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
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stringlengths 73
2.09M
|
|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | use zp, zi | case right.intro
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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
zq : Complex.abs z ≤ q
zp : Complex.abs z ≤ p + 1
zi : ¬z ∈ t
⊢ Complex.abs z ≤ p + 1 ∧ z ∉ t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro
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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
zq : Complex.abs z ≤ q
zp : Complex.abs z ≤ p + 1
zi : ¬z ∈ t
⊢ Complex.abs z ≤ p + 1 ∧ z ∉ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | refine ⟨p + 1, by bound, ?_⟩ | 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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u = ∅
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t | 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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u = ∅
⊢ closedBall 0 (p + 1) ⊆ t | 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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u = ∅
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | rw [image_eq_empty, diff_eq_empty] at ne | 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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u = ∅
⊢ closedBall 0 (p + 1) ⊆ t | 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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : closedBall 0 (p + 1) ⊆ t
⊢ closedBall 0 (p + 1) ⊆ t | 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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u = ∅
⊢ closedBall 0 (p + 1) ⊆ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | exact ne | 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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : closedBall 0 (p + 1) ⊆ t
⊢ closedBall 0 (p + 1) ⊆ t | 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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : closedBall 0 (p + 1) ⊆ t
⊢ closedBall 0 (p + 1) ⊆ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | bound | 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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u = ∅
⊢ p < p + 1 | 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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u = ∅
⊢ p < p + 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | intro x 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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
⊢ ∀ x ∈ u, p < x | 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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
x : ℝ
m : x ∈ u
⊢ p < x | 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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
⊢ ∀ x ∈ u, p < x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | rcases m with ⟨z, ⟨_, mt⟩, e⟩ | 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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
x : ℝ
m : x ∈ u
⊢ p < x | case intro.intro.intro
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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
x : ℝ
z : ℂ
e : Complex.abs z = x
left✝ : z ∈ closedBall 0 (p + 1)
mt : z ∉ t
⊢ p < x | 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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
x : ℝ
m : x ∈ u
⊢ p < x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | rw [← e] | case intro.intro.intro
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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
x : ℝ
z : ℂ
e : Complex.abs z = x
left✝ : z ∈ closedBall 0 (p + 1)
mt : z ∉ t
⊢ p < x | case intro.intro.intro
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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
x : ℝ
z : ℂ
e : Complex.abs z = x
left✝ : z ∈ closedBall 0 (p + 1)
mt : z ∉ t
⊢ p < Complex.abs z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
x : ℝ
z : ℂ
e : Complex.abs z = x
left✝ : z ∈ closedBall 0 (p + 1)
mt : z ∉ t
⊢ p < x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | contrapose mt | case intro.intro.intro
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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
x : ℝ
z : ℂ
e : Complex.abs z = x
left✝ : z ∈ closedBall 0 (p + 1)
mt : z ∉ t
⊢ p < Complex.abs z | case intro.intro.intro
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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
x : ℝ
z : ℂ
e : Complex.abs z = x
left✝ : z ∈ closedBall 0 (p + 1)
mt : ¬p < Complex.abs z
⊢ ¬z ∉ t | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
x : ℝ
z : ℂ
e : Complex.abs z = x
left✝ : z ∈ closedBall 0 (p + 1)
mt : z ∉ t
⊢ p < Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | simp only [not_not, not_lt] at mt ⊢ | case intro.intro.intro
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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
x : ℝ
z : ℂ
e : Complex.abs z = x
left✝ : z ∈ closedBall 0 (p + 1)
mt : ¬p < Complex.abs z
⊢ ¬z ∉ t | case intro.intro.intro
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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
x : ℝ
z : ℂ
e : Complex.abs z = x
left✝ : z ∈ closedBall 0 (p + 1)
mt : Complex.abs z ≤ p
⊢ z ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
x : ℝ
z : ℂ
e : Complex.abs z = x
left✝ : z ∈ closedBall 0 (p + 1)
mt : ¬p < Complex.abs z
⊢ ¬z ∉ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | apply sub | case intro.intro.intro
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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
x : ℝ
z : ℂ
e : Complex.abs z = x
left✝ : z ∈ closedBall 0 (p + 1)
mt : Complex.abs z ≤ p
⊢ z ∈ t | case intro.intro.intro.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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
x : ℝ
z : ℂ
e : Complex.abs z = x
left✝ : z ∈ closedBall 0 (p + 1)
mt : Complex.abs z ≤ p
⊢ z ∈ closedBall 0 p | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
x : ℝ
z : ℂ
e : Complex.abs z = x
left✝ : z ∈ closedBall 0 (p + 1)
mt : Complex.abs z ≤ p
⊢ z ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | simp only [mem_closedBall, Complex.dist_eq, sub_zero, mt] | case intro.intro.intro.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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
x : ℝ
z : ℂ
e : Complex.abs z = x
left✝ : z ∈ closedBall 0 (p + 1)
mt : Complex.abs z ≤ p
⊢ z ∈ closedBall 0 p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
x : ℝ
z : ℂ
e : Complex.abs z = x
left✝ : z ∈ closedBall 0 (p + 1)
mt : Complex.abs z ≤ p
⊢ z ∈ closedBall 0 p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | linarith | 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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
⊢ p < p + 1 | 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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
⊢ p < p + 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open | [150, 1] | [156, 64] | have sub : closedBall 0 p ⊆ {b | (c, b) ∈ t} := by
intro z m; simp only [mem_setOf]; apply sub; exact ⟨mem_singleton _, 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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
⊢ ∃ q, p < q ∧ {c} ×ˢ closedBall 0 q ⊆ t | 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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub✝ : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
sub : closedBall 0 p ⊆ {b | (c, b) ∈ t}
⊢ ∃ q, p < q ∧ {c} ×ˢ closedBall 0 q ⊆ t | 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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
⊢ ∃ q, p < q ∧ {c} ×ˢ closedBall 0 q ⊆ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open | [150, 1] | [156, 64] | rcases domain_open' sub (o.snd_preimage c) with ⟨q, pq, sub⟩ | 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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub✝ : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
sub : closedBall 0 p ⊆ {b | (c, b) ∈ t}
⊢ ∃ q, p < q ∧ {c} ×ˢ closedBall 0 q ⊆ t | case intro.intro
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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub✝¹ : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
sub✝ : closedBall 0 p ⊆ {b | (c, b) ∈ t}
q : ℝ
pq : p < q
sub : closedBall 0 q ⊆ {b | (c, b) ∈ t}
⊢ ∃ q, p < q ∧ {c} ×ˢ closedBall 0 q ⊆ t | 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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub✝ : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
sub : closedBall 0 p ⊆ {b | (c, b) ∈ t}
⊢ ∃ q, p < q ∧ {c} ×ˢ closedBall 0 q ⊆ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open | [150, 1] | [156, 64] | use q, pq | case intro.intro
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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub✝¹ : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
sub✝ : closedBall 0 p ⊆ {b | (c, b) ∈ t}
q : ℝ
pq : p < q
sub : closedBall 0 q ⊆ {b | (c, b) ∈ t}
⊢ ∃ q, p < q ∧ {c} ×ˢ closedBall 0 q ⊆ t | case right
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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub✝¹ : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
sub✝ : closedBall 0 p ⊆ {b | (c, b) ∈ t}
q : ℝ
pq : p < q
sub : closedBall 0 q ⊆ {b | (c, b) ∈ t}
⊢ {c} ×ˢ closedBall 0 q ⊆ t | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub✝¹ : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
sub✝ : closedBall 0 p ⊆ {b | (c, b) ∈ t}
q : ℝ
pq : p < q
sub : closedBall 0 q ⊆ {b | (c, b) ∈ t}
⊢ ∃ q, p < q ∧ {c} ×ˢ closedBall 0 q ⊆ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open | [150, 1] | [156, 64] | intro ⟨e, z⟩ ⟨ec, m⟩ | case right
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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub✝¹ : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
sub✝ : closedBall 0 p ⊆ {b | (c, b) ∈ t}
q : ℝ
pq : p < q
sub : closedBall 0 q ⊆ {b | (c, b) ∈ t}
⊢ {c} ×ˢ closedBall 0 q ⊆ t | case right
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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub✝¹ : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
sub✝ : closedBall 0 p ⊆ {b | (c, b) ∈ t}
q : ℝ
pq : p < q
sub : closedBall 0 q ⊆ {b | (c, b) ∈ t}
e z : ℂ
ec : (e, z).1 ∈ {c}
m : (e, z).2 ∈ closedBall 0 q
⊢ (e, z) ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case right
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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub✝¹ : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
sub✝ : closedBall 0 p ⊆ {b | (c, b) ∈ t}
q : ℝ
pq : p < q
sub : closedBall 0 q ⊆ {b | (c, b) ∈ t}
⊢ {c} ×ˢ closedBall 0 q ⊆ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open | [150, 1] | [156, 64] | simp only [mem_singleton_iff] at ec | case right
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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub✝¹ : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
sub✝ : closedBall 0 p ⊆ {b | (c, b) ∈ t}
q : ℝ
pq : p < q
sub : closedBall 0 q ⊆ {b | (c, b) ∈ t}
e z : ℂ
ec : (e, z).1 ∈ {c}
m : (e, z).2 ∈ closedBall 0 q
⊢ (e, z) ∈ t | case right
