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 |
|---|---|---|---|---|---|---|---|---|
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
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
dg : DifferentiableAt ℂ g (↑(extChartAt I a) z)
d0 : ∀ (z : ℂ), DifferentiableAt ℂ (fun z => z - ↑(extChartAt I a) a) z
⊢ DifferentiableAt ℂ (fun z => z + ↑(extChartAt I a) a) (↑(extChartAt I a) z - ↑(extChartAt I a) a) | no goals |
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
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
dg : DifferentiableAt ℂ g (↑(extChartAt I a) z)
d0 : ∀ (z : ℂ), DifferentiableAt ℂ (fun z => z - ↑(extChartAt I a) a) z
d1 : DifferentiableAt ℂ (g ∘ fun z => z + ↑(extChartAt I a) a) (↑(extChartAt I a) z - ↑(extChartAt I a) a)
⊢ deriv g (↑(extChartAt I a) z - ↑(extChartAt I a) a + ↑(extChartAt I a) a) *
deriv (fun z => z + ↑(extChartAt I a) a) (↑(extChartAt I a) z - ↑(extChartAt I a) a) =
0 ↔
deriv g (↑(extChartAt I a) z) = 0 | no goals |
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
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
dg : DifferentiableAt ℂ g (↑(extChartAt I a) z)
d0 : ∀ (z : ℂ), DifferentiableAt ℂ (fun z => z - ↑(extChartAt I a) a) z
d1 : DifferentiableAt ℂ (g ∘ fun z => z + ↑(extChartAt I a) a) (↑(extChartAt I a) z - ↑(extChartAt I a) a)
⊢ DifferentiableAt ℂ g (↑(extChartAt I a) z - ↑(extChartAt I a) a + ↑(extChartAt I a) a) | no goals |
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
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
dg : DifferentiableAt ℂ g (↑(extChartAt I a) z)
d0 : ∀ (z : ℂ), DifferentiableAt ℂ (fun z => z - ↑(extChartAt I a) a) z
d1 : DifferentiableAt ℂ (g ∘ fun z => z + ↑(extChartAt I a) a) (↑(extChartAt I a) z - ↑(extChartAt I a) a)
⊢ DifferentiableAt ℂ (fun z => z + ↑(extChartAt I a) a) (↑(extChartAt I a) z - ↑(extChartAt I a) a) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.isClosed_critical_not_a | [592, 1] | [600, 91] | rw [← isOpen_compl_iff] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ IsClosed {p | Critical (f p.1) p.2 ∧ p.2 ≠ a} | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ IsOpen {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ |
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
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ IsOpen {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ ∀ x ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ, ∀ᶠ (y : ℂ × S) in 𝓝 x, y ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.isClosed_critical_not_a | [592, 1] | [600, 91] | intro ⟨c, z⟩ m | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ ∀ x ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ, ∀ᶠ (y : ℂ × S) in 𝓝 x, y ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ
⊢ ∀ᶠ (y : ℂ × S) in 𝓝 (c, z), y ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ |
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
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ
⊢ ∀ᶠ (y : ℂ × S) in 𝓝 (c, z), y ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ | case pos
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ
za : z = a
⊢ ∀ᶠ (y : ℂ × S) in 𝓝 (c, z), y ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ
case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ
za : ¬z = a
⊢ ∀ᶠ (y : ℂ × S) in 𝓝 (c, z), y ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ |
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
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ
za : z = a
⊢ ∀ᶠ (y : ℂ × S) in 𝓝 (c, z), y ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ | case pos
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ
za : z = a
⊢ ∀ᶠ (y : ℂ × S) in 𝓝 (c, a), y ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ |
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
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ
za : z = a
⊢ ∀ᶠ (y : ℂ × S) in 𝓝 (c, a), y ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ | case pos
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ
za : z = a
⊢ ∀ (x : ℂ × S), (Critical (f x.1) x.2 ↔ x.2 = a) → x ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.isClosed_critical_not_a | [592, 1] | [600, 91] | intro ⟨e, w⟩ h | case pos
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ
za : z = a
⊢ ∀ (x : ℂ × S), (Critical (f x.1) x.2 ↔ x.2 = a) → x ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ | case pos
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ
za : z = a
e : ℂ
w : S
h : Critical (f (e, w).1) (e, w).2 ↔ (e, w).2 = a
⊢ (e, w) ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ |
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 ⊢ | case pos
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ
za : z = a
e : ℂ
w : S
h : Critical (f (e, w).1) (e, w).2 ↔ (e, w).2 = a
⊢ (e, w) ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ | case pos
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ
za : z = a
e : ℂ
