url stringclasses 147 values | commit stringclasses 147 values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | exact differentiableAt_id.add (differentiableAt_const _) | case hf
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ : S
d n : β
s : Super f d a
c : β
am : a β (extChartAt I a).source
ezmβ : βαΆ (p : β Γ S) in π (c, a), f p.1 p.2 β (extChartAt I a).source
e : β
z : S
ezm : f e z β (extChartAt I a).source
zm : z β (extChartAt I a).source
g : β β β
hg : (fun w => β(extChartAt I (f c a)) (f e (β(extChartAt I a).symm w))) = g
dg : DifferentiableAt β g (β(extChartAt I a) z)
d0 : β (z : β), DifferentiableAt β (fun z => z - β(extChartAt I a) a) z
β’ DifferentiableAt β (fun z => z + β(extChartAt I a) a) (β(extChartAt I a) z - β(extChartAt I a) a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ : S
d n : β
s : Super f d a
c : β
am : a β (extChartAt I a).source
ezmβ : βαΆ (p : β Γ S) in π (c, a), f p.1 p.2 β (extChartAt I a).source
e : β
z : S
ezm : f e z β (extChartAt I a).source
zm : z β (extChartAt I a).source
g : β β β
hg : (fun w => β(extChartAt I (f c a)) (f e (β(extChartAt I a).symm w))) = g
dg : DifferentiableAt β g (β(extChartAt I a) z)
d0 : β (z : β), DifferentiableAt β (fun z => z - β(extChartAt I a) a) z
β’ DifferentiableAt β (fun z => z + β(extChartAt I a) a) (β(extChartAt I a) z - β(extChartAt I a) a)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | simp only [deriv_add_const, deriv_sub_const, deriv_id'', mul_one, sub_add_cancel, Function.comp] | case hp
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ : S
d n : β
s : Super f d a
c : β
am : a β (extChartAt I a).source
ezmβ : βαΆ (p : β Γ S) in π (c, a), f p.1 p.2 β (extChartAt I a).source
e : β
z : S
ezm : f e z β (extChartAt I a).source
zm : z β (extChartAt I a).source
g : β β β
hg : (fun w => β(extChartAt I (f c a)) (f e (β(extChartAt I a).symm w))) = g
dg : DifferentiableAt β g (β(extChartAt I a) z)
d0 : β (z : β), DifferentiableAt β (fun z => z - β(extChartAt I a) a) z
d1 : DifferentiableAt β (g β fun z => z + β(extChartAt I a) a) (β(extChartAt I a) z - β(extChartAt I a) a)
β’ deriv g (β(extChartAt I a) z - β(extChartAt I a) a + β(extChartAt I a) a) *
deriv (fun z => z + β(extChartAt I a) a) (β(extChartAt I a) z - β(extChartAt I a) a) =
0 β
deriv g (β(extChartAt I a) z) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hp
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ : S
d n : β
s : Super f d a
c : β
am : a β (extChartAt I a).source
ezmβ : βαΆ (p : β Γ S) in π (c, a), f p.1 p.2 β (extChartAt I a).source
e : β
z : S
ezm : f e z β (extChartAt I a).source
zm : z β (extChartAt I a).source
g : β β β
hg : (fun w => β(extChartAt I (f c a)) (f e (β(extChartAt I a).symm w))) = g
dg : DifferentiableAt β g (β(extChartAt I a) z)
d0 : β (z : β), DifferentiableAt β (fun z => z - β(extChartAt I a) a) z
d1 : DifferentiableAt β (g β fun z => z + β(extChartAt I a) a) (β(extChartAt I a) z - β(extChartAt I a) a)
β’ deriv g (β(extChartAt I a) z - β(extChartAt I a) a + β(extChartAt I a) a) *
deriv (fun z => z + β(extChartAt I a) a) (β(extChartAt I a) z - β(extChartAt I a) a) =
0 β
deriv g (β(extChartAt I a) z) = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | simp only [sub_add_cancel, dg] | S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ : S
d n : β
s : Super f d a
c : β
am : a β (extChartAt I a).source
ezmβ : βαΆ (p : β Γ S) in π (c, a), f p.1 p.2 β (extChartAt I a).source
e : β
z : S
ezm : f e z β (extChartAt I a).source
zm : z β (extChartAt I a).source
g : β β β
hg : (fun w => β(extChartAt I (f c a)) (f e (β(extChartAt I a).symm w))) = g
dg : DifferentiableAt β g (β(extChartAt I a) z)
d0 : β (z : β), DifferentiableAt β (fun z => z - β(extChartAt I a) a) z
d1 : DifferentiableAt β (g β fun z => z + β(extChartAt I a) a) (β(extChartAt I a) z - β(extChartAt I a) a)
β’ DifferentiableAt β g (β(extChartAt I a) z - β(extChartAt I a) a + β(extChartAt I a) a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ : S
d n : β
s : Super f d a
c : β
am : a β (extChartAt I a).source
ezmβ : βαΆ (p : β Γ S) in π (c, a), f p.1 p.2 β (extChartAt I a).source
e : β
z : S
ezm : f e z β (extChartAt I a).source
zm : z β (extChartAt I a).source
g : β β β
hg : (fun w => β(extChartAt I (f c a)) (f e (β(extChartAt I a).symm w))) = g
dg : DifferentiableAt β g (β(extChartAt I a) z)
d0 : β (z : β), DifferentiableAt β (fun z => z - β(extChartAt I a) a) z
d1 : DifferentiableAt β (g β fun z => z + β(extChartAt I a) a) (β(extChartAt I a) z - β(extChartAt I a) a)
β’ DifferentiableAt β g (β(extChartAt I a) z - β(extChartAt I a) a + β(extChartAt I a) a)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | exact differentiableAt_id.add (differentiableAt_const _) | S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ : S
d n : β
s : Super f d a
c : β
am : a β (extChartAt I a).source
ezmβ : βαΆ (p : β Γ S) in π (c, a), f p.1 p.2 β (extChartAt I a).source
e : β
z : S
ezm : f e z β (extChartAt I a).source
zm : z β (extChartAt I a).source
g : β β β
hg : (fun w => β(extChartAt I (f c a)) (f e (β(extChartAt I a).symm w))) = g
dg : DifferentiableAt β g (β(extChartAt I a) z)
d0 : β (z : β), DifferentiableAt β (fun z => z - β(extChartAt I a) a) z
d1 : DifferentiableAt β (g β fun z => z + β(extChartAt I a) a) (β(extChartAt I a) z - β(extChartAt I a) a)
β’ DifferentiableAt β (fun z => z + β(extChartAt I a) a) (β(extChartAt I a) z - β(extChartAt I a) a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ : S
d n : β
s : Super f d a
c : β
am : a β (extChartAt I a).source
ezmβ : βαΆ (p : β Γ S) in π (c, a), f p.1 p.2 β (extChartAt I a).source
e : β
z : S
ezm : f e z β (extChartAt I a).source
zm : z β (extChartAt I a).source
g : β β β
hg : (fun w => β(extChartAt I (f c a)) (f e (β(extChartAt I a).symm w))) = g
dg : DifferentiableAt β g (β(extChartAt I a) z)
d0 : β (z : β), DifferentiableAt β (fun z => z - β(extChartAt I a) a) z
d1 : DifferentiableAt β (g β fun z => z + β(extChartAt I a) a) (β(extChartAt I a) z - β(extChartAt I a) a)
β’ DifferentiableAt β (fun z => z + β(extChartAt I a) a) (β(extChartAt I a) z - β(extChartAt I a) a)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.isClosed_critical_not_a | [592, 1] | [600, 91] | rw [β isOpen_compl_iff] | S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d n : β
s : Super f d a
β’ IsClosed {p | Critical (f p.1) p.2 β§ p.2 β a} | S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d n : β
s : Super f d a
β’ IsOpen {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d n : β
s : Super f d a
β’ IsClosed {p | Critical (f p.1) p.2 β§ p.2 β a}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.isClosed_critical_not_a | [592, 1] | [600, 91] | rw [isOpen_iff_eventually] | S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d n : β
s : Super f d a
β’ IsOpen {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ | S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d n : β
s : Super f d a
β’ β x β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ, βαΆ (y : β Γ S) in π x, y β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d n : β
s : Super f d a
β’ IsOpen {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.isClosed_critical_not_a | [592, 1] | [600, 91] | intro β¨c, zβ© m | S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d n : β
s : Super f d a
β’ β x β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ, βαΆ (y : β Γ S) in π x, y β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ | S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ : S
d n : β
s : Super f d a
c : β
z : S
m : (c, z) β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ
β’ βαΆ (y : β Γ S) in π (c, z), y β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d n : β
s : Super f d a
β’ β x β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ, βαΆ (y : β Γ S) in π x, y β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.isClosed_critical_not_a | [592, 1] | [600, 91] | by_cases za : z = a | S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ : S
d n : β
s : Super f d a
c : β
z : S
m : (c, z) β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ
