url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | exact differentiableAt_id.add (differentiableAt_const _) | case hf
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) i... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | simp only [deriv_add_const, deriv_sub_const, deriv_id'', mul_one, sub_add_cancel, Function.comp] | case hp
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hp
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) i... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | simp only [sub_add_cancel, dg] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c,... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | exact differentiableAt_id.add (differentiableAt_const _) | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c,... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.isClosed_critical_not_a | [592, 1] | [600, 91] | rw [← isOpen_compl_iff] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ IsClosed {p | Critical (f p.1) p.2 ∧ p.2 ≠ a} | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ IsOpen {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ IsClosed {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.isClosed_critical_not_a | [592, 1] | [600, 91] | rw [isOpen_iff_eventually] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ IsOpen {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ ∀ x ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ, ∀ᶠ (y : ℂ × S) in 𝓝 x, y ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ IsOpen {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.isClosed_critical_not_a | [592, 1] | [600, 91] | intro ⟨c, z⟩ m | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ ∀ x ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ, ∀ᶠ (y : ℂ × S) in 𝓝 x, y ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ
⊢ ∀ᶠ (y : ℂ × S) in 𝓝 (c, z), y ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ ∀ x ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ, ∀ᶠ (y : ℂ × S) in 𝓝 x, y... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.isClosed_critical_not_a | [592, 1] | [600, 91] | by_cases za : z = a | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ
⊢ ∀ᶠ (y : ℂ × S) in 𝓝 (c, z), y ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠... | case pos
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ
za : z = a
⊢ ∀ᶠ (y : ℂ × S) in 𝓝 (c, z), y ∈ {p | Critical... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ
⊢ ∀ᶠ (y... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.isClosed_critical_not_a | [592, 1] | [600, 91] | rw [za] | case pos
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ
za : z = a
⊢ ∀ᶠ (y : ℂ × S) in 𝓝 (c, z), y ∈ {p | Critical... | case pos
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ
za : z = a
⊢ ∀ᶠ (y : ℂ × S) in 𝓝 (c, a), y ∈ {p | Critical... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.isClosed_critical_not_a | [592, 1] | [600, 91] | refine (s.f_noncritical_near_a c).mp (eventually_of_forall ?_) | case pos
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ
za : z = a
⊢ ∀ᶠ (y : ℂ × S) in 𝓝 (c, a), y ∈ {p | Critical... | case pos
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ
za : z = a
⊢ ∀ (x : ℂ × S), (Critical (f x.1) x.2 ↔ x.2 = a... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.isClosed_critical_not_a | [592, 1] | [600, 91] | intro ⟨e, w⟩ h | case pos
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ
za : z = a
⊢ ∀ (x : ℂ × S), (Critical (f x.1) x.2 ↔ x.2 = a... | case pos
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ
za : z = a
e : ℂ
w : S
h : Critical (f (e, w).1) (e, w).2 ↔... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.isClosed_critical_not_a | [592, 1] | [600, 91] | simp only [mem_compl_iff, mem_setOf, not_and, not_not] at h ⊢ | case pos
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ
za : z = a
e : ℂ
w : S
h : Critical (f (e, w).1) (e, w).2 ↔... | case pos
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ
za : z = a
e : ℂ
w : S
h : Critical (f e) w ↔ w = a
⊢ Criti... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.isClosed_critical_not_a | [592, 1] | [600, 91] | exact h.1 | case pos
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ
za : z = a
e : ℂ
w : S
h : Critical (f e) w ↔ w = a
⊢ Criti... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.isClosed_critical_not_a | [592, 1] | [600, 91] | have o := isOpen_iff_eventually.mp (isOpen_noncritical s.fa) | case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ
za : ¬z = a
⊢ ∀ᶠ (y : ℂ × S) in 𝓝 (c, z), y ∈ {p | Critica... | case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ
