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stringlengths 1
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stringlengths 6
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https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.ball_diff_center | [177, 1] | [196, 44] | simp only [Metric.ball_eq_empty.mpr rp, empty_diff] | case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : r ≤ 0
⊢ IsPreconnected (ball a r \ {a})
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : ¬r ≤ 0
⊢ IsPreconnected (ball a r \ {a}) | case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : r ≤ 0
⊢ IsPreconnected ∅
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : ¬r ≤ 0
⊢ IsPreconnected (ball a r \ {a}) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : r ≤ 0
⊢ IsPreconnected (ball a r \ {a})
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : ¬r ≤ 0
⊢ IsPreconnected (ball a r \ {a})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.ball_diff_center | [177, 1] | [196, 44] | exact isPreconnected_empty | case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : r ≤ 0
⊢ IsPreconnected ∅
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : ¬r ≤ 0
⊢ IsPreconnected (ball a r \ {a}) | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : ¬r ≤ 0
⊢ IsPreconnected (ball a r \ {a}) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : r ≤ 0
⊢ IsPreconnected ∅
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : ¬r ≤ 0
⊢ IsPreconnected (ball a r \ {a})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.ball_diff_center | [177, 1] | [196, 44] | simp only [not_le] at rp | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : ¬r ≤ 0
⊢ IsPreconnected (ball a r \ {a}) | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
⊢ IsPreconnected (ball a r \ {a}) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : ¬r ≤ 0
⊢ IsPreconnected (ball a r \ {a})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.ball_diff_center | [177, 1] | [196, 44] | rw [e] | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
e : ball a r \ {a} = (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ
⊢ IsPreconnected (ball a r \ {a}) | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
e : ball a r \ {a} = (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ
⊢ IsPreconnected ((fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
e : ball a r \ {a} = (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ
⊢ IsPreconnected (ball a r \ {a})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.ball_diff_center | [177, 1] | [196, 44] | apply IsPreconnected.image | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
e : ball a r \ {a} = (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ
⊢ IsPreconnected ((fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ) | case neg.H
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
e : ball a r \ {a} = (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ
⊢ IsPreconnected (Ioo 0 r ×ˢ univ)
case neg.hf
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
e : ball a r \ {a} = (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ
⊢ ContinuousOn (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) (Ioo 0 r ×ˢ univ) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
e : ball a r \ {a} = (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ
⊢ IsPreconnected ((fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.ball_diff_center | [177, 1] | [196, 44] | exact isPreconnected_Ioo.prod isPreconnected_univ | case neg.H
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
e : ball a r \ {a} = (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ
⊢ IsPreconnected (Ioo 0 r ×ˢ univ)
case neg.hf
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
e : ball a r \ {a} = (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ
⊢ ContinuousOn (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) (Ioo 0 r ×ˢ univ) | case neg.hf
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
e : ball a r \ {a} = (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ
⊢ ContinuousOn (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) (Ioo 0 r ×ˢ univ) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.H
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
e : ball a r \ {a} = (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ
⊢ IsPreconnected (Ioo 0 r ×ˢ univ)
case neg.hf
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
e : ball a r \ {a} = (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ
⊢ ContinuousOn (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) (Ioo 0 r ×ˢ univ)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.ball_diff_center | [177, 1] | [196, 44] | apply Continuous.continuousOn | case neg.hf
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
e : ball a r \ {a} = (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ
⊢ ContinuousOn (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) (Ioo 0 r ×ˢ univ) | case neg.hf.h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
e : ball a r \ {a} = (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ
⊢ Continuous fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.hf
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
e : ball a r \ {a} = (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ
⊢ ContinuousOn (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) (Ioo 0 r ×ˢ univ)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.ball_diff_center | [177, 1] | [196, 44] | continuity | case neg.hf.h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
e : ball a r \ {a} = (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ
⊢ Continuous fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.hf.h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
e : ball a r \ {a} = (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ
⊢ Continuous fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.ball_diff_center | [177, 1] | [196, 44] | apply Set.ext | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
⊢ ball a r \ {a} = (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
⊢ ∀ (x : ℂ), x ∈ ball a r \ {a} ↔ x ∈ (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
⊢ ball a r \ {a} = (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.ball_diff_center | [177, 1] | [196, 44] | intro z | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
⊢ ∀ (x : ℂ), x ∈ ball a r \ {a} ↔ x ∈ (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
⊢ z ∈ ball a r \ {a} ↔ z ∈ (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
⊢ ∀ (x : ℂ), x ∈ ball a r \ {a} ↔ x ∈ (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.ball_diff_center | [177, 1] | [196, 44] | simp only [mem_diff, mem_ball, Complex.dist_eq, mem_singleton_iff, mem_image, Prod.exists,
mem_prod_eq, mem_Ioo, mem_univ, and_true_iff] | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
⊢ z ∈ ball a r \ {a} ↔ z ∈ (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
⊢ Complex.abs (z - a) < r ∧ ¬z = a ↔ ∃ a_1 b, (0 < a_1 ∧ a_1 < r) ∧ a + ↑a_1 * (↑b * Complex.I).exp = z | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
⊢ z ∈ ball a r \ {a} ↔ z ∈ (fun p => a + ↑p.1 * (↑p.2 * Complex.I).exp) '' Ioo 0 r ×ˢ univ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.ball_diff_center | [177, 1] | [196, 44] | constructor | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
⊢ Complex.abs (z - a) < r ∧ ¬z = a ↔ ∃ a_1 b, (0 < a_1 ∧ a_1 < r) ∧ a + ↑a_1 * (↑b * Complex.I).exp = z | case h.mp