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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub✝¹ : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
sub✝ : closedBall 0 p ⊆ {b | (c, b) ∈ t}
q : ℝ
pq : p < q
sub : closedBall 0 q ⊆ {b | (c, b) ∈ t}
e z : ℂ
m : (e, z).2 ∈ closedBall 0 q
ec : e = c
⊢ (e, z) ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case right
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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub✝¹ : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
sub✝ : closedBall 0 p ⊆ {b | (c, b) ∈ t}
q : ℝ
pq : p < q
sub : closedBall 0 q ⊆ {b | (c, b) ∈ t}
e z : ℂ
ec : (e, z).1 ∈ {c}
m : (e, z).2 ∈ closedBall 0 q
⊢ (e, z) ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open | [150, 1] | [156, 64] | replace m := sub m | case right
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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub✝¹ : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
sub✝ : closedBall 0 p ⊆ {b | (c, b) ∈ t}
q : ℝ
pq : p < q
sub : closedBall 0 q ⊆ {b | (c, b) ∈ t}
e z : ℂ
m : (e, z).2 ∈ closedBall 0 q
ec : e = c
⊢ (e, z) ∈ t | case right
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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub✝¹ : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
sub✝ : closedBall 0 p ⊆ {b | (c, b) ∈ t}
q : ℝ
pq : p < q
sub : closedBall 0 q ⊆ {b | (c, b) ∈ t}
e z : ℂ
ec : e = c
m : (e, z).2 ∈ {b | (c, b) ∈ t}
⊢ (e, z) ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case right
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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub✝¹ : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
sub✝ : closedBall 0 p ⊆ {b | (c, b) ∈ t}
q : ℝ
pq : p < q
sub : closedBall 0 q ⊆ {b | (c, b) ∈ t}
e z : ℂ
m : (e, z).2 ∈ closedBall 0 q
ec : e = c
⊢ (e, z) ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open | [150, 1] | [156, 64] | simp only [← ec, mem_setOf] at m | case right
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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub✝¹ : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
sub✝ : closedBall 0 p ⊆ {b | (c, b) ∈ t}
q : ℝ
pq : p < q
sub : closedBall 0 q ⊆ {b | (c, b) ∈ t}
e z : ℂ
ec : e = c
m : (e, z).2 ∈ {b | (c, b) ∈ t}
⊢ (e, z) ∈ t | case right
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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub✝¹ : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
sub✝ : closedBall 0 p ⊆ {b | (c, b) ∈ t}
q : ℝ
pq : p < q
sub : closedBall 0 q ⊆ {b | (c, b) ∈ t}
e z : ℂ
ec : e = c
m : (e, z) ∈ t
⊢ (e, z) ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case right
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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub✝¹ : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
sub✝ : closedBall 0 p ⊆ {b | (c, b) ∈ t}
q : ℝ
pq : p < q
sub : closedBall 0 q ⊆ {b | (c, b) ∈ t}
e z : ℂ
ec : e = c
m : (e, z).2 ∈ {b | (c, b) ∈ t}
⊢ (e, z) ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open | [150, 1] | [156, 64] | exact m | case right
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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub✝¹ : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
sub✝ : closedBall 0 p ⊆ {b | (c, b) ∈ t}
q : ℝ
pq : p < q
sub : closedBall 0 q ⊆ {b | (c, b) ∈ t}
e z : ℂ
ec : e = c
m : (e, z) ∈ t
⊢ (e, z) ∈ t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub✝¹ : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
sub✝ : closedBall 0 p ⊆ {b | (c, b) ∈ t}
q : ℝ
pq : p < q
sub : closedBall 0 q ⊆ {b | (c, b) ∈ t}
e z : ℂ
ec : e = c
m : (e, z) ∈ t
⊢ (e, z) ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open | [150, 1] | [156, 64] | intro 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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
⊢ closedBall 0 p ⊆ {b | (c, b) ∈ t} | 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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
z : ℂ
m : z ∈ closedBall 0 p
⊢ z ∈ {b | (c, b) ∈ t} | 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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
⊢ closedBall 0 p ⊆ {b | (c, b) ∈ t}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open | [150, 1] | [156, 64] | simp only [mem_setOf] | 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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
z : ℂ
m : z ∈ closedBall 0 p
⊢ z ∈ {b | (c, b) ∈ t} | 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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
z : ℂ
m : z ∈ closedBall 0 p
⊢ (c, z) ∈ t | 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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
z : ℂ
m : z ∈ closedBall 0 p
⊢ z ∈ {b | (c, b) ∈ t}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open | [150, 1] | [156, 64] | apply sub | 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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
z : ℂ
m : z ∈ closedBall 0 p
⊢ (c, z) ∈ t | case 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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
z : ℂ
m : z ∈ closedBall 0 p
⊢ (c, z) ∈ {c} ×ˢ closedBall 0 p | 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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
z : ℂ
m : z ∈ closedBall 0 p
⊢ (c, z) ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open | [150, 1] | [156, 64] | exact ⟨mem_singleton _, m⟩ | case 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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
z : ℂ
m : z ∈ closedBall 0 p
⊢ (c, z) ∈ {c} ×ˢ closedBall 0 p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case 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
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
z : ℂ
m : z ∈ closedBall 0 p
⊢ (c, z) ∈ {c} ×ˢ closedBall 0 p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.congr | [159, 1] | [173, 26] | have e := e.self_of_nhdsSet (mem_domain c g.nonneg) | 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
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
e : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
⊢ r1 c 0 = 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
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
e✝ : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
e : uncurry r0 (c, 0) = uncurry r1 (c, 0)
⊢ r1 c 0 = 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
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
e : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
⊢ r1 c 0 = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.congr | [159, 1] | [173, 26] | simp only [uncurry] at e | 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
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
e✝ : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
e : uncurry r0 (c, 0) = uncurry r1 (c, 0)
⊢ r1 c 0 = 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
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
e✝ : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
e : r0 c 0 = r1 c 0
⊢ r1 c 0 = 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
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
e✝ : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
e : uncurry r0 (c, 0) = uncurry r1 (c, 0)
⊢ r1 c 0 = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.congr | [159, 1] | [173, 26] | rw [← e] | 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
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
e✝ : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
e : r0 c 0 = r1 c 0
⊢ r1 c 0 = 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
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
e✝ : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
e : r0 c 0 = r1 c 0
⊢ r0 c 0 = 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
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
e✝ : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
e : r0 c 0 = r1 c 0
⊢ r1 c 0 = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.congr | [159, 1] | [173, 26] | exact g.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
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
e✝ : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
e : r0 c 0 = r1 c 0
⊢ r0 c 0 = 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
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
e✝ : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
e : r0 c 0 = r1 c 0
⊢ r0 c 0 = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.congr | [159, 1] | [173, 26] | refine g.start.mp ((e.filter_mono (nhds_le_nhdsSet (mem_domain c g.nonneg))).mp ?_) | 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
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
e : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r1 x.1 x.2) = x.2 | 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
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
e : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0),
uncurry r0 x = uncurry r1 x → s.bottcherNear x.1 (r0 x.1 x.2) = x.2 → s.bottcherNear x.1 (r1 x.1 x.2) = x.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
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
e : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r1 x.1 x.2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.congr | [159, 1] | [173, 26] | refine eventually_of_forall fun x e s ↦ ?_ | 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
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
e : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0),
uncurry r0 x = uncurry r1 x → s.bottcherNear x.1 (r0 x.1 x.2) = x.2 → s.bottcherNear x.1 (r1 x.1 x.2) = x.2 | 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
p : ℝ
s✝ : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s✝ c p n r0
e✝ : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
x : ℂ × ℂ
e : uncurry r0 x = uncurry r1 x
s : s✝.bottcherNear x.1 (r0 x.1 x.2) = x.2
⊢ s✝.bottcherNear x.1 (r1 x.1 x.2) = x.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
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
e : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0),
uncurry r0 x = uncurry r1 x → s.bottcherNear x.1 (r0 x.1 x.2) = x.2 → s.bottcherNear x.1 (r1 x.1 x.2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.congr | [159, 1] | [173, 26] | simp only [uncurry] at e | 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
p : ℝ
s✝ : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s✝ c p n r0
e✝ : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
x : ℂ × ℂ
e : uncurry r0 x = uncurry r1 x
s : s✝.bottcherNear x.1 (r0 x.1 x.2) = x.2
⊢ s✝.bottcherNear x.1 (r1 x.1 x.2) = x.2 | 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
p : ℝ
s✝ : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s✝ c p n r0
e✝ : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
x : ℂ × ℂ
e : r0 x.1 x.2 = r1 x.1 x.2
s : s✝.bottcherNear x.1 (r0 x.1 x.2) = x.2
⊢ s✝.bottcherNear x.1 (r1 x.1 x.2) = x.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
p : ℝ
s✝ : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s✝ c p n r0
e✝ : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
x : ℂ × ℂ
e : uncurry r0 x = uncurry r1 x