za : ¬z = a
o : ∀ x ∈ {p | ¬Critical (f p.1) p.2}, ∀ᶠ (y : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.isClosed_critical_not_a | [592, 1] | [600, 91] | simp only [za, mem_compl_iff, mem_setOf, not_and, not_not, imp_false] at m o ⊢ | case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}ᶜ
za : ¬z = a
o : ∀ x ∈ {p | ¬Critical (f p.1) p.2}, ∀ᶠ (y : ... | case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
za : ¬z = a
m : ¬Critical (f c) z
o : ∀ (x : ℂ × S), ¬Critical (f x.1) x.2 → ∀ᶠ (y : ℂ × S) in 𝓝 x, ¬Critical... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ {p | Critical (f p.1) p.2 ∧ p.2 ≠ a}... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.isClosed_critical_not_a | [592, 1] | [600, 91] | refine (o (c, z) m).mp (eventually_of_forall ?_) | case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
za : ¬z = a
m : ¬Critical (f c) z
o : ∀ (x : ℂ × S), ¬Critical (f x.1) x.2 → ∀ᶠ (y : ℂ × S) in 𝓝 x, ¬Critical... | case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
za : ¬z = a
m : ¬Critical (f c) z
o : ∀ (x : ℂ × S), ¬Critical (f x.1) x.2 → ∀ᶠ (y : ℂ × S) in 𝓝 x, ¬Critical... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
za : ¬z = a
m : ¬Critical (f c) z
o : ∀ (x : ℂ × ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.isClosed_critical_not_a | [592, 1] | [600, 91] | intro ⟨e, w⟩ a b | case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
za : ¬z = a
m : ¬Critical (f c) z
o : ∀ (x : ℂ × S), ¬Critical (f x.1) x.2 → ∀ᶠ (y : ℂ × S) in 𝓝 x, ¬Critical... | case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a✝ z✝ : S
d n : ℕ
s : Super f d a✝
c : ℂ
z : S
za : ¬z = a✝
m : ¬Critical (f c) z
o : ∀ (x : ℂ × S), ¬Critical (f x.1) x.2 → ∀ᶠ (y : ℂ × S) in 𝓝 x, ¬Criti... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
za : ¬z = a
m : ¬Critical (f c) z
o : ∀ (x : ℂ × ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.isClosed_critical_not_a | [592, 1] | [600, 91] | exfalso | case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a✝ z✝ : S
d n : ℕ
s : Super f d a✝
c : ℂ
z : S
za : ¬z = a✝
m : ¬Critical (f c) z
o : ∀ (x : ℂ × S), ¬Critical (f x.1) x.2 → ∀ᶠ (y : ℂ × S) in 𝓝 x, ¬Criti... | case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a✝ z✝ : S
d n : ℕ
s : Super f d a✝
c : ℂ
z : S
za : ¬z = a✝
m : ¬Critical (f c) z
o : ∀ (x : ℂ × S), ¬Critical (f x.1) x.2 → ∀ᶠ (y : ℂ × S) in 𝓝 x, ¬Criti... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a✝ z✝ : S
d n : ℕ
s : Super f d a✝
c : ℂ
z : S
za : ¬z = a✝
m : ¬Critical (f c) z
o : ∀ (x : ℂ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.isClosed_critical_not_a | [592, 1] | [600, 91] | exact a b | case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a✝ z✝ : S
d n : ℕ
s : Super f d a✝
c : ℂ
z : S
za : ¬z = a✝
m : ¬Critical (f c) z
o : ∀ (x : ℂ × S), ¬Critical (f x.1) x.2 → ∀ᶠ (y : ℂ × S) in 𝓝 x, ¬Criti... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a✝ z✝ : S
d n : ℕ
s : Super f d a✝
c : ℂ
z : S
za : ¬z = a✝
m : ¬Critical (f c) z
o : ∀ (x : ℂ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNearIter_mfderiv_ne_zero | [610, 1] | [614, 69] | apply mderiv_comp_ne_zero' b0 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
b0 : mfderiv I I (s.bottcherNear c) ((f c)^[n] z) ≠ 0
f0 : ¬Precritical (f c) z
⊢ mfderiv I I (s.bottcherNearIter n c) z ≠ 0 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
b0 : mfderiv I I (s.bottcherNear c) ((f c)^[n] z) ≠ 0
f0 : ¬Precritical (f c) z
⊢ mfderiv I I (f c)^[n] z ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
b0 : mfderiv I I (s.bottcherNear c) ((f c)^[n] z) ≠ 0
f0 : ¬Precritical ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNearIter_mfderiv_ne_zero | [610, 1] | [614, 69] | contrapose f0 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
b0 : mfderiv I I (s.bottcherNear c) ((f c)^[n] z) ≠ 0
f0 : ¬Precritical (f c) z
⊢ mfderiv I I (f c)^[n] z ≠ 0 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
b0 : mfderiv I I (s.bottcherNear c) ((f c)^[n] z) ≠ 0
f0 : ¬mfderiv I I (f c)^[n] z ≠ 0
⊢ ¬¬Precritical (f c) z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
b0 : mfderiv I I (s.bottcherNear c) ((f c)^[n] z) ≠ 0
f0 : ¬Precritical ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNearIter_mfderiv_ne_zero | [610, 1] | [614, 69] | simp only [not_not] at f0 ⊢ | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
b0 : mfderiv I I (s.bottcherNear c) ((f c)^[n] z) ≠ 0
f0 : ¬mfderiv I I (f c)^[n] z ≠ 0
⊢ ¬¬Precritical (f c) z | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
b0 : mfderiv I I (s.bottcherNear c) ((f c)^[n] z) ≠ 0