w : S
h : Critical (f e) w ↔ w = a
⊢ Critical (f e) w → w = a |
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
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ
za : z = a
e : ℂ
w : S
h : Critical (f e) w ↔ w = a
⊢ Critical (f e) w → w = a | no goals |
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
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ
za : ¬z = a
⊢ ∀ᶠ (y : ℂ × S) in 𝓝 (c, z), y ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ | case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ
za : ¬z = a
o : ∀ x ∈ {p | ¬Critical (f p.1) p.2}, ∀ᶠ (y : ℂ × S) in 𝓝 x, y ∈ {p | ¬Critical (f p.1) p.2}
⊢ ∀ᶠ (y : ℂ × S) in 𝓝 (c, z), y ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ |
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 ⊢ | case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ
za : ¬z = a
o : ∀ x ∈ {p | ¬Critical (f p.1) p.2}, ∀ᶠ (y : ℂ × S) in 𝓝 x, y ∈ {p | ¬Critical (f p.1) p.2}
⊢ ∀ᶠ (y : ℂ × S) in 𝓝 (c, z), y ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ | case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
za : ¬z = a
m : ¬Critical (f c) z
o : ∀ (x : ℂ × S), ¬Critical (f x.1) x.2 → ∀ᶠ (y : ℂ × S) in 𝓝 x, ¬Critical (f y.1) y.2
⊢ ∀ᶠ (y : ℂ × S) in 𝓝 (c, z), Critical (f y.1) y.2 → y.2 = a |
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
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
za : ¬z = a
m : ¬Critical (f c) z
o : ∀ (x : ℂ × S), ¬Critical (f x.1) x.2 → ∀ᶠ (y : ℂ × S) in 𝓝 x, ¬Critical (f y.1) y.2
⊢ ∀ᶠ (y : ℂ × S) in 𝓝 (c, z), Critical (f y.1) y.2 → y.2 = a | case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
za : ¬z = a
m : ¬Critical (f c) z
o : ∀ (x : ℂ × S), ¬Critical (f x.1) x.2 → ∀ᶠ (y : ℂ × S) in 𝓝 x, ¬Critical (f y.1) y.2
⊢ ∀ (x : ℂ × S), ¬Critical (f x.1) x.2 → Critical (f x.1) x.2 → x.2 = a |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.isClosed_critical_not_a | [592, 1] | [600, 91] | intro ⟨e, w⟩ a b | case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
za : ¬z = a
m : ¬Critical (f c) z
o : ∀ (x : ℂ × S), ¬Critical (f x.1) x.2 → ∀ᶠ (y : ℂ × S) in 𝓝 x, ¬Critical (f y.1) y.2
⊢ ∀ (x : ℂ × S), ¬Critical (f x.1) x.2 → Critical (f x.1) x.2 → x.2 = a | case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a✝ z✝ : S
d n : ℕ
s : Super f d a✝
c : ℂ
z : S
za : ¬z = a✝
m : ¬Critical (f c) z
o : ∀ (x : ℂ × S), ¬Critical (f x.1) x.2 → ∀ᶠ (y : ℂ × S) in 𝓝 x, ¬Critical (f y.1) y.2
e : ℂ
w : S
a : ¬Critical (f (e, w).1) (e, w).2
b : Critical (f (e, w).1) (e, w).2
⊢ (e, w).2 = a✝ |
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
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a✝ z✝ : S
d n : ℕ
s : Super f d a✝
c : ℂ
z : S
za : ¬z = a✝
m : ¬Critical (f c) z
o : ∀ (x : ℂ × S), ¬Critical (f x.1) x.2 → ∀ᶠ (y : ℂ × S) in 𝓝 x, ¬Critical (f y.1) y.2
e : ℂ
w : S
a : ¬Critical (f (e, w).1) (e, w).2
b : Critical (f (e, w).1) (e, w).2
⊢ (e, w).2 = a✝ | case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a✝ z✝ : S
d n : ℕ
s : Super f d a✝
c : ℂ
z : S
za : ¬z = a✝
m : ¬Critical (f c) z
o : ∀ (x : ℂ × S), ¬Critical (f x.1) x.2 → ∀ᶠ (y : ℂ × S) in 𝓝 x, ¬Critical (f y.1) y.2
e : ℂ
w : S
a : ¬Critical (f (e, w).1) (e, w).2
b : Critical (f (e, w).1) (e, w).2
⊢ False |
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
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a✝ z✝ : S
d n : ℕ
s : Super f d a✝
c : ℂ
z : S
za : ¬z = a✝
m : ¬Critical (f c) z
o : ∀ (x : ℂ × S), ¬Critical (f x.1) x.2 → ∀ᶠ (y : ℂ × S) in 𝓝 x, ¬Critical (f y.1) y.2
e : ℂ
w : S
a : ¬Critical (f (e, w).1) (e, w).2
b : Critical (f (e, w).1) (e, w).2
⊢ False | no goals |
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
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
b0 : mfderiv I I (s.bottcherNear c) ((f c)^[n] z) ≠ 0
f0 : ¬Precritical (f c) z
⊢ mfderiv I I (s.bottcherNearIter n c) z ≠ 0 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
b0 : mfderiv I I (s.bottcherNear c) ((f c)^[n] z) ≠ 0
f0 : ¬Precritical (f c) z
⊢ mfderiv I I (f c)^[n] z ≠ 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNearIter_mfderiv_ne_zero | [610, 1] | [614, 69] | contrapose f0 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
b0 : mfderiv I I (s.bottcherNear c) ((f c)^[n] z) ≠ 0
f0 : ¬Precritical (f c) z
⊢ mfderiv I I (f c)^[n] z ≠ 0 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
b0 : mfderiv I I (s.bottcherNear c) ((f c)^[n] z) ≠ 0
f0 : ¬mfderiv I I (f c)^[n] z ≠ 0
⊢ ¬¬Precritical (f c) z |
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 ⊢ | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
b0 : mfderiv I I (s.bottcherNear c) ((f c)^[n] z) ≠ 0
f0 : ¬mfderiv I I (f c)^[n] z ≠ 0
⊢ ¬¬Precritical (f c) z | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
b0 : mfderiv I I (s.bottcherNear c) ((f c)^[n] z) ≠ 0
f0 : mfderiv I I (f c)^[n] z = 0
⊢ Precritical (f c) z |
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