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
⊢ Complex.abs (z - a) < r ∧ ¬z = a → ∃ a_2 b, (0 < a_2 ∧ a_2 < r) ∧ a + ↑a_2 * (↑b * Complex.I).exp = z
case h.mpr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
⊢ (∃ a_1 b, (0 < a_1 ∧ a_1 < r) ∧ a + ↑a_1 * (↑b * Complex.I).exp = z) → Complex.abs (z - a) < r ∧ ¬z = a | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
⊢ Complex.abs (z - a) < r ∧ ¬z = a ↔ ∃ a_1 b, (0 < a_1 ∧ a_1 < r) ∧ a + ↑a_1 * (↑b * Complex.I).exp = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.ball_diff_center | [177, 1] | [196, 44] | intro ⟨zr, za⟩ | case h.mp
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
⊢ Complex.abs (z - a) < r ∧ ¬z = a → ∃ a_2 b, (0 < a_2 ∧ a_2 < r) ∧ a + ↑a_2 * (↑b * Complex.I).exp = z | case h.mp
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
zr : Complex.abs (z - a) < r
za : ¬z = a
⊢ ∃ a_1 b, (0 < a_1 ∧ a_1 < r) ∧ a + ↑a_1 * (↑b * Complex.I).exp = z | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
⊢ Complex.abs (z - a) < r ∧ ¬z = a → ∃ a_2 b, (0 < a_2 ∧ a_2 < r) ∧ a + ↑a_2 * (↑b * Complex.I).exp = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.ball_diff_center | [177, 1] | [196, 44] | use abs (z - a), Complex.arg (z - a) | case h.mp
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
zr : Complex.abs (z - a) < r
za : ¬z = a
⊢ ∃ a_1 b, (0 < a_1 ∧ a_1 < r) ∧ a + ↑a_1 * (↑b * Complex.I).exp = z | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
zr : Complex.abs (z - a) < r
za : ¬z = a
⊢ (0 < Complex.abs (z - a) ∧ Complex.abs (z - a) < r) ∧ a + ↑(Complex.abs (z - a)) * (↑(z - a).arg * Complex.I).exp = z | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
zr : Complex.abs (z - a) < r
za : ¬z = a
⊢ ∃ a_1 b, (0 < a_1 ∧ a_1 < r) ∧ a + ↑a_1 * (↑b * Complex.I).exp = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.ball_diff_center | [177, 1] | [196, 44] | simp only [AbsoluteValue.pos_iff, Ne, Complex.abs_mul_exp_arg_mul_I,
add_sub_cancel, eq_self_iff_true, sub_eq_zero, za, zr, not_false_iff, and_true_iff] | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
zr : Complex.abs (z - a) < r
za : ¬z = a
⊢ (0 < Complex.abs (z - a) ∧ Complex.abs (z - a) < r) ∧ a + ↑(Complex.abs (z - a)) * (↑(z - a).arg * Complex.I).exp = z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
zr : Complex.abs (z - a) < r
za : ¬z = a
⊢ (0 < Complex.abs (z - a) ∧ Complex.abs (z - a) < r) ∧ a + ↑(Complex.abs (z - a)) * (↑(z - a).arg * Complex.I).exp = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.ball_diff_center | [177, 1] | [196, 44] | intro ⟨s, t, ⟨s0, sr⟩, e⟩ | case h.mpr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
⊢ (∃ a_1 b, (0 < a_1 ∧ a_1 < r) ∧ a + ↑a_1 * (↑b * Complex.I).exp = z) → Complex.abs (z - a) < r ∧ ¬z = a | case h.mpr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
s t : ℝ
s0 : 0 < s
sr : s < r
e : a + ↑s * (↑t * Complex.I).exp = z
⊢ Complex.abs (z - a) < r ∧ ¬z = a | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
⊢ (∃ a_1 b, (0 < a_1 ∧ a_1 < r) ∧ a + ↑a_1 * (↑b * Complex.I).exp = z) → Complex.abs (z - a) < r ∧ ¬z = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.ball_diff_center | [177, 1] | [196, 44] | simp only [← e, add_sub_cancel_left, Complex.abs.map_mul, Complex.abs_ofReal, abs_of_pos s0,
Complex.abs_exp_ofReal_mul_I, mul_one, sr, true_and_iff, add_right_eq_self, mul_eq_zero,
Complex.exp_ne_zero, or_false_iff, Complex.ofReal_eq_zero] | case h.mpr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
s t : ℝ
s0 : 0 < s
sr : s < r
e : a + ↑s * (↑t * Complex.I).exp = z
⊢ Complex.abs (z - a) < r ∧ ¬z = a | case h.mpr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
s t : ℝ
s0 : 0 < s
sr : s < r
e : a + ↑s * (↑t * Complex.I).exp = z
⊢ ¬s = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
s t : ℝ
s0 : 0 < s
sr : s < r
e : a + ↑s * (↑t * Complex.I).exp = z
⊢ Complex.abs (z - a) < r ∧ ¬z = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.ball_diff_center | [177, 1] | [196, 44] | exact s0.ne' | case h.mpr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
s t : ℝ
s0 : 0 < s
sr : s < r
e : a + ↑s * (↑t * Complex.I).exp = z
⊢ ¬s = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
r : ℝ
rp : 0 < r
z : ℂ
s t : ℝ
s0 : 0 < s
sr : s < r
e : a + ↑s * (↑t * Complex.I).exp = z
⊢ ¬s = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Complex.nonseparating_singleton | [199, 1] | [217, 70] | rw [dense_iff_inter_open] | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
⊢ Dense {a}ᶜ | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
⊢ ∀ (U : Set ℂ), IsOpen U → U.Nonempty → (U ∩ {a}ᶜ).Nonempty | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
⊢ Dense {a}ᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Complex.nonseparating_singleton | [199, 1] | [217, 70] | intro u uo ⟨z, m⟩ | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
⊢ ∀ (U : Set ℂ), IsOpen U → U.Nonempty → (U ∩ {a}ᶜ).Nonempty | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
z : ℂ
m : z ∈ u
⊢ (u ∩ {a}ᶜ).Nonempty | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
⊢ ∀ (U : Set ℂ), IsOpen U → U.Nonempty → (U ∩ {a}ᶜ).Nonempty
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Complex.nonseparating_singleton | [199, 1] | [217, 70] | by_cases za : z ≠ a | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
z : ℂ
m : z ∈ u
⊢ (u ∩ {a}ᶜ).Nonempty | case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
z : ℂ
m : z ∈ u
za : z ≠ a
⊢ (u ∩ {a}ᶜ).Nonempty
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
z : ℂ
m : z ∈ u
za : ¬z ≠ a
⊢ (u ∩ {a}ᶜ).Nonempty | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
z : ℂ
m : z ∈ u
⊢ (u ∩ {a}ᶜ).Nonempty
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Complex.nonseparating_singleton | [199, 1] | [217, 70] | simp only [not_not] at za | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
z : ℂ
m : z ∈ u
za : ¬z ≠ a
⊢ (u ∩ {a}ᶜ).Nonempty | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
z : ℂ
m : z ∈ u
za : z = a
⊢ (u ∩ {a}ᶜ).Nonempty | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
z : ℂ
m : z ∈ u
za : ¬z ≠ a
⊢ (u ∩ {a}ᶜ).Nonempty
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Complex.nonseparating_singleton | [199, 1] | [217, 70] | rw [za] at m | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
z : ℂ
m : z ∈ u
za : z = a
⊢ (u ∩ {a}ᶜ).Nonempty | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
z : ℂ
m : a ∈ u
za : z = a
⊢ (u ∩ {a}ᶜ).Nonempty | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
z : ℂ
m : z ∈ u
za : z = a
⊢ (u ∩ {a}ᶜ).Nonempty
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Complex.nonseparating_singleton | [199, 1] | [217, 70] | clear za z | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
z : ℂ
m : a ∈ u
za : z = a
⊢ (u ∩ {a}ᶜ).Nonempty | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
m : a ∈ u
⊢ (u ∩ {a}ᶜ).Nonempty | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
z : ℂ
m : a ∈ u
za : z = a
⊢ (u ∩ {a}ᶜ).Nonempty
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Complex.nonseparating_singleton | [199, 1] | [217, 70] | rcases Metric.isOpen_iff.mp uo a m with ⟨r, rp, rs⟩ | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
m : a ∈ u
⊢ (u ∩ {a}ᶜ).Nonempty | case neg.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
m : a ∈ u
r : ℝ
rp : r > 0
rs : ball a r ⊆ u
⊢ (u ∩ {a}ᶜ).Nonempty | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
m : a ∈ u
⊢ (u ∩ {a}ᶜ).Nonempty
