).symm
(β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) w)))))
w =
f w | 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 : S β T
z : S
n : NontrivialHolomorphicAt f z
g : β β β
hg : (fun x => β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β g (β(extChartAt π(β, β) z) z)
h :
π (f z) β€
Filter.map
(fun x =>
β(extChartAt π(β, β) (f z)).symm
(β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x)))))
(π z)
w : S
fm : f w β (extChartAt π(β, β) (f z)).source
m : w β (extChartAt π(β, β) z).source
β’ (fun w =>
β(extChartAt π(β, β) (f z)).symm
(β(extChartAt π(β, β) (f z)) (f (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) w)))))
w =
f w
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | refine le_antisymm ?_ (continuousAt_fst.prod fa.continuousAt) | 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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
β’ π (c, f c z) = Filter.map (fun p => (p.1, f p.1 p.2)) (π (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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
β’ π (c, f c z) β€ Filter.map (fun p => (p.1, f p.1 p.2)) (π (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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
β’ π (c, f c z) = Filter.map (fun p => (p.1, f p.1 p.2)) (π (c, z))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | generalize hg : (fun e x β¦ extChartAt I (f c z) (f e ((extChartAt I z).symm x))) = g | 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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
β’ π (c, f c z) β€ Filter.map (fun p => (p.1, f p.1 p.2)) (π (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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
β’ π (c, f c z) β€ Filter.map (fun p => (p.1, f p.1 p.2)) (π (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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
β’ π (c, f c z) β€ Filter.map (fun p => (p.1, f p.1 p.2)) (π (c, z))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | have ga : AnalyticAt β (uncurry g) (c, extChartAt I z z) := by
rw [β hg]; exact (holomorphicAt_iff.mp fa).2 | 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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
β’ π (c, f c z) β€ Filter.map (fun p => (p.1, f p.1 p.2)) (π (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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
β’ π (c, f c z) β€ Filter.map (fun p => (p.1, f p.1 p.2)) (π (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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
β’ π (c, f c z) β€ Filter.map (fun p => (p.1, f p.1 p.2)) (π (c, z))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | have gn : NontrivialHolomorphicAt (g c) (extChartAt I z z) := by rw [β hg]; exact n.inCharts | 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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
β’ π (c, f c z) β€ Filter.map (fun p => (p.1, f p.1 p.2)) (π (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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
β’ π (c, f c z) β€ Filter.map (fun p => (p.1, f p.1 p.2)) (π (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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
β’ π (c, f c z) β€ Filter.map (fun p => (p.1, f p.1 p.2)) (π (c, z))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | have h := gn.nhds_le_map_nhds_param' ga | 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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
β’ π (c, f c z) β€ Filter.map (fun p => (p.1, f p.1 p.2)) (π (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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
h : π (c, g c (β(extChartAt π(β, β) z) z)) β€ Filter.map (fun p => (p.1, g p.1 p.2)) (π (c, β(extChartAt π(β, β) z) z))
β’ π (c, f c z) β€ Filter.map (fun p => (p.1, f p.1 p.2)) (π (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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
β’ π (c, f c z) β€ Filter.map (fun p => (p.1, f p.1 p.2)) (π (c, z))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | simp only [nhds_prod_eq, β extChartAt_map_nhds' I z, Filter.map_map, Filter.prod_map_id_map_eq,
Function.comp] at 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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
h : π (c, g c (β(extChartAt π(β, β) z) z)) β€ Filter.map (fun p => (p.1, g p.1 p.2)) (π (c, β(extChartAt π(β, β) z) z))
β’ π (c, f c z) β€ Filter.map (fun p => (p.1, f p.1 p.2)) (π (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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
h :
π c ΓΛ’ π (g c (β(extChartAt π(β, β) z) z)) β€
Filter.map (fun x => (x.1, g x.1 (β(extChartAt π(β, β) z) x.2))) (π c ΓΛ’ π z)
β’ π (c, f c z) β€ Filter.map (fun p => (p.1, f p.1 p.2)) (π (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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
h : π (c, g c (β(extChartAt π(β, β) z) z)) β€ Filter.map (fun p => (p.1, g p.1 p.2)) (π (c, β(extChartAt π(β, β) z) z))
β’ π (c, f c z) β€ Filter.map (fun p => (p.1, f p.1 p.2)) (π (c, z))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | replace h := @Filter.map_mono _ _ (fun p : β Γ β β¦ (p.1, (extChartAt I (f c z)).symm p.2)) _ _ 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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
h :
π c ΓΛ’ π (g c (β(extChartAt π(β, β) z) z)) β€
Filter.map (fun x => (x.1, g x.1 (β(extChartAt π(β, β) z) x.2))) (π c ΓΛ’ π z)
β’ π (c, f c z) β€ Filter.map (fun p => (p.1, f p.1 p.2)) (π (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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
h :
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2)) (π c ΓΛ’ π (g c (β(extChartAt π(β, β) z) z))) β€
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(Filter.map (fun x => (x.1, g x.1 (β(extChartAt π(β, β) z) x.2))) (π c ΓΛ’ π z))
β’ π (c, f c z) β€ Filter.map (fun p => (p.1, f p.1 p.2)) (π (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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
h :
π c ΓΛ’ π (g c (β(extChartAt π(β, β) z) z)) β€
Filter.map (fun x => (x.1, g x.1 (β(extChartAt π(β, β) z) x.2))) (π c ΓΛ’ π z)
β’ π (c, f c z) β€ Filter.map (fun p => (p.1, f p.1 p.2)) (π (c, z))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | simp only [β hg] at 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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
h :
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2)) (π c ΓΛ’ π (g c (β(extChartAt π(β, β) z) z))) β€
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(Filter.map (fun x => (x.1, g x.1 (β(extChartAt π(β, β) z) x.2))) (π c ΓΛ’ π z))
β’ π (c, f c z) β€ Filter.map (fun p => (p.1, f p.1 p.2)) (π (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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
h :