s : s✝.bottcherNear x.1 (r0 x.1 x.2) = x.2
⊢ s✝.bottcherNear x.1 (r1 x.1 x.2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.congr | [159, 1] | [173, 26] | rw [← e] | 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
p : ℝ
s✝ : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s✝ c p n r0
e✝ : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
x : ℂ × ℂ
e : r0 x.1 x.2 = r1 x.1 x.2
s : s✝.bottcherNear x.1 (r0 x.1 x.2) = x.2
⊢ s✝.bottcherNear x.1 (r1 x.1 x.2) = x.2 | 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
p : ℝ
s✝ : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s✝ c p n r0
e✝ : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
x : ℂ × ℂ
e : r0 x.1 x.2 = r1 x.1 x.2
s : s✝.bottcherNear x.1 (r0 x.1 x.2) = x.2
⊢ s✝.bottcherNear x.1 (r0 x.1 x.2) = x.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
p : ℝ
s✝ : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s✝ c p n r0
e✝ : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
x : ℂ × ℂ
e : r0 x.1 x.2 = r1 x.1 x.2
s : s✝.bottcherNear x.1 (r0 x.1 x.2) = x.2
⊢ s✝.bottcherNear x.1 (r1 x.1 x.2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.congr | [159, 1] | [173, 26] | exact s | 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
p : ℝ
s✝ : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s✝ c p n r0
e✝ : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
x : ℂ × ℂ
e : r0 x.1 x.2 = r1 x.1 x.2
s : s✝.bottcherNear x.1 (r0 x.1 x.2) = x.2
⊢ s✝.bottcherNear x.1 (r0 x.1 x.2) = x.2 | 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
p : ℝ
s✝ : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s✝ c p n r0
e✝ : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
x : ℂ × ℂ
e : r0 x.1 x.2 = r1 x.1 x.2
s : s✝.bottcherNear x.1 (r0 x.1 x.2) = x.2
⊢ s✝.bottcherNear x.1 (r0 x.1 x.2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.congr | [159, 1] | [173, 26] | have eqn := g.eqn | 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
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
e : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ ({c} ×ˢ closedBall 0 p), Eqn s n r1 x | 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
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
e : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
eqn : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ ({c} ×ˢ closedBall 0 p), Eqn s n r0 x
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ ({c} ×ˢ closedBall 0 p), Eqn s n r1 x | 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
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
e : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ ({c} ×ˢ closedBall 0 p), Eqn s n r1 x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.congr | [159, 1] | [173, 26] | simp only [Filter.EventuallyEq, eventually_nhdsSet_iff_forall] at eqn e ⊢ | 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
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
e : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
eqn : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ ({c} ×ˢ closedBall 0 p), Eqn s n r0 x
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ ({c} ×ˢ closedBall 0 p), Eqn s n r1 x | 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
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
eqn : ∀ x ∈ {c} ×ˢ closedBall 0 p, ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, Eqn s n r0 x
e : ∀ x ∈ {c} ×ˢ closedBall 0 p, ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, uncurry r0 x = uncurry r1 x
⊢ ∀ x ∈ {c} ×ˢ closedBall 0 p, ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, Eqn s n r1 x | 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
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
e : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
eqn : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ ({c} ×ˢ closedBall 0 p), Eqn s n r0 x
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ ({c} ×ˢ closedBall 0 p), Eqn s n r1 x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.congr | [159, 1] | [173, 26] | intro x 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
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
eqn : ∀ x ∈ {c} ×ˢ closedBall 0 p, ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, Eqn s n r0 x
e : ∀ x ∈ {c} ×ˢ closedBall 0 p, ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, uncurry r0 x = uncurry r1 x
⊢ ∀ x ∈ {c} ×ˢ closedBall 0 p, ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, Eqn s n r1 x | 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
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
eqn : ∀ x ∈ {c} ×ˢ closedBall 0 p, ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, Eqn s n r0 x
e : ∀ x ∈ {c} ×ˢ closedBall 0 p, ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, uncurry r0 x = uncurry r1 x
x : ℂ × ℂ
m : x ∈ {c} ×ˢ closedBall 0 p
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, Eqn s n r1 x | 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
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
eqn : ∀ x ∈ {c} ×ˢ closedBall 0 p, ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, Eqn s n r0 x
e : ∀ x ∈ {c} ×ˢ closedBall 0 p, ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, uncurry r0 x = uncurry r1 x
⊢ ∀ x ∈ {c} ×ˢ closedBall 0 p, ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, Eqn s n r1 x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.congr | [159, 1] | [173, 26] | refine (eqn x m).mp ((e x m).eventually_nhds.mp (eventually_of_forall fun y e eqn ↦ ?_)) | 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
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
eqn : ∀ x ∈ {c} ×ˢ closedBall 0 p, ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, Eqn s n r0 x
e : ∀ x ∈ {c} ×ˢ closedBall 0 p, ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, uncurry r0 x = uncurry r1 x
x : ℂ × ℂ
m : x ∈ {c} ×ˢ closedBall 0 p
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, Eqn s n r1 x | 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
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
eqn✝ : ∀ x ∈ {c} ×ˢ closedBall 0 p, ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, Eqn s n r0 x
e✝ : ∀ x ∈ {c} ×ˢ closedBall 0 p, ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, uncurry r0 x = uncurry r1 x
x : ℂ × ℂ
m : x ∈ {c} ×ˢ closedBall 0 p
y : ℂ × ℂ
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝 y, uncurry r0 x = uncurry r1 x
eqn : Eqn s n r0 y
⊢ Eqn s n r1 y | 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
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
eqn : ∀ x ∈ {c} ×ˢ closedBall 0 p, ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, Eqn s n r0 x
e : ∀ x ∈ {c} ×ˢ closedBall 0 p, ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, uncurry r0 x = uncurry r1 x
x : ℂ × ℂ
m : x ∈ {c} ×ˢ closedBall 0 p
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, Eqn s n r1 x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.congr | [159, 1] | [173, 26] | exact eqn.congr e | 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
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
eqn✝ : ∀ x ∈ {c} ×ˢ closedBall 0 p, ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, Eqn s n r0 x
e✝ : ∀ x ∈ {c} ×ˢ closedBall 0 p, ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, uncurry r0 x = uncurry r1 x
x : ℂ × ℂ
m : x ∈ {c} ×ˢ closedBall 0 p
y : ℂ × ℂ
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝 y, uncurry r0 x = uncurry r1 x
eqn : Eqn s n r0 y
⊢ Eqn s n r1 y | 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
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
eqn✝ : ∀ x ∈ {c} ×ˢ closedBall 0 p, ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, Eqn s n r0 x
e✝ : ∀ x ∈ {c} ×ˢ closedBall 0 p, ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, uncurry r0 x = uncurry r1 x
x : ℂ × ℂ
m : x ∈ {c} ×ˢ closedBall 0 p
y : ℂ × ℂ
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝 y, uncurry r0 x = uncurry r1 x
eqn : Eqn s n r0 y
⊢ Eqn s n r1 y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Eqn.potential | [176, 1] | [178, 75] | simp only [s.potential_eq e.near, Super.potential', e.eqn, Complex.abs.map_pow, ← Nat.cast_pow,
Real.pow_rpow_inv_natCast (Complex.abs.nonneg _) (pow_ne_zero _ s.d0)] | 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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
e : Eqn s n r x
⊢ s.potential x.1 (r x.1 x.2) = Complex.abs x.2 | 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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
e : Eqn s n r x
⊢ s.potential x.1 (r x.1 x.2) = Complex.abs x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | eqn_noncritical | [181, 1] | [198, 64] | rcases x with ⟨c, x⟩ | 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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
e : ∀ᶠ (y : ℂ × ℂ) in 𝓝 x, Eqn s n r y
x0 : x.2 ≠ 0
⊢ mfderiv I I (s.bottcherNearIter n x.1) (r x.1 x.2) ≠ 0 | case mk
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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
c x : ℂ
e : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y
x0 : (c, x).2 ≠ 0
⊢ mfderiv I I (s.bottcherNearIter n (c, x).1) (r (c, x).1 (c, x).2) ≠ 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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
e : ∀ᶠ (y : ℂ × ℂ) in 𝓝 x, Eqn s n r y
x0 : x.2 ≠ 0
⊢ mfderiv I I (s.bottcherNearIter n x.1) (r x.1 x.2) ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | eqn_noncritical | [181, 1] | [198, 64] | contrapose x0 | case mk
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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
c x : ℂ
e : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y
x0 : (c, x).2 ≠ 0
⊢ mfderiv I I (s.bottcherNearIter n (c, x).1) (r (c, x).1 (c, x).2) ≠ 0 | case mk
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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
c x : ℂ
e : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y
x0 : ¬mfderiv I I (s.bottcherNearIter n (c, x).1) (r (c, x).1 (c, x).2) ≠ 0
⊢ ¬(c, x).2 ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
c x : ℂ
e : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y
x0 : (c, x).2 ≠ 0
⊢ mfderiv I I (s.bottcherNearIter n (c, x).1) (r (c, x).1 (c, x).2) ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | eqn_noncritical | [181, 1] | [198, 64] | simp only [not_not] at x0 ⊢ | case mk
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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
c x : ℂ
e : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y
x0 : ¬mfderiv I I (s.bottcherNearIter n (c, x).1) (r (c, x).1 (c, x).2) ≠ 0
⊢ ¬(c, x).2 ≠ 0 | case mk
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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
c x : ℂ
e : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y
x0 : mfderiv I I (s.bottcherNearIter n c) (r c x) = 0
⊢ x = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
c x : ℂ
e : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y
x0 : ¬mfderiv I I (s.bottcherNearIter n (c, x).1) (r (c, x).1 (c, x).2) ≠ 0
⊢ ¬(c, x).2 ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | eqn_noncritical | [181, 1] | [198, 64] | replace x0 : mfderiv I I (fun y ↦ s.bottcherNearIter n c (r c y)) x = 0 := by