f0 : mfderiv I I (f c)^[n] z = 0
⊢ Precritical (f c) z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
b0 : mfderiv I I (s.bottcherNear c) ((f c)^[n] z) ≠ 0
f0 : ¬mfderiv I I ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNearIter_mfderiv_ne_zero | [610, 1] | [614, 69] | exact critical_iter s.fa.along_snd f0 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
b0 : mfderiv I I (s.bottcherNear c) ((f c)^[n] z) ≠ 0
f0 : mfderiv I I (f c)^[n] z = 0
⊢ Precritical (f c) z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
b0 : mfderiv I I (s.bottcherNear c) ((f c)^[n] z) ≠ 0
f0 : mfderiv I I (... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.iter_nontrivial_a | [617, 1] | [621, 47] | induction' n with n h | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ NontrivialHolomorphicAt (fun z => (f c)^[n] z) a | case zero
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ NontrivialHolomorphicAt (fun z => (f c)^[0] z) a
case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ NontrivialHolomorphicAt (fun z => (f c)^[n] z) a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.iter_nontrivial_a | [617, 1] | [621, 47] | simp only [Function.iterate_zero_apply] | case zero
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ NontrivialHolomorphicAt (fun z => (f c)^[0] z) a
case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
... | case zero
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ NontrivialHolomorphicAt (fun z => z) a
case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T... | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ NontrivialHolomorphicAt (fun z => (f c)^[0] z) a
case succ
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.iter_nontrivial_a | [617, 1] | [621, 47] | apply nontrivialHolomorphicAt_id | case zero
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ NontrivialHolomorphicAt (fun z => z) a
case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T... | case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
h : NontrivialHolomorphicAt (fun z => (f c)^[n] z) a
⊢ NontrivialHolomorphicAt (fun z => (f c)^[n + 1] z) a | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ NontrivialHolomorphicAt (fun z => z) a
case succ
S : Type
i... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.iter_nontrivial_a | [617, 1] | [621, 47] | simp only [Function.iterate_succ_apply'] | case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
h : NontrivialHolomorphicAt (fun z => (f c)^[n] z) a
⊢ NontrivialHolomorphicAt (fun z => (f c)^[n + 1] z) a | case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
h : NontrivialHolomorphicAt (fun z => (f c)^[n] z) a
⊢ NontrivialHolomorphicAt (fun z => f c ((f c)^[n] z)) a | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
h : NontrivialHolomorphicAt (fun z => (f c)^[n] z) a
⊢ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.iter_nontrivial_a | [617, 1] | [621, 47] | refine NontrivialHolomorphicAt.comp ?_ h | case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
h : NontrivialHolomorphicAt (fun z => (f c)^[n] z) a
⊢ NontrivialHolomorphicAt (fun z => f c ((f c)^[n] z)) a | case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
h : NontrivialHolomorphicAt (fun z => (f c)^[n] z) a
⊢ NontrivialHolomorphicAt (f c) ((f c)^[n] a) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
h : NontrivialHolomorphicAt (fun z => (f c)^[n] z) a
⊢ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.iter_nontrivial_a | [617, 1] | [621, 47] | simp only [s.iter_a] | case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
h : NontrivialHolomorphicAt (fun z => (f c)^[n] z) a
⊢ NontrivialHolomorphicAt (f c) ((f c)^[n] a) | case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
h : NontrivialHolomorphicAt (fun z => (f c)^[n] z) a
⊢ NontrivialHolomorphicAt (f c) a | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
h : NontrivialHolomorphicAt (fun z => (f c)^[n] z) a
⊢ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.iter_nontrivial_a | [617, 1] | [621, 47] | exact s.f_nontrivial c | case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
h : NontrivialHolomorphicAt (fun z => (f c)^[n] z) a
⊢ NontrivialHolomorphicAt (f c) a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
h : NontrivialHolomorphicAt (fun z => (f c)^[n] z) a