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(π c ΓΛ’ π (β(extChartAt π(β, β) (f c z)) (f c (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) z))))) β€
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(Filter.map
(fun x =>
(x.1, β(extChartAt π(β, β) (f c z)) (f x.1 (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x.2)))))
(π c ΓΛ’ π z))
β’ π (c, f c z) β€ Filter.map (fun p => (p.1, f p.1 p.2)) (π (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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
h :
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2)) (π c ΓΛ’ π (g c (β(extChartAt π(β, β) z) z))) β€
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(Filter.map (fun x => (x.1, g x.1 (β(extChartAt π(β, β) z) x.2))) (π c ΓΛ’ π z))
β’ π (c, f c z) β€ Filter.map (fun p => (p.1, f p.1 p.2)) (π (c, z))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | rw [PartialEquiv.left_inv _ (mem_extChartAt_source I z)] at 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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
h :
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(π c ΓΛ’ π (β(extChartAt π(β, β) (f c z)) (f c (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) z))))) β€
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(Filter.map
(fun x =>
(x.1, β(extChartAt π(β, β) (f c z)) (f x.1 (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x.2)))))
(π c ΓΛ’ π z))
β’ π (c, f c z) β€ Filter.map (fun p => (p.1, f p.1 p.2)) (π (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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
h :
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(π c ΓΛ’ π (β(extChartAt π(β, β) (f c z)) (f c z))) β€
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(Filter.map
(fun x =>
(x.1, β(extChartAt π(β, β) (f c z)) (f x.1 (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x.2)))))
(π c ΓΛ’ π z))
β’ π (c, f c z) β€ Filter.map (fun p => (p.1, f p.1 p.2)) (π (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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
h :
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(π c ΓΛ’ π (β(extChartAt π(β, β) (f c z)) (f c (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) z))))) β€
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(Filter.map
(fun x =>
(x.1, β(extChartAt π(β, β) (f c z)) (f x.1 (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x.2)))))
(π c ΓΛ’ π z))
β’ π (c, f c z) β€ Filter.map (fun p => (p.1, f p.1 p.2)) (π (c, z))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | have pe := Filter.prod_map_id_map_eq (f := π c) (g := π (extChartAt I (f c z) (f c z)))
(m := fun x β¦ (extChartAt I (f c z)).symm x) | 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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
h :
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(π c ΓΛ’ π (β(extChartAt π(β, β) (f c z)) (f c z))) β€
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(Filter.map
(fun x =>
(x.1, β(extChartAt π(β, β) (f c z)) (f x.1 (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x.2)))))
(π c ΓΛ’ π z))
β’ π (c, f c z) β€ Filter.map (fun p => (p.1, f p.1 p.2)) (π (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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
h :
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(π c ΓΛ’ π (β(extChartAt π(β, β) (f c z)) (f c z))) β€
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(Filter.map
(fun x =>
(x.1, β(extChartAt π(β, β) (f c z)) (f x.1 (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x.2)))))
(π c ΓΛ’ π z))
pe :
π c ΓΛ’ Filter.map (fun x => β(extChartAt π(β, β) (f c z)).symm x) (π (β(extChartAt π(β, β) (f c z)) (f c z))) =
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(π c ΓΛ’ π (β(extChartAt π(β, β) (f c z)) (f c z)))
β’ π (c, f c z) β€ Filter.map (fun p => (p.1, f p.1 p.2)) (π (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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
h :
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(π c ΓΛ’ π (β(extChartAt π(β, β) (f c z)) (f c z))) β€
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(Filter.map
(fun x =>
(x.1, β(extChartAt π(β, β) (f c z)) (f x.1 (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x.2)))))
(π c ΓΛ’ π z))
β’ π (c, f c z) β€ Filter.map (fun p => (p.1, f p.1 p.2)) (π (c, z))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | rw [extChartAt_symm_map_nhds', βnhds_prod_eq] at pe | 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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
h :
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(π c ΓΛ’ π (β(extChartAt π(β, β) (f c z)) (f c z))) β€
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(Filter.map
(fun x =>
(x.1, β(extChartAt π(β, β) (f c z)) (f x.1 (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x.2)))))
(π c ΓΛ’ π z))
pe :
π c ΓΛ’ Filter.map (fun x => β(extChartAt π(β, β) (f c z)).symm x) (π (β(extChartAt π(β, β) (f c z)) (f c z))) =
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(π c ΓΛ’ π (β(extChartAt π(β, β) (f c z)) (f c z)))
β’ π (c, f c z) β€ Filter.map (fun p => (p.1, f p.1 p.2)) (π (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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
h :
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(π c ΓΛ’ π (β(extChartAt π(β, β) (f c z)) (f c z))) β€
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(Filter.map
(fun x =>
(x.1, β(extChartAt π(β, β) (f c z)) (f x.1 (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x.2)))))
(π c ΓΛ’ π z))
pe :
π (c, f c z) =
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(π c ΓΛ’ π (β(extChartAt π(β, β) (f c z)) (f c z)))
β’ π (c, f c z) β€ Filter.map (fun p => (p.1, f p.1 p.2)) (π (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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
h :
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(π c ΓΛ’ π (β(extChartAt π(β, β) (f c z)) (f c z))) β€
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(Filter.map
(fun x =>
(x.1, β(extChartAt π(β, β) (f c z)) (f x.1 (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x.2)))))
(π c ΓΛ’ π z))
pe :
π c ΓΛ’ Filter.map (fun x => β(extChartAt π(β, β) (f c z)).symm x) (π (β(extChartAt π(β, β) (f c z)) (f c z))) =
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(π c ΓΛ’ π (β(extChartAt π(β, β) (f c z)) (f c z)))
β’ π (c, f c z) β€ Filter.map (fun p => (p.1, f p.1 p.2)) (π (c, z))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | refine _root_.trans (le_of_eq pe) (_root_.trans h (le_of_eq ?_)) | 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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
h :
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(π c ΓΛ’ π (β(extChartAt π(β, β) (f c z)) (f c z))) β€
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(Filter.map
(fun x =>