rw [←Function.comp_def,
mfderiv_comp x (s.bottcherNearIter_holomorphic e.self_of_nhds.near).along_snd.mdifferentiableAt
e.self_of_nhds.holo.along_snd.mdifferentiableAt,
x0, ContinuousLinearMap.zero_comp] | case mk
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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
c x : ℂ
e : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y
x0 : mfderiv I I (s.bottcherNearIter n c) (r c x) = 0
⊢ x = 0 | case mk
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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
c x : ℂ
e : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y
x0 : mfderiv I I (fun y => s.bottcherNearIter n c (r c y)) x = 0
⊢ x = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
c x : ℂ
e : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y
x0 : mfderiv I I (s.bottcherNearIter n c) (r c x) = 0
⊢ x = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | eqn_noncritical | [181, 1] | [198, 64] | have loc : (fun y ↦ s.bottcherNearIter n c (r c y)) =ᶠ[𝓝 x] fun y ↦ y ^ d ^ n :=
((continuousAt_const.prod continuousAt_id).eventually e).mp
(eventually_of_forall fun _ e ↦ e.eqn) | case mk
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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
c x : ℂ
e : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y
x0 : mfderiv I I (fun y => s.bottcherNearIter n c (r c y)) x = 0
⊢ x = 0 | case mk
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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
c x : ℂ
e : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y
x0 : mfderiv I I (fun y => s.bottcherNearIter n c (r c y)) x = 0
loc : (𝓝 x).EventuallyEq (fun y => s.bottcherNearIter n c (r c y)) fun y => y ^ d ^ n
⊢ x = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
c x : ℂ
e : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y
x0 : mfderiv I I (fun y => s.bottcherNearIter n c (r c y)) x = 0
⊢ x = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | eqn_noncritical | [181, 1] | [198, 64] | rw [mfderiv_eq_fderiv, loc.fderiv_eq] at x0 | case mk
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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
c x : ℂ
e : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y
x0 : mfderiv I I (fun y => s.bottcherNearIter n c (r c y)) x = 0
loc : (𝓝 x).EventuallyEq (fun y => s.bottcherNearIter n c (r c y)) fun y => y ^ d ^ n
⊢ x = 0 | case mk
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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
c x : ℂ
e : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y
x0 : fderiv ℂ (fun y => y ^ d ^ n) x = 0
loc : (𝓝 x).EventuallyEq (fun y => s.bottcherNearIter n c (r c y)) fun y => y ^ d ^ n
⊢ x = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
c x : ℂ
e : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y
x0 : mfderiv I I (fun y => s.bottcherNearIter n c (r c y)) x = 0
loc : (𝓝 x).EventuallyEq (fun y => s.bottcherNearIter n c (r c y)) fun y => y ^ d ^ n
⊢ x = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | eqn_noncritical | [181, 1] | [198, 64] | have d := (differentiableAt_pow (𝕜 := ℂ) (x := x) (d ^ n)).hasFDerivAt.hasDerivAt.deriv | case mk
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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
c x : ℂ
e : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y
x0 : fderiv ℂ (fun y => y ^ d ^ n) x = 0
loc : (𝓝 x).EventuallyEq (fun y => s.bottcherNearIter n c (r c y)) fun y => y ^ d ^ n
⊢ x = 0 | case mk
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
p : ℝ
s : Super f d✝ a
r : ℂ → ℂ → S
c x : ℂ
e : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y
x0 : fderiv ℂ (fun y => y ^ d✝ ^ n) x = 0
loc : (𝓝 x).EventuallyEq (fun y => s.bottcherNearIter n c (r c y)) fun y => y ^ d✝ ^ n
d : deriv (fun x => x ^ d✝ ^ n) x = (fderiv ℂ (fun x => x ^ d✝ ^ n) x) 1
⊢ x = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
c x : ℂ
e : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y
x0 : fderiv ℂ (fun y => y ^ d ^ n) x = 0
loc : (𝓝 x).EventuallyEq (fun y => s.bottcherNearIter n c (r c y)) fun y => y ^ d ^ n
⊢ x = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | eqn_noncritical | [181, 1] | [198, 64] | apply_fun (fun x ↦ x 1) at x0 | case mk
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
p : ℝ
s : Super f d✝ a
r : ℂ → ℂ → S
c x : ℂ
e : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y
x0 : fderiv ℂ (fun y => y ^ d✝ ^ n) x = 0
loc : (𝓝 x).EventuallyEq (fun y => s.bottcherNearIter n c (r c y)) fun y => y ^ d✝ ^ n
d : deriv (fun x => x ^ d✝ ^ n) x = (fderiv ℂ (fun x => x ^ d✝ ^ n) x) 1
⊢ x = 0 | case mk
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
p : ℝ
s : Super f d✝ a
r : ℂ → ℂ → S
c x : ℂ
e : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y
loc : (𝓝 x).EventuallyEq (fun y => s.bottcherNearIter n c (r c y)) fun y => y ^ d✝ ^ n
d : deriv (fun x => x ^ d✝ ^ n) x = (fderiv ℂ (fun x => x ^ d✝ ^ n) x) 1
x0 : (fderiv ℂ (fun y => y ^ d✝ ^ n) x) 1 = 0 1
⊢ x = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
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
p : ℝ
s : Super f d✝ a
r : ℂ → ℂ → S
c x : ℂ
e : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y
x0 : fderiv ℂ (fun y => y ^ d✝ ^ n) x = 0
loc : (𝓝 x).EventuallyEq (fun y => s.bottcherNearIter n c (r c y)) fun y => y ^ d✝ ^ n
d : deriv (fun x => x ^ d✝ ^ n) x = (fderiv ℂ (fun x => x ^ d✝ ^ n) x) 1
⊢ x = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | eqn_noncritical | [181, 1] | [198, 64] | rw [x0] at d | case mk
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
p : ℝ
s : Super f d✝ a
r : ℂ → ℂ → S
c x : ℂ
e : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y
loc : (𝓝 x).EventuallyEq (fun y => s.bottcherNearIter n c (r c y)) fun y => y ^ d✝ ^ n
d : deriv (fun x => x ^ d✝ ^ n) x = (fderiv ℂ (fun x => x ^ d✝ ^ n) x) 1
x0 : (fderiv ℂ (fun y => y ^ d✝ ^ n) x) 1 = 0 1
⊢ x = 0 | case mk
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
p : ℝ
s : Super f d✝ a
r : ℂ → ℂ → S
c x : ℂ
e : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y
loc : (𝓝 x).EventuallyEq (fun y => s.bottcherNearIter n c (r c y)) fun y => y ^ d✝ ^ n
d : deriv (fun x => x ^ d✝ ^ n) x = 0 1
x0 : (fderiv ℂ (fun y => y ^ d✝ ^ n) x) 1 = 0 1
⊢ x = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
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
p : ℝ
s : Super f d✝ a
r : ℂ → ℂ → S
c x : ℂ
e : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y
loc : (𝓝 x).EventuallyEq (fun y => s.bottcherNearIter n c (r c y)) fun y => y ^ d✝ ^ n
d : deriv (fun x => x ^ d✝ ^ n) x = (fderiv ℂ (fun x => x ^ d✝ ^ n) x) 1
x0 : (fderiv ℂ (fun y => y ^ d✝ ^ n) x) 1 = 0 1
⊢ x = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | eqn_noncritical | [181, 1] | [198, 64] | replace d := Eq.trans d (ContinuousLinearMap.zero_apply _) | case mk
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
p : ℝ
s : Super f d✝ a
r : ℂ → ℂ → S
c x : ℂ
e : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y
loc : (𝓝 x).EventuallyEq (fun y => s.bottcherNearIter n c (r c y)) fun y => y ^ d✝ ^ n
d : deriv (fun x => x ^ d✝ ^ n) x = 0 1
x0 : (fderiv ℂ (fun y => y ^ d✝ ^ n) x) 1 = 0 1
⊢ x = 0 | case mk
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
p : ℝ
s : Super f d✝ a
r : ℂ → ℂ → S
c x : ℂ
e : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y
loc : (𝓝 x).EventuallyEq (fun y => s.bottcherNearIter n c (r c y)) fun y => y ^ d✝ ^ n
x0 : (fderiv ℂ (fun y => y ^ d✝ ^ n) x) 1 = 0 1
d : deriv (fun x => x ^ d✝ ^ n) x = 0
⊢ x = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
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
p : ℝ
s : Super f d✝ a
r : ℂ → ℂ → S
c x : ℂ
e : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y
loc : (𝓝 x).EventuallyEq (fun y => s.bottcherNearIter n c (r c y)) fun y => y ^ d✝ ^ n
d : deriv (fun x => x ^ d✝ ^ n) x = 0 1
x0 : (fderiv ℂ (fun y => y ^ d✝ ^ n) x) 1 = 0 1
⊢ x = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | eqn_noncritical | [181, 1] | [198, 64] | rw [deriv_pow, mul_eq_zero, Nat.cast_eq_zero, pow_eq_zero_iff', pow_eq_zero_iff'] at d | case mk
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
p : ℝ
s : Super f d✝ a
r : ℂ → ℂ → S
c x : ℂ
e : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y
loc : (𝓝 x).EventuallyEq (fun y => s.bottcherNearIter n c (r c y)) fun y => y ^ d✝ ^ n
x0 : (fderiv ℂ (fun y => y ^ d✝ ^ n) x) 1 = 0 1
d : deriv (fun x => x ^ d✝ ^ n) x = 0
⊢ x = 0 | case mk
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
p : ℝ
s : Super f d✝ a
r : ℂ → ℂ → S
c x : ℂ
e : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y
loc : (𝓝 x).EventuallyEq (fun y => s.bottcherNearIter n c (r c y)) fun y => y ^ d✝ ^ n
x0 : (fderiv ℂ (fun y => y ^ d✝ ^ n) x) 1 = 0 1
d : d✝ = 0 ∧ n ≠ 0 ∨ x = 0 ∧ d✝ ^ n - 1 ≠ 0
⊢ x = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
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
p : ℝ
s : Super f d✝ a
r : ℂ → ℂ → S
c x : ℂ
e : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y
loc : (𝓝 x).EventuallyEq (fun y => s.bottcherNearIter n c (r c y)) fun y => y ^ d✝ ^ n
x0 : (fderiv ℂ (fun y => y ^ d✝ ^ n) x) 1 = 0 1
d : deriv (fun x => x ^ d✝ ^ n) x = 0
⊢ x = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | eqn_noncritical | [181, 1] | [198, 64] | simp only [s.d0, false_and_iff, false_or_iff] at d | case mk
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
p : ℝ
s : Super f d✝ a
r : ℂ → ℂ → S
c x : ℂ
e : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y
loc : (𝓝 x).EventuallyEq (fun y => s.bottcherNearIter n c (r c y)) fun y => y ^ d✝ ^ n
x0 : (fderiv ℂ (fun y => y ^ d✝ ^ n) x) 1 = 0 1
d : d✝ = 0 ∧ n ≠ 0 ∨ x = 0 ∧ d✝ ^ n - 1 ≠ 0
⊢ x = 0 | case mk
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
p : ℝ
s : Super f d✝ a
r : ℂ → ℂ → S
c x : ℂ
e : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y
loc : (𝓝 x).EventuallyEq (fun y => s.bottcherNearIter n c (r c y)) fun y => y ^ d✝ ^ n
x0 : (fderiv ℂ (fun y => y ^ d✝ ^ n) x) 1 = 0 1
d : x = 0 ∧ d✝ ^ n - 1 ≠ 0
⊢ x = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
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
p : ℝ
s : Super f d✝ a
r : ℂ → ℂ → S
c x : ℂ
e : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y
loc : (𝓝 x).EventuallyEq (fun y => s.bottcherNearIter n c (r c y)) fun y => y ^ d✝ ^ n
x0 : (fderiv ℂ (fun y => y ^ d✝ ^ n) x) 1 = 0 1
d : d✝ = 0 ∧ n ≠ 0 ∨ x = 0 ∧ d✝ ^ n - 1 ≠ 0
⊢ x = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | eqn_noncritical | [181, 1] | [198, 64] | exact d.1 | case mk
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
p : ℝ
s : Super f d✝ a
r : ℂ → ℂ → S
c x : ℂ
e : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y
loc : (𝓝 x).EventuallyEq (fun y => s.bottcherNearIter n c (r c y)) fun y => y ^ d✝ ^ n
x0 : (fderiv ℂ (fun y => y ^ d✝ ^ n) x) 1 = 0 1
d : x = 0 ∧ d✝ ^ n - 1 ≠ 0
⊢ x = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
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
p : ℝ
s : Super f d✝ a
r : ℂ → ℂ → S
c x : ℂ
e : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y
loc : (𝓝 x).EventuallyEq (fun y => s.bottcherNearIter n c (r c y)) fun y => y ^ d✝ ^ n