⊢ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNearIter_nontrivial_a | [624, 1] | [631, 29] | simp only [s.iter_a] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ NontrivialHolomorphicAt (s.bottcherNear c) ((f c)^[n] a) | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ NontrivialHolomorphicAt (s.bottcherNear c) a | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ NontrivialHolomorphicAt (s.bottcherNear c) ((f c)^[n] a)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNearIter_nontrivial_a | [624, 1] | [631, 29] | exact nontrivialHolomorphicAt_of_mfderiv_ne_zero
(s.bottcherNear_holomorphic _ (s.mem_near c)).along_snd
(s.bottcherNear_mfderiv_ne_zero c) | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ NontrivialHolomorphicAt (s.bottcherNear c) a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ NontrivialHolomorphicAt (s.bottcherNear c) a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | have e : ((univ : Set X) ×ˢ t)ᶜ = univ ×ˢ tᶜ := by
apply Set.ext; intro ⟨a, x⟩; rw [mem_compl_iff]
simp only [prod_mk_mem_set_prod_eq, mem_univ, mem_compl_iff, true_and_iff] | X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
⊢ Nonseparating (univ ×ˢ t) | X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
⊢ Nonseparating (univ ×ˢ t) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
⊢ Nonseparating (univ ×ˢ t)
TA... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | constructor | X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
⊢ Nonseparating (univ ×ˢ t) | case dense
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
⊢ Dense (univ ×ˢ t)ᶜ
case loc
X : Type
inst✝⁵ : T... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | apply Set.ext | X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
⊢ (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ | case h
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
⊢ ∀ (x : X × Y), x ∈ (univ ×ˢ t)ᶜ ↔ x ∈ univ ×ˢ tᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
⊢ (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
TA... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | intro ⟨a, x⟩ | case h
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
⊢ ∀ (x : X × Y), x ∈ (univ ×ˢ t)ᶜ ↔ x ∈ univ ×ˢ tᶜ | case h
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
a : X
x : Y
⊢ (a, x) ∈ (univ ×ˢ t)ᶜ ↔ (a, x) ∈ univ ×ˢ tᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
⊢ ∀ (x : X × Y), x ∈ (u... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | rw [mem_compl_iff] | case h
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
a : X
x : Y
⊢ (a, x) ∈ (univ ×ˢ t)ᶜ ↔ (a, x) ∈ univ ×ˢ tᶜ | case h
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
a : X
x : Y
⊢ (a, x) ∉ univ ×ˢ t ↔ (a, x) ∈ univ ×ˢ tᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
a : X
x : Y
⊢ (a, x) ∈ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | simp only [prod_mk_mem_set_prod_eq, mem_univ, mem_compl_iff, true_and_iff] | case h
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
a : X
x : Y
⊢ (a, x) ∉ univ ×ˢ t ↔ (a, x) ∈ univ ×ˢ tᶜ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
a : X
x : Y
⊢ (a, x) ∉ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | rw [e] | case dense
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
⊢ Dense (univ ×ˢ t)ᶜ | case dense
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
⊢ Dense (univ ×ˢ tᶜ) | Please generate a tactic in lean4 to solve the state.
STATE:
case dense
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | exact dense_univ.prod n.dense | case dense
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
⊢ Dense (univ ×ˢ tᶜ) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case dense
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | intro ⟨a, x⟩ u m un | case loc
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
⊢ ∀ (x : X × Y) (u : Set (X × Y)),
x ∈ univ ×ˢ t... | case loc
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
m : (a, x) ∈ univ ×ˢ t
u... | Please generate a tactic in lean4 to solve the state.
STATE:
case loc
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = un... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | simp only [mem_prod_eq, mem_univ, true_and_iff] at m | case loc
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
m : (a, x) ∈ univ ×ˢ t
u... | case loc
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
un : u ∈ 𝓝 (a, x)
m : x... | Please generate a tactic in lean4 to solve the state.
STATE:
case loc
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = un... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | rcases mem_nhds_prod_iff.mp un with ⟨u0, n0, u1, n1, uu⟩ | case loc
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
un : u ∈ 𝓝 (a, x)
m : x... | case loc.intro.intro.intro.intro
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
... | Please generate a tactic in lean4 to solve the state.