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
b0 : mfderiv I I (s.bottcherNear c) ((f c)^[n] z) ≠ 0
f0 : mfderiv I I (f c)^[n] z = 0
⊢ Precritical (f c) z | no goals |
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
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ NontrivialHolomorphicAt (fun z => (f c)^[n] z) a | case zero
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ NontrivialHolomorphicAt (fun z => (f c)^[0] z) a
case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
h : NontrivialHolomorphicAt (fun z => (f c)^[n] z) a
⊢ NontrivialHolomorphicAt (fun z => (f c)^[n + 1] z) a |
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
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ NontrivialHolomorphicAt (fun z => (f c)^[0] z) a
case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
h : NontrivialHolomorphicAt (fun z => (f c)^[n] z) a
⊢ NontrivialHolomorphicAt (fun z => (f c)^[n + 1] z) a | case zero
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ NontrivialHolomorphicAt (fun z => z) a
case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
h : NontrivialHolomorphicAt (fun z => (f c)^[n] z) a
⊢ NontrivialHolomorphicAt (fun z => (f c)^[n + 1] z) a |
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
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ NontrivialHolomorphicAt (fun z => z) a
case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
h : NontrivialHolomorphicAt (fun z => (f c)^[n] z) a
⊢ NontrivialHolomorphicAt (fun z => (f c)^[n + 1] z) a | case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
h : NontrivialHolomorphicAt (fun z => (f c)^[n] z) a
⊢ NontrivialHolomorphicAt (fun z => (f c)^[n + 1] z) a |
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
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
h : NontrivialHolomorphicAt (fun z => (f c)^[n] z) a
⊢ NontrivialHolomorphicAt (fun z => (f c)^[n + 1] z) a | case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
h : NontrivialHolomorphicAt (fun z => (f c)^[n] z) a
⊢ NontrivialHolomorphicAt (fun z => f c ((f c)^[n] z)) a |
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
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
h : NontrivialHolomorphicAt (fun z => (f c)^[n] z) a
⊢ NontrivialHolomorphicAt (fun z => f c ((f c)^[n] z)) a | case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
h : NontrivialHolomorphicAt (fun z => (f c)^[n] z) a
⊢ NontrivialHolomorphicAt (f c) ((f c)^[n] a) |
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
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
h : NontrivialHolomorphicAt (fun z => (f c)^[n] z) a
⊢ NontrivialHolomorphicAt (f c) ((f c)^[n] a) | case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
h : NontrivialHolomorphicAt (fun z => (f c)^[n] z) a
⊢ NontrivialHolomorphicAt (f c) a |
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
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
h : NontrivialHolomorphicAt (fun z => (f c)^[n] z) a
⊢ NontrivialHolomorphicAt (f c) a | no goals |
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
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ NontrivialHolomorphicAt (s.bottcherNear c) ((f c)^[n] a) | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ NontrivialHolomorphicAt (s.bottcherNear c) a |
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
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ NontrivialHolomorphicAt (s.bottcherNear c) a | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_summable | [47, 1] | [54, 55] | rw [summable_iff_vanishing_norm] | f : ℕ → ℂ → ℂ
s : Set ℂ
z : ℂ
zs : z ∈ s
h : UniformVanishing f s
⊢ Summable fun n => f n z | f : ℕ → ℂ → ℂ
s : Set ℂ
z : ℂ
zs : z ∈ s
h : UniformVanishing f s
⊢ ∀ ε > 0, ∃ s, ∀ (t : Finset ℕ), Disjoint t s → ‖t.sum fun i => f i z‖ < ε |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_summable | [47, 1] | [54, 55] | intro e ep | f : ℕ → ℂ → ℂ
s : Set ℂ
z : ℂ
zs : z ∈ s
h : UniformVanishing f s
⊢ ∀ ε > 0, ∃ s, ∀ (t : Finset ℕ), Disjoint t s → ‖t.sum fun i => f i z‖ < ε | f : ℕ → ℂ → ℂ
s : Set ℂ
z : ℂ
zs : z ∈ s
h : UniformVanishing f s
e : ℝ
ep : e > 0
⊢ ∃ s, ∀ (t : Finset ℕ), Disjoint t s → ‖t.sum fun i => f i z‖ < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_summable | [47, 1] | [54, 55] | rcases h e ep with ⟨m, hm⟩ | f : ℕ → ℂ → ℂ
s : Set ℂ
z : ℂ
zs : z ∈ s
h : UniformVanishing f s
e : ℝ
ep : e > 0
⊢ ∃ s, ∀ (t : Finset ℕ), Disjoint t s → ‖t.sum fun i => f i z‖ < e | case intro
f : ℕ → ℂ → ℂ
s : Set ℂ
z : ℂ
zs : z ∈ s
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e
⊢ ∃ s, ∀ (t : Finset ℕ), Disjoint t s → ‖t.sum fun i => f i z‖ < e |
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 : ℕ → ℂ → ℂ
s : Set ℂ
z : ℂ
zs : z ∈ s
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e
⊢ ∃ s, ∀ (t : Finset ℕ), Disjoint t s → ‖t.sum fun i => f i z‖ < e | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
z : ℂ
zs : z ∈ s