x0 : (fderiv ℂ (fun y => y ^ d✝ ^ n) x) 1 = 0 1
d : x = 0 ∧ d✝ ^ n - 1 ≠ 0
⊢ x = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | eqn_noncritical | [181, 1] | [198, 64] | rw [←Function.comp_def,
mfderiv_comp x (s.bottcherNearIter_holomorphic e.self_of_nhds.near).along_snd.mdifferentiableAt
e.self_of_nhds.holo.along_snd.mdifferentiableAt,
x0, ContinuousLinearMap.zero_comp] | 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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
c x : ℂ
e : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y
x0 : mfderiv I I (s.bottcherNearIter n c) (r c x) = 0
⊢ mfderiv I I (fun y => s.bottcherNearIter n c (r c y)) x = 0 | 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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
c x : ℂ
e : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y
x0 : mfderiv I I (s.bottcherNearIter n c) (r c x) = 0
⊢ mfderiv I I (fun y => s.bottcherNearIter n c (r c y)) x = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.p1 | [201, 1] | [208, 93] | by_contra p1 | 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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
g : Grow s c p n r
⊢ p < 1 | 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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
g : Grow s c p n r
p1 : ¬p < 1
⊢ False | 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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
g : Grow s c p n r
⊢ p < 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.p1 | [201, 1] | [208, 93] | simp only [not_lt] at p1 | 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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
g : Grow s c p n r
p1 : ¬p < 1
⊢ False | 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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
g : Grow s c p n r
p1 : 1 ≤ p
⊢ False | 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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
g : Grow s c p n r
p1 : ¬p < 1
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.p1 | [201, 1] | [208, 93] | have e := (g.eqn.filter_mono (nhds_le_nhdsSet (x := (c, 1)) ?_)).self_of_nhds | 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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
g : Grow s c p n r
p1 : 1 ≤ p
⊢ False | case refine_2
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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
g : Grow s c p n r
p1 : 1 ≤ p
e : Eqn s n r (c, 1)
⊢ False
case refine_1
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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
g : Grow s c p n r
p1 : 1 ≤ p
⊢ (c, 1) ∈ {c} ×ˢ closedBall 0 p | 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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
g : Grow s c p n r
p1 : 1 ≤ p
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.p1 | [201, 1] | [208, 93] | have lt := s.potential_lt_one ⟨_, e.near⟩ | case refine_2
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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
g : Grow s c p n r
p1 : 1 ≤ p
e : Eqn s n r (c, 1)
⊢ False | case refine_2
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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
g : Grow s c p n r
p1 : 1 ≤ p
e : Eqn s n r (c, 1)
lt : s.potential (c, 1).1 (r (c, 1).1 (c, 1).2) < 1
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
g : Grow s c p n r
p1 : 1 ≤ p
e : Eqn s n r (c, 1)
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.p1 | [201, 1] | [208, 93] | rw [e.potential, Complex.abs.map_one, lt_self_iff_false] at lt | case refine_2
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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
g : Grow s c p n r
p1 : 1 ≤ p
e : Eqn s n r (c, 1)
lt : s.potential (c, 1).1 (r (c, 1).1 (c, 1).2) < 1
⊢ False | case refine_2
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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
g : Grow s c p n r
p1 : 1 ≤ p
e : Eqn s n r (c, 1)
lt : False
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
g : Grow s c p n r
p1 : 1 ≤ p
e : Eqn s n r (c, 1)
lt : s.potential (c, 1).1 (r (c, 1).1 (c, 1).2) < 1
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.p1 | [201, 1] | [208, 93] | exact lt | case refine_2
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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
g : Grow s c p n r
p1 : 1 ≤ p
e : Eqn s n r (c, 1)
lt : False
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
g : Grow s c p n r
p1 : 1 ≤ p
e : Eqn s n r (c, 1)
lt : False
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.p1 | [201, 1] | [208, 93] | simp only [p1, singleton_prod, mem_image, mem_closedBall_zero_iff, Complex.norm_eq_abs,
Prod.mk.inj_iff, eq_self_iff_true, true_and_iff, exists_eq_right, Complex.abs.map_one] | case refine_1
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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
g : Grow s c p n r
p1 : 1 ≤ p
⊢ (c, 1) ∈ {c} ×ˢ closedBall 0 p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
g : Grow s c p n r
p1 : 1 ≤ p
⊢ (c, 1) ∈ {c} ×ˢ closedBall 0 p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Super.grow_start | [215, 1] | [241, 101] | have ba := s.bottcherNear_holomorphic _ (s.mem_near 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
p : ℝ
s✝ : Super f d a
r : ℂ → ℂ → S
s : Super f d a
c : ℂ
⊢ ∃ p r, 0 < p ∧ Grow s c p 0 r | 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
p : ℝ
s✝ : Super f d a
r : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
⊢ ∃ p r, 0 < p ∧ Grow s c p 0 r | 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
p : ℝ
s✝ : Super f d a
r : ℂ → ℂ → S
s : Super f d a
c : ℂ
⊢ ∃ p r, 0 < p ∧ Grow s c p 0 r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Super.grow_start | [215, 1] | [241, 101] | have nc := 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
p : ℝ
s✝ : Super f d a
r : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
⊢ ∃ p r, 0 < p ∧ Grow s c p 0 r | 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
p : ℝ
s✝ : Super f d a
r : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
⊢ ∃ p r, 0 < p ∧ Grow s c p 0 r | 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
p : ℝ
s✝ : Super f d a
r : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
⊢ ∃ p r, 0 < p ∧ Grow s c p 0 r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Super.grow_start | [215, 1] | [241, 101] | rcases complex_inverse_fun ba nc with ⟨r, ra, rb, br⟩ | 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
p : ℝ
s✝ : Super f d a
r : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
⊢ ∃ p r, 0 < p ∧ Grow s c p 0 r | case intro.intro.intro
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
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, s.bottcherNear c a)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, s.bottcherNear c a), s.bottcherNear x.1 (r x.1 x.2) = x.2
⊢ ∃ p r, 0 < p ∧ Grow s c p 0 r | 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
p : ℝ
s✝ : Super f d a
r : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
⊢ ∃ p r, 0 < p ∧ Grow s c p 0 r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Super.grow_start | [215, 1] | [241, 101] | rw [s.bottcherNear_a] at ra br | case intro.intro.intro
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
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, s.bottcherNear c a)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, s.bottcherNear c a), s.bottcherNear x.1 (r x.1 x.2) = x.2
⊢ ∃ p r, 0 < p ∧ Grow s c p 0 r | case intro.intro.intro
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
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
⊢ ∃ p r, 0 < p ∧ Grow s c p 0 r | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
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
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, s.bottcherNear c a)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, s.bottcherNear c a), s.bottcherNear x.1 (r x.1 x.2) = x.2
⊢ ∃ p r, 0 < p ∧ Grow s c p 0 r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Super.grow_start | [215, 1] | [241, 101] | have rm : ∀ᶠ x : ℂ × ℂ in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near := by
refine (continuousAt_fst.prod ra.continuousAt).eventually_mem (s.isOpen_near.mem_nhds ?_)
have r0 := rb.self_of_nhds; simp only [s.bottcherNear_a] at r0
simp only [uncurry, r0]; exact s.mem_near c | case intro.intro.intro
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
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
⊢ ∃ p r, 0 < p ∧ Grow s c p 0 r | case intro.intro.intro
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
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
⊢ ∃ p r, 0 < p ∧ Grow s c p 0 r | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
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
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
⊢ ∃ p r, 0 < p ∧ Grow s c p 0 r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Super.grow_start | [215, 1] | [241, 101] | rcases eventually_nhds_iff.mp (ra.eventually.and (br.and rm)) with ⟨t, h, o, m⟩ | case intro.intro.intro
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
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
⊢ ∃ p r, 0 < p ∧ Grow s c p 0 r | case intro.intro.intro.intro.intro.intro
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
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
h : ∀ x ∈ t, HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
o : IsOpen t
m : (c, 0) ∈ t
⊢ ∃ p r, 0 < p ∧ Grow s c p 0 r | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
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
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
⊢ ∃ p r, 0 < p ∧ Grow s c p 0 r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Super.grow_start | [215, 1] | [241, 101] | rcases Metric.isOpen_iff.mp o _ m with ⟨p, pp, sub⟩ | case intro.intro.intro.intro.intro.intro
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
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
h : ∀ x ∈ t, HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
o : IsOpen t
m : (c, 0) ∈ t
⊢ ∃ p r, 0 < p ∧ Grow s c p 0 r | case intro.intro.intro.intro.intro.intro.intro.intro
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
p✝ : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
h : ∀ x ∈ t, HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
o : IsOpen t
m : (c, 0) ∈ t
p : ℝ
pp : p > 0
sub : ball (c, 0) p ⊆ t
⊢ ∃ p r, 0 < p ∧ Grow s c p 0 r | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro
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
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
h : ∀ x ∈ t, HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
o : IsOpen t
m : (c, 0) ∈ t
⊢ ∃ p r, 0 < p ∧ Grow s c p 0 r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Super.grow_start | [215, 1] | [241, 101] | replace h := fun (x : ℂ × ℂ) m ↦ h x (sub m) | case intro.intro.intro.intro.intro.intro.intro.intro
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
p✝ : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
h : ∀ x ∈ t, HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
o : IsOpen t
m : (c, 0) ∈ t
p : ℝ
pp : p > 0
sub : ball (c, 0) p ⊆ t