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Complex.nonseparating_singleton | [199, 1] | [217, 70] | use a + r / 2 | case neg.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
m : a ∈ u
r : ℝ
rp : r > 0
rs : ball a r ⊆ u
⊢ (u ∩ {a}ᶜ).Nonempty | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
m : a ∈ u
r : ℝ
rp : r > 0
rs : ball a r ⊆ u
⊢ a + ↑r / 2 ∈ u ∩ {a}ᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
m : a ∈ u
r : ℝ
rp : r > 0
rs : ball a r ⊆ u
⊢ (u ∩ {a}ᶜ).Nonempty
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Complex.nonseparating_singleton | [199, 1] | [217, 70] | simp only [mem_inter_iff, mem_compl_iff, mem_singleton_iff, add_right_eq_self,
div_eq_zero_iff, Complex.ofReal_eq_zero, bit0_eq_zero, one_ne_zero, or_false_iff,
rp.ne', not_false_iff, and_true_iff, false_or, two_ne_zero] | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
m : a ∈ u
r : ℝ
rp : r > 0
rs : ball a r ⊆ u
⊢ a + ↑r / 2 ∈ u ∩ {a}ᶜ | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
m : a ∈ u
r : ℝ
rp : r > 0
rs : ball a r ⊆ u
⊢ a + ↑r / 2 ∈ u | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
m : a ∈ u
r : ℝ
rp : r > 0
rs : ball a r ⊆ u
⊢ a + ↑r / 2 ∈ u ∩ {a}ᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Complex.nonseparating_singleton | [199, 1] | [217, 70] | apply rs | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
m : a ∈ u
r : ℝ
rp : r > 0
rs : ball a r ⊆ u
⊢ a + ↑r / 2 ∈ u | case h.a
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
m : a ∈ u
r : ℝ
rp : r > 0
rs : ball a r ⊆ u
⊢ a + ↑r / 2 ∈ ball a r | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
m : a ∈ u
r : ℝ
rp : r > 0
rs : ball a r ⊆ u
⊢ a + ↑r / 2 ∈ u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Complex.nonseparating_singleton | [199, 1] | [217, 70] | simp only [mem_ball, dist_self_add_left, Complex.norm_eq_abs, map_div₀, Complex.abs_ofReal,
Complex.abs_two, abs_of_pos rp, half_lt_self rp] | case h.a
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
m : a ∈ u
r : ℝ
rp : r > 0
rs : ball a r ⊆ u
⊢ a + ↑r / 2 ∈ ball a r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
m : a ∈ u
r : ℝ
rp : r > 0
rs : ball a r ⊆ u
⊢ a + ↑r / 2 ∈ ball a r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Complex.nonseparating_singleton | [199, 1] | [217, 70] | use z | case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
z : ℂ
m : z ∈ u
za : z ≠ a
⊢ (u ∩ {a}ᶜ).Nonempty | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
z : ℂ
m : z ∈ u
za : z ≠ a
⊢ z ∈ u ∩ {a}ᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
z : ℂ
m : z ∈ u
za : z ≠ a
⊢ (u ∩ {a}ᶜ).Nonempty
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Complex.nonseparating_singleton | [199, 1] | [217, 70] | use m | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
z : ℂ
m : z ∈ u
za : z ≠ a
⊢ z ∈ u ∩ {a}ᶜ | case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
z : ℂ
m : z ∈ u
za : z ≠ a
⊢ z ∈ {a}ᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
z : ℂ
m : z ∈ u
za : z ≠ a
⊢ z ∈ u ∩ {a}ᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Complex.nonseparating_singleton | [199, 1] | [217, 70] | exact za | case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
z : ℂ
m : z ∈ u
za : z ≠ a
⊢ z ∈ {a}ᶜ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
uo : IsOpen u
z : ℂ
m : z ∈ u
za : z ≠ a
⊢ z ∈ {a}ᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Complex.nonseparating_singleton | [199, 1] | [217, 70] | intro z u m n | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
⊢ ∀ (x : ℂ) (u : Set ℂ), x ∈ {a} → u ∈ 𝓝 x → ∃ c ⊆ u \ {a}, c ∈ 𝓝[{a}ᶜ] x ∧ IsPreconnected c | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : ℂ
u : Set ℂ
m : z ∈ {a}
n : u ∈ 𝓝 z
⊢ ∃ c ⊆ u \ {a}, c ∈ 𝓝[{a}ᶜ] z ∧ IsPreconnected c | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
⊢ ∀ (x : ℂ) (u : Set ℂ), x ∈ {a} → u ∈ 𝓝 x → ∃ c ⊆ u \ {a}, c ∈ 𝓝[{a}ᶜ] x ∧ IsPreconnected c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Complex.nonseparating_singleton | [199, 1] | [217, 70] | simp only [mem_singleton_iff] at m | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : ℂ
u : Set ℂ
m : z ∈ {a}
n : u ∈ 𝓝 z
⊢ ∃ c ⊆ u \ {a}, c ∈ 𝓝[{a}ᶜ] z ∧ IsPreconnected c | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : ℂ
u : Set ℂ
n : u ∈ 𝓝 z
m : z = a
⊢ ∃ c ⊆ u \ {a}, c ∈ 𝓝[{a}ᶜ] z ∧ IsPreconnected c | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : ℂ
u : Set ℂ
m : z ∈ {a}
n : u ∈ 𝓝 z
⊢ ∃ c ⊆ u \ {a}, c ∈ 𝓝[{a}ᶜ] z ∧ IsPreconnected c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Complex.nonseparating_singleton | [199, 1] | [217, 70] | simp only [m] at n ⊢ | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : ℂ
u : Set ℂ
n : u ∈ 𝓝 z
m : z = a
⊢ ∃ c ⊆ u \ {a}, c ∈ 𝓝[{a}ᶜ] z ∧ IsPreconnected c | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : ℂ
u : Set ℂ
m : z = a
n : u ∈ 𝓝 a
⊢ ∃ c ⊆ u \ {a}, c ∈ 𝓝[≠] a ∧ IsPreconnected c | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : ℂ
u : Set ℂ
n : u ∈ 𝓝 z
m : z = a
⊢ ∃ c ⊆ u \ {a}, c ∈ 𝓝[{a}ᶜ] z ∧ IsPreconnected c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Complex.nonseparating_singleton | [199, 1] | [217, 70] | clear m z | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : ℂ
u : Set ℂ
m : z = a
n : u ∈ 𝓝 a
⊢ ∃ c ⊆ u \ {a}, c ∈ 𝓝[≠] a ∧ IsPreconnected c | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
n : u ∈ 𝓝 a
⊢ ∃ c ⊆ u \ {a}, c ∈ 𝓝[≠] a ∧ IsPreconnected c | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : ℂ
u : Set ℂ
m : z = a
n : u ∈ 𝓝 a
⊢ ∃ c ⊆ u \ {a}, c ∈ 𝓝[≠] a ∧ IsPreconnected c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Complex.nonseparating_singleton | [199, 1] | [217, 70] | rcases Metric.mem_nhds_iff.mp n with ⟨r, rp, rs⟩ | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
n : u ∈ 𝓝 a
⊢ ∃ c ⊆ u \ {a}, c ∈ 𝓝[≠] a ∧ IsPreconnected c | case intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
n : u ∈ 𝓝 a
r : ℝ
rp : r > 0
rs : ball a r ⊆ u
⊢ ∃ c ⊆ u \ {a}, c ∈ 𝓝[≠] a ∧ IsPreconnected c | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
n : u ∈ 𝓝 a
⊢ ∃ c ⊆ u \ {a}, c ∈ 𝓝[≠] a ∧ IsPreconnected c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Complex.nonseparating_singleton | [199, 1] | [217, 70] | use ball a r \ {a} | case intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
n : u ∈ 𝓝 a
r : ℝ
rp : r > 0
rs : ball a r ⊆ u
⊢ ∃ c ⊆ u \ {a}, c ∈ 𝓝[≠] a ∧ IsPreconnected c | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
n : u ∈ 𝓝 a
r : ℝ
rp : r > 0
rs : ball a r ⊆ u
⊢ ball a r \ {a} ⊆ u \ {a} ∧ ball a r \ {a} ∈ 𝓝[≠] a ∧ IsPreconnected (ball a r \ {a}) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
n : u ∈ 𝓝 a
r : ℝ
rp : r > 0
rs : ball a r ⊆ u
⊢ ∃ c ⊆ u \ {a}, c ∈ 𝓝[≠] a ∧ IsPreconnected c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Complex.nonseparating_singleton | [199, 1] | [217, 70] | refine ⟨diff_subset_diff_left rs, ?_, IsPreconnected.ball_diff_center⟩ | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
n : u ∈ 𝓝 a
r : ℝ
rp : r > 0
rs : ball a r ⊆ u
⊢ ball a r \ {a} ⊆ u \ {a} ∧ ball a r \ {a} ∈ 𝓝[≠] a ∧ IsPreconnected (ball a r \ {a}) | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
n : u ∈ 𝓝 a
r : ℝ
rp : r > 0
rs : ball a r ⊆ u
⊢ ball a r \ {a} ∈ 𝓝[≠] a | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
n : u ∈ 𝓝 a
r : ℝ
rp : r > 0
rs : ball a r ⊆ u
⊢ ball a r \ {a} ⊆ u \ {a} ∧ ball a r \ {a} ∈ 𝓝[≠] a ∧ IsPreconnected (ball a r \ {a})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Complex.nonseparating_singleton | [199, 1] | [217, 70] | exact diff_mem_nhdsWithin_compl (Metric.ball_mem_nhds _ rp) _ | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