β’ βαΆ (y : β Γ S) in π (c, z), y β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ | case pos
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ : S
d n : β
s : Super f d a
c : β
z : S
m : (c, z) β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ
za : z = a
β’ βαΆ (y : β Γ S) in π (c, z), y β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ
case neg
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ : S
d n : β
s : Super f d a
c : β
z : S
m : (c, z) β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ
za : Β¬z = a
β’ βαΆ (y : β Γ S) in π (c, z), y β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ : S
d n : β
s : Super f d a
c : β
z : S
m : (c, z) β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ
β’ βαΆ (y : β Γ S) in π (c, z), y β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.isClosed_critical_not_a | [592, 1] | [600, 91] | rw [za] | case pos
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ : S
d n : β
s : Super f d a
c : β
z : S
m : (c, z) β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ
za : z = a
β’ βαΆ (y : β Γ S) in π (c, z), y β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ | case pos
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ : S
d n : β
s : Super f d a
c : β
z : S
m : (c, z) β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ
za : z = a
β’ βαΆ (y : β Γ S) in π (c, a), y β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ : S
d n : β
s : Super f d a
c : β
z : S
m : (c, z) β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ
za : z = a
β’ βαΆ (y : β Γ S) in π (c, z), y β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.isClosed_critical_not_a | [592, 1] | [600, 91] | refine (s.f_noncritical_near_a c).mp (eventually_of_forall ?_) | case pos
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ : S
d n : β
s : Super f d a
c : β
z : S
m : (c, z) β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ
za : z = a
β’ βαΆ (y : β Γ S) in π (c, a), y β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ | case pos
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ : S
d n : β
s : Super f d a
c : β
z : S
m : (c, z) β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ
za : z = a
β’ β (x : β Γ S), (Critical (f x.1) x.2 β x.2 = a) β x β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ : S
d n : β
s : Super f d a
c : β
z : S
m : (c, z) β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ
za : z = a
β’ βαΆ (y : β Γ S) in π (c, a), y β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.isClosed_critical_not_a | [592, 1] | [600, 91] | intro β¨e, wβ© h | case pos
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ : S
d n : β
s : Super f d a
c : β
z : S
m : (c, z) β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ
za : z = a
β’ β (x : β Γ S), (Critical (f x.1) x.2 β x.2 = a) β x β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ | case pos
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ : S
d n : β
s : Super f d a
c : β
z : S
m : (c, z) β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ
za : z = a
e : β
w : S
h : Critical (f (e, w).1) (e, w).2 β (e, w).2 = a
β’ (e, w) β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ : S
d n : β
s : Super f d a
c : β
z : S
m : (c, z) β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ
za : z = a
β’ β (x : β Γ S), (Critical (f x.1) x.2 β x.2 = a) β x β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.isClosed_critical_not_a | [592, 1] | [600, 91] | simp only [mem_compl_iff, mem_setOf, not_and, not_not] at h β’ | case pos
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ : S
d n : β
s : Super f d a
c : β
z : S
m : (c, z) β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ
za : z = a
e : β
w : S
h : Critical (f (e, w).1) (e, w).2 β (e, w).2 = a
β’ (e, w) β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ | case pos
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ : S
d n : β
s : Super f d a
c : β
z : S
m : (c, z) β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ
za : z = a
e : β
w : S
h : Critical (f e) w β w = a
β’ Critical (f e) w β w = a | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ : S
d n : β
s : Super f d a
c : β
z : S
m : (c, z) β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ
za : z = a
e : β
w : S
h : Critical (f (e, w).1) (e, w).2 β (e, w).2 = a
β’ (e, w) β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.isClosed_critical_not_a | [592, 1] | [600, 91] | exact h.1 | case pos
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ : S
d n : β
s : Super f d a
c : β
z : S
m : (c, z) β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ
za : z = a
e : β
w : S
h : Critical (f e) w β w = a
β’ Critical (f e) w β w = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ : S
d n : β
s : Super f d a
c : β
z : S
m : (c, z) β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ
za : z = a
e : β
w : S
h : Critical (f e) w β w = a
β’ Critical (f e) w β w = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.isClosed_critical_not_a | [592, 1] | [600, 91] | have o := isOpen_iff_eventually.mp (isOpen_noncritical s.fa) | case neg
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ : S
d n : β
s : Super f d a
c : β
z : S
m : (c, z) β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ
za : Β¬z = a
β’ βαΆ (y : β Γ S) in π (c, z), y β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ | case neg
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ : S
d n : β
s : Super f d a
c : β
z : S
m : (c, z) β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ
za : Β¬z = a
o : β x β {p | Β¬Critical (f p.1) p.2}, βαΆ (y : β Γ S) in π x, y β {p | Β¬Critical (f p.1) p.2}
β’ βαΆ (y : β Γ S) in π (c, z), y β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ : S
d n : β
s : Super f d a
c : β
z : S
m : (c, z) β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ
za : Β¬z = a
β’ βαΆ (y : β Γ S) in π (c, z), y β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.isClosed_critical_not_a | [592, 1] | [600, 91] | simp only [za, mem_compl_iff, mem_setOf, not_and, not_not, imp_false] at m o β’ | case neg
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ : S
d n : β
s : Super f d a
c : β
z : S
m : (c, z) β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ
za : Β¬z = a
o : β x β {p | Β¬Critical (f p.1) p.2}, βαΆ (y : β Γ S) in π x, y β {p | Β¬Critical (f p.1) p.2}
β’ βαΆ (y : β Γ S) in π (c, z), y β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ | case neg
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ : S
d n : β
s : Super f d a
c : β
z : S
za : Β¬z = a
m : Β¬Critical (f c) z
o : β (x : β Γ S), Β¬Critical (f x.1) x.2 β βαΆ (y : β Γ S) in π x, Β¬Critical (f y.1) y.2
β’ βαΆ (y : β Γ S) in π (c, z), Critical (f y.1) y.2 β y.2 = a | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ : S
d n : β
s : Super f d a
c : β
z : S
m : (c, z) β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ
za : Β¬z = a
o : β x β {p | Β¬Critical (f p.1) p.2}, βαΆ (y : β Γ S) in π x, y β {p | Β¬Critical (f p.1) p.2}
β’ βαΆ (y : β Γ S) in π (c, z), y β {p | Critical (f p.1) p.2 β§ p.2 β a}αΆ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.isClosed_critical_not_a | [592, 1] | [600, 91] | refine (o (c, z) m).mp (eventually_of_forall ?_) | case neg
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ : S
d n : β
s : Super f d a
c : β
z : S
za : Β¬z = a
m : Β¬Critical (f c) z
o : β (x : β Γ S), Β¬Critical (f x.1) x.2 β βαΆ (y : β Γ S) in π x, Β¬Critical (f y.1) y.2
β’ βαΆ (y : β Γ S) in π (c, z), Critical (f y.1) y.2 β y.2 = a | case neg
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ : S
d n : β
s : Super f d a
c : β
z : S
za : Β¬z = a
m : Β¬Critical (f c) z
o : β (x : β Γ S), Β¬Critical (f x.1) x.2 β βαΆ (y : β Γ S) in π x, Β¬Critical (f y.1) y.2
β’ β (x : β Γ S), Β¬Critical (f x.1) x.2 β Critical (f x.1) x.2 β x.2 = a | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ : S
d n : β
s : Super f d a
c : β
z : S
za : Β¬z = a
m : Β¬Critical (f c) z
o : β (x : β Γ S), Β¬Critical (f x.1) x.2 β βαΆ (y : β Γ S) in π x, Β¬Critical (f y.1) y.2