STATE:
case loc
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = un... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | rcases n.loc x u1 m n1 with ⟨c1, cs1, cn1, cp1⟩ | case loc.intro.intro.intro.intro
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
... | case loc.intro.intro.intro.intro.intro.intro.intro
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case loc.intro.intro.intro.intro
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | rcases locallyConnectedSpace_iff_open_connected_subsets.mp (by infer_instance) a u0 n0 with
⟨c0, cs0, co0, cm0, cc0⟩ | case loc.intro.intro.intro.intro.intro.intro.intro
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : ... | case loc.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ... | Please generate a tactic in lean4 to solve the state.
STATE:
case loc.intro.intro.intro.intro.intro.intro.intro
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | use c0 ×ˢ c1 | case loc.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ... | case h
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
un : u ∈ 𝓝 (a, x)
m : x ∈... | Please generate a tactic in lean4 to solve the state.
STATE:
case loc.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyCo... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | refine ⟨?_, ?_, ?_⟩ | case h
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
un : u ∈ 𝓝 (a, x)
m : x ∈... | case h.refine_1
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
un : u ∈ 𝓝 (a, x... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | infer_instance | X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
un : u ∈ 𝓝 (a, x)
m : x ∈ t
u0 :... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | intro ⟨b, y⟩ m' | case h.refine_1
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
un : u ∈ 𝓝 (a, x... | case h.refine_1
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
un : u ∈ 𝓝 (a, x... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_1
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | simp only [mem_prod_eq, mem_diff, mem_univ, true_and_iff] at m' ⊢ | case h.refine_1
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
un : u ∈ 𝓝 (a, x... | case h.refine_1
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
un : u ∈ 𝓝 (a, x... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_1
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | refine ⟨?_, (cs1 m'.2).2⟩ | case h.refine_1
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
un : u ∈ 𝓝 (a, x... | case h.refine_1
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
un : u ∈ 𝓝 (a, x... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_1
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | apply uu | case h.refine_1
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
un : u ∈ 𝓝 (a, x... | case h.refine_1.a
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
un : u ∈ 𝓝 (a,... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_1
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | use cs0 m'.1, (cs1 m'.2).1 | case h.refine_1.a
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
un : u ∈ 𝓝 (a,... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_1.a
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | rw [e, nhdsWithin_prod_eq, nhdsWithin_univ] | case h.refine_2
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
un : u ∈ 𝓝 (a, x... | case h.refine_2
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
un : u ∈ 𝓝 (a, x... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_2
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | exact Filter.prod_mem_prod (co0.mem_nhds cm0) cn1 | case h.refine_2
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
un : u ∈ 𝓝 (a, x... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_2
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | exact cc0.isPreconnected.prod cp1 | case h.refine_3
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
un : u ∈ 𝓝 (a, x... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_3
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
⊢ Dense tᶜ | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
⊢ ∀ (U : Set S), IsOpen U → U.Nonempty → (U ∩ tᶜ).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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
⊢ ∀ (U : Set S), IsOpen U → U.Nonempty → (U ∩ tᶜ).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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u
⊢ (u ∩ tᶜ).Nonempt... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | by_cases zt : z ∉ t | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u
⊢ (u ∩ tᶜ).Nonempt... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u
zt : z ∉ ... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | use z, m, zt | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u
zt : 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u
zt : ¬z ∉... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | simp only [not_not] at zt | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u
zt : ¬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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u
zt : z ∈ ... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | generalize hv : (extChartAt I z).target ∩ (extChartAt I z).symm ⁻¹' u = v | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u
zt : 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u
zt : z ∈ ... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | have vo : IsOpen v := by
rw [← hv]
exact (continuousOn_extChartAt_symm I z).isOpen_inter_preimage
(isOpen_extChartAt_target I z) uo | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u
zt : 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u
zt : z ∈ ... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | have vn : v.Nonempty := by
use extChartAt I z z
simp only [mem_inter_iff, mem_extChartAt_target, true_and_iff, mem_preimage,
PartialEquiv.left_inv _ (mem_extChartAt_source I z), m, ← hv] | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u
zt : 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u
zt : z ∈ ... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | rcases dense_iff_inter_open.mp (h z).dense v vo vn with ⟨y, 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u
zt : z ∈ ... | case neg.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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m✝ : z ∈ u
zt... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | use(extChartAt I z).symm y | case neg.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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m✝ : z ∈ u
zt... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m✝ : z ∈ u
zt : z ∈ t... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | simp only [mem_inter_iff, mem_preimage, mem_compl_iff, not_and, ← hv] at 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m✝ : z ∈ u
zt : z ∈ t... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m✝ : z ∈ u
zt : z ∈ t... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).sy... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | rcases m with ⟨⟨ym, yu⟩, yt⟩ | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m✝ : z ∈ u
zt : z ∈ t... | case h.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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).sy... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | simp only [mem_inter_iff, ym, yu, true_and_iff, mem_compl_iff] | case h.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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u... | case h.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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extCha... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | exact yt ym | case h.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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extCha... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | rw [← hv] | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u