n : u ∈ 𝓝 a
r : ℝ
rp : r > 0
rs : ball a r ⊆ u
⊢ ball a r \ {a} ∈ 𝓝[≠] a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : ℂ
u : Set ℂ
n : u ∈ 𝓝 a
r : ℝ
rp : r > 0
rs : ball a r ⊆ u
⊢ ball a r \ {a} ∈ 𝓝[≠] a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | AnalyticManifold.nonseparating_singleton | [220, 1] | [230, 38] | apply Nonseparating.complexManifold | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : S
⊢ Nonseparating {a} | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : S
⊢ ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' {a}) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : S
⊢ Nonseparating {a}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | AnalyticManifold.nonseparating_singleton | [220, 1] | [230, 38] | intro z | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : S
⊢ ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' {a}) | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
⊢ Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' {a}) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a : S
⊢ ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' {a})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | AnalyticManifold.nonseparating_singleton | [220, 1] | [230, 38] | by_cases az : a ∈ (extChartAt I z).source | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
⊢ Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' {a}) | case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
az : a ∈ (extChartAt I z).source
⊢ Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' {a})
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
az : a ∉ (extChartAt I z).source
⊢ Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' {a}) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
⊢ Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' {a})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | AnalyticManifold.nonseparating_singleton | [220, 1] | [230, 38] | convert Complex.nonseparating_singleton (extChartAt I z a) | case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
az : a ∈ (extChartAt I z).source
⊢ Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' {a}) | case h.e'_3
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
az : a ∈ (extChartAt I z).source
⊢ (extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' {a} = {↑(extChartAt I z) a} | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
az : a ∈ (extChartAt I z).source
⊢ Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' {a})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | AnalyticManifold.nonseparating_singleton | [220, 1] | [230, 38] | simp only [eq_singleton_iff_unique_mem, mem_inter_iff, PartialEquiv.map_source _ az, true_and_iff,
mem_preimage, mem_singleton_iff, PartialEquiv.left_inv _ az, eq_self_iff_true] | case h.e'_3
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
az : a ∈ (extChartAt I z).source
⊢ (extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' {a} = {↑(extChartAt I z) a} | case h.e'_3
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
az : a ∈ (extChartAt I z).source
⊢ ∀ (x : ℂ), x ∈ (extChartAt I z).target ∧ ↑(extChartAt I z).symm x = a → x = ↑(extChartAt I z) a | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
az : a ∈ (extChartAt I z).source
⊢ (extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' {a} = {↑(extChartAt I z) a}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | AnalyticManifold.nonseparating_singleton | [220, 1] | [230, 38] | intro x ⟨m, e⟩ | case h.e'_3
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
az : a ∈ (extChartAt I z).source
⊢ ∀ (x : ℂ), x ∈ (extChartAt I z).target ∧ ↑(extChartAt I z).symm x = a → x = ↑(extChartAt I z) a | case h.e'_3
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
az : a ∈ (extChartAt I z).source
x : ℂ
m : x ∈ (extChartAt I z).target
e : ↑(extChartAt I z).symm x = a
⊢ x = ↑(extChartAt I z) a | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
az : a ∈ (extChartAt I z).source
⊢ ∀ (x : ℂ), x ∈ (extChartAt I z).target ∧ ↑(extChartAt I z).symm x = a → x = ↑(extChartAt I z) a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | AnalyticManifold.nonseparating_singleton | [220, 1] | [230, 38] | simp only [← e, PartialEquiv.right_inv _ m] | case h.e'_3
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
az : a ∈ (extChartAt I z).source
x : ℂ
m : x ∈ (extChartAt I z).target
e : ↑(extChartAt I z).symm x = a
⊢ x = ↑(extChartAt I z) a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
az : a ∈ (extChartAt I z).source
x : ℂ
m : x ∈ (extChartAt I z).target
e : ↑(extChartAt I z).symm x = a
⊢ x = ↑(extChartAt I z) a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | AnalyticManifold.nonseparating_singleton | [220, 1] | [230, 38] | convert Nonseparating.empty | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
az : a ∉ (extChartAt I z).source
⊢ Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' {a}) | case h.e'_3
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
az : a ∉ (extChartAt I z).source
⊢ (extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' {a} = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
az : a ∉ (extChartAt I z).source
⊢ Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' {a})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | AnalyticManifold.nonseparating_singleton | [220, 1] | [230, 38] | simp only [eq_empty_iff_forall_not_mem, mem_inter_iff, mem_preimage, mem_singleton_iff, not_and] | case h.e'_3
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
az : a ∉ (extChartAt I z).source
⊢ (extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' {a} = ∅ | case h.e'_3
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
az : a ∉ (extChartAt I z).source
⊢ ∀ x ∈ (extChartAt I z).target, ¬↑(extChartAt I z).symm x = a | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
az : a ∉ (extChartAt I z).source
⊢ (extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' {a} = ∅
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | AnalyticManifold.nonseparating_singleton | [220, 1] | [230, 38] | intro x m | case h.e'_3
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
az : a ∉ (extChartAt I z).source
⊢ ∀ x ∈ (extChartAt I z).target, ¬↑(extChartAt I z).symm x = a | case h.e'_3
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
az : a ∉ (extChartAt I z).source
x : ℂ
m : x ∈ (extChartAt I z).target
⊢ ¬↑(extChartAt I z).symm x = a | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
az : a ∉ (extChartAt I z).source
⊢ ∀ x ∈ (extChartAt I z).target, ¬↑(extChartAt I z).symm x = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | AnalyticManifold.nonseparating_singleton | [220, 1] | [230, 38] | contrapose az | case h.e'_3
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
az : a ∉ (extChartAt I z).source
x : ℂ
m : x ∈ (extChartAt I z).target
⊢ ¬↑(extChartAt I z).symm x = a | case h.e'_3
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
x : ℂ
m : x ∈ (extChartAt I z).target
az : ¬¬↑(extChartAt I z).symm x = a
⊢ ¬a ∉ (extChartAt I z).source | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
az : a ∉ (extChartAt I z).source
x : ℂ
m : x ∈ (extChartAt I z).target
⊢ ¬↑(extChartAt I z).symm x = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | AnalyticManifold.nonseparating_singleton | [220, 1] | [230, 38] | simp only [not_not] at az ⊢ | case h.e'_3