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e
⊢ ∀ (t : Finset ℕ), Disjoint t (Finset.range m) → ‖t.sum fun i => f i z‖ < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_summable | [47, 1] | [54, 55] | intro A d | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
z : ℂ
zs : z ∈ s
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e
⊢ ∀ (t : Finset ℕ), Disjoint t (Finset.range m) → ‖t.sum fun i => f i z‖ < e | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
z : ℂ
zs : z ∈ s
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e
A : Finset ℕ
d : Disjoint A (Finset.range m)
⊢ ‖A.sum fun i => f i z‖ < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_summable | [47, 1] | [54, 55] | calc ‖A.sum (fun n ↦ f n z)‖
_ ≤ A.sum (fun n ↦ ‖f n z‖) := by bound
_ < e := hm _ _ (late_iff_disjoint_range.mpr d) zs | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
z : ℂ
zs : z ∈ s
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e
A : Finset ℕ
d : Disjoint A (Finset.range m)
⊢ ‖A.sum fun i => f i z‖ < e | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_summable | [47, 1] | [54, 55] | bound | f : ℕ → ℂ → ℂ
s : Set ℂ
z : ℂ
zs : z ∈ s
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e
A : Finset ℕ
d : Disjoint A (Finset.range m)
⊢ ‖A.sum fun n => f n z‖ ≤ A.sum fun n => ‖f n z‖ | no goals |
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 : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
⊢ UniformCauchySeqOn (fun N z => N.sum fun n => f n z) atTop s | f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
⊢ ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (m.sum fun n => f n x) (n.sum fun n => f n x) < ε |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_uniform_cauchy_series | [57, 1] | [67, 51] | intro e ep | f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
⊢ ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (m.sum fun n => f n x) (n.sum fun n => f n x) < ε | f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
⊢ ∃ N, ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (m.sum fun n => f n x) (n.sum fun n => f n x) < e |
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 ⟨m, hm⟩ | f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
⊢ ∃ N, ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (m.sum fun n => f n x) (n.sum fun n => f n x) < e | case intro
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e
⊢ ∃ N, ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (m.sum fun n => f n x) (n.sum fun n => f n x) < e |
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 : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e
⊢ ∃ N, ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (m.sum fun n => f n x) (n.sum fun n => f n x) < e | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e
⊢ ∀ m_1 ≥ Finset.range m, ∀ n ≥ Finset.range m, ∀ x ∈ s, dist (m_1.sum fun n => f n x) (n.sum fun n => f n x) < e |
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 : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e
⊢ ∀ m_1 ≥ Finset.range m, ∀ n ≥ Finset.range m, ∀ x ∈ s, dist (m_1.sum fun n => f n x) (n.sum fun n => f n x) < e | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e
A : Finset ℕ
HA : A ≥ Finset.range m
B : Finset ℕ
HB : B ≥ Finset.range m
z : ℂ
zs : z ∈ s
⊢ dist (A.sum fun n => f n z) (B.sum fun n => f n z) < e |
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 ↦ f n z) (B.sum fun n ↦ f n z)
_ ≤ (A ∆ B).sum fun n ↦ abs (f n z) := symmDiff_bound _ _ _
_ < e := hm (A ∆ B) z (symmDiff_late HA HB) zs | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e
A : Finset ℕ
HA : A ≥ Finset.range m
B : Finset ℕ
HB : B ≥ Finset.range m
z : ℂ
zs : z ∈ s
⊢ dist (A.sum fun n => f n z) (B.sum fun n => f n z) < e | no goals |
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 : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
⊢ HasUniformSum f (tsumOn f) s | f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
⊢ ∀ ε > 0, ∀ᶠ (n : Finset ℕ) in atTop, ∀ x ∈ s, dist (tsumOn f x) (n.sum fun n => f n x) < ε |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | intro e ep | f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
⊢ ∀ ε > 0, ∀ᶠ (n : Finset ℕ) in atTop, ∀ x ∈ s, dist (tsumOn f x) (n.sum fun n => f n x) < ε | f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
⊢ ∀ᶠ (n : Finset ℕ) in atTop, ∀ x ∈ s, dist (tsumOn f x) (n.sum fun n => f n x) < e |
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 ⟨m, hm⟩ | f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
⊢ ∀ᶠ (n : Finset ℕ) in atTop, ∀ x ∈ s, dist (tsumOn f x) (n.sum fun n => f n x) < e | case intro
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
⊢ ∀ᶠ (n : Finset ℕ) in atTop, ∀ x ∈ s, dist (tsumOn f x) (n.sum fun n => f n x) < e |
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 : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
⊢ ∀ᶠ (n : Finset ℕ) in atTop, ∀ x ∈ s, dist (tsumOn f x) (n.sum fun n => f n x) < e | case intro