(x.1, β(extChartAt π(β, β) (f c z)) (f x.1 (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x.2)))))
(π c ΓΛ’ π z))
pe :
π (c, f c z) =
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(π c ΓΛ’ π (β(extChartAt π(β, β) (f c z)) (f c z)))
β’ π (c, f c z) β€ Filter.map (fun p => (p.1, f p.1 p.2)) (π (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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
h :
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(π c ΓΛ’ π (β(extChartAt π(β, β) (f c z)) (f c z))) β€
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(Filter.map
(fun x =>
(x.1, β(extChartAt π(β, β) (f c z)) (f x.1 (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x.2)))))
(π c ΓΛ’ π z))
pe :
π (c, f c z) =
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(π c ΓΛ’ π (β(extChartAt π(β, β) (f c z)) (f c z)))
β’ Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(Filter.map
(fun x =>
(x.1, β(extChartAt π(β, β) (f c z)) (f x.1 (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x.2)))))
(π c ΓΛ’ π z)) =
Filter.map (fun p => (p.1, f p.1 p.2)) (π (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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
h :
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(π c ΓΛ’ π (β(extChartAt π(β, β) (f c z)) (f c z))) β€
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(Filter.map
(fun x =>
(x.1, β(extChartAt π(β, β) (f c z)) (f x.1 (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x.2)))))
(π c ΓΛ’ π z))
pe :
π (c, f c z) =
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(π c ΓΛ’ π (β(extChartAt π(β, β) (f c z)) (f c z)))
β’ π (c, f c z) β€ Filter.map (fun p => (p.1, f p.1 p.2)) (π (c, z))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | clear h pe | 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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
h :
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(π c ΓΛ’ π (β(extChartAt π(β, β) (f c z)) (f c z))) β€
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(Filter.map
(fun x =>
(x.1, β(extChartAt π(β, β) (f c z)) (f x.1 (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x.2)))))
(π c ΓΛ’ π z))
pe :
π (c, f c z) =
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(π c ΓΛ’ π (β(extChartAt π(β, β) (f c z)) (f c z)))
β’ Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(Filter.map
(fun x =>
(x.1, β(extChartAt π(β, β) (f c z)) (f x.1 (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x.2)))))
(π c ΓΛ’ π z)) =
Filter.map (fun p => (p.1, f p.1 p.2)) (π (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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
β’ Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(Filter.map
(fun x =>
(x.1, β(extChartAt π(β, β) (f c z)) (f x.1 (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x.2)))))
(π c ΓΛ’ π z)) =
Filter.map (fun p => (p.1, f p.1 p.2)) (π (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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
h :
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(π c ΓΛ’ π (β(extChartAt π(β, β) (f c z)) (f c z))) β€
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(Filter.map
(fun x =>
(x.1, β(extChartAt π(β, β) (f c z)) (f x.1 (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x.2)))))
(π c ΓΛ’ π z))
pe :
π (c, f c z) =
Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(π c ΓΛ’ π (β(extChartAt π(β, β) (f c z)) (f c z)))
β’ Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(Filter.map
(fun x =>
(x.1, β(extChartAt π(β, β) (f c z)) (f x.1 (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x.2)))))
(π c ΓΛ’ π z)) =
Filter.map (fun p => (p.1, f p.1 p.2)) (π (c, z))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | rw [βnhds_prod_eq, Filter.map_map] | 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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
β’ Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(Filter.map
(fun x =>
(x.1, β(extChartAt π(β, β) (f c z)) (f x.1 (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x.2)))))
(π c ΓΛ’ π z)) =
Filter.map (fun p => (p.1, f p.1 p.2)) (π (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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
β’ Filter.map
((fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2)) β fun x =>
(x.1, β(extChartAt π(β, β) (f c z)) (f x.1 (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x.2)))))
(π (c, z)) =
Filter.map (fun p => (p.1, f p.1 p.2)) (π (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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
β’ Filter.map (fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2))
(Filter.map
(fun x =>
(x.1, β(extChartAt π(β, β) (f c z)) (f x.1 (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x.2)))))
(π c ΓΛ’ π z)) =
Filter.map (fun p => (p.1, f p.1 p.2)) (π (c, z))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | apply Filter.map_congr | 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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
β’ Filter.map
((fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2)) β fun x =>
(x.1, β(extChartAt π(β, β) (f c z)) (f x.1 (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x.2)))))
(π (c, z)) =
Filter.map (fun p => (p.1, f p.1 p.2)) (π (c, z)) | 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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
β’ (π (c, z)).EventuallyEq
((fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2)) β fun x =>
(x.1, β(extChartAt π(β, β) (f c z)) (f x.1 (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x.2)))))
fun p => (p.1, f p.1 p.2) | 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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
β’ Filter.map
((fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2)) β fun x =>
(x.1, β(extChartAt π(β, β) (f c z)) (f x.1 (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x.2)))))
(π (c, z)) =
Filter.map (fun p => (p.1, f p.1 p.2)) (π (c, z))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | apply ((isOpen_extChartAt_source II (c, z)).eventually_mem (mem_extChartAt_source II (c, z))).mp | 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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
β’ (π (c, z)).EventuallyEq
((fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2)) β fun x =>
(x.1, β(extChartAt π(β, β) (f c z)) (f x.1 (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x.2)))))
fun p => (p.1, f p.1 p.2) | 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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
β’ βαΆ (x : β Γ S) in π (c, z),
x β (extChartAt (π(β, β).prod π(β, β)) (c, z)).source β
((fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2)) β fun x =>
(x.1, β(extChartAt π(β, β) (f c z)) (f x.1 (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x.2)))))
x =