β’ βαΆ (y : β Γ S) in π (c, z), Critical (f y.1) y.2 β y.2 = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.isClosed_critical_not_a | [592, 1] | [600, 91] | intro β¨e, wβ© a b | case neg
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ : S
d n : β
s : Super f d a
c : β
z : S
za : Β¬z = a
m : Β¬Critical (f c) z
o : β (x : β Γ S), Β¬Critical (f x.1) x.2 β βαΆ (y : β Γ S) in π x, Β¬Critical (f y.1) y.2
β’ β (x : β Γ S), Β¬Critical (f x.1) x.2 β Critical (f x.1) x.2 β x.2 = a | case neg
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
aβ zβ : S
d n : β
s : Super f d aβ
c : β
z : S
za : Β¬z = aβ
m : Β¬Critical (f c) z
o : β (x : β Γ S), Β¬Critical (f x.1) x.2 β βαΆ (y : β Γ S) in π x, Β¬Critical (f y.1) y.2
e : β
w : S
a : Β¬Critical (f (e, w).1) (e, w).2
b : Critical (f (e, w).1) (e, w).2
β’ (e, w).2 = aβ | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ : S
d n : β
s : Super f d a
c : β
z : S
za : Β¬z = a
m : Β¬Critical (f c) z
o : β (x : β Γ S), Β¬Critical (f x.1) x.2 β βαΆ (y : β Γ S) in π x, Β¬Critical (f y.1) y.2
β’ β (x : β Γ S), Β¬Critical (f x.1) x.2 β Critical (f x.1) x.2 β x.2 = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.isClosed_critical_not_a | [592, 1] | [600, 91] | exfalso | case neg
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
aβ zβ : S
d n : β
s : Super f d aβ
c : β
z : S
za : Β¬z = aβ
m : Β¬Critical (f c) z
o : β (x : β Γ S), Β¬Critical (f x.1) x.2 β βαΆ (y : β Γ S) in π x, Β¬Critical (f y.1) y.2
e : β
w : S
a : Β¬Critical (f (e, w).1) (e, w).2
b : Critical (f (e, w).1) (e, w).2
β’ (e, w).2 = aβ | case neg
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
aβ zβ : S
d n : β
s : Super f d aβ
c : β
z : S
za : Β¬z = aβ
m : Β¬Critical (f c) z
o : β (x : β Γ S), Β¬Critical (f x.1) x.2 β βαΆ (y : β Γ S) in π x, Β¬Critical (f y.1) y.2
e : β
w : S
a : Β¬Critical (f (e, w).1) (e, w).2
b : Critical (f (e, w).1) (e, w).2
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
aβ zβ : S
d n : β
s : Super f d aβ
c : β
z : S
za : Β¬z = aβ
m : Β¬Critical (f c) z
o : β (x : β Γ S), Β¬Critical (f x.1) x.2 β βαΆ (y : β Γ S) in π x, Β¬Critical (f y.1) y.2
e : β
w : S
a : Β¬Critical (f (e, w).1) (e, w).2
b : Critical (f (e, w).1) (e, w).2
β’ (e, w).2 = aβ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.isClosed_critical_not_a | [592, 1] | [600, 91] | exact a b | case neg
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
aβ zβ : S
d n : β
s : Super f d aβ
c : β
z : S
za : Β¬z = aβ
m : Β¬Critical (f c) z
o : β (x : β Γ S), Β¬Critical (f x.1) x.2 β βαΆ (y : β Γ S) in π x, Β¬Critical (f y.1) y.2
e : β
w : S
a : Β¬Critical (f (e, w).1) (e, w).2
b : Critical (f (e, w).1) (e, w).2
β’ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
aβ zβ : S
d n : β
s : Super f d aβ
c : β
z : S
za : Β¬z = aβ
m : Β¬Critical (f c) z
o : β (x : β Γ S), Β¬Critical (f x.1) x.2 β βαΆ (y : β Γ S) in π x, Β¬Critical (f y.1) y.2
e : β
w : S
a : Β¬Critical (f (e, w).1) (e, w).2
b : Critical (f (e, w).1) (e, w).2
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNearIter_mfderiv_ne_zero | [610, 1] | [614, 69] | apply mderiv_comp_ne_zero' b0 | S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d n : β
s : Super f d a
b0 : mfderiv I I (s.bottcherNear c) ((f c)^[n] z) β 0
f0 : Β¬Precritical (f c) z
β’ mfderiv I I (s.bottcherNearIter n c) z β 0 | S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d n : β
s : Super f d a
b0 : mfderiv I I (s.bottcherNear c) ((f c)^[n] z) β 0
f0 : Β¬Precritical (f c) z
β’ mfderiv I I (f c)^[n] z β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d n : β
s : Super f d a
b0 : mfderiv I I (s.bottcherNear c) ((f c)^[n] z) β 0
f0 : Β¬Precritical (f c) z
β’ mfderiv I I (s.bottcherNearIter n c) z β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNearIter_mfderiv_ne_zero | [610, 1] | [614, 69] | contrapose f0 | S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d n : β
s : Super f d a
b0 : mfderiv I I (s.bottcherNear c) ((f c)^[n] z) β 0
f0 : Β¬Precritical (f c) z
β’ mfderiv I I (f c)^[n] z β 0 | S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d n : β
s : Super f d a
b0 : mfderiv I I (s.bottcherNear c) ((f c)^[n] z) β 0
f0 : Β¬mfderiv I I (f c)^[n] z β 0
⒠¬¬Precritical (f c) z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d n : β
s : Super f d a
b0 : mfderiv I I (s.bottcherNear c) ((f c)^[n] z) β 0
f0 : Β¬Precritical (f c) z
β’ mfderiv I I (f c)^[n] z β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNearIter_mfderiv_ne_zero | [610, 1] | [614, 69] | simp only [not_not] at f0 β’ | S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d n : β
s : Super f d a
b0 : mfderiv I I (s.bottcherNear c) ((f c)^[n] z) β 0
f0 : Β¬mfderiv I I (f c)^[n] z β 0
⒠¬¬Precritical (f c) z | S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d n : β
s : Super f d a
b0 : mfderiv I I (s.bottcherNear c) ((f c)^[n] z) β 0
f0 : mfderiv I I (f c)^[n] z = 0
β’ Precritical (f c) z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d n : β
s : Super f d a
b0 : mfderiv I I (s.bottcherNear c) ((f c)^[n] z) β 0
f0 : Β¬mfderiv I I (f c)^[n] z β 0
⒠¬¬Precritical (f c) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNearIter_mfderiv_ne_zero | [610, 1] | [614, 69] | exact critical_iter s.fa.along_snd f0 | S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d n : β
s : Super f d a
b0 : mfderiv I I (s.bottcherNear c) ((f c)^[n] z) β 0
f0 : mfderiv I I (f c)^[n] z = 0
β’ Precritical (f c) z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d n : β
s : Super f d a
b0 : mfderiv I I (s.bottcherNear c) ((f c)^[n] z) β 0
f0 : mfderiv I I (f c)^[n] z = 0
β’ Precritical (f c) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.iter_nontrivial_a | [617, 1] | [621, 47] | induction' n with n h | S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d n : β
s : Super f d a
β’ NontrivialHolomorphicAt (fun z => (f c)^[n] z) a | case zero
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d n : β
s : Super f d a
β’ NontrivialHolomorphicAt (fun z => (f c)^[0] z) a
case succ
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d nβ : β
s : Super f d a
n : β
h : NontrivialHolomorphicAt (fun z => (f c)^[n] z) a
β’ NontrivialHolomorphicAt (fun z => (f c)^[n + 1] z) a | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d n : β
s : Super f d a
β’ NontrivialHolomorphicAt (fun z => (f c)^[n] z) a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.iter_nontrivial_a | [617, 1] | [621, 47] | simp only [Function.iterate_zero_apply] | case zero
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d n : β
s : Super f d a
β’ NontrivialHolomorphicAt (fun z => (f c)^[0] z) a
case succ
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d nβ : β
s : Super f d a
n : β
h : NontrivialHolomorphicAt (fun z => (f c)^[n] z) a
β’ NontrivialHolomorphicAt (fun z => (f c)^[n + 1] z) a | case zero
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d n : β
s : Super f d a
β’ NontrivialHolomorphicAt (fun z => z) a
case succ
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d nβ : β
s : Super f d a
n : β
h : NontrivialHolomorphicAt (fun z => (f c)^[n] z) a
β’ NontrivialHolomorphicAt (fun z => (f c)^[n + 1] z) a | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d n : β
s : Super f d a
β’ NontrivialHolomorphicAt (fun z => (f c)^[0] z) a
case succ
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d nβ : β
s : Super f d a
n : β
h : NontrivialHolomorphicAt (fun z => (f c)^[n] z) a
β’ NontrivialHolomorphicAt (fun z => (f c)^[n + 1] z) a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.iter_nontrivial_a | [617, 1] | [621, 47] | apply nontrivialHolomorphicAt_id | case zero
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d n : β
s : Super f d a
β’ NontrivialHolomorphicAt (fun z => z) a
case succ
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d nβ : β
s : Super f d a
n : β
h : NontrivialHolomorphicAt (fun z => (f c)^[n] z) a
β’ NontrivialHolomorphicAt (fun z => (f c)^[n + 1] z) a | case succ
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d nβ : β
s : Super f d a
n : β
h : NontrivialHolomorphicAt (fun z => (f c)^[n] z) a
β’ NontrivialHolomorphicAt (fun z => (f c)^[n + 1] z) a | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d n : β