⊢ ∃ p r, 0 < p ∧ Grow s c p 0 r | case intro.intro.intro.intro.intro.intro.intro.intro
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
p✝ : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
o : IsOpen t
m : (c, 0) ∈ t
p : ℝ
pp : p > 0
sub : ball (c, 0) p ⊆ t
h :
∀ x ∈ ball (c, 0) p,
HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
⊢ ∃ p r, 0 < p ∧ Grow s c p 0 r | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro
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
p✝ : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
h : ∀ x ∈ t, HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
o : IsOpen t
m : (c, 0) ∈ t
p : ℝ
pp : p > 0
sub : ball (c, 0) p ⊆ t
⊢ ∃ p r, 0 < p ∧ Grow s c p 0 r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Super.grow_start | [215, 1] | [241, 101] | have nb : ball (c, (0 : ℂ)) p ∈ 𝓝ˢ ({c} ×ˢ closedBall (0 : ℂ) (p / 2)) := by
rw [isOpen_ball.mem_nhdsSet, ← ball_prod_same]; apply prod_mono
rw [singleton_subset_iff]; exact mem_ball_self pp
apply Metric.closedBall_subset_ball; exact half_lt_self pp | case intro.intro.intro.intro.intro.intro.intro.intro
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
p✝ : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
o : IsOpen t
m : (c, 0) ∈ t
p : ℝ
pp : p > 0
sub : ball (c, 0) p ⊆ t
h :
∀ x ∈ ball (c, 0) p,
HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
⊢ ∃ p r, 0 < p ∧ Grow s c p 0 r | case intro.intro.intro.intro.intro.intro.intro.intro
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
p✝ : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
o : IsOpen t
m : (c, 0) ∈ t
p : ℝ
pp : p > 0
sub : ball (c, 0) p ⊆ t
h :
∀ x ∈ ball (c, 0) p,
HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
nb : ball (c, 0) p ∈ 𝓝ˢ ({c} ×ˢ closedBall 0 (p / 2))
⊢ ∃ p r, 0 < p ∧ Grow s c p 0 r | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro
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
p✝ : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
o : IsOpen t
m : (c, 0) ∈ t
p : ℝ
pp : p > 0
sub : ball (c, 0) p ⊆ t
h :
∀ x ∈ ball (c, 0) p,
HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
⊢ ∃ p r, 0 < p ∧ Grow s c p 0 r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Super.grow_start | [215, 1] | [241, 101] | use p / 2, r, half_pos pp | case intro.intro.intro.intro.intro.intro.intro.intro
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
p✝ : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
o : IsOpen t
m : (c, 0) ∈ t
p : ℝ
pp : p > 0
sub : ball (c, 0) p ⊆ t
h :
∀ x ∈ ball (c, 0) p,
HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
nb : ball (c, 0) p ∈ 𝓝ˢ ({c} ×ˢ closedBall 0 (p / 2))
⊢ ∃ p r, 0 < p ∧ Grow s c p 0 r | case right
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
p✝ : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
o : IsOpen t
m : (c, 0) ∈ t
p : ℝ
pp : p > 0
sub : ball (c, 0) p ⊆ t
h :
∀ x ∈ ball (c, 0) p,
HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
nb : ball (c, 0) p ∈ 𝓝ˢ ({c} ×ˢ closedBall 0 (p / 2))
⊢ Grow s c (p / 2) 0 r | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro
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
p✝ : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
o : IsOpen t
m : (c, 0) ∈ t
p : ℝ
pp : p > 0
sub : ball (c, 0) p ⊆ t
h :
∀ x ∈ ball (c, 0) p,
HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
nb : ball (c, 0) p ∈ 𝓝ˢ ({c} ×ˢ closedBall 0 (p / 2))
⊢ ∃ p r, 0 < p ∧ Grow s c p 0 r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Super.grow_start | [215, 1] | [241, 101] | exact
{ nonneg := (half_pos pp).le
zero := by convert rb.self_of_nhds; simp only [s.bottcherNear_a]
start := Filter.eventually_iff_exists_mem.mpr ⟨_, ball_mem_nhds _ pp, fun _ m ↦ (h _ m).2.1⟩
eqn :=
Filter.eventually_iff_exists_mem.mpr
⟨_, nb, fun _ m ↦
{ holo := (h _ m).1
near := (h _ m).2.2
eqn := by simp only [Function.iterate_zero_apply, pow_zero, pow_one, (h _ m).2.1] }⟩ } | case right
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
p✝ : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
o : IsOpen t
m : (c, 0) ∈ t
p : ℝ
pp : p > 0
sub : ball (c, 0) p ⊆ t
h :
∀ x ∈ ball (c, 0) p,
HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
nb : ball (c, 0) p ∈ 𝓝ˢ ({c} ×ˢ closedBall 0 (p / 2))
⊢ Grow s c (p / 2) 0 r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
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
p✝ : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
o : IsOpen t
m : (c, 0) ∈ t
p : ℝ
pp : p > 0
sub : ball (c, 0) p ⊆ t
h :
∀ x ∈ ball (c, 0) p,
HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
nb : ball (c, 0) p ∈ 𝓝ˢ ({c} ×ˢ closedBall 0 (p / 2))
⊢ Grow s c (p / 2) 0 r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Super.grow_start | [215, 1] | [241, 101] | refine (continuousAt_fst.prod ra.continuousAt).eventually_mem (s.isOpen_near.mem_nhds ?_) | 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
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near | 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
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
⊢ ((c, 0).1, uncurry r (c, 0)) ∈ s.near | 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
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Super.grow_start | [215, 1] | [241, 101] | have r0 := rb.self_of_nhds | 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
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
⊢ ((c, 0).1, uncurry r (c, 0)) ∈ s.near | 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
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
r0 : r (c, a).1 (s.bottcherNear (c, a).1 (c, a).2) = (c, a).2
⊢ ((c, 0).1, uncurry r (c, 0)) ∈ s.near | 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
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
⊢ ((c, 0).1, uncurry r (c, 0)) ∈ s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Super.grow_start | [215, 1] | [241, 101] | simp only [s.bottcherNear_a] at r0 | 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
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
r0 : r (c, a).1 (s.bottcherNear (c, a).1 (c, a).2) = (c, a).2
⊢ ((c, 0).1, uncurry r (c, 0)) ∈ s.near | 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
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
r0 : r c 0 = a
⊢ ((c, 0).1, uncurry r (c, 0)) ∈ s.near | 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
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
r0 : r (c, a).1 (s.bottcherNear (c, a).1 (c, a).2) = (c, a).2
⊢ ((c, 0).1, uncurry r (c, 0)) ∈ s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Super.grow_start | [215, 1] | [241, 101] | simp only [uncurry, r0] | 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
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
r0 : r c 0 = a
⊢ ((c, 0).1, uncurry r (c, 0)) ∈ s.near | 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
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
r0 : r c 0 = a
⊢ (c, a) ∈ s.near | 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
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
r0 : r c 0 = a
⊢ ((c, 0).1, uncurry r (c, 0)) ∈ s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Super.grow_start | [215, 1] | [241, 101] | exact s.mem_near 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
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
r0 : r c 0 = a
⊢ (c, a) ∈ s.near | 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
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
r0 : r c 0 = a
⊢ (c, a) ∈ s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Super.grow_start | [215, 1] | [241, 101] | rw [isOpen_ball.mem_nhdsSet, ← ball_prod_same] | 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
p✝ : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
o : IsOpen t
m : (c, 0) ∈ t
p : ℝ
pp : p > 0
sub : ball (c, 0) p ⊆ t
h :
∀ x ∈ ball (c, 0) p,
HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
⊢ ball (c, 0) p ∈ 𝓝ˢ ({c} ×ˢ closedBall 0 (p / 2)) | 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
p✝ : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
o : IsOpen t
m : (c, 0) ∈ t
p : ℝ
pp : p > 0
sub : ball (c, 0) p ⊆ t
h :
∀ x ∈ ball (c, 0) p,
HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
⊢ {c} ×ˢ closedBall 0 (p / 2) ⊆ ball c p ×ˢ ball 0 p | 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
p✝ : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
o : IsOpen t
m : (c, 0) ∈ t
p : ℝ
pp : p > 0
sub : ball (c, 0) p ⊆ t
h :
∀ x ∈ ball (c, 0) p,
HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
⊢ ball (c, 0) p ∈ 𝓝ˢ ({c} ×ˢ closedBall 0 (p / 2))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Super.grow_start | [215, 1] | [241, 101] | apply prod_mono | 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
p✝ : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
o : IsOpen t
m : (c, 0) ∈ t
p : ℝ
pp : p > 0
sub : ball (c, 0) p ⊆ t
h :
∀ x ∈ ball (c, 0) p,
HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
⊢ {c} ×ˢ closedBall 0 (p / 2) ⊆ ball c p ×ˢ ball 0 p | case hs
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
p✝ : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
o : IsOpen t
m : (c, 0) ∈ t
p : ℝ
pp : p > 0
sub : ball (c, 0) p ⊆ t
h :
∀ x ∈ ball (c, 0) p,
HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
⊢ {c} ⊆ ball c p
case ht
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
p✝ : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
o : IsOpen t
m : (c, 0) ∈ t
p : ℝ
pp : p > 0
sub : ball (c, 0) p ⊆ t
h :
∀ x ∈ ball (c, 0) p,
HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
⊢ closedBall 0 (p / 2) ⊆ ball 0 p | 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
p✝ : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
o : IsOpen t
m : (c, 0) ∈ t
p : ℝ
pp : p > 0
sub : ball (c, 0) p ⊆ t
h :
∀ x ∈ ball (c, 0) p,
HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
⊢ {c} ×ˢ closedBall 0 (p / 2) ⊆ ball c p ×ˢ ball 0 p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Super.grow_start | [215, 1] | [241, 101] | rw [singleton_subset_iff] | case hs
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
p✝ : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
o : IsOpen t
m : (c, 0) ∈ t
p : ℝ
pp : p > 0
sub : ball (c, 0) p ⊆ t
h :
∀ x ∈ ball (c, 0) p,
HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
⊢ {c} ⊆ ball c p
case ht
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
p✝ : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
o : IsOpen t
m : (c, 0) ∈ t
p : ℝ
pp : p > 0
sub : ball (c, 0) p ⊆ t
h :
∀ x ∈ ball (c, 0) p,
HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
⊢ closedBall 0 (p / 2) ⊆ ball 0 p | case hs
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
p✝ : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
o : IsOpen t
m : (c, 0) ∈ t
p : ℝ
pp : p > 0
sub : ball (c, 0) p ⊆ t
h :
∀ x ∈ ball (c, 0) p,
HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
⊢ c ∈ ball c p
case ht
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
p✝ : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
o : IsOpen t
m : (c, 0) ∈ t
p : ℝ
pp : p > 0
sub : ball (c, 0) p ⊆ t
h :
∀ x ∈ ball (c, 0) p,
HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
⊢ closedBall 0 (p / 2) ⊆ ball 0 p | Please generate a tactic in lean4 to solve the state.