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
x : ℂ
m : x ∈ (extChartAt I z).target
az : ¬¬↑(extChartAt I z).symm x = a
⊢ ¬a ∉ (extChartAt I z).source | case h.e'_3
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
x : ℂ
m : x ∈ (extChartAt I z).target
az : ↑(extChartAt I z).symm x = a
⊢ a ∈ (extChartAt I z).source | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
x : ℂ
m : x ∈ (extChartAt I z).target
az : ¬¬↑(extChartAt I z).symm x = a
⊢ ¬a ∉ (extChartAt I z).source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | AnalyticManifold.nonseparating_singleton | [220, 1] | [230, 38] | rw [← az] | case h.e'_3
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
x : ℂ
m : x ∈ (extChartAt I z).target
az : ↑(extChartAt I z).symm x = a
⊢ a ∈ (extChartAt I z).source | case h.e'_3
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
x : ℂ
m : x ∈ (extChartAt I z).target
az : ↑(extChartAt I z).symm x = a
⊢ ↑(extChartAt I z).symm x ∈ (extChartAt I z).source | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
x : ℂ
m : x ∈ (extChartAt I z).target
az : ↑(extChartAt I z).symm x = a
⊢ a ∈ (extChartAt I z).source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | AnalyticManifold.nonseparating_singleton | [220, 1] | [230, 38] | exact PartialEquiv.map_target _ m | case h.e'_3
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
x : ℂ
m : x ∈ (extChartAt I z).target
az : ↑(extChartAt I z).symm x = a
⊢ ↑(extChartAt I z).symm x ∈ (extChartAt I z).source | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
a z : S
x : ℂ
m : x ∈ (extChartAt I z).target
az : ↑(extChartAt I z).symm x = a
⊢ ↑(extChartAt I z).symm x ∈ (extChartAt I z).source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff_line | [238, 1] | [244, 85] | apply IsPreconnected.open_diff sc so | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s : Set (ℂ × S)
sc : IsPreconnected s
so : IsOpen s
a : S
⊢ IsPreconnected (s \ {p | p.2 = a}) | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s : Set (ℂ × S)
sc : IsPreconnected s
so : IsOpen s
a : S
⊢ Nonseparating {p | p.2 = a} | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s : Set (ℂ × S)
sc : IsPreconnected s
so : IsOpen s
a : S
⊢ IsPreconnected (s \ {p | p.2 = a})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff_line | [238, 1] | [244, 85] | have e : {p : ℂ × S | p.2 = a} = univ ×ˢ {a} := by
apply Set.ext; intro ⟨c, z⟩
simp only [mem_prod_eq, mem_setOf, mem_univ, true_and_iff, mem_singleton_iff] | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s : Set (ℂ × S)
sc : IsPreconnected s
so : IsOpen s
a : S
⊢ Nonseparating {p | p.2 = a} | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s : Set (ℂ × S)
sc : IsPreconnected s
so : IsOpen s
a : S
e : {p | p.2 = a} = univ ×ˢ {a}
⊢ Nonseparating {p | p.2 = a} | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s : Set (ℂ × S)
sc : IsPreconnected s
so : IsOpen s
a : S
⊢ Nonseparating {p | p.2 = a}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff_line | [238, 1] | [244, 85] | rw [e] | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s : Set (ℂ × S)
sc : IsPreconnected s
so : IsOpen s
a : S
e : {p | p.2 = a} = univ ×ˢ {a}
⊢ Nonseparating {p | p.2 = a} | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s : Set (ℂ × S)
sc : IsPreconnected s
so : IsOpen s
a : S
e : {p | p.2 = a} = univ ×ˢ {a}
⊢ Nonseparating (univ ×ˢ {a}) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s : Set (ℂ × S)
sc : IsPreconnected s
so : IsOpen s
a : S
e : {p | p.2 = a} = univ ×ˢ {a}
⊢ Nonseparating {p | p.2 = a}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff_line | [238, 1] | [244, 85] | exact Nonseparating.univ_prod (AnalyticManifold.nonseparating_singleton _) | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s : Set (ℂ × S)
sc : IsPreconnected s
so : IsOpen s
a : S
e : {p | p.2 = a} = univ ×ˢ {a}
⊢ Nonseparating (univ ×ˢ {a}) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s : Set (ℂ × S)
sc : IsPreconnected s
so : IsOpen s
a : S
e : {p | p.2 = a} = univ ×ˢ {a}
⊢ Nonseparating (univ ×ˢ {a})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff_line | [238, 1] | [244, 85] | apply Set.ext | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s : Set (ℂ × S)
sc : IsPreconnected s
so : IsOpen s
a : S
⊢ {p | p.2 = a} = univ ×ˢ {a} | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s : Set (ℂ × S)
sc : IsPreconnected s
so : IsOpen s
a : S
⊢ ∀ (x : ℂ × S), x ∈ {p | p.2 = a} ↔ x ∈ univ ×ˢ {a} | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s : Set (ℂ × S)
sc : IsPreconnected s
so : IsOpen s
a : S
⊢ {p | p.2 = a} = univ ×ˢ {a}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff_line | [238, 1] | [244, 85] | intro ⟨c, z⟩ | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s : Set (ℂ × S)
sc : IsPreconnected s
so : IsOpen s
a : S
⊢ ∀ (x : ℂ × S), x ∈ {p | p.2 = a} ↔ x ∈ univ ×ˢ {a} | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s : Set (ℂ × S)
sc : IsPreconnected s
so : IsOpen s
a : S
c : ℂ
z : S
⊢ (c, z) ∈ {p | p.2 = a} ↔ (c, z) ∈ univ ×ˢ {a} | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s : Set (ℂ × S)
sc : IsPreconnected s
so : IsOpen s
a : S
⊢ ∀ (x : ℂ × S), x ∈ {p | p.2 = a} ↔ x ∈ univ ×ˢ {a}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff_line | [238, 1] | [244, 85] | simp only [mem_prod_eq, mem_setOf, mem_univ, true_and_iff, mem_singleton_iff] | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s : Set (ℂ × S)
sc : IsPreconnected s
so : IsOpen s
a : S
c : ℂ
z : S
⊢ (c, z) ∈ {p | p.2 = a} ↔ (c, z) ∈ univ ×ˢ {a} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s : Set (ℂ × S)
sc : IsPreconnected s
so : IsOpen s
a : S
c : ℂ
z : S
⊢ (c, z) ∈ {p | p.2 = a} ↔ (c, z) ∈ univ ×ˢ {a}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_on_cball_radius | [32, 1] | [38, 53] | have hd : DifferentiableOn ℂ f (closedBall z r) := by
intro x H; exact AnalyticAt.differentiableWithinAt (h x H) | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (closedBall z ↑r)
⊢ HasFPowerSeriesOnBall f (cauchyPowerSeries f z ↑r) z ↑r | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (closedBall z ↑r)
hd : DifferentiableOn ℂ f (closedBall z ↑r)
⊢ HasFPowerSeriesOnBall f (cauchyPowerSeries f z ↑r) z ↑r | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (closedBall z ↑r)
⊢ HasFPowerSeriesOnBall f (cauchyPowerSeries f z ↑r) z ↑r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_on_cball_radius | [32, 1] | [38, 53] | set p : FormalMultilinearSeries ℂ ℂ ℂ := cauchyPowerSeries f z r | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (closedBall z ↑r)
hd : DifferentiableOn ℂ f (closedBall z ↑r)
⊢ HasFPowerSeriesOnBall f (cauchyPowerSeries f z ↑r) z ↑r | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (closedBall z ↑r)
hd : DifferentiableOn ℂ f (closedBall z ↑r)
p : FormalMultilinearSeries ℂ ℂ ℂ := cauchyPowerSeries f z ↑r
⊢ HasFPowerSeriesOnBall f p z ↑r | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (closedBall z ↑r)
hd : DifferentiableOn ℂ f (closedBall z ↑r)
⊢ HasFPowerSeriesOnBall f (cauchyPowerSeries f z ↑r) z ↑r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_on_cball_radius | [32, 1] | [38, 53] | exact DifferentiableOn.hasFPowerSeriesOnBall hd rp | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (closedBall z ↑r)
hd : DifferentiableOn ℂ f (closedBall z ↑r)
p : FormalMultilinearSeries ℂ ℂ ℂ := cauchyPowerSeries f z ↑r