zt : z ∈ t
v : Set... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u
zt : z ∈ t
v : Set... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | exact (continuousOn_extChartAt_symm I z).isOpen_inter_preimage
(isOpen_extChartAt_target I z) uo | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u
zt : z ∈ t
v : Set... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | use extChartAt I z 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u
zt : z ∈ t
v : Set... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u
zt : z ∈ t
... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | simp only [mem_inter_iff, mem_extChartAt_target, true_and_iff, mem_preimage,
PartialEquiv.left_inv _ (mem_extChartAt_source I z), m, ← hv] | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u
zt : z ∈ t
... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).sy... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | intro z u zt un | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
⊢ ∀ (x : S) (u : Set S), x ∈ t → u ∈ 𝓝 x → ∃ c ⊆ u \ t, 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
⊢ ∃ c ⊆ u \ t, 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | have m : extChartAt I z z ∈ (extChartAt I z).target ∩ (extChartAt I z).symm ⁻¹' t := by
simp only [mem_inter_iff, mem_extChartAt_target I z, true_and_iff, mem_preimage,
PartialEquiv.left_inv _ (mem_extChartAt_source I z), zt] | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
⊢ ∃ c ⊆ u \ t, 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(extChartAt ... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | have n : (extChartAt I z).target ∩ (extChartAt I z).symm ⁻¹' u ∈ 𝓝 (extChartAt I z z) := by
apply Filter.inter_mem
exact (isOpen_extChartAt_target I z).mem_nhds (mem_extChartAt_target I z)
exact extChartAt_preimage_mem_nhds _ un | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(extChartAt ... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(extChartAt ... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | rcases (h z).loc _ _ m n with ⟨c, cs, cn, cp⟩ | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(extChartAt ... | case intro.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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | use(extChartAt I z).source ∩ extChartAt I z ⁻¹' c | case intro.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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(extC... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(ex... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | refine ⟨?_, ?_, ?_⟩ | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(extC... | case h.refine_1
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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).sy... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | simp only [mem_inter_iff, mem_extChartAt_target I z, true_and_iff, mem_preimage,
PartialEquiv.left_inv _ (mem_extChartAt_source I z), zt] | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
⊢ ↑(extChartAt I ... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | apply Filter.inter_mem | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(extChartAt ... | case hs
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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(ext... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | exact (isOpen_extChartAt_target I z).mem_nhds (mem_extChartAt_target I z) | case hs
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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(ext... | case ht
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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(ext... | Please generate a tactic in lean4 to solve the state.
STATE:
case hs
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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).s... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | exact extChartAt_preimage_mem_nhds _ un | case ht
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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(ext... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case ht
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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).s... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(extChartAt ... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(extC... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | intro x | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(extC... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(extC... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).sy... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | simp only [mem_inter_iff, mem_preimage, mem_image] | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(extC... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(extC... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).sy... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(extC... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(e... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).sy... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | intro ⟨xz, xc⟩ | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(e... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(e... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | refine ⟨_, xc, ?_⟩ | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(e... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(e... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | simp only [PartialEquiv.left_inv _ xz] | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(e... | no goals | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | intro ⟨y, yc, yx⟩ | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | rw [← yx] | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | have xc := cs yc | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | simp only [mem_diff, mem_inter_iff, mem_preimage] at xc | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | have yz := xc.1.1 | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | use PartialEquiv.map_target _ yz | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | simp only [PartialEquiv.right_inv _ yz, yc] | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(... | 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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | intro x xm | case h.refine_1
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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m... | case h.refine_1
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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_1
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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartA... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | simp only [mem_inter_iff, mem_preimage] at xm | case h.refine_1
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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m... | case h.refine_1
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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_1
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
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartA... |
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