(fun p => (p.1, f p.1 p.2)) x | 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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
β’ (π (c, z)).EventuallyEq
((fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2)) β fun x =>
(x.1, β(extChartAt π(β, β) (f c z)) (f x.1 (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x.2)))))
fun p => (p.1, f p.1 p.2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | apply (fa.continuousAt.eventually_mem (extChartAt_source_mem_nhds I (f c z))).mp | 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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
β’ βαΆ (x : β Γ S) in π (c, z),
x β (extChartAt (π(β, β).prod π(β, β)) (c, z)).source β
((fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2)) β fun x =>
(x.1, β(extChartAt π(β, β) (f c z)) (f x.1 (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x.2)))))
x =
(fun p => (p.1, f p.1 p.2)) x | 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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
β’ βαΆ (x : β Γ S) in π (c, z),
uncurry f x β (extChartAt π(β, β) (f c z)).source β
x β (extChartAt (π(β, β).prod π(β, β)) (c, z)).source β
((fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2)) β fun x =>
(x.1, β(extChartAt π(β, β) (f c z)) (f x.1 (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x.2)))))
x =
(fun p => (p.1, f p.1 p.2)) x | 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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
β’ βαΆ (x : β Γ S) in π (c, z),
x β (extChartAt (π(β, β).prod π(β, β)) (c, z)).source β
((fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2)) β fun x =>
(x.1, β(extChartAt π(β, β) (f c z)) (f x.1 (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x.2)))))
x =
(fun p => (p.1, f p.1 p.2)) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | apply eventually_of_forall | 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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
β’ βαΆ (x : β Γ S) in π (c, z),
uncurry f x β (extChartAt π(β, β) (f c z)).source β
x β (extChartAt (π(β, β).prod π(β, β)) (c, z)).source β
((fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2)) β fun x =>
(x.1, β(extChartAt π(β, β) (f c z)) (f x.1 (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x.2)))))
x =
(fun p => (p.1, f p.1 p.2)) x | case h.hp
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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
β’ β (x : β Γ S),
uncurry f x β (extChartAt π(β, β) (f c z)).source β
x β (extChartAt (π(β, β).prod π(β, β)) (c, z)).source β
((fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2)) β fun x =>
(x.1, β(extChartAt π(β, β) (f c z)) (f x.1 (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x.2)))))
x =
(fun p => (p.1, f p.1 p.2)) x | 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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
β’ βαΆ (x : β Γ S) in π (c, z),
uncurry f x β (extChartAt π(β, β) (f c z)).source β
x β (extChartAt (π(β, β).prod π(β, β)) (c, z)).source β
((fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2)) β fun x =>
(x.1, β(extChartAt π(β, β) (f c z)) (f x.1 (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x.2)))))
x =
(fun p => (p.1, f p.1 p.2)) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | intro β¨e, wβ© fm m | case h.hp
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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
β’ β (x : β Γ S),
uncurry f x β (extChartAt π(β, β) (f c z)).source β
x β (extChartAt (π(β, β).prod π(β, β)) (c, z)).source β
((fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2)) β fun x =>
(x.1, β(extChartAt π(β, β) (f c z)) (f x.1 (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x.2)))))
x =
(fun p => (p.1, f p.1 p.2)) x | case h.hp
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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
e : β
w : S
fm : uncurry f (e, w) β (extChartAt π(β, β) (f c z)).source
m : (e, w) β (extChartAt (π(β, β).prod π(β, β)) (c, z)).source
β’ ((fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2)) β fun x =>
(x.1, β(extChartAt π(β, β) (f c z)) (f x.1 (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x.2)))))
(e, w) =
(fun p => (p.1, f p.1 p.2)) (e, w) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hp
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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
β’ β (x : β Γ S),
uncurry f x β (extChartAt π(β, β) (f c z)).source β
x β (extChartAt (π(β, β).prod π(β, β)) (c, z)).source β
((fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2)) β fun x =>
(x.1, β(extChartAt π(β, β) (f c z)) (f x.1 (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x.2)))))
x =
(fun p => (p.1, f p.1 p.2)) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | simp only [Function.comp, uncurry, extChartAt_prod, PartialEquiv.prod_source, mem_prod_eq] at fm m | case h.hp
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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
e : β
w : S
fm : uncurry f (e, w) β (extChartAt π(β, β) (f c z)).source
m : (e, w) β (extChartAt (π(β, β).prod π(β, β)) (c, z)).source
β’ ((fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2)) β fun x =>
(x.1, β(extChartAt π(β, β) (f c z)) (f x.1 (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x.2)))))
(e, w) =
(fun p => (p.1, f p.1 p.2)) (e, w) | case h.hp
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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
e : β
w : S
fm : f e w β (extChartAt π(β, β) (f c z)).source
m : e β (extChartAt π(β, β) c).source β§ w β (extChartAt π(β, β) z).source
β’ ((fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2)) β fun x =>
(x.1, β(extChartAt π(β, β) (f c z)) (f x.1 (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x.2)))))
(e, w) =
(fun p => (p.1, f p.1 p.2)) (e, w) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hp
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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
e : β
w : S
fm : uncurry f (e, w) β (extChartAt π(β, β) (f c z)).source
m : (e, w) β (extChartAt (π(β, β).prod π(β, β)) (c, z)).source
β’ ((fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2)) β fun x =>
(x.1, β(extChartAt π(β, β) (f c z)) (f x.1 (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x.2)))))
(e, w) =
(fun p => (p.1, f p.1 p.2)) (e, w)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | simp only [Function.comp, PartialEquiv.left_inv _ m.2, PartialEquiv.left_inv _ fm] | case h.hp
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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
e : β
w : S
fm : f e w β (extChartAt π(β, β) (f c z)).source
m : e β (extChartAt π(β, β) c).source β§ w β (extChartAt π(β, β) z).source
β’ ((fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2)) β fun x =>