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
⊢ ∃ a, ∀ b ≥ a, ∀ x ∈ s, dist (tsumOn f x) (b.sum fun n => f n x) < e |
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 : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
⊢ ∃ a, ∀ b ≥ a, ∀ x ∈ s, dist (tsumOn f x) (b.sum fun n => f n x) < e | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
⊢ ∀ b ≥ Finset.range m, ∀ x ∈ s, dist (tsumOn f x) (b.sum fun n => f n x) < e |
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 : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
⊢ ∀ b ≥ Finset.range m, ∀ x ∈ s, dist (tsumOn f x) (b.sum fun n => f n x) < e | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
⊢ dist (tsumOn f z) (N.sum fun n => f n z) < e |
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 : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
⊢ dist (tsumOn f z) (N.sum fun n => f n z) < e | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
⊢ dist (∑' (n : ℕ), f n z) (N.sum fun n => f n z) < e |
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 ↦ f n z) = g | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
⊢ dist (∑' (n : ℕ), f n z) (N.sum fun n => f n z) < e | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
G : ∑' (n : ℕ), f n z = g
⊢ dist g (N.sum fun n => f n z) < e |
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 ↦ f n z) := uniformVanishing_to_summable zs h | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
G : ∑' (n : ℕ), f n z = g
⊢ dist g (N.sum fun n => f n z) < e | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
G : ∑' (n : ℕ), f n z = g
S : Summable fun n => f n z
⊢ dist g (N.sum fun n => f n z) < e |
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 ↦ f n z) g := by rw [← G]; exact Summable.hasSum S | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
G : ∑' (n : ℕ), f n z = g
S : Summable fun n => f n z
⊢ dist g (N.sum fun n => f n z) < e | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
G : ∑' (n : ℕ), f n z = g
S : Summable fun n => f n z
GS : HasSum (fun n => f n z) g
⊢ dist g (N.sum fun n => f n z) < e |
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 : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
G : ∑' (n : ℕ), f n z = g
S : Summable fun n => f n z
GS : HasSum (fun n => f n z) g
⊢ dist g (N.sum fun n => f n z) < e | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
G : ∑' (n : ℕ), f n z = g
GS : HasSum (fun n => f n z) g
⊢ dist g (N.sum fun n => f n z) < e |
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 : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
G : ∑' (n : ℕ), f n z = g
GS : HasSum (fun n => f n z) g
⊢ dist g (N.sum fun n => f n z) < e | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
G : ∑' (n : ℕ), f n z = g
GS : Filter.Tendsto (fun s => s.sum fun b => f b z) atTop (𝓝 g)
⊢ dist g (N.sum fun n => f n z) < e |
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 : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
G : ∑' (n : ℕ), f n z = g
GS : Filter.Tendsto (fun s => s.sum fun b => f b z) atTop (𝓝 g)
⊢ dist g (N.sum fun n => f n z) < e | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
G : ∑' (n : ℕ), f n z = g
GS : ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (n.sum fun b => f b z) g < ε
⊢ dist g (N.sum fun n => f n z) < e |
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 ⟨M, HM⟩ | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
G : ∑' (n : ℕ), f n z = g
GS : ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (n.sum fun b => f b z) g < ε
⊢ dist g (N.sum fun n => f n z) < e | case h.intro
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
G : ∑' (n : ℕ), f n z = g
GS : ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (n.sum fun b => f b z) g < ε
M : Finset ℕ
HM : ∀ n ≥ M, dist (n.sum fun b => f b z) g < e / 4
⊢ dist g (N.sum fun n => f n z) < e |
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 : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
G : ∑' (n : ℕ), f n z = g
GS : ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (n.sum fun b => f b z) g < ε
M : Finset ℕ
HM : ∀ n ≥ M, dist (n.sum fun b => f b z) g < e / 4
⊢ dist g (N.sum fun n => f n z) < e | case h.intro
f : ℕ → ℂ → ℂ
s : Set ℂ
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
M : Finset ℕ
HM : ∀ n ≥ M, dist (n.sum fun b => f b z) g < e / 4
⊢ dist g (N.sum fun n => f n z) < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | set A := N ∪ M \ N | case h.intro
f : ℕ → ℂ → ℂ
s : Set ℂ
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
M : Finset ℕ
HM : ∀ n ≥ M, dist (n.sum fun b => f b z) g < e / 4
⊢ dist g (N.sum fun n => f n z) < e | case h.intro
f : ℕ → ℂ → ℂ
s : Set ℂ
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
M : Finset ℕ
HM : ∀ n ≥ M, dist (n.sum fun b => f b z) g < e / 4
A : Finset ℕ := N ∪ M \ N
⊢ dist g (N.sum fun n => f n z) < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | have AM : M ⊆ A := subset_union_sdiff _ _ | case h.intro