⊢ HasFPowerSeriesOnBall f p z ↑r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (closedBall z ↑r)
hd : DifferentiableOn ℂ f (closedBall z ↑r)
p : FormalMultilinearSeries ℂ ℂ ℂ := cauchyPowerSeries f z ↑r
⊢ HasFPowerSeriesOnBall f p z ↑r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_on_cball_radius | [32, 1] | [38, 53] | intro x H | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (closedBall z ↑r)
⊢ DifferentiableOn ℂ f (closedBall z ↑r) | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (closedBall z ↑r)
x : ℂ
H : x ∈ closedBall z ↑r
⊢ DifferentiableWithinAt ℂ f (closedBall z ↑r) x | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (closedBall z ↑r)
⊢ DifferentiableOn ℂ f (closedBall z ↑r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_on_cball_radius | [32, 1] | [38, 53] | exact AnalyticAt.differentiableWithinAt (h x H) | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (closedBall z ↑r)
x : ℂ
H : x ∈ closedBall z ↑r
⊢ DifferentiableWithinAt ℂ f (closedBall z ↑r) x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (closedBall z ↑r)
x : ℂ
H : x ∈ closedBall z ↑r
⊢ DifferentiableWithinAt ℂ f (closedBall z ↑r) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_small_cball | [45, 1] | [51, 15] | intro x hx | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
h : AnalyticOn ℂ f (ball z ↑r)
s : ℝ≥0
sr : s < r
⊢ AnalyticOn ℂ f (closedBall z ↑s) | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
h : AnalyticOn ℂ f (ball z ↑r)
s : ℝ≥0
sr : s < r
x : ℂ
hx : x ∈ closedBall z ↑s
⊢ AnalyticAt ℂ f x | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
h : AnalyticOn ℂ f (ball z ↑r)
s : ℝ≥0
sr : s < r
⊢ AnalyticOn ℂ f (closedBall z ↑s)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_small_cball | [45, 1] | [51, 15] | rw [closedBall] at hx | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
h : AnalyticOn ℂ f (ball z ↑r)
s : ℝ≥0
sr : s < r
x : ℂ
hx : x ∈ closedBall z ↑s
⊢ AnalyticAt ℂ f x | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
h : AnalyticOn ℂ f (ball z ↑r)
s : ℝ≥0
sr : s < r
x : ℂ
hx : x ∈ {y | dist y z ≤ ↑s}
⊢ AnalyticAt ℂ f x | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
h : AnalyticOn ℂ f (ball z ↑r)
s : ℝ≥0
sr : s < r
x : ℂ
hx : x ∈ closedBall z ↑s
⊢ AnalyticAt ℂ f x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_small_cball | [45, 1] | [51, 15] | simp at hx | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
h : AnalyticOn ℂ f (ball z ↑r)
s : ℝ≥0
sr : s < r
x : ℂ
hx : x ∈ {y | dist y z ≤ ↑s}
⊢ AnalyticAt ℂ f x | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
h : AnalyticOn ℂ f (ball z ↑r)
s : ℝ≥0
sr : s < r
x : ℂ
hx : nndist x z ≤ s
⊢ AnalyticAt ℂ f x | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
h : AnalyticOn ℂ f (ball z ↑r)
s : ℝ≥0
sr : s < r
x : ℂ
hx : x ∈ {y | dist y z ≤ ↑s}
⊢ AnalyticAt ℂ f x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_small_cball | [45, 1] | [51, 15] | have hb : x ∈ ball z r := by
rw [ball]; simp only [dist_lt_coe, Set.mem_setOf_eq]; exact lt_of_le_of_lt hx sr | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
h : AnalyticOn ℂ f (ball z ↑r)
s : ℝ≥0
sr : s < r
x : ℂ
hx : nndist x z ≤ s
⊢ AnalyticAt ℂ f x | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
h : AnalyticOn ℂ f (ball z ↑r)
s : ℝ≥0
sr : s < r
x : ℂ
hx : nndist x z ≤ s
hb : x ∈ ball z ↑r
⊢ AnalyticAt ℂ f x | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
h : AnalyticOn ℂ f (ball z ↑r)
s : ℝ≥0
sr : s < r
x : ℂ
hx : nndist x z ≤ s
⊢ AnalyticAt ℂ f x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_small_cball | [45, 1] | [51, 15] | exact h x hb | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
h : AnalyticOn ℂ f (ball z ↑r)
s : ℝ≥0
sr : s < r
x : ℂ
hx : nndist x z ≤ s
hb : x ∈ ball z ↑r
⊢ AnalyticAt ℂ f x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
h : AnalyticOn ℂ f (ball z ↑r)
s : ℝ≥0
sr : s < r
x : ℂ
hx : nndist x z ≤ s
hb : x ∈ ball z ↑r
⊢ AnalyticAt ℂ f x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_small_cball | [45, 1] | [51, 15] | rw [ball] | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
h : AnalyticOn ℂ f (ball z ↑r)
s : ℝ≥0
sr : s < r
x : ℂ
hx : nndist x z ≤ s
⊢ x ∈ ball z ↑r | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
h : AnalyticOn ℂ f (ball z ↑r)
s : ℝ≥0
sr : s < r
x : ℂ
hx : nndist x z ≤ s
⊢ x ∈ {y | dist y z < ↑r} | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
h : AnalyticOn ℂ f (ball z ↑r)
s : ℝ≥0
sr : s < r
x : ℂ
hx : nndist x z ≤ s
⊢ x ∈ ball z ↑r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_small_cball | [45, 1] | [51, 15] | simp only [dist_lt_coe, Set.mem_setOf_eq] | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
h : AnalyticOn ℂ f (ball z ↑r)
s : ℝ≥0
sr : s < r
x : ℂ
hx : nndist x z ≤ s
⊢ x ∈ {y | dist y z < ↑r} | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
h : AnalyticOn ℂ f (ball z ↑r)
s : ℝ≥0
sr : s < r
x : ℂ
hx : nndist x z ≤ s
⊢ nndist x z < r | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
h : AnalyticOn ℂ f (ball z ↑r)
s : ℝ≥0
sr : s < r
x : ℂ
hx : nndist x z ≤ s
⊢ x ∈ {y | dist y z < ↑r}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_small_cball | [45, 1] | [51, 15] | exact lt_of_le_of_lt hx sr | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
h : AnalyticOn ℂ f (ball z ↑r)
s : ℝ≥0
sr : s < r
x : ℂ
hx : nndist x z ≤ s
⊢ nndist x z < r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
h : AnalyticOn ℂ f (ball z ↑r)
s : ℝ≥0
sr : s < r
x : ℂ
hx : nndist x z ≤ s
⊢ nndist x z < r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | have h0 := analyticOn_small_cball h (r / 2) (NNReal.half_lt_self <| rp.ne') | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
⊢ ∃ p, HasFPowerSeriesOnBall f p z ↑r | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
⊢ ∃ p, HasFPowerSeriesOnBall f p z ↑r | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
⊢ ∃ p, HasFPowerSeriesOnBall f p z ↑r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | rcases analyticOn_cball_radius (half_pos rp) h0 with ⟨p, ph⟩ | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
⊢ ∃ p, HasFPowerSeriesOnBall f p z ↑r | case intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
⊢ ∃ p, HasFPowerSeriesOnBall f p z ↑r | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
⊢ ∃ p, HasFPowerSeriesOnBall f p z ↑r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | set R := FormalMultilinearSeries.radius p | case intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
⊢ ∃ p, HasFPowerSeriesOnBall f p z ↑r | case intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
⊢ ∃ p, HasFPowerSeriesOnBall f p z ↑r | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
⊢ ∃ p, HasFPowerSeriesOnBall f p z ↑r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | refine
⟨p, {
r_le := ?_
r_pos := ENNReal.coe_pos.mpr rp
hasSum := ?_ }⟩ | case intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
⊢ ∃ p, HasFPowerSeriesOnBall f p z ↑r | case intro.refine_1
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
⊢ ↑r ≤ p.radius
case intro.refine_2
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
⊢ ∀ {y : ℂ}, y ∈ EMetric.ball 0 ↑r → HasSum (fun n => (p n) fun x => y) (f (z + y)) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
⊢ ∃ p, HasFPowerSeriesOnBall f p z ↑r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | apply ENNReal.le_of_forall_pos_nnreal_lt | case intro.refine_1
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
⊢ ↑r ≤ p.radius | case intro.refine_1.h