(x.1, β(extChartAt π(β, β) (f c z)) (f x.1 (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x.2)))))
(e, w) =
(fun p => (p.1, f p.1 p.2)) (e, w) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hp
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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
gn : NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
e : β
w : S
fm : f e w β (extChartAt π(β, β) (f c z)).source
m : e β (extChartAt π(β, β) c).source β§ w β (extChartAt π(β, β) z).source
β’ ((fun p => (p.1, β(extChartAt π(β, β) (f c z)).symm p.2)) β fun x =>
(x.1, β(extChartAt π(β, β) (f c z)) (f x.1 (β(extChartAt π(β, β) z).symm (β(extChartAt π(β, β) z) x.2)))))
(e, w) =
(fun p => (p.1, f p.1 p.2)) (e, w)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | rw [β hg] | 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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
β’ AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) 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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
β’ AnalyticAt β (uncurry fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x)))
(c, β(extChartAt π(β, β) z) 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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
β’ AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | exact (holomorphicAt_iff.mp fa).2 | 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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
β’ AnalyticAt β (uncurry fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x)))
(c, β(extChartAt π(β, β) z) z) | 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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
β’ AnalyticAt β (uncurry fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x)))
(c, β(extChartAt π(β, β) z) z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | rw [β hg] | 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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
β’ NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) 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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
β’ NontrivialHolomorphicAt ((fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) c)
(β(extChartAt π(β, β) z) 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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
β’ NontrivialHolomorphicAt (g c) (β(extChartAt π(β, β) z) z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | NontrivialHolomorphicAt.nhds_eq_map_nhds_param | [234, 1] | [258, 85] | exact n.inCharts | 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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
β’ NontrivialHolomorphicAt ((fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) c)
(β(extChartAt π(β, β) z) z) | 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 S β T
c : β
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (π(β, β).prod π(β, β)) π(β, β) (uncurry f) (c, z)
g : β β β β β
hg : (fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) = g
ga : AnalyticAt β (uncurry g) (c, β(extChartAt π(β, β) z) z)
β’ NontrivialHolomorphicAt ((fun e x => β(extChartAt π(β, β) (f c z)) (f e (β(extChartAt π(β, β) z).symm x))) c)
(β(extChartAt π(β, β) z) z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | set fl := fun n z β¦ log (f n z) | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0 | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | have near1 : β n z, z β s β abs (f n z - 1) β€ 1 / 2 := by
intro n z zs
calc abs (f n z - 1)
_ β€ c * a ^ n := hf n z zs
_ β€ (1 / 2 : β) * (1:β) ^ n := by bound
_ = 1 / 2 := by norm_num | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0 | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | have near1' : β n z, z β s β abs (f n z - 1) < 1 := fun n z zs β¦
lt_of_le_of_lt (near1 n z zs) (by linarith) | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0 | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | have expfl : β n z, z β s β exp (fl n z) = f n z := by
intro n z zs; refine Complex.exp_log ?_
exact near_one_avoids_zero (near1' n z zs) | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0 | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | have hl : β n, AnalyticOn β (fl n) s := fun n β¦
(h n).log (fun z m β¦ mem_slitPlane_of_near_one (near1' n z m)) | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0 | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | set c2 := 2 * c | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0 | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | have hfl : β n z, z β s β abs (fl n z) β€ c2 * a ^ n := by
intro n z zs
calc abs (fl n z)
_ = abs (log (f n z)) := rfl
_ β€ 2 * abs (f n z - 1) := (log_small (near1 n z zs))
_ β€ 2 * (c * a ^ n) := by linarith [hf n z zs]
_ = 2 * c * a ^ n := by ring
_ = c2 * a ^ n := rfl | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0 | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | rcases fast_series_converge o a0 a1 hl hfl with β¨gl, gla, usβ© | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0 | case intro.intro
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | generalize hg : (fun z β¦ exp (gl z)) = g | case intro.intro
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0 | case intro.intro
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
g : β β β
hg : (fun z => (gl z).exp) = g
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | use g | case intro.intro
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
g : β β β
hg : (fun z => (gl z).exp) = g
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0 | case h
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
g : β β β
hg : (fun z => (gl z).exp) = g
β’ HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
g : β β β
hg : (fun z => (gl z).exp) = g
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | refine β¨?_, ?_, ?_β© | case h
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
g : β β β
hg : (fun z => (gl z).exp) = g
β’ HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0 | case h.refine_1
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
g : β β β
hg : (fun z => (gl z).exp) = g
β’ HasProdOn f g s
case h.refine_2
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
g : β β β
hg : (fun z => (gl z).exp) = g
β’ AnalyticOn β g s
case h.refine_3
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
g : β β β
hg : (fun z => (gl z).exp) = g
β’ β z β s, g z β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
g : β β β
hg : (fun z => (gl z).exp) = g
β’ HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | intro n z zs | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
β’ β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2 | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
n : β
z : β
zs : z β s
β’ Complex.abs (f n z - 1) β€ 1 / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