f : ℕ → ℂ → ℂ
s : Set ℂ
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
M : Finset ℕ
HM : ∀ n ≥ M, dist (n.sum fun b => f b z) g < e / 4
A : Finset ℕ := N ∪ M \ N
⊢ dist g (N.sum fun n => f n z) < e | case h.intro
f : ℕ → ℂ → ℂ
s : Set ℂ
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
M : Finset ℕ
HM : ∀ n ≥ M, dist (n.sum fun b => f b z) g < e / 4
A : Finset ℕ := N ∪ M \ N
AM : M ⊆ A
⊢ dist g (N.sum fun n => f n z) < e |
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 : ℕ → ℂ → ℂ
s : Set ℂ
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
M : Finset ℕ
HM : ∀ n ≥ M, dist (n.sum fun b => f b z) g < e / 4
A : Finset ℕ := N ∪ M \ N
AM : M ⊆ A
⊢ dist g (N.sum fun n => f n z) < e | case h.intro
f : ℕ → ℂ → ℂ
s : Set ℂ
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
M : Finset ℕ
A : Finset ℕ := N ∪ M \ N
AM : M ⊆ A
HM : ∀ (n : Finset ℕ), M ⊆ n → dist (n.sum fun b => f b z) g < e / 4
⊢ dist g (N.sum fun n => f n z) < e |
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 : ℕ → ℂ → ℂ
s : Set ℂ
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
M : Finset ℕ
A : Finset ℕ := N ∪ M \ N
AM : M ⊆ A
HM : ∀ (n : Finset ℕ), M ⊆ n → dist (n.sum fun b => f b z) g < e / 4
⊢ dist g (N.sum fun n => f n z) < e | case h.intro
f : ℕ → ℂ → ℂ
s : Set ℂ
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
M : Finset ℕ
A : Finset ℕ := N ∪ M \ N
AM : M ⊆ A
HM : dist (A.sum fun b => f b z) g < e / 4
⊢ dist g (N.sum fun n => f n z) < e |
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 : ℕ → ℂ → ℂ
s : Set ℂ
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
M : Finset ℕ
A : Finset ℕ := N ∪ M \ N
AM : M ⊆ A
HM : dist (A.sum fun b => f b z) g < e / 4
⊢ dist g (N.sum fun n => f n z) < e | case h.intro
f : ℕ → ℂ → ℂ
s : Set ℂ
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
M : Finset ℕ
A : Finset ℕ := N ∪ M \ N
AM : M ⊆ A
HM : dist g (A.sum fun b => f b z) < e / 4
⊢ dist g (N.sum fun n => f n z) < e |
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 ↦ f n z)
_ ≤ dist g (A.sum fun n ↦ f n z) + dist (A.sum fun n ↦ f n z) (N.sum fun n ↦ f n z) := by bound
_ ≤ e / 4 + dist (A.sum fun n ↦ f n z) (N.sum fun n ↦ f n z) := by linarith
_ = 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) := by rw [Finset.sum_union Finset.disjoint_sdiff]
_ = e / 4 + abs (((N.sum fun n ↦ f n z) + (M \ N).sum fun n ↦ f n z) -
N.sum fun n ↦ f n z) := by rw [Complex.dist_eq]
_ = e / 4 + abs ((M \ N).sum fun n ↦ f n z) := by ring_nf
_ ≤ e / 4 + (M \ N).sum fun n ↦ abs (f n z) := by
linarith [finset_complex_abs_sum_le (M \ N) fun n ↦ f n z]
_ ≤ 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 : ℕ → ℂ → ℂ
s : Set ℂ
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
M : Finset ℕ
A : Finset ℕ := N ∪ M \ N
AM : M ⊆ A
HM : dist g (A.sum fun b => f b z) < e / 4
⊢ dist g (N.sum fun n => f n z) < e | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | linarith | f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
⊢ e / 4 > 0 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | rw [← G] | f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
G : ∑' (n : ℕ), f n z = g
S : Summable fun n => f n z
⊢ HasSum (fun n => f n z) g | f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
G : ∑' (n : ℕ), f n z = g
S : Summable fun n => f n z
⊢ HasSum (fun n => f n z) (∑' (n : ℕ), f n z) |
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 : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
G : ∑' (n : ℕ), f n z = g
S : Summable fun n => f n z
⊢ HasSum (fun n => f n z) (∑' (n : ℕ), f n z) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | linarith | f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
G : ∑' (n : ℕ), f n z = g
GS : ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (n.sum fun b => f b z) g < ε
⊢ e / 4 > 0 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | bound | f : ℕ → ℂ → ℂ
s : Set ℂ
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
M : Finset ℕ
A : Finset ℕ := N ∪ M \ N
AM : M ⊆ A
HM : dist g (A.sum fun b => f b z) < e / 4
⊢ dist g (N.sum fun n => f n z) ≤ 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 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | linarith | f : ℕ → ℂ → ℂ
s : Set ℂ
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
M : Finset ℕ
A : Finset ℕ := N ∪ M \ N
AM : M ⊆ A
HM : dist g (A.sum fun b => f b z) < e / 4
⊢ dist g (A.sum fun n => f n z) + dist (A.sum fun n => f n z) (N.sum fun n => f n z) ≤
e / 4 + dist (A.sum fun n => f n z) (N.sum fun n => f n z) | no goals |
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 : ℕ → ℂ → ℂ
s : Set ℂ
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
M : Finset ℕ
A : Finset ℕ := N ∪ M \ N
AM : M ⊆ A
HM : dist g (A.sum fun b => f b z) < e / 4
⊢ 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 |
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 : ℕ → ℂ → ℂ