STATE:
case hs
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
p✝ : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
o : IsOpen t
m : (c, 0) ∈ t
p : ℝ
pp : p > 0
sub : ball (c, 0) p ⊆ t
h :
∀ x ∈ ball (c, 0) p,
HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
⊢ {c} ⊆ ball c p
case ht
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
p✝ : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
o : IsOpen t
m : (c, 0) ∈ t
p : ℝ
pp : p > 0
sub : ball (c, 0) p ⊆ t
h :
∀ x ∈ ball (c, 0) p,
HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
⊢ closedBall 0 (p / 2) ⊆ ball 0 p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Super.grow_start | [215, 1] | [241, 101] | exact mem_ball_self pp | case hs
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
p✝ : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
o : IsOpen t
m : (c, 0) ∈ t
p : ℝ
pp : p > 0
sub : ball (c, 0) p ⊆ t
h :
∀ x ∈ ball (c, 0) p,
HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
⊢ c ∈ ball c p
case ht
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
p✝ : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
o : IsOpen t
m : (c, 0) ∈ t
p : ℝ
pp : p > 0
sub : ball (c, 0) p ⊆ t
h :
∀ x ∈ ball (c, 0) p,
HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
⊢ closedBall 0 (p / 2) ⊆ ball 0 p | case ht
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
p✝ : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
o : IsOpen t
m : (c, 0) ∈ t
p : ℝ
pp : p > 0
sub : ball (c, 0) p ⊆ t
h :
∀ x ∈ ball (c, 0) p,
HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
⊢ closedBall 0 (p / 2) ⊆ ball 0 p | Please generate a tactic in lean4 to solve the state.
STATE:
case hs
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
p✝ : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
o : IsOpen t
m : (c, 0) ∈ t
p : ℝ
pp : p > 0
sub : ball (c, 0) p ⊆ t
h :
∀ x ∈ ball (c, 0) p,
HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
⊢ c ∈ ball c p
case ht
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
p✝ : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
o : IsOpen t
m : (c, 0) ∈ t
p : ℝ
pp : p > 0
sub : ball (c, 0) p ⊆ t
h :
∀ x ∈ ball (c, 0) p,
HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
⊢ closedBall 0 (p / 2) ⊆ ball 0 p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Super.grow_start | [215, 1] | [241, 101] | apply Metric.closedBall_subset_ball | case ht
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
p✝ : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
o : IsOpen t
m : (c, 0) ∈ t
p : ℝ
pp : p > 0
sub : ball (c, 0) p ⊆ t
h :
∀ x ∈ ball (c, 0) p,
HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
⊢ closedBall 0 (p / 2) ⊆ ball 0 p | case ht.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
p✝ : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
o : IsOpen t
m : (c, 0) ∈ t
p : ℝ
pp : p > 0
sub : ball (c, 0) p ⊆ t
h :
∀ x ∈ ball (c, 0) p,
HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
⊢ p / 2 < p | Please generate a tactic in lean4 to solve the state.
STATE:
case ht
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
p✝ : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
o : IsOpen t
m : (c, 0) ∈ t
p : ℝ
pp : p > 0
sub : ball (c, 0) p ⊆ t
h :
∀ x ∈ ball (c, 0) p,
HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
⊢ closedBall 0 (p / 2) ⊆ ball 0 p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Super.grow_start | [215, 1] | [241, 101] | exact half_lt_self pp | case ht.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
p✝ : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
o : IsOpen t
m : (c, 0) ∈ t
p : ℝ
pp : p > 0
sub : ball (c, 0) p ⊆ t
h :
∀ x ∈ ball (c, 0) p,
HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
⊢ p / 2 < p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case ht.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
p✝ : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
o : IsOpen t
m : (c, 0) ∈ t
p : ℝ
pp : p > 0
sub : ball (c, 0) p ⊆ t
h :
∀ x ∈ ball (c, 0) p,
HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
⊢ p / 2 < p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Super.grow_start | [215, 1] | [241, 101] | convert rb.self_of_nhds | 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
p✝ : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
o : IsOpen t
m : (c, 0) ∈ t
p : ℝ
pp : p > 0
sub : ball (c, 0) p ⊆ t
h :
∀ x ∈ ball (c, 0) p,
HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
nb : ball (c, 0) p ∈ 𝓝ˢ ({c} ×ˢ closedBall 0 (p / 2))
⊢ r c 0 = a | case h.e'_2.h.e'_2
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
p✝ : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
o : IsOpen t
m : (c, 0) ∈ t
p : ℝ
pp : p > 0
sub : ball (c, 0) p ⊆ t
h :
∀ x ∈ ball (c, 0) p,
HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
nb : ball (c, 0) p ∈ 𝓝ˢ ({c} ×ˢ closedBall 0 (p / 2))
⊢ 0 = s.bottcherNear (c, a).1 (c, a).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
p✝ : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
o : IsOpen t
m : (c, 0) ∈ t
p : ℝ
pp : p > 0
sub : ball (c, 0) p ⊆ t
h :
∀ x ∈ ball (c, 0) p,
HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
nb : ball (c, 0) p ∈ 𝓝ˢ ({c} ×ˢ closedBall 0 (p / 2))
⊢ r c 0 = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Super.grow_start | [215, 1] | [241, 101] | simp only [s.bottcherNear_a] | case h.e'_2.h.e'_2
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
p✝ : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
o : IsOpen t
m : (c, 0) ∈ t
p : ℝ
pp : p > 0
sub : ball (c, 0) p ⊆ t
h :
∀ x ∈ ball (c, 0) p,
HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
nb : ball (c, 0) p ∈ 𝓝ˢ ({c} ×ˢ closedBall 0 (p / 2))
⊢ 0 = s.bottcherNear (c, a).1 (c, a).2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_2
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
p✝ : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
o : IsOpen t
m : (c, 0) ∈ t
p : ℝ
pp : p > 0
sub : ball (c, 0) p ⊆ t
h :
∀ x ∈ ball (c, 0) p,
HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
nb : ball (c, 0) p ∈ 𝓝ˢ ({c} ×ˢ closedBall 0 (p / 2))
⊢ 0 = s.bottcherNear (c, a).1 (c, a).2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Super.grow_start | [215, 1] | [241, 101] | simp only [Function.iterate_zero_apply, pow_zero, pow_one, (h _ m).2.1] | 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
p✝ : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
o : IsOpen t
m✝ : (c, 0) ∈ t
p : ℝ
pp : p > 0
sub : ball (c, 0) p ⊆ t
h :
∀ x ∈ ball (c, 0) p,
HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
nb : ball (c, 0) p ∈ 𝓝ˢ ({c} ×ˢ closedBall 0 (p / 2))
x✝ : ℂ × ℂ
m : x✝ ∈ ball (c, 0) p
⊢ s.bottcherNear x✝.1 ((f x✝.1)^[0] (r x✝.1 x✝.2)) = x✝.2 ^ d ^ 0 | 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
p✝ : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
c : ℂ
ba : HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) (c, a)
nc : mfderiv I I (s.bottcherNear c) a ≠ 0
r : ℂ → ℂ → S
ra : HolomorphicAt (I.prod I) I (uncurry r) (c, 0)
rb : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), r x.1 (s.bottcherNear x.1 x.2) = x.2
br : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
rm : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), (x.1, r x.1 x.2) ∈ s.near
t : Set (ℂ × ℂ)
o : IsOpen t
m✝ : (c, 0) ∈ t
p : ℝ
pp : p > 0
sub : ball (c, 0) p ⊆ t
h :
∀ x ∈ ball (c, 0) p,
HolomorphicAt (I.prod I) I (uncurry r) x ∧ s.bottcherNear x.1 (r x.1 x.2) = x.2 ∧ (x.1, r x.1 x.2) ∈ s.near
nb : ball (c, 0) p ∈ 𝓝ˢ ({c} ×ˢ closedBall 0 (p / 2))
x✝ : ℂ × ℂ
m : x✝ ∈ ball (c, 0) p
⊢ s.bottcherNear x✝.1 ((f x✝.1)^[0] (r x✝.1 x✝.2)) = x✝.2 ^ d ^ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.open | [244, 1] | [268, 61] | have e := g.eqn | 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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
g : Grow s c p n r
⊢ ∃ p', p < p' ∧ ∀ᶠ (c' : ℂ) in 𝓝 c, Grow s c' p' n r | 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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
g : Grow s c p n r
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ ({c} ×ˢ closedBall 0 p), Eqn s n r x