β’ β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | calc abs (f n z - 1)
_ β€ c * a ^ n := hf n z zs
_ β€ (1 / 2 : β) * (1:β) ^ n := by bound
_ = 1 / 2 := by norm_num | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
n : β
z : β
zs : z β s
β’ Complex.abs (f n z - 1) β€ 1 / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
n : β
z : β
zs : z β s
β’ Complex.abs (f n z - 1) β€ 1 / 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | bound | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
n : β
z : β
zs : z β s
β’ c * a ^ n β€ 1 / 2 * 1 ^ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
n : β
z : β
zs : z β s
β’ c * a ^ n β€ 1 / 2 * 1 ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | norm_num | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
n : β
z : β
zs : z β s
β’ 1 / 2 * 1 ^ n = 1 / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
n : β
z : β
zs : z β s
β’ 1 / 2 * 1 ^ n = 1 / 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | linarith | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
n : β
z : β
zs : z β s
β’ 1 / 2 < 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
n : β
z : β
zs : z β s
β’ 1 / 2 < 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | intro n z zs | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
β’ β (n : β), β z β s, (fl n z).exp = f n z | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
n : β
z : β
zs : z β s
β’ (fl n z).exp = f n z | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
β’ β (n : β), β z β s, (fl n z).exp = f n z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | refine Complex.exp_log ?_ | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
n : β
z : β
zs : z β s
β’ (fl n z).exp = f n z | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
n : β
z : β
zs : z β s
β’ f n z β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
n : β
z : β
zs : z β s
β’ (fl n z).exp = f n z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | exact near_one_avoids_zero (near1' n z zs) | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
n : β
z : β
zs : z β s
β’ f n z β 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
n : β
z : β
zs : z β s
β’ f n z β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | intro n z zs | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
β’ β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
n : β
z : β
zs : z β s
β’ Complex.abs (fl n z) β€ c2 * a ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
β’ β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | calc abs (fl n z)
_ = abs (log (f n z)) := rfl
_ β€ 2 * abs (f n z - 1) := (log_small (near1 n z zs))
_ β€ 2 * (c * a ^ n) := by linarith [hf n z zs]
_ = 2 * c * a ^ n := by ring
_ = c2 * a ^ n := rfl | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
n : β
z : β
zs : z β s
β’ Complex.abs (fl n z) β€ c2 * a ^ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
n : β
z : β
zs : z β s
β’ Complex.abs (fl n z) β€ c2 * a ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | linarith [hf n z zs] | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
n : β
z : β
zs : z β s
β’ 2 * Complex.abs (f n z - 1) β€ 2 * (c * a ^ n) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
n : β
z : β
zs : z β s
β’ 2 * Complex.abs (f n z - 1) β€ 2 * (c * a ^ n)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | ring | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
n : β
z : β
zs : z β s
β’ 2 * (c * a ^ n) = 2 * c * a ^ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
n : β
z : β
zs : z β s
β’ 2 * (c * a ^ n) = 2 * c * a ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | intro z zs | case h.refine_1
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
g : β β β
hg : (fun z => (gl z).exp) = g
β’ HasProdOn f g s | case h.refine_1
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
β’ HasProd (fun n => f n z) (g z) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_1
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
g : β β β
hg : (fun z => (gl z).exp) = g
β’ HasProdOn f g s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | specialize us z zs | case h.refine_1
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
β’ HasProd (fun n => f n z) (g z) | case h.refine_1
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => (fun n => fl n) n z) (gl z)
β’ HasProd (fun n => f n z) (g z) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_1
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
β’ HasProd (fun n => f n z) (g z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | simp at us | case h.refine_1
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => (fun n => fl n) n z) (gl z)
β’ HasProd (fun n => f n z) (g z) | case h.refine_1
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
β’ HasProd (fun n => f n z) (g z) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_1
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => (fun n => fl n) n z) (gl z)
β’ HasProd (fun n => f n z) (g z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | have comp :
Filter.Tendsto (exp β fun N : Finset β β¦ N.sum fun n β¦ fl n z) atTop (π (exp (gl z))) :=
Filter.Tendsto.comp (Continuous.tendsto Complex.continuous_exp _) us | case h.refine_1
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
β’ HasProd (fun n => f n z) (g z) | case h.refine_1
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp β fun N => N.sum fun n => fl n z) atTop (π (gl z).exp)
β’ HasProd (fun n => f n z) (g z) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_1
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
β’ HasProd (fun n => f n z) (g z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | have expsum0 : (exp β fun N : Finset β β¦ N.sum fun n β¦ fl n z) = fun N : Finset β β¦
N.prod fun n β¦ f n z := by
apply funext; intro N; simp; rw [Complex.exp_sum]; simp_rw [expfl _ z zs] | case h.refine_1
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp β fun N => N.sum fun n => fl n z) atTop (π (gl z).exp)
β’ HasProd (fun n => f n z) (g z) | case h.refine_1
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp β fun N => N.sum fun n => fl n z) atTop (π (gl z).exp)
expsum0 : (exp β fun N => N.sum fun n => fl n z) = fun N => N.prod fun n => f n z
β’ HasProd (fun n => f n z) (g z) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_1
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp β fun N => N.sum fun n => fl n z) atTop (π (gl z).exp)
β’ HasProd (fun n => f n z) (g z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | rw [expsum0] at comp | case h.refine_1
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp β fun N => N.sum fun n => fl n z) atTop (π (gl z).exp)
expsum0 : (exp β fun N => N.sum fun n => fl n z) = fun N => N.prod fun n => f n z
β’ HasProd (fun n => f n z) (g z) | case h.refine_1
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (fun N => N.prod fun n => f n z) atTop (π (gl z).exp)