s : Super f d a
β’ NontrivialHolomorphicAt (fun z => z) a
case succ
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d nβ : β
s : Super f d a
n : β
h : NontrivialHolomorphicAt (fun z => (f c)^[n] z) a
β’ NontrivialHolomorphicAt (fun z => (f c)^[n + 1] z) a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.iter_nontrivial_a | [617, 1] | [621, 47] | simp only [Function.iterate_succ_apply'] | case succ
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d nβ : β
s : Super f d a
n : β
h : NontrivialHolomorphicAt (fun z => (f c)^[n] z) a
β’ NontrivialHolomorphicAt (fun z => (f c)^[n + 1] z) a | case succ
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d nβ : β
s : Super f d a
n : β
h : NontrivialHolomorphicAt (fun z => (f c)^[n] z) a
β’ NontrivialHolomorphicAt (fun z => f c ((f c)^[n] z)) a | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d nβ : β
s : Super f d a
n : β
h : NontrivialHolomorphicAt (fun z => (f c)^[n] z) a
β’ NontrivialHolomorphicAt (fun z => (f c)^[n + 1] z) a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.iter_nontrivial_a | [617, 1] | [621, 47] | refine NontrivialHolomorphicAt.comp ?_ h | case succ
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d nβ : β
s : Super f d a
n : β
h : NontrivialHolomorphicAt (fun z => (f c)^[n] z) a
β’ NontrivialHolomorphicAt (fun z => f c ((f c)^[n] z)) a | case succ
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d nβ : β
s : Super f d a
n : β
h : NontrivialHolomorphicAt (fun z => (f c)^[n] z) a
β’ NontrivialHolomorphicAt (f c) ((f c)^[n] a) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d nβ : β
s : Super f d a
n : β
h : NontrivialHolomorphicAt (fun z => (f c)^[n] z) a
β’ NontrivialHolomorphicAt (fun z => f c ((f c)^[n] z)) a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.iter_nontrivial_a | [617, 1] | [621, 47] | simp only [s.iter_a] | case succ
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d nβ : β
s : Super f d a
n : β
h : NontrivialHolomorphicAt (fun z => (f c)^[n] z) a
β’ NontrivialHolomorphicAt (f c) ((f c)^[n] a) | case succ
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d nβ : β
s : Super f d a
n : β
h : NontrivialHolomorphicAt (fun z => (f c)^[n] z) a
β’ NontrivialHolomorphicAt (f c) a | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d nβ : β
s : Super f d a
n : β
h : NontrivialHolomorphicAt (fun z => (f c)^[n] z) a
β’ NontrivialHolomorphicAt (f c) ((f c)^[n] a)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.iter_nontrivial_a | [617, 1] | [621, 47] | exact s.f_nontrivial c | case succ
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d nβ : β
s : Super f d a
n : β
h : NontrivialHolomorphicAt (fun z => (f c)^[n] z) a
β’ NontrivialHolomorphicAt (f c) a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d nβ : β
s : Super f d a
n : β
h : NontrivialHolomorphicAt (fun z => (f c)^[n] z) a
β’ NontrivialHolomorphicAt (f c) a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNearIter_nontrivial_a | [624, 1] | [631, 29] | simp only [s.iter_a] | S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d n : β
s : Super f d a
β’ NontrivialHolomorphicAt (s.bottcherNear c) ((f c)^[n] a) | S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d n : β
s : Super f d a
β’ NontrivialHolomorphicAt (s.bottcherNear c) a | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d n : β
s : Super f d a
β’ NontrivialHolomorphicAt (s.bottcherNear c) ((f c)^[n] a)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNearIter_nontrivial_a | [624, 1] | [631, 29] | exact nontrivialHolomorphicAt_of_mfderiv_ne_zero
(s.bottcherNear_holomorphic _ (s.mem_near c)).along_snd
(s.bottcherNear_mfderiv_ne_zero c) | S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d n : β
s : Super f d a
β’ NontrivialHolomorphicAt (s.bottcherNear c) a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z : S
d n : β
s : Super f d a
β’ NontrivialHolomorphicAt (s.bottcherNear c) a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_summable | [47, 1] | [54, 55] | rw [summable_iff_vanishing_norm] | f : β β β β β
s : Set β
z : β
zs : z β s
h : UniformVanishing f s
β’ Summable fun n => f n z | f : β β β β β
s : Set β
z : β
zs : z β s
h : UniformVanishing f s
β’ β Ξ΅ > 0, β s, β (t : Finset β), Disjoint t s β βt.sum fun i => f i zβ < Ξ΅ | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
z : β
zs : z β s
h : UniformVanishing f s
β’ Summable fun n => f n z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_summable | [47, 1] | [54, 55] | intro e ep | f : β β β β β
s : Set β
z : β
zs : z β s
h : UniformVanishing f s
β’ β Ξ΅ > 0, β s, β (t : Finset β), Disjoint t s β βt.sum fun i => f i zβ < Ξ΅ | f : β β β β β
s : Set β
z : β
zs : z β s
h : UniformVanishing f s
e : β
ep : e > 0
β’ β s, β (t : Finset β), Disjoint t s β βt.sum fun i => f i zβ < e | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
z : β
zs : z β s
h : UniformVanishing f s
β’ β Ξ΅ > 0, β s, β (t : Finset β), Disjoint t s β βt.sum fun i => f i zβ < Ξ΅
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_summable | [47, 1] | [54, 55] | rcases h e ep with β¨m, hmβ© | f : β β β β β
s : Set β
z : β
zs : z β s
h : UniformVanishing f s
e : β
ep : e > 0
β’ β s, β (t : Finset β), Disjoint t s β βt.sum fun i => f i zβ < e | case intro
f : β β β β β
s : Set β
z : β
zs : z β s
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e
β’ β s, β (t : Finset β), Disjoint t s β βt.sum fun i => f i zβ < e | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
z : β
zs : z β s
h : UniformVanishing f s
e : β
ep : e > 0
β’ β s, β (t : Finset β), Disjoint t s β βt.sum fun i => f i zβ < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_summable | [47, 1] | [54, 55] | use Finset.range m | case intro
f : β β β β β
s : Set β
z : β
zs : z β s
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e
β’ β s, β (t : Finset β), Disjoint t s β βt.sum fun i => f i zβ < e | case h
f : β β β β β
s : Set β
z : β
zs : z β s
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e
β’ β (t : Finset β), Disjoint t (Finset.range m) β βt.sum fun i => f i zβ < e | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : β β β β β
s : Set β
z : β
zs : z β s
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e
β’ β s, β (t : Finset β), Disjoint t s β βt.sum fun i => f i zβ < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_summable | [47, 1] | [54, 55] | intro A d | case h
f : β β β β β
s : Set β
z : β
zs : z β s
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e
β’ β (t : Finset β), Disjoint t (Finset.range m) β βt.sum fun i => f i zβ < e | case h
f : β β β β β
s : Set β
z : β
zs : z β s
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e
A : Finset β
d : Disjoint A (Finset.range m)
β’ βA.sum fun i => f i zβ < e | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β β β
s : Set β
z : β
zs : z β s
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e
β’ β (t : Finset β), Disjoint t (Finset.range m) β βt.sum fun i => f i zβ < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_summable | [47, 1] | [54, 55] | calc βA.sum (fun n β¦ f n z)β
_ β€ A.sum (fun n β¦ βf n zβ) := by bound
_ < e := hm _ _ (late_iff_disjoint_range.mpr d) zs | case h
f : β β β β β
s : Set β
z : β
zs : z β s
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e
A : Finset β
d : Disjoint A (Finset.range m)
β’ βA.sum fun i => f i zβ < e | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β β β
s : Set β
z : β
zs : z β s
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e
A : Finset β
d : Disjoint A (Finset.range m)
β’ βA.sum fun i => f i zβ < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_summable | [47, 1] | [54, 55] | bound | f : β β β β β
s : Set β
z : β
zs : z β s
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e
A : Finset β
d : Disjoint A (Finset.range m)
β’ βA.sum fun n => f n zβ β€ A.sum fun n => βf n zβ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
z : β
zs : z β s
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e
A : Finset β
d : Disjoint A (Finset.range m)