s : Set ℂ
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
M : Finset ℕ
A : Finset ℕ := N ∪ M \ N
AM : M ⊆ A
HM : dist g (A.sum fun b => f b z) < e / 4
⊢ 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 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | ring_nf | f : ℕ → ℂ → ℂ
s : Set ℂ
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
M : Finset ℕ
A : Finset ℕ := N ∪ M \ N
AM : M ⊆ A
HM : dist g (A.sum fun b => f b z) < e / 4
⊢ 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 |
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 ↦ f n z] | f : ℕ → ℂ → ℂ
s : Set ℂ
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
M : Finset ℕ
A : Finset ℕ := N ∪ M \ N
AM : M ⊆ A
HM : dist g (A.sum fun b => f b z) < e / 4
⊢ e / 4 + Complex.abs ((M \ N).sum fun n => f n z) ≤ e / 4 + (M \ N).sum fun n => Complex.abs (f n z) | no goals |
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 : ℕ → ℂ → ℂ
s : Set ℂ
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
M : Finset ℕ
A : Finset ℕ := N ∪ M \ N
AM : M ⊆ A
HM : dist g (A.sum fun b => f b z) < e / 4
⊢ (e / 4 + (M \ N).sum fun n => Complex.abs (f n z)) ≤ e / 4 + e / 4 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | ring | f : ℕ → ℂ → ℂ
s : Set ℂ
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
M : Finset ℕ
A : Finset ℕ := N ∪ M \ N
AM : M ⊆ A
HM : dist g (A.sum fun b => f b z) < e / 4
⊢ e / 4 + e / 4 = e / 2 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | linarith | f : ℕ → ℂ → ℂ
s : Set ℂ
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
M : Finset ℕ
A : Finset ℕ := N ∪ M \ N
AM : M ⊆ A
HM : dist g (A.sum fun b => f b z) < e / 4
⊢ e / 2 < e | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | CNonpos.degenerate | [106, 1] | [110, 85] | intro n z zs | f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
c0 : c ≤ 0
a0 : 0 ≤ a
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
⊢ ∀ (n : ℕ), ∀ z ∈ s, f n z = 0 | f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
c0 : c ≤ 0
a0 : 0 ≤ a
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
n : ℕ
z : ℂ
zs : z ∈ s
⊢ f n z = 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | CNonpos.degenerate | [106, 1] | [110, 85] | specialize hf n z zs | f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
c0 : c ≤ 0
a0 : 0 ≤ a
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
n : ℕ
z : ℂ
zs : z ∈ s
⊢ f n z = 0 | f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
c0 : c ≤ 0
a0 : 0 ≤ a
n : ℕ
z : ℂ
zs : z ∈ s
hf : Complex.abs (f n z) ≤ c * a ^ n
⊢ f n z = 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | CNonpos.degenerate | [106, 1] | [110, 85] | have ca : c * a ^ n ≤ 0 := mul_nonpos_iff.mpr (Or.inr ⟨c0, by bound⟩) | f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
c0 : c ≤ 0
a0 : 0 ≤ a
n : ℕ
z : ℂ
zs : z ∈ s
hf : Complex.abs (f n z) ≤ c * a ^ n
⊢ f n z = 0 | f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
c0 : c ≤ 0
a0 : 0 ≤ a
n : ℕ
z : ℂ
zs : z ∈ s
hf : Complex.abs (f n z) ≤ c * a ^ n
ca : c * a ^ n ≤ 0
⊢ f n z = 0 |
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 : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
c0 : c ≤ 0
a0 : 0 ≤ a
n : ℕ
z : ℂ
zs : z ∈ s
hf : Complex.abs (f n z) ≤ c * a ^ n
ca : c * a ^ n ≤ 0
⊢ f n z = 0 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | CNonpos.degenerate | [106, 1] | [110, 85] | bound | f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
c0 : c ≤ 0
a0 : 0 ≤ a
n : ℕ
z : ℂ
zs : z ∈ s
hf : Complex.abs (f n z) ≤ c * a ^ n
⊢ 0 ≤ a ^ n | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | by_cases c0 : c ≤ 0 | f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
⊢ HasUniformSum f (tsumOn f) s | case pos
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : c ≤ 0
⊢ HasUniformSum f (tsumOn f) s
case neg
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : ¬c ≤ 0
⊢ HasUniformSum f (tsumOn f) s |
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 : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : c ≤ 0
⊢ HasUniformSum f (tsumOn f) s | case pos
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : c ≤ 0
fz : ∀ (n : ℕ), ∀ z ∈ s, f n z = 0
⊢ HasUniformSum f (tsumOn f) s |
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 : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : c ≤ 0
fz : ∀ (n : ℕ), ∀ z ∈ s, f n z = 0
⊢ HasUniformSum f (tsumOn f) s | case pos
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : c ≤ 0
fz : ∀ (n : ℕ), ∀ z ∈ s, f n z = 0
⊢ ∀ ε > 0, ∀ᶠ (n : Finset ℕ) in atTop, ∀ x ∈ s, dist (tsumOn f x) (n.sum fun n => f n x) < ε |
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 : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : c ≤ 0
fz : ∀ (n : ℕ), ∀ z ∈ s, f n z = 0
⊢ ∀ ε > 0, ∀ᶠ (n : Finset ℕ) in atTop, ∀ x ∈ s, dist (tsumOn f x) (n.sum fun n => f n x) < ε | case pos
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : c ≤ 0
fz : ∀ (n : ℕ), ∀ z ∈ s, f n z = 0
e : ℝ
ep : e > 0