expsum0 : (exp β fun N => N.sum fun n => fl n z) = fun N => N.prod fun n => f n z
β’ HasProd (fun n => f n z) (g z) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_1
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp β fun N => N.sum fun n => fl n z) atTop (π (gl z).exp)
expsum0 : (exp β fun N => N.sum fun n => fl n z) = fun N => N.prod fun n => f n z
β’ HasProd (fun n => f n z) (g z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | rw [β hg] | case h.refine_1
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (fun N => N.prod fun n => f n z) atTop (π (gl z).exp)
expsum0 : (exp β fun N => N.sum fun n => fl n z) = fun N => N.prod fun n => f n z
β’ HasProd (fun n => f n z) (g z) | case h.refine_1
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (fun N => N.prod fun n => f n z) atTop (π (gl z).exp)
expsum0 : (exp β fun N => N.sum fun n => fl n z) = fun N => N.prod fun n => f n z
β’ HasProd (fun n => f n z) ((fun z => (gl z).exp) z) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_1
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (fun N => N.prod fun n => f n z) atTop (π (gl z).exp)
expsum0 : (exp β fun N => N.sum fun n => fl n z) = fun N => N.prod fun n => f n z
β’ HasProd (fun n => f n z) (g z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | assumption | case h.refine_1
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (fun N => N.prod fun n => f n z) atTop (π (gl z).exp)
expsum0 : (exp β fun N => N.sum fun n => fl n z) = fun N => N.prod fun n => f n z
β’ HasProd (fun n => f n z) ((fun z => (gl z).exp) z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_1
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (fun N => N.prod fun n => f n z) atTop (π (gl z).exp)
expsum0 : (exp β fun N => N.sum fun n => fl n z) = fun N => N.prod fun n => f n z
β’ HasProd (fun n => f n z) ((fun z => (gl z).exp) z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | apply funext | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp β fun N => N.sum fun n => fl n z) atTop (π (gl z).exp)
β’ (exp β fun N => N.sum fun n => fl n z) = fun N => N.prod fun n => f n z | case h
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp β fun N => N.sum fun n => fl n z) atTop (π (gl z).exp)
β’ β (x : Finset β), (exp β fun N => N.sum fun n => fl n z) x = x.prod fun n => f n z | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp β fun N => N.sum fun n => fl n z) atTop (π (gl z).exp)
β’ (exp β fun N => N.sum fun n => fl n z) = fun N => N.prod fun n => f n z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | intro N | case h
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp β fun N => N.sum fun n => fl n z) atTop (π (gl z).exp)
β’ β (x : Finset β), (exp β fun N => N.sum fun n => fl n z) x = x.prod fun n => f n z | case h
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp β fun N => N.sum fun n => fl n z) atTop (π (gl z).exp)
N : Finset β
β’ (exp β fun N => N.sum fun n => fl n z) N = N.prod fun n => f n z | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp β fun N => N.sum fun n => fl n z) atTop (π (gl z).exp)
β’ β (x : Finset β), (exp β fun N => N.sum fun n => fl n z) x = x.prod fun n => f n z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | simp | case h
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp β fun N => N.sum fun n => fl n z) atTop (π (gl z).exp)
N : Finset β
β’ (exp β fun N => N.sum fun n => fl n z) N = N.prod fun n => f n z | case h
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp β fun N => N.sum fun n => fl n z) atTop (π (gl z).exp)
N : Finset β
β’ (N.sum fun n => fl n z).exp = N.prod fun n => f n z | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp β fun N => N.sum fun n => fl n z) atTop (π (gl z).exp)
N : Finset β
β’ (exp β fun N => N.sum fun n => fl n z) N = N.prod fun n => f n z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | rw [Complex.exp_sum] | case h
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp β fun N => N.sum fun n => fl n z) atTop (π (gl z).exp)
N : Finset β
β’ (N.sum fun n => fl n z).exp = N.prod fun n => f n z | case h
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp β fun N => N.sum fun n => fl n z) atTop (π (gl z).exp)
N : Finset β
β’ (N.prod fun x => (fl x z).exp) = N.prod fun n => f n z | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp β fun N => N.sum fun n => fl n z) atTop (π (gl z).exp)
N : Finset β
β’ (N.sum fun n => fl n z).exp = N.prod fun n => f n z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | simp_rw [expfl _ z zs] | case h
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp β fun N => N.sum fun n => fl n z) atTop (π (gl z).exp)
N : Finset β
β’ (N.prod fun x => (fl x z).exp) = N.prod fun n => f n z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp β fun N => N.sum fun n => fl n z) atTop (π (gl z).exp)
N : Finset β
β’ (N.prod fun x => (fl x z).exp) = N.prod fun n => f n z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | rw [β hg] | case h.refine_2
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
g : β β β
hg : (fun z => (gl z).exp) = g
β’ AnalyticOn β g s | case h.refine_2
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
g : β β β
hg : (fun z => (gl z).exp) = g
β’ AnalyticOn β (fun z => (gl z).exp) s | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_2
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
g : β β β
hg : (fun z => (gl z).exp) = g
β’ AnalyticOn β g s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | exact fun z zs β¦ AnalyticAt.exp.comp (gla z zs) | case h.refine_2
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
g : β β β
hg : (fun z => (gl z).exp) = g
β’ AnalyticOn β (fun z => (gl z).exp) s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_2
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
g : β β β
hg : (fun z => (gl z).exp) = g
β’ AnalyticOn β (fun z => (gl z).exp) s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | simp only [Complex.exp_ne_zero, Ne, not_false_iff, imp_true_iff, β hg] | case h.refine_3
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
g : β β β
hg : (fun z => (gl z).exp) = g
β’ β z β s, g z β 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_3
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
g : β β β
hg : (fun z => (gl z).exp) = g
β’ β z β s, g z β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge' | [101, 1] | [109, 55] | rcases fast_products_converge o c12 a0 a1 h hf with β¨g, gp, ga, g0β© | f : β β β β β
s : Set β
c a : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
β’ ProdExistsOn f s β§ AnalyticOn β (tprodOn f) s β§ β z β s, tprodOn f z β 0 | case intro.intro.intro
f : β β β β β
s : Set β
c a : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
g : β β β
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn β g s
g0 : β z β s, g z β 0