β’ βA.sum fun n => f n zβ β€ A.sum fun n => βf n zβ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_uniform_cauchy_series | [57, 1] | [67, 51] | rw [Metric.uniformCauchySeqOn_iff] | f : β β β β β
s : Set β
h : UniformVanishing f s
β’ UniformCauchySeqOn (fun N z => N.sum fun n => f n z) atTop s | f : β β β β β
s : Set β
h : UniformVanishing f s
β’ β Ξ΅ > 0, β N, β m β₯ N, β n β₯ N, β x β s, dist (m.sum fun n => f n x) (n.sum fun n => f n x) < Ξ΅ | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
h : UniformVanishing f s
β’ UniformCauchySeqOn (fun N z => N.sum fun n => f n z) atTop s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_uniform_cauchy_series | [57, 1] | [67, 51] | intro e ep | f : β β β β β
s : Set β
h : UniformVanishing f s
β’ β Ξ΅ > 0, β N, β m β₯ N, β n β₯ N, β x β s, dist (m.sum fun n => f n x) (n.sum fun n => f n x) < Ξ΅ | f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
β’ β N, β m β₯ N, β n β₯ N, β x β s, dist (m.sum fun n => f n x) (n.sum fun n => f n x) < e | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
h : UniformVanishing f s
β’ β Ξ΅ > 0, β N, β m β₯ N, β n β₯ N, β x β s, dist (m.sum fun n => f n x) (n.sum fun n => f n x) < Ξ΅
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_uniform_cauchy_series | [57, 1] | [67, 51] | rcases h e ep with β¨m, hmβ© | f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
β’ β N, β m β₯ N, β n β₯ N, β x β s, dist (m.sum fun n => f n x) (n.sum fun n => f n x) < e | case intro
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e
β’ β N, β m β₯ N, β n β₯ N, β x β s, dist (m.sum fun n => f n x) (n.sum fun n => f n x) < e | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
β’ β N, β m β₯ N, β n β₯ N, β x β s, dist (m.sum fun n => f n x) (n.sum fun n => f n x) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_uniform_cauchy_series | [57, 1] | [67, 51] | use Finset.range m | case intro
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e
β’ β N, β m β₯ N, β n β₯ N, β x β s, dist (m.sum fun n => f n x) (n.sum fun n => f n x) < e | case h
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e
β’ β m_1 β₯ Finset.range m, β n β₯ Finset.range m, β x β s, dist (m_1.sum fun n => f n x) (n.sum fun n => f n x) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e
β’ β N, β m β₯ N, β n β₯ N, β x β s, dist (m.sum fun n => f n x) (n.sum fun n => f n x) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_uniform_cauchy_series | [57, 1] | [67, 51] | intro A HA B HB z zs | case h
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e
β’ β m_1 β₯ Finset.range m, β n β₯ Finset.range m, β x β s, dist (m_1.sum fun n => f n x) (n.sum fun n => f n x) < e | case h
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e
A : Finset β
HA : A β₯ Finset.range m
B : Finset β
HB : B β₯ Finset.range m
z : β
zs : z β s
β’ dist (A.sum fun n => f n z) (B.sum fun n => f n z) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e
β’ β m_1 β₯ Finset.range m, β n β₯ Finset.range m, β x β s, dist (m_1.sum fun n => f n x) (n.sum fun n => f n x) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_uniform_cauchy_series | [57, 1] | [67, 51] | calc dist (A.sum fun n β¦ f n z) (B.sum fun n β¦ f n z)
_ β€ (A β B).sum fun n β¦ abs (f n z) := symmDiff_bound _ _ _
_ < e := hm (A β B) z (symmDiff_late HA HB) zs | case h
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e
A : Finset β
HA : A β₯ Finset.range m
B : Finset β
HB : B β₯ Finset.range m
z : β
zs : z β s
β’ dist (A.sum fun n => f n z) (B.sum fun n => f n z) < e | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e
A : Finset β
HA : A β₯ Finset.range m
B : Finset β
HB : B β₯ Finset.range m
z : β
zs : z β s
β’ dist (A.sum fun n => f n z) (B.sum fun n => f n z) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | rw [HasUniformSum, Metric.tendstoUniformlyOn_iff] | f : β β β β β
s : Set β
h : UniformVanishing f s
β’ HasUniformSum f (tsumOn f) s | f : β β β β β
s : Set β
h : UniformVanishing f s
β’ β Ξ΅ > 0, βαΆ (n : Finset β) in atTop, β x β s, dist (tsumOn f x) (n.sum fun n => f n x) < Ξ΅ | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
h : UniformVanishing f s
β’ HasUniformSum f (tsumOn f) s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | intro e ep | f : β β β β β
s : Set β
h : UniformVanishing f s
β’ β Ξ΅ > 0, βαΆ (n : Finset β) in atTop, β x β s, dist (tsumOn f x) (n.sum fun n => f n x) < Ξ΅ | f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
β’ βαΆ (n : Finset β) in atTop, β x β s, dist (tsumOn f x) (n.sum fun n => f n x) < e | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
h : UniformVanishing f s
β’ β Ξ΅ > 0, βαΆ (n : Finset β) in atTop, β x β s, dist (tsumOn f x) (n.sum fun n => f n x) < Ξ΅
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | rcases h (e / 4) (by linarith) with β¨m, hmβ© | f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
β’ βαΆ (n : Finset β) in atTop, β x β s, dist (tsumOn f x) (n.sum fun n => f n x) < e | case intro
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
β’ βαΆ (n : Finset β) in atTop, β x β s, dist (tsumOn f x) (n.sum fun n => f n x) < e | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
β’ βαΆ (n : Finset β) in atTop, β x β s, dist (tsumOn f x) (n.sum fun n => f n x) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | rw [Filter.eventually_atTop] | case intro
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
β’ βαΆ (n : Finset β) in atTop, β x β s, dist (tsumOn f x) (n.sum fun n => f n x) < e | case intro
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
β’ β a, β b β₯ a, β x β s, dist (tsumOn f x) (b.sum fun n => f n x) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
β’ βαΆ (n : Finset β) in atTop, β x β s, dist (tsumOn f x) (n.sum fun n => f n x) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | use Finset.range m | case intro
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
β’ β a, β b β₯ a, β x β s, dist (tsumOn f x) (b.sum fun n => f n x) < e | case h
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
β’ β b β₯ Finset.range m, β x β s, dist (tsumOn f x) (b.sum fun n => f n x) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
β’ β a, β b β₯ a, β x β s, dist (tsumOn f x) (b.sum fun n => f n x) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | intro N Nm z zs | case h
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
β’ β b β₯ Finset.range m, β x β s, dist (tsumOn f x) (b.sum fun n => f n x) < e | case h
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
β’ dist (tsumOn f z) (N.sum fun n => f n z) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
β’ β b β₯ Finset.range m, β x β s, dist (tsumOn f x) (b.sum fun n => f n x) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | rw [tsumOn] | case h
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
β’ dist (tsumOn f z) (N.sum fun n => f n z) < e | case h
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
β’ dist (β' (n : β), f n z) (N.sum fun n => f n z) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
β’ dist (tsumOn f z) (N.sum fun n => f n z) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | generalize G : tsum (fun n β¦ f n z) = g | case h
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
β’ dist (β' (n : β), f n z) (N.sum fun n => f n z) < e | case h
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
G : β' (n : β), f n z = g
β’ dist g (N.sum fun n => f n z) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
β’ dist (β' (n : β), f n z) (N.sum fun n => f n z) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | have S : Summable (fun n β¦ f n z) := uniformVanishing_to_summable zs h | case h
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
G : β' (n : β), f n z = g
β’ dist g (N.sum fun n => f n z) < e | case h
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
G : β' (n : β), f n z = g
S : Summable fun n => f n z
β’ dist g (N.sum fun n => f n z) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
G : β' (n : β), f n z = g
β’ dist g (N.sum fun n => f n z) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | have GS : HasSum (fun n β¦ f n z) g := by rw [β G]; exact Summable.hasSum S | case h
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