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
⊢ ∀ (r_1 : ℝ≥0), 0 < r_1 → ↑r_1 < ↑r → ↑r_1 ≤ p.radius | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_1
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
⊢ ↑r ≤ p.radius
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | intro t tp tr | case intro.refine_1.h
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
⊢ ∀ (r_1 : ℝ≥0), 0 < r_1 → ↑r_1 < ↑r → ↑r_1 ≤ p.radius | case intro.refine_1.h
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
t : ℝ≥0
tp : 0 < t
tr : ↑t < ↑r
⊢ ↑t ≤ p.radius | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_1.h
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
⊢ ∀ (r_1 : ℝ≥0), 0 < r_1 → ↑r_1 < ↑r → ↑r_1 ≤ p.radius
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | have ht := analyticOn_small_cball h t (ENNReal.coe_lt_coe.mp tr) | case intro.refine_1.h
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
t : ℝ≥0
tp : 0 < t
tr : ↑t < ↑r
⊢ ↑t ≤ p.radius | case intro.refine_1.h
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
t : ℝ≥0
tp : 0 < t
tr : ↑t < ↑r
ht : AnalyticOn ℂ f (closedBall z ↑t)
⊢ ↑t ≤ p.radius | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_1.h
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
t : ℝ≥0
tp : 0 < t
tr : ↑t < ↑r
⊢ ↑t ≤ p.radius
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | rcases analyticOn_cball_radius tp ht with ⟨p', hp'⟩ | case intro.refine_1.h
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
t : ℝ≥0
tp : 0 < t
tr : ↑t < ↑r
ht : AnalyticOn ℂ f (closedBall z ↑t)
⊢ ↑t ≤ p.radius | case intro.refine_1.h.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
t : ℝ≥0
tp : 0 < t
tr : ↑t < ↑r
ht : AnalyticOn ℂ f (closedBall z ↑t)
p' : FormalMultilinearSeries ℂ ℂ ℂ
hp' : HasFPowerSeriesOnBall f p' z ↑t
⊢ ↑t ≤ p.radius | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_1.h
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
t : ℝ≥0
tp : 0 < t
tr : ↑t < ↑r
ht : AnalyticOn ℂ f (closedBall z ↑t)
⊢ ↑t ≤ p.radius
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | have pp : p = p' := HasFPowerSeriesAt.eq_formalMultilinearSeries ⟨↑(r / 2), ph⟩ ⟨t, hp'⟩ | case intro.refine_1.h.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
t : ℝ≥0
tp : 0 < t
tr : ↑t < ↑r
ht : AnalyticOn ℂ f (closedBall z ↑t)
p' : FormalMultilinearSeries ℂ ℂ ℂ
hp' : HasFPowerSeriesOnBall f p' z ↑t
⊢ ↑t ≤ p.radius | case intro.refine_1.h.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
t : ℝ≥0
tp : 0 < t
tr : ↑t < ↑r
ht : AnalyticOn ℂ f (closedBall z ↑t)
p' : FormalMultilinearSeries ℂ ℂ ℂ
hp' : HasFPowerSeriesOnBall f p' z ↑t
pp : p = p'
⊢ ↑t ≤ p.radius | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_1.h.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
t : ℝ≥0
tp : 0 < t
tr : ↑t < ↑r
ht : AnalyticOn ℂ f (closedBall z ↑t)
p' : FormalMultilinearSeries ℂ ℂ ℂ
hp' : HasFPowerSeriesOnBall f p' z ↑t
⊢ ↑t ≤ p.radius
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | rw [← pp] at hp' | case intro.refine_1.h.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
t : ℝ≥0
tp : 0 < t
tr : ↑t < ↑r
ht : AnalyticOn ℂ f (closedBall z ↑t)
p' : FormalMultilinearSeries ℂ ℂ ℂ
hp' : HasFPowerSeriesOnBall f p' z ↑t
pp : p = p'
⊢ ↑t ≤ p.radius | case intro.refine_1.h.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
t : ℝ≥0
tp : 0 < t
tr : ↑t < ↑r
ht : AnalyticOn ℂ f (closedBall z ↑t)
p' : FormalMultilinearSeries ℂ ℂ ℂ
hp' : HasFPowerSeriesOnBall f p z ↑t
pp : p = p'
⊢ ↑t ≤ p.radius | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_1.h.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
t : ℝ≥0
tp : 0 < t
tr : ↑t < ↑r
ht : AnalyticOn ℂ f (closedBall z ↑t)
p' : FormalMultilinearSeries ℂ ℂ ℂ
hp' : HasFPowerSeriesOnBall f p' z ↑t
pp : p = p'
⊢ ↑t ≤ p.radius
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | refine hp'.r_le | case intro.refine_1.h.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
t : ℝ≥0
tp : 0 < t
tr : ↑t < ↑r
ht : AnalyticOn ℂ f (closedBall z ↑t)
p' : FormalMultilinearSeries ℂ ℂ ℂ
hp' : HasFPowerSeriesOnBall f p z ↑t
pp : p = p'
⊢ ↑t ≤ p.radius | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_1.h.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
t : ℝ≥0
tp : 0 < t
tr : ↑t < ↑r
ht : AnalyticOn ℂ f (closedBall z ↑t)
p' : FormalMultilinearSeries ℂ ℂ ℂ
hp' : HasFPowerSeriesOnBall f p z ↑t
pp : p = p'
⊢ ↑t ≤ p.radius
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | intro y yr | case intro.refine_2
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
⊢ ∀ {y : ℂ}, y ∈ EMetric.ball 0 ↑r → HasSum (fun n => (p n) fun x => y) (f (z + y)) | case intro.refine_2
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : y ∈ EMetric.ball 0 ↑r
⊢ HasSum (fun n => (p n) fun x => y) (f (z + y)) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_2
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
⊢ ∀ {y : ℂ}, y ∈ EMetric.ball 0 ↑r → HasSum (fun n => (p n) fun x => y) (f (z + y))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | rw [EMetric.ball, Set.mem_setOf] at yr | case intro.refine_2
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : y ∈ EMetric.ball 0 ↑r
⊢ HasSum (fun n => (p n) fun x => y) (f (z + y)) | case intro.refine_2
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
⊢ HasSum (fun n => (p n) fun x => y) (f (z + y)) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_2
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : y ∈ EMetric.ball 0 ↑r
⊢ HasSum (fun n => (p n) fun x => y) (f (z + y))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | rcases exists_between yr with ⟨t, t0, t1⟩ | case intro.refine_2
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
⊢ HasSum (fun n => (p n) fun x => y) (f (z + y)) | case intro.refine_2.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
⊢ HasSum (fun n => (p n) fun x => y) (f (z + y)) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_2
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
⊢ HasSum (fun n => (p n) fun x => y) (f (z + y))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | have t1' : t.toNNReal < r := by
rw [← WithTop.coe_lt_coe]; exact lt_of_le_of_lt ENNReal.coe_toNNReal_le_self t1 | case intro.refine_2.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
⊢ HasSum (fun n => (p n) fun x => y) (f (z + y)) | case intro.refine_2.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
t1' : t.toNNReal < r
⊢ HasSum (fun n => (p n) fun x => y) (f (z + y)) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_2.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
⊢ HasSum (fun n => (p n) fun x => y) (f (z + y))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | have ht := analyticOn_small_cball h t.toNNReal t1' | case intro.refine_2.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
t1' : t.toNNReal < r
⊢ HasSum (fun n => (p n) fun x => y) (f (z + y)) | case intro.refine_2.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
t1' : t.toNNReal < r
ht : AnalyticOn ℂ f (closedBall z ↑t.toNNReal)
⊢ HasSum (fun n => (p n) fun x => y) (f (z + y)) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_2.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
t1' : t.toNNReal < r
⊢ HasSum (fun n => (p n) fun x => y) (f (z + y))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | have tp : 0 < ENNReal.toNNReal t :=
ENNReal.toNNReal_pos (ne_of_gt <| pos_of_gt t0) (ne_top_of_lt t1) | case intro.refine_2.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
t1' : t.toNNReal < r