β’ ProdExistsOn f s β§ AnalyticOn β (tprodOn f) s β§ β z β s, tprodOn f z β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
c a : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
β’ ProdExistsOn f s β§ AnalyticOn β (tprodOn f) s β§ β z β s, tprodOn f z β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge' | [101, 1] | [109, 55] | refine β¨?_, ?_, ?_β© | case intro.intro.intro
f : β β β β β
s : Set β
c a : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
g : β β β
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn β g s
g0 : β z β s, g z β 0
β’ ProdExistsOn f s β§ AnalyticOn β (tprodOn f) s β§ β z β s, tprodOn f z β 0 | case intro.intro.intro.refine_1
f : β β β β β
s : Set β
c a : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
g : β β β
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn β g s
g0 : β z β s, g z β 0
β’ ProdExistsOn f s
case intro.intro.intro.refine_2
f : β β β β β
s : Set β
c a : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
g : β β β
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn β g s
g0 : β z β s, g z β 0
β’ AnalyticOn β (tprodOn f) s
case intro.intro.intro.refine_3
f : β β β β β
s : Set β
c a : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
g : β β β
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn β g s
g0 : β z β s, g z β 0
β’ β z β s, tprodOn f z β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
f : β β β β β
s : Set β
c a : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
g : β β β
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn β g s
g0 : β z β s, g z β 0
β’ ProdExistsOn f s β§ AnalyticOn β (tprodOn f) s β§ β z β s, tprodOn f z β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge' | [101, 1] | [109, 55] | exact fun z zs β¦ β¨g z, gp z zsβ© | case intro.intro.intro.refine_1
f : β β β β β
s : Set β
c a : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
g : β β β
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn β g s
g0 : β z β s, g z β 0
β’ ProdExistsOn f s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_1
f : β β β β β
s : Set β
c a : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
g : β β β
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn β g s
g0 : β z β s, g z β 0
β’ ProdExistsOn f s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge' | [101, 1] | [109, 55] | rwa [β analyticOn_congr o fun z zs β¦ (gp.tprodOn_eq z zs).symm] | case intro.intro.intro.refine_2
f : β β β β β
s : Set β
c a : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
g : β β β
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn β g s
g0 : β z β s, g z β 0
β’ AnalyticOn β (tprodOn f) s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_2
f : β β β β β
s : Set β
c a : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
g : β β β
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn β g s
g0 : β z β s, g z β 0
β’ AnalyticOn β (tprodOn f) s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge' | [101, 1] | [109, 55] | intro z zs | case intro.intro.intro.refine_3
f : β β β β β
s : Set β
c a : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
g : β β β
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn β g s
g0 : β z β s, g z β 0
β’ β z β s, tprodOn f z β 0 | case intro.intro.intro.refine_3
f : β β β β β
s : Set β
c a : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
g : β β β
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn β g s
g0 : β z β s, g z β 0
z : β
zs : z β s
β’ tprodOn f z β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_3
f : β β β β β
s : Set β
c a : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
g : β β β
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn β g s
g0 : β z β s, g z β 0
β’ β z β s, tprodOn f z β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge' | [101, 1] | [109, 55] | rw [gp.tprodOn_eq z zs] | case intro.intro.intro.refine_3
f : β β β β β
s : Set β
c a : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
g : β β β
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn β g s
g0 : β z β s, g z β 0
z : β
zs : z β s
β’ tprodOn f z β 0 | case intro.intro.intro.refine_3
f : β β β β β
s : Set β
c a : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
g : β β β
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn β g s
g0 : β z β s, g z β 0
z : β
zs : z β s
β’ g z β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_3
f : β β β β β
s : Set β
c a : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
g : β β β
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn β g s
g0 : β z β s, g z β 0
z : β
zs : z β s
β’ tprodOn f z β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge' | [101, 1] | [109, 55] | exact g0 z zs | case intro.intro.intro.refine_3
f : β β β β β
s : Set β
c a : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
g : β β β
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn β g s
g0 : β z β s, g z β 0
z : β
zs : z β s
β’ g z β 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_3
f : β β β β β
s : Set β
c a : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
g : β β β
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn β g s
g0 : β z β s, g z β 0
z : β
zs : z β s
β’ g z β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_pow | [112, 1] | [115, 72] | rw [HasProd] | f : β β β
g : β
p : β
h : HasProd f g
β’ HasProd (fun n => f n ^ p) (g ^ p) | f : β β β
g : β
p : β
h : HasProd f g
β’ Filter.Tendsto (fun s => s.prod fun b => f b ^ p) atTop (π (g ^ p)) | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
g : β
p : β
h : HasProd f g
β’ HasProd (fun n => f n ^ p) (g ^ p)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_pow | [112, 1] | [115, 72] | simp_rw [Finset.prod_pow] | f : β β β
g : β
p : β
h : HasProd f g
β’ Filter.Tendsto (fun s => s.prod fun b => f b ^ p) atTop (π (g ^ p)) | f : β β β
g : β
p : β
h : HasProd f g
β’ Filter.Tendsto (fun s => (s.prod fun x => f x) ^ p) atTop (π (g ^ p)) | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
g : β
p : β
h : HasProd f g
β’ Filter.Tendsto (fun s => s.prod fun b => f b ^ p) atTop (π (g ^ p))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_pow | [112, 1] | [115, 72] | exact Filter.Tendsto.comp (Continuous.tendsto (continuous_pow p) g) h | f : β β β
g : β
p : β
h : HasProd f g
β’ Filter.Tendsto (fun s => (s.prod fun x => f x) ^ p) atTop (π (g ^ p)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
g : β
p : β
h : HasProd f g
β’ Filter.Tendsto (fun s => (s.prod fun x => f x) ^ p) atTop (π (g ^ p))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_pow' | [118, 1] | [120, 96] | rcases h with β¨g, hβ© | f : β β β
p : β
h : ProdExists f
β’ tprod f ^ p = β' (n : β), f n ^ p | case intro
f : β β β
p : β
g : β
h : HasProd f g
β’ tprod f ^ p = β' (n : β), f n ^ p | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
p : β
h : ProdExists f
β’ tprod f ^ p = β' (n : β), f n ^ p
TACTIC:
|
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