G : β' (n : β), f n z = g
S : Summable fun n => f n z
β’ dist g (N.sum fun n => f n z) < e | case h
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
G : β' (n : β), f n z = g
S : Summable fun n => f n z
GS : HasSum (fun n => f n z) g
β’ dist g (N.sum fun n => f n z) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
G : β' (n : β), f n z = g
S : Summable fun n => f n z
β’ dist g (N.sum fun n => f n z) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | clear S | case h
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
G : β' (n : β), f n z = g
S : Summable fun n => f n z
GS : HasSum (fun n => f n z) g
β’ dist g (N.sum fun n => f n z) < e | case h
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
G : β' (n : β), f n z = g
GS : HasSum (fun n => f n z) g
β’ dist g (N.sum fun n => f n z) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
G : β' (n : β), f n z = g
S : Summable fun n => f n z
GS : HasSum (fun n => f n z) g
β’ dist g (N.sum fun n => f n z) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | rw [HasSum] at GS | case h
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
G : β' (n : β), f n z = g
GS : HasSum (fun n => f n z) g
β’ dist g (N.sum fun n => f n z) < e | case h
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
G : β' (n : β), f n z = g
GS : Filter.Tendsto (fun s => s.sum fun b => f b z) atTop (π g)
β’ dist g (N.sum fun n => f n z) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
G : β' (n : β), f n z = g
GS : HasSum (fun n => f n z) g
β’ dist g (N.sum fun n => f n z) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | rw [Metric.tendsto_atTop] at GS | case h
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
G : β' (n : β), f n z = g
GS : Filter.Tendsto (fun s => s.sum fun b => f b z) atTop (π g)
β’ dist g (N.sum fun n => f n z) < e | case h
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
G : β' (n : β), f n z = g
GS : β Ξ΅ > 0, β N, β n β₯ N, dist (n.sum fun b => f b z) g < Ξ΅
β’ dist g (N.sum fun n => f n z) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
G : β' (n : β), f n z = g
GS : Filter.Tendsto (fun s => s.sum fun b => f b z) atTop (π g)
β’ dist g (N.sum fun n => f n z) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | rcases GS (e / 4) (by linarith) with β¨M, HMβ© | case h
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
G : β' (n : β), f n z = g
GS : β Ξ΅ > 0, β N, β n β₯ N, dist (n.sum fun b => f b z) g < Ξ΅
β’ dist g (N.sum fun n => f n z) < e | case h.intro
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
G : β' (n : β), f n z = g
GS : β Ξ΅ > 0, β N, β n β₯ N, dist (n.sum fun b => f b z) g < Ξ΅
M : Finset β
HM : β n β₯ M, dist (n.sum fun b => f b z) g < e / 4
β’ dist g (N.sum fun n => f n z) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
G : β' (n : β), f n z = g
GS : β Ξ΅ > 0, β N, β n β₯ N, dist (n.sum fun b => f b z) g < Ξ΅
β’ dist g (N.sum fun n => f n z) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | clear GS G h | case h.intro
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
G : β' (n : β), f n z = g
GS : β Ξ΅ > 0, β N, β n β₯ N, dist (n.sum fun b => f b z) g < Ξ΅
M : Finset β
HM : β n β₯ M, dist (n.sum fun b => f b z) g < e / 4
β’ dist g (N.sum fun n => f n z) < e | case h.intro
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
HM : β n β₯ M, dist (n.sum fun b => f b z) g < e / 4
β’ dist g (N.sum fun n => f n z) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
G : β' (n : β), f n z = g
GS : β Ξ΅ > 0, β N, β n β₯ N, dist (n.sum fun b => f b z) g < Ξ΅
M : Finset β
HM : β n β₯ M, dist (n.sum fun b => f b z) g < e / 4
β’ dist g (N.sum fun n => f n z) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | set A := N βͺ M \ N | case h.intro
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
HM : β n β₯ M, dist (n.sum fun b => f b z) g < e / 4
β’ dist g (N.sum fun n => f n z) < e | case h.intro
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
HM : β n β₯ M, dist (n.sum fun b => f b z) g < e / 4
A : Finset β := N βͺ M \ N
β’ dist g (N.sum fun n => f n z) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
HM : β n β₯ M, dist (n.sum fun b => f b z) g < e / 4
β’ dist g (N.sum fun n => f n z) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | have AM : M β A := subset_union_sdiff _ _ | case h.intro
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
HM : β n β₯ M, dist (n.sum fun b => f b z) g < e / 4
A : Finset β := N βͺ M \ N
β’ dist g (N.sum fun n => f n z) < e | case h.intro
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
HM : β n β₯ M, dist (n.sum fun b => f b z) g < e / 4
A : Finset β := N βͺ M \ N
AM : M β A
β’ dist g (N.sum fun n => f n z) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
HM : β n β₯ M, dist (n.sum fun b => f b z) g < e / 4
A : Finset β := N βͺ M \ N
β’ dist g (N.sum fun n => f n z) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | simp at HM | case h.intro
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
HM : β n β₯ M, dist (n.sum fun b => f b z) g < e / 4
A : Finset β := N βͺ M \ N
AM : M β A
β’ dist g (N.sum fun n => f n z) < e | case h.intro
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : β (n : Finset β), M β n β dist (n.sum fun b => f b z) g < e / 4
β’ dist g (N.sum fun n => f n z) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
HM : β n β₯ M, dist (n.sum fun b => f b z) g < e / 4
A : Finset β := N βͺ M \ N
AM : M β A
β’ dist g (N.sum fun n => f n z) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | specialize HM A AM | case h.intro
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : β (n : Finset β), M β n β dist (n.sum fun b => f b z) g < e / 4
β’ dist g (N.sum fun n => f n z) < e | case h.intro
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist (A.sum fun b => f b z) g < e / 4
β’ dist g (N.sum fun n => f n z) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : β (n : Finset β), M β n β dist (n.sum fun b => f b z) g < e / 4
β’ dist g (N.sum fun n => f n z) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | rw [dist_comm] at HM | case h.intro
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist (A.sum fun b => f b z) g < e / 4
β’ dist g (N.sum fun n => f n z) < e | case h.intro
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist g (A.sum fun b => f b z) < e / 4
β’ dist g (N.sum fun n => f n z) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist (A.sum fun b => f b z) g < e / 4
β’ dist g (N.sum fun n => f n z) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | calc dist g (N.sum fun n β¦ f n z)
_ β€ dist g (A.sum fun n β¦ f n z) + dist (A.sum fun n β¦ f n z) (N.sum fun n β¦ f n z) := by bound
_ β€ e / 4 + dist (A.sum fun n β¦ f n z) (N.sum fun n β¦ f n z) := by linarith
_ = e / 4 + dist ((N.sum fun n β¦ f n z) + (M \ N).sum fun n β¦ f n z)
(N.sum fun n β¦ f n z) := by rw [Finset.sum_union Finset.disjoint_sdiff]
_ = e / 4 + abs (((N.sum fun n β¦ f n z) + (M \ N).sum fun n β¦ f n z) -
N.sum fun n β¦ f n z) := by rw [Complex.dist_eq]
_ = e / 4 + abs ((M \ N).sum fun n β¦ f n z) := by ring_nf
_ β€ e / 4 + (M \ N).sum fun n β¦ abs (f n z) := by
linarith [finset_complex_abs_sum_le (M \ N) fun n β¦ f n z]
_ β€ e / 4 + e / 4 := by linarith [hm (M \ N) z (sdiff_late M Nm) zs]
_ = e / 2 := by ring
_ < e := by linarith | case h.intro
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist g (A.sum fun b => f b z) < e / 4
β’ dist g (N.sum fun n => f n z) < e | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist g (A.sum fun b => f b z) < e / 4
β’ dist g (N.sum fun n => f n z) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | linarith | f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
β’ e / 4 > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
β’ e / 4 > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | rw [β G] | f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
G : β' (n : β), f n z = g
S : Summable fun n => f n z
β’ HasSum (fun n => f n z) g | f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
G : β' (n : β), f n z = g
S : Summable fun n => f n z
β’ HasSum (fun n => f n z) (β' (n : β), f n z) | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