⊢ ∀ᶠ (n : Finset ℕ) in atTop, ∀ x ∈ s, dist (tsumOn f x) (n.sum fun n => f n x) < e |
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 : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : c ≤ 0
fz : ∀ (n : ℕ), ∀ z ∈ s, f n z = 0
e : ℝ
ep : e > 0
⊢ ∀ᶠ (n : Finset ℕ) in atTop, ∀ x ∈ s, dist (tsumOn f x) (n.sum fun n => f n x) < e | case pos.hp
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : c ≤ 0
fz : ∀ (n : ℕ), ∀ z ∈ s, f n z = 0
e : ℝ
ep : e > 0
⊢ ∀ (x : Finset ℕ), ∀ x_1 ∈ s, dist (tsumOn f x_1) (x.sum fun n => f n x_1) < e |
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 : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : c ≤ 0
fz : ∀ (n : ℕ), ∀ z ∈ s, f n z = 0
e : ℝ
ep : e > 0
⊢ ∀ (x : Finset ℕ), ∀ x_1 ∈ s, dist (tsumOn f x_1) (x.sum fun n => f n x_1) < e | case pos.hp
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : c ≤ 0
fz : ∀ (n : ℕ), ∀ z ∈ s, f n z = 0
e : ℝ
ep : e > 0
n : Finset ℕ
z : ℂ
zs : z ∈ s
⊢ dist (tsumOn f z) (n.sum fun n => f n z) < e |
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 : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : c ≤ 0
fz : ∀ (n : ℕ), ∀ z ∈ s, f n z = 0
e : ℝ
ep : e > 0
n : Finset ℕ
z : ℂ
zs : z ∈ s
⊢ dist (tsumOn f z) (n.sum fun n => f n z) < e | case pos.hp
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : c ≤ 0
fz : ∀ (n : ℕ), ∀ z ∈ s, f n z = 0
e : ℝ
ep : e > 0
n : Finset ℕ
z : ℂ
zs : z ∈ s
⊢ dist (∑' (n : ℕ), f n z) (n.sum fun n => f n z) < e |
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 : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : c ≤ 0
fz : ∀ (n : ℕ), ∀ z ∈ s, f n z = 0
e : ℝ
ep : e > 0
n : Finset ℕ
z : ℂ
zs : z ∈ s
⊢ dist (∑' (n : ℕ), f n z) (n.sum fun n => f n z) < e | case pos.hp
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : c ≤ 0
fz : ∀ (n : ℕ), ∀ z ∈ s, f n z = 0
e : ℝ
ep : e > 0
n : Finset ℕ
z : ℂ
zs : z ∈ s
⊢ dist (∑' (n : ℕ), 0) (n.sum fun n => 0) < e |
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 : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : c ≤ 0
fz : ∀ (n : ℕ), ∀ z ∈ s, f n z = 0
e : ℝ
ep : e > 0
n : Finset ℕ
z : ℂ
zs : z ∈ s
⊢ dist (∑' (n : ℕ), 0) (n.sum fun n => 0) < e | case pos.hp
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : c ≤ 0
fz : ∀ (n : ℕ), ∀ z ∈ s, f n z = 0
e : ℝ
ep : e > 0
n : Finset ℕ
z : ℂ
zs : z ∈ s
⊢ 0 < e |
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 : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : c ≤ 0
fz : ∀ (n : ℕ), ∀ z ∈ s, f n z = 0
e : ℝ
ep : e > 0
n : Finset ℕ
z : ℂ
zs : z ∈ s
⊢ 0 < e | no goals |
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 : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : ¬c ≤ 0
⊢ HasUniformSum f (tsumOn f) s | case neg
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : 0 < c
⊢ HasUniformSum f (tsumOn f) s |
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 : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : 0 < c
⊢ HasUniformSum f (tsumOn f) s | case neg.h
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : 0 < c
⊢ UniformVanishing f s |
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 : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : 0 < c
⊢ UniformVanishing f s | case neg.h
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : 0 < c
e : ℝ
ep : e > 0
⊢ ∃ n, ∀ (N : Finset ℕ) (z : ℂ), Late N n → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | set t := (1 - ↑a) / ↑c * (e / 2) | case neg.h
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : 0 < c
e : ℝ
ep : e > 0
⊢ ∃ n, ∀ (N : Finset ℕ) (z : ℂ), Late N n → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e | case neg.h
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : 0 < c
e : ℝ
ep : e > 0
t : ℝ := (1 - a) / c * (e / 2)
⊢ ∃ n, ∀ (N : Finset ℕ) (z : ℂ), Late N n → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e |
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 : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : 0 < c
e : ℝ
ep : e > 0
t : ℝ := (1 - a) / c * (e / 2)
⊢ ∃ n, ∀ (N : Finset ℕ) (z : ℂ), Late N n → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e | case neg.h
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : 0 < c
e : ℝ
ep : e > 0
t : ℝ := (1 - a) / c * (e / 2)
tp : t > 0
⊢ ∃ n, ∀ (N : Finset ℕ) (z : ℂ), Late N n → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e |
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 ⟨n, nt⟩ | case neg.h
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : 0 < c
e : ℝ
ep : e > 0
t : ℝ := (1 - a) / c * (e / 2)
tp : t > 0
⊢ ∃ n, ∀ (N : Finset ℕ) (z : ℂ), Late N n → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e | case neg.h.intro
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : 0 < c
e : ℝ
ep : e > 0
t : ℝ := (1 - a) / c * (e / 2)
tp : t > 0
n : ℕ
nt : a ^ n < t
⊢ ∃ n, ∀ (N : Finset ℕ) (z : ℂ), Late N n → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e |
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