ht : AnalyticOn ℂ f (closedBall z ↑t.toNNReal)
⊢ HasSum (fun n => (p n) fun x => y) (f (z + y)) | case intro.refine_2.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
t1' : t.toNNReal < r
ht : AnalyticOn ℂ f (closedBall z ↑t.toNNReal)
tp : 0 < t.toNNReal
⊢ HasSum (fun n => (p n) fun x => y) (f (z + y)) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_2.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
t1' : t.toNNReal < r
ht : AnalyticOn ℂ f (closedBall z ↑t.toNNReal)
⊢ HasSum (fun n => (p n) fun x => y) (f (z + y))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | rcases analyticOn_cball_radius tp ht with ⟨p', hp'⟩ | case intro.refine_2.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
t1' : t.toNNReal < r
ht : AnalyticOn ℂ f (closedBall z ↑t.toNNReal)
tp : 0 < t.toNNReal
⊢ HasSum (fun n => (p n) fun x => y) (f (z + y)) | case intro.refine_2.intro.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
t1' : t.toNNReal < r
ht : AnalyticOn ℂ f (closedBall z ↑t.toNNReal)
tp : 0 < t.toNNReal
p' : FormalMultilinearSeries ℂ ℂ ℂ
hp' : HasFPowerSeriesOnBall f p' z ↑t.toNNReal
⊢ HasSum (fun n => (p n) fun x => y) (f (z + y)) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_2.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
t1' : t.toNNReal < r
ht : AnalyticOn ℂ f (closedBall z ↑t.toNNReal)
tp : 0 < t.toNNReal
⊢ HasSum (fun n => (p n) fun x => y) (f (z + y))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | have pp : p = p' :=
HasFPowerSeriesAt.eq_formalMultilinearSeries ⟨↑(r / 2), ph⟩ ⟨t.toNNReal, hp'⟩ | case intro.refine_2.intro.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
t1' : t.toNNReal < r
ht : AnalyticOn ℂ f (closedBall z ↑t.toNNReal)
tp : 0 < t.toNNReal
p' : FormalMultilinearSeries ℂ ℂ ℂ
hp' : HasFPowerSeriesOnBall f p' z ↑t.toNNReal
⊢ HasSum (fun n => (p n) fun x => y) (f (z + y)) | case intro.refine_2.intro.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
t1' : t.toNNReal < r
ht : AnalyticOn ℂ f (closedBall z ↑t.toNNReal)
tp : 0 < t.toNNReal
p' : FormalMultilinearSeries ℂ ℂ ℂ
hp' : HasFPowerSeriesOnBall f p' z ↑t.toNNReal
pp : p = p'
⊢ HasSum (fun n => (p n) fun x => y) (f (z + y)) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_2.intro.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
t1' : t.toNNReal < r
ht : AnalyticOn ℂ f (closedBall z ↑t.toNNReal)
tp : 0 < t.toNNReal
p' : FormalMultilinearSeries ℂ ℂ ℂ
hp' : HasFPowerSeriesOnBall f p' z ↑t.toNNReal
⊢ HasSum (fun n => (p n) fun x => y) (f (z + y))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | rw [← pp] at hp' | case intro.refine_2.intro.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
t1' : t.toNNReal < r
ht : AnalyticOn ℂ f (closedBall z ↑t.toNNReal)
tp : 0 < t.toNNReal
p' : FormalMultilinearSeries ℂ ℂ ℂ
hp' : HasFPowerSeriesOnBall f p' z ↑t.toNNReal
pp : p = p'
⊢ HasSum (fun n => (p n) fun x => y) (f (z + y)) | case intro.refine_2.intro.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
t1' : t.toNNReal < r
ht : AnalyticOn ℂ f (closedBall z ↑t.toNNReal)
tp : 0 < t.toNNReal
p' : FormalMultilinearSeries ℂ ℂ ℂ
hp' : HasFPowerSeriesOnBall f p z ↑t.toNNReal
pp : p = p'
⊢ HasSum (fun n => (p n) fun x => y) (f (z + y)) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_2.intro.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
t1' : t.toNNReal < r
ht : AnalyticOn ℂ f (closedBall z ↑t.toNNReal)
tp : 0 < t.toNNReal
p' : FormalMultilinearSeries ℂ ℂ ℂ
hp' : HasFPowerSeriesOnBall f p' z ↑t.toNNReal
pp : p = p'
⊢ HasSum (fun n => (p n) fun x => y) (f (z + y))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | refine hp'.hasSum ?_ | case intro.refine_2.intro.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
t1' : t.toNNReal < r
ht : AnalyticOn ℂ f (closedBall z ↑t.toNNReal)
tp : 0 < t.toNNReal
p' : FormalMultilinearSeries ℂ ℂ ℂ
hp' : HasFPowerSeriesOnBall f p z ↑t.toNNReal
pp : p = p'
⊢ HasSum (fun n => (p n) fun x => y) (f (z + y)) | case intro.refine_2.intro.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
t1' : t.toNNReal < r
ht : AnalyticOn ℂ f (closedBall z ↑t.toNNReal)
tp : 0 < t.toNNReal
p' : FormalMultilinearSeries ℂ ℂ ℂ
hp' : HasFPowerSeriesOnBall f p z ↑t.toNNReal
pp : p = p'
⊢ y ∈ EMetric.ball 0 ↑t.toNNReal | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_2.intro.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
t1' : t.toNNReal < r
ht : AnalyticOn ℂ f (closedBall z ↑t.toNNReal)
tp : 0 < t.toNNReal
p' : FormalMultilinearSeries ℂ ℂ ℂ
hp' : HasFPowerSeriesOnBall f p z ↑t.toNNReal
pp : p = p'
⊢ HasSum (fun n => (p n) fun x => y) (f (z + y))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | rw [EMetric.ball, Set.mem_setOf] | case intro.refine_2.intro.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
t1' : t.toNNReal < r
ht : AnalyticOn ℂ f (closedBall z ↑t.toNNReal)
tp : 0 < t.toNNReal
p' : FormalMultilinearSeries ℂ ℂ ℂ
hp' : HasFPowerSeriesOnBall f p z ↑t.toNNReal
pp : p = p'
⊢ y ∈ EMetric.ball 0 ↑t.toNNReal | case intro.refine_2.intro.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
t1' : t.toNNReal < r
ht : AnalyticOn ℂ f (closedBall z ↑t.toNNReal)
tp : 0 < t.toNNReal
p' : FormalMultilinearSeries ℂ ℂ ℂ
hp' : HasFPowerSeriesOnBall f p z ↑t.toNNReal
pp : p = p'
⊢ edist y 0 < ↑t.toNNReal | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_2.intro.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
t1' : t.toNNReal < r
ht : AnalyticOn ℂ f (closedBall z ↑t.toNNReal)
tp : 0 < t.toNNReal
p' : FormalMultilinearSeries ℂ ℂ ℂ
hp' : HasFPowerSeriesOnBall f p z ↑t.toNNReal
pp : p = p'
⊢ y ∈ EMetric.ball 0 ↑t.toNNReal
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | calc edist y 0
_ < t := t0
_ = ↑t.toNNReal := (ENNReal.coe_toNNReal <| ne_top_of_lt t1).symm | case intro.refine_2.intro.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
t1' : t.toNNReal < r
ht : AnalyticOn ℂ f (closedBall z ↑t.toNNReal)
tp : 0 < t.toNNReal
p' : FormalMultilinearSeries ℂ ℂ ℂ
hp' : HasFPowerSeriesOnBall f p z ↑t.toNNReal
pp : p = p'
⊢ edist y 0 < ↑t.toNNReal | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_2.intro.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
t1' : t.toNNReal < r
ht : AnalyticOn ℂ f (closedBall z ↑t.toNNReal)
tp : 0 < t.toNNReal
p' : FormalMultilinearSeries ℂ ℂ ℂ
hp' : HasFPowerSeriesOnBall f p z ↑t.toNNReal
pp : p = p'
⊢ edist y 0 < ↑t.toNNReal
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | rw [← WithTop.coe_lt_coe] | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
⊢ t.toNNReal < r | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
⊢ ↑t.toNNReal < ↑r | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
⊢ t.toNNReal < r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | exact lt_of_le_of_lt ENNReal.coe_toNNReal_le_self t1 | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
⊢ ↑t.toNNReal < ↑r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
⊢ ↑t.toNNReal < ↑r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_bound | [89, 1] | [121, 92] | set wr := abs w ^ n * r⁻¹ ^ n * d | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
⊢ Complex.abs ((cauchyPowerSeries f c (↑r) n) fun x => w) ≤ Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
⊢ Complex.abs ((cauchyPowerSeries f c (↑r) n) fun x => w) ≤ wr | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
⊢ Complex.abs ((cauchyPowerSeries f c (↑r) n) fun x => w) ≤ Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
TACTIC:
|
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