G : β' (n : β), f n z = g
S : Summable fun n => f n z
β’ HasSum (fun n => f n z) g
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | exact Summable.hasSum S | f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
G : β' (n : β), f n z = g
S : Summable fun n => f n z
β’ HasSum (fun n => f n z) (β' (n : β), f n z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
G : β' (n : β), f n z = g
S : Summable fun n => f n z
β’ HasSum (fun n => f n z) (β' (n : β), f n z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | linarith | f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
G : β' (n : β), f n z = g
GS : β Ξ΅ > 0, β N, β n β₯ N, dist (n.sum fun b => f b z) g < Ξ΅
β’ e / 4 > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
h : UniformVanishing f s
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
G : β' (n : β), f n z = g
GS : β Ξ΅ > 0, β N, β n β₯ N, dist (n.sum fun b => f b z) g < Ξ΅
β’ e / 4 > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | bound | f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist g (A.sum fun b => f b z) < e / 4
β’ dist g (N.sum fun n => f n z) β€ dist g (A.sum fun n => f n z) + dist (A.sum fun n => f n z) (N.sum fun n => f n z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist g (A.sum fun b => f b z) < e / 4
β’ dist g (N.sum fun n => f n z) β€ dist g (A.sum fun n => f n z) + dist (A.sum fun n => f n z) (N.sum fun n => f n z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | linarith | f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist g (A.sum fun b => f b z) < e / 4
β’ dist g (A.sum fun n => f n z) + dist (A.sum fun n => f n z) (N.sum fun n => f n z) β€
e / 4 + dist (A.sum fun n => f n z) (N.sum fun n => f n z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist g (A.sum fun b => f b z) < e / 4
β’ dist g (A.sum fun n => f n z) + dist (A.sum fun n => f n z) (N.sum fun n => f n z) β€
e / 4 + dist (A.sum fun n => f n z) (N.sum fun n => f n z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | rw [Finset.sum_union Finset.disjoint_sdiff] | f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist g (A.sum fun b => f b z) < e / 4
β’ e / 4 + dist (A.sum fun n => f n z) (N.sum fun n => f n z) =
e / 4 + dist ((N.sum fun n => f n z) + (M \ N).sum fun n => f n z) (N.sum fun n => f n z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist g (A.sum fun b => f b z) < e / 4
β’ e / 4 + dist (A.sum fun n => f n z) (N.sum fun n => f n z) =
e / 4 + dist ((N.sum fun n => f n z) + (M \ N).sum fun n => f n z) (N.sum fun n => f n z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | rw [Complex.dist_eq] | f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist g (A.sum fun b => f b z) < e / 4
β’ e / 4 + dist ((N.sum fun n => f n z) + (M \ N).sum fun n => f n z) (N.sum fun n => f n z) =
e / 4 + Complex.abs (((N.sum fun n => f n z) + (M \ N).sum fun n => f n z) - N.sum fun n => f n z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist g (A.sum fun b => f b z) < e / 4
β’ e / 4 + dist ((N.sum fun n => f n z) + (M \ N).sum fun n => f n z) (N.sum fun n => f n z) =
e / 4 + Complex.abs (((N.sum fun n => f n z) + (M \ N).sum fun n => f n z) - N.sum fun n => f n z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | ring_nf | f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist g (A.sum fun b => f b z) < e / 4
β’ e / 4 + Complex.abs (((N.sum fun n => f n z) + (M \ N).sum fun n => f n z) - N.sum fun n => f n z) =
e / 4 + Complex.abs ((M \ N).sum fun n => f n z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist g (A.sum fun b => f b z) < e / 4
β’ e / 4 + Complex.abs (((N.sum fun n => f n z) + (M \ N).sum fun n => f n z) - N.sum fun n => f n z) =
e / 4 + Complex.abs ((M \ N).sum fun n => f n z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | linarith [finset_complex_abs_sum_le (M \ N) fun n β¦ f n z] | f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist g (A.sum fun b => f b z) < e / 4
β’ e / 4 + Complex.abs ((M \ N).sum fun n => f n z) β€ e / 4 + (M \ N).sum fun n => Complex.abs (f n z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist g (A.sum fun b => f b z) < e / 4
β’ e / 4 + Complex.abs ((M \ N).sum fun n => f n z) β€ e / 4 + (M \ N).sum fun n => Complex.abs (f n z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | linarith [hm (M \ N) z (sdiff_late M Nm) zs] | f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist g (A.sum fun b => f b z) < e / 4
β’ (e / 4 + (M \ N).sum fun n => Complex.abs (f n z)) β€ e / 4 + e / 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist g (A.sum fun b => f b z) < e / 4
β’ (e / 4 + (M \ N).sum fun n => Complex.abs (f n z)) β€ e / 4 + e / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | ring | f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist g (A.sum fun b => f b z) < e / 4
β’ e / 4 + e / 4 = e / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist g (A.sum fun b => f b z) < e / 4
β’ e / 4 + e / 4 = e / 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | linarith | f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist g (A.sum fun b => f b z) < e / 4
β’ e / 2 < e | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
e : β
ep : e > 0
m : β
hm : β (N : Finset β) (z : β), Late N m β z β s β (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset β
Nm : N β₯ Finset.range m
z : β
zs : z β s
g : β
M : Finset β
A : Finset β := N βͺ M \ N
AM : M β A
HM : dist g (A.sum fun b => f b z) < e / 4
β’ e / 2 < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | CNonpos.degenerate | [106, 1] | [110, 85] | intro n z zs | f : β β β β β
s : Set β
c a : β
c0 : c β€ 0
a0 : 0 β€ a
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
β’ β (n : β), β z β s, f n z = 0 | f : β β β β β
s : Set β
c a : β
c0 : c β€ 0
a0 : 0 β€ a
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
n : β
z : β
zs : z β s
β’ f n z = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
c a : β
c0 : c β€ 0
a0 : 0 β€ a
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
β’ β (n : β), β z β s, f n z = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | CNonpos.degenerate | [106, 1] | [110, 85] | specialize hf n z zs | f : β β β β β
s : Set β
c a : β
c0 : c β€ 0
a0 : 0 β€ a
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
n : β
z : β
zs : z β s
β’ f n z = 0 | f : β β β β β
s : Set β
c a : β
c0 : c β€ 0
a0 : 0 β€ a
n : β
z : β
zs : z β s
hf : Complex.abs (f n z) β€ c * a ^ n
β’ f n z = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
c a : β
c0 : c β€ 0
a0 : 0 β€ a
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
n : β
z : β
zs : z β s
β’ f n z = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | CNonpos.degenerate | [106, 1] | [110, 85] | have ca : c * a ^ n β€ 0 := mul_nonpos_iff.mpr (Or.inr β¨c0, by boundβ©) | f : β β β β β
s : Set β
c a : β
c0 : c β€ 0
a0 : 0 β€ a
n : β
z : β
zs : z β s
hf : Complex.abs (f n z) β€ c * a ^ n
β’ f n z = 0 | f : β β β β β
s : Set β
c a : β
c0 : c β€ 0
a0 : 0 β€ a
n : β
z : β
zs : z β s
hf : Complex.abs (f n z) β€ c * a ^ n
ca : c * a ^ n β€ 0
β’ f n z = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
c a : β
c0 : c β€ 0
a0 : 0 β€ a
n : β
z : β
zs : z β s
hf : Complex.abs (f n z) β€ c * a ^ n
β’ f n z = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | CNonpos.degenerate | [106, 1] | [110, 85] | exact Complex.abs.eq_zero.mp (le_antisymm (le_trans hf ca) (Complex.abs.nonneg _)) | f : β β β β β
s : Set β
c a : β
c0 : c β€ 0
a0 : 0 β€ a
n : β
z : β
zs : z β s
hf : Complex.abs (f n z) β€ c * a ^ n
ca : c * a ^ n β€ 0
β’ f n z = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
c a : β
c0 : c β€ 0
a0 : 0 β€ a
n : β
z : β
zs : z β s
hf : Complex.abs (f n z) β€ c * a ^ n
ca : c * a ^ n β€ 0
β’ f n z = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | CNonpos.degenerate | [106, 1] | [110, 85] | bound | f : β β β β β
s : Set β
c a : β
c0 : c β€ 0
a0 : 0 β€ a
n : β
z : β
zs : z β s
hf : Complex.abs (f n z) β€ c * a ^ n
β’ 0 β€ a ^ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
c a : β
c0 : c β€ 0
a0 : 0 β€ a
n : β
z : β
zs : z β s
hf : Complex.abs (f n z) β€ c * a ^ n
β’ 0 β€ a ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | by_cases c0 : c β€ 0 | f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
β’ HasUniformSum f (tsumOn f) s | case pos
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
β’ HasUniformSum f (tsumOn f) s
case neg
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : Β¬c β€ 0
β’ HasUniformSum f (tsumOn f) s | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
β’ HasUniformSum f (tsumOn f) s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | have fz := CNonpos.degenerate c0 a0 hf | case pos
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
β’ HasUniformSum f (tsumOn f) s | case pos
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
β’ HasUniformSum f (tsumOn f) s | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
β’ HasUniformSum f (tsumOn f) s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | rw [HasUniformSum, Metric.tendstoUniformlyOn_iff] | case pos
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
β’ HasUniformSum f (tsumOn f) s | case pos
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
β’ β Ξ΅ > 0, βαΆ (n : Finset β) in atTop, β x β s, dist (tsumOn f x) (n.sum fun n => f n x) < Ξ΅ | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
β’ HasUniformSum f (tsumOn f) s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | intro e ep | case pos
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
β’ β Ξ΅ > 0, βαΆ (n : Finset β) in atTop, β x β s, dist (tsumOn f x) (n.sum fun n => f n x) < Ξ΅ | case pos
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
e : β
ep : e > 0
β’ βαΆ (n : Finset β) in atTop, β x β s, dist (tsumOn f x) (n.sum fun n => f n x) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
β’ β Ξ΅ > 0, βαΆ (n : Finset β) in atTop, β x β s, dist (tsumOn f x) (n.sum fun n => f n x) < Ξ΅
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | apply Filter.eventually_of_forall | case pos
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
e : β
ep : e > 0
β’ βαΆ (n : Finset β) in atTop, β x β s, dist (tsumOn f x) (n.sum fun n => f n x) < e | case pos.hp
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
e : β
ep : e > 0
β’ β (x : Finset β), β x_1 β s, dist (tsumOn f x_1) (x.sum fun n => f n x_1) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
e : β
ep : e > 0
β’ βαΆ (n : Finset β) in atTop, β x β s, dist (tsumOn f x) (n.sum fun n => f n x) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | intro n z zs | case pos.hp
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
e : β
ep : e > 0
β’ β (x : Finset β), β x_1 β s, dist (tsumOn f x_1) (x.sum fun n => f n x_1) < e | case pos.hp
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
e : β
ep : e > 0
n : Finset β
z : β
zs : z β s
β’ dist (tsumOn f z) (n.sum fun n => f n z) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.hp
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
e : β
ep : e > 0
β’ β (x : Finset β), β x_1 β s, dist (tsumOn f x_1) (x.sum fun n => f n x_1) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | rw [tsumOn] | case pos.hp
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
e : β
ep : e > 0
n : Finset β
z : β
zs : z β s
β’ dist (tsumOn f z) (n.sum fun n => f n z) < e | case pos.hp
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
e : β
ep : e > 0
n : Finset β
z : β
zs : z β s
β’ dist (β' (n : β), f n z) (n.sum fun n => f n z) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.hp
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
e : β
ep : e > 0
n : Finset β
z : β
zs : z β s
β’ dist (tsumOn f z) (n.sum fun n => f n z) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | simp_rw [fz _ z zs] | case pos.hp
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
e : β
ep : e > 0
n : Finset β
z : β
zs : z β s
β’ dist (β' (n : β), f n z) (n.sum fun n => f n z) < e | case pos.hp
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
e : β
ep : e > 0
n : Finset β
z : β
zs : z β s
β’ dist (β' (n : β), 0) (n.sum fun n => 0) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.hp
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
e : β
ep : e > 0
n : Finset β
z : β
zs : z β s
β’ dist (β' (n : β), f n z) (n.sum fun n => f n z) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | simp only [tsum_zero, Finset.sum_const_zero, dist_zero_left, Complex.norm_eq_abs,
AbsoluteValue.map_zero] | case pos.hp
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
e : β
ep : e > 0
n : Finset β
z : β
zs : z β s
β’ dist (β' (n : β), 0) (n.sum fun n => 0) < e | case pos.hp
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
e : β
ep : e > 0
n : Finset β
z : β
zs : z β s
β’ 0 < e | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.hp
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
e : β
ep : e > 0
n : Finset β
z : β
zs : z β s
β’ dist (β' (n : β), 0) (n.sum fun n => 0) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | assumption | case pos.hp
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
e : β
ep : e > 0
n : Finset β
z : β
zs : z β s
β’ 0 < e | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.hp
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : c β€ 0
fz : β (n : β), β z β s, f n z = 0
e : β
ep : e > 0
n : Finset β
z : β
zs : z β s
β’ 0 < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | simp only [not_le] at c0 | case neg
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : Β¬c β€ 0
β’ HasUniformSum f (tsumOn f) s | case neg
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
β’ HasUniformSum f (tsumOn f) s | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : Β¬c β€ 0
β’ HasUniformSum f (tsumOn f) s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | apply uniformVanishing_to_tendsto_uniformly_on | case neg
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
β’ HasUniformSum f (tsumOn f) s | case neg.h
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
β’ UniformVanishing f s | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
β’ HasUniformSum f (tsumOn f) s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | intro e ep | case neg.h
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
β’ UniformVanishing f s | case neg.h
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
β’ β n, β (N : Finset β) (z : β), Late N n β z β s β (N.sum fun n => Complex.abs (f n z)) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
β’ UniformVanishing f s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | set t := (1 - βa) / βc * (e / 2) | case neg.h
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
β’ β n, β (N : Finset β) (z : β), Late N n β z β s β (N.sum fun n => Complex.abs (f n z)) < e | case neg.h
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
β’ β n, β (N : Finset β) (z : β), Late N n β z β s β (N.sum fun n => Complex.abs (f n z)) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
β’ β n, β (N : Finset β) (z : β), Late N n β z β s β (N.sum fun n => Complex.abs (f n z)) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | have tp : t > 0 := by bound | case neg.h
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
β’ β n, β (N : Finset β) (z : β), Late N n β z β s β (N.sum fun n => Complex.abs (f n z)) < e | case neg.h
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
β’ β n, β (N : Finset β) (z : β), Late N n β z β s β (N.sum fun n => Complex.abs (f n z)) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
β’ β n, β (N : Finset β) (z : β), Late N n β z β s β (N.sum fun n => Complex.abs (f n z)) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | rcases exists_pow_lt_of_lt_one tp a1 with β¨n, ntβ© | case neg.h
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
β’ β n, β (N : Finset β) (z : β), Late N n β z β s β (N.sum fun n => Complex.abs (f n z)) < e | case neg.h.intro
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
β’ β n, β (N : Finset β) (z : β), Late N n β z β s β (N.sum fun n => Complex.abs (f n z)) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
β’ β n, β (N : Finset β) (z : β), Late N n β z β s β (N.sum fun n => Complex.abs (f n z)) < e
TACTIC:
|
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