⊢ ∃ p', p < p' ∧ ∀ᶠ (c' : ℂ) in 𝓝 c, Grow s c' p' n r | 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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
g : Grow s c p n r
⊢ ∃ p', p < p' ∧ ∀ᶠ (c' : ℂ) in 𝓝 c, Grow s c' p' n r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.open | [244, 1] | [268, 61] | simp only [isCompact_singleton.nhdsSet_prod_eq (isCompact_closedBall _ _)] at e | 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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
g : Grow s c p n r
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ ({c} ×ˢ closedBall 0 p), Eqn s n r x
⊢ ∃ p', p < p' ∧ ∀ᶠ (c' : ℂ) in 𝓝 c, Grow s c' p' n r | 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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
g : Grow s c p n r
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ {c} ×ˢ 𝓝ˢ (closedBall 0 p), Eqn s n r x
⊢ ∃ p', p < p' ∧ ∀ᶠ (c' : ℂ) in 𝓝 c, Grow s c' p' n r | 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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
g : Grow s c p n r
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ ({c} ×ˢ closedBall 0 p), Eqn s n r x
⊢ ∃ p', p < p' ∧ ∀ᶠ (c' : ℂ) in 𝓝 c, Grow s c' p' n r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.open | [244, 1] | [268, 61] | rcases Filter.mem_prod_iff.mp e with ⟨a', an, b', bn, sub⟩ | 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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
g : Grow s c p n r
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ {c} ×ˢ 𝓝ˢ (closedBall 0 p), Eqn s n r x
⊢ ∃ p', p < p' ∧ ∀ᶠ (c' : ℂ) in 𝓝 c, Grow s c' p' n r | case intro.intro.intro.intro
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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
g : Grow s c p n r
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ {c} ×ˢ 𝓝ˢ (closedBall 0 p), Eqn s n r x
a' : Set ℂ
an : a' ∈ 𝓝ˢ {c}
b' : Set ℂ
bn : b' ∈ 𝓝ˢ (closedBall 0 p)
sub : a' ×ˢ b' ⊆ {x | (fun x => Eqn s n r x) x}
⊢ ∃ p', p < p' ∧ ∀ᶠ (c' : ℂ) in 𝓝 c, Grow s c' p' n r | 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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
g : Grow s c p n r
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ {c} ×ˢ 𝓝ˢ (closedBall 0 p), Eqn s n r x
⊢ ∃ p', p < p' ∧ ∀ᶠ (c' : ℂ) in 𝓝 c, Grow s c' p' n r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.open | [244, 1] | [268, 61] | simp only [subset_setOf] at sub | case intro.intro.intro.intro
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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
g : Grow s c p n r
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ {c} ×ˢ 𝓝ˢ (closedBall 0 p), Eqn s n r x
a' : Set ℂ
an : a' ∈ 𝓝ˢ {c}
b' : Set ℂ
bn : b' ∈ 𝓝ˢ (closedBall 0 p)
sub : a' ×ˢ b' ⊆ {x | (fun x => Eqn s n r x) x}
⊢ ∃ p', p < p' ∧ ∀ᶠ (c' : ℂ) in 𝓝 c, Grow s c' p' n r | case intro.intro.intro.intro
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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
g : Grow s c p n r
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ {c} ×ˢ 𝓝ˢ (closedBall 0 p), Eqn s n r x
a' : Set ℂ
an : a' ∈ 𝓝ˢ {c}
b' : Set ℂ
bn : b' ∈ 𝓝ˢ (closedBall 0 p)
sub : ∀ x ∈ a' ×ˢ b', Eqn s n r x
⊢ ∃ p', p < p' ∧ ∀ᶠ (c' : ℂ) in 𝓝 c, Grow s c' p' n r | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
g : Grow s c p n r
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ {c} ×ˢ 𝓝ˢ (closedBall 0 p), Eqn s n r x
a' : Set ℂ
an : a' ∈ 𝓝ˢ {c}
b' : Set ℂ
bn : b' ∈ 𝓝ˢ (closedBall 0 p)
sub : a' ×ˢ b' ⊆ {x | (fun x => Eqn s n r x) x}
⊢ ∃ p', p < p' ∧ ∀ᶠ (c' : ℂ) in 𝓝 c, Grow s c' p' n r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.open | [244, 1] | [268, 61] | rcases eventually_nhds_iff.mp (nhdsSet_singleton.subst an) with ⟨a, aa, ao, am⟩ | case intro.intro.intro.intro
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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
g : Grow s c p n r
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ {c} ×ˢ 𝓝ˢ (closedBall 0 p), Eqn s n r x
a' : Set ℂ
an : a' ∈ 𝓝ˢ {c}
b' : Set ℂ
bn : b' ∈ 𝓝ˢ (closedBall 0 p)
sub : ∀ x ∈ a' ×ˢ b', Eqn s n r x
⊢ ∃ p', p < p' ∧ ∀ᶠ (c' : ℂ) in 𝓝 c, Grow s c' p' n r | case intro.intro.intro.intro.intro.intro.intro
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
p : ℝ
s : Super f d a✝
r : ℂ → ℂ → S
g : Grow s c p n r
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ {c} ×ˢ 𝓝ˢ (closedBall 0 p), Eqn s n r x
a' : Set ℂ
an : a' ∈ 𝓝ˢ {c}
b' : Set ℂ
bn : b' ∈ 𝓝ˢ (closedBall 0 p)
sub : ∀ x ∈ a' ×ˢ b', Eqn s n r x
a : Set ℂ
aa : ∀ x ∈ a, a' x
ao : IsOpen a
am : c ∈ a
⊢ ∃ p', p < p' ∧ ∀ᶠ (c' : ℂ) in 𝓝 c, Grow s c' p' n r | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
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
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
g : Grow s c p n r
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ {c} ×ˢ 𝓝ˢ (closedBall 0 p), Eqn s n r x
a' : Set ℂ
an : a' ∈ 𝓝ˢ {c}
b' : Set ℂ
bn : b' ∈ 𝓝ˢ (closedBall 0 p)
sub : ∀ x ∈ a' ×ˢ b', Eqn s n r x
⊢ ∃ p', p < p' ∧ ∀ᶠ (c' : ℂ) in 𝓝 c, Grow s c' p' n r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.open | [244, 1] | [268, 61] | rcases eventually_nhdsSet_iff_exists.mp bn with ⟨b, bo, bp, bb⟩ | case intro.intro.intro.intro.intro.intro.intro
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
p : ℝ
s : Super f d a✝
r : ℂ → ℂ → S
g : Grow s c p n r
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ {c} ×ˢ 𝓝ˢ (closedBall 0 p), Eqn s n r x
a' : Set ℂ
an : a' ∈ 𝓝ˢ {c}
b' : Set ℂ
bn : b' ∈ 𝓝ˢ (closedBall 0 p)
sub : ∀ x ∈ a' ×ˢ b', Eqn s n r x
a : Set ℂ
aa : ∀ x ∈ a, a' x
ao : IsOpen a
am : c ∈ a
⊢ ∃ p', p < p' ∧ ∀ᶠ (c' : ℂ) in 𝓝 c, Grow s c' p' n r | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
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
p : ℝ
s : Super f d a✝
r : ℂ → ℂ → S
g : Grow s c p n r
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ {c} ×ˢ 𝓝ˢ (closedBall 0 p), Eqn s n r x
a' : Set ℂ
an : a' ∈ 𝓝ˢ {c}
b' : Set ℂ
bn : b' ∈ 𝓝ˢ (closedBall 0 p)
sub : ∀ x ∈ a' ×ˢ b', Eqn s n r x
a : Set ℂ
aa : ∀ x ∈ a, a' x
ao : IsOpen a
am : c ∈ a
b : Set ℂ
bo : IsOpen b
bp : closedBall 0 p ⊆ b
bb : ∀ x ∈ b, b' x
⊢ ∃ p', p < p' ∧ ∀ᶠ (c' : ℂ) in 𝓝 c, Grow s c' p' n r | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro
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
p : ℝ
s : Super f d a✝
r : ℂ → ℂ → S
g : Grow s c p n r
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ {c} ×ˢ 𝓝ˢ (closedBall 0 p), Eqn s n r x
a' : Set ℂ
an : a' ∈ 𝓝ˢ {c}
b' : Set ℂ
bn : b' ∈ 𝓝ˢ (closedBall 0 p)
sub : ∀ x ∈ a' ×ˢ b', Eqn s n r x
a : Set ℂ
aa : ∀ x ∈ a, a' x
ao : IsOpen a
am : c ∈ a
⊢ ∃ p', p < p' ∧ ∀ᶠ (c' : ℂ) in 𝓝 c, Grow s c' p' n r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.open | [244, 1] | [268, 61] | rcases domain_open' bp bo with ⟨q, pq, qb⟩ | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
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
p : ℝ
s : Super f d a✝
r : ℂ → ℂ → S
g : Grow s c p n r
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ {c} ×ˢ 𝓝ˢ (closedBall 0 p), Eqn s n r x
a' : Set ℂ
an : a' ∈ 𝓝ˢ {c}
b' : Set ℂ
bn : b' ∈ 𝓝ˢ (closedBall 0 p)
sub : ∀ x ∈ a' ×ˢ b', Eqn s n r x
a : Set ℂ
aa : ∀ x ∈ a, a' x
ao : IsOpen a
am : c ∈ a
b : Set ℂ
bo : IsOpen b
bp : closedBall 0 p ⊆ b
bb : ∀ x ∈ b, b' x
⊢ ∃ p', p < p' ∧ ∀ᶠ (c' : ℂ) in 𝓝 c, Grow s c' p' n r | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
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
p : ℝ
s : Super f d a✝
r : ℂ → ℂ → S
g : Grow s c p n r
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ {c} ×ˢ 𝓝ˢ (closedBall 0 p), Eqn s n r x
a' : Set ℂ
an : a' ∈ 𝓝ˢ {c}
b' : Set ℂ
bn : b' ∈ 𝓝ˢ (closedBall 0 p)
sub : ∀ x ∈ a' ×ˢ b', Eqn s n r x
a : Set ℂ
aa : ∀ x ∈ a, a' x
ao : IsOpen a
am : c ∈ a
b : Set ℂ
bo : IsOpen b
bp : closedBall 0 p ⊆ b
bb : ∀ x ∈ b, b' x
q : ℝ
pq : p < q
qb : closedBall 0 q ⊆ b
⊢ ∃ p', p < p' ∧ ∀ᶠ (c' : ℂ) in 𝓝 c, Grow s c' p' n r | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
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
p : ℝ
s : Super f d a✝
r : ℂ → ℂ → S
g : Grow s c p n r
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ {c} ×ˢ 𝓝ˢ (closedBall 0 p), Eqn s n r x
a' : Set ℂ
an : a' ∈ 𝓝ˢ {c}
b' : Set ℂ
bn : b' ∈ 𝓝ˢ (closedBall 0 p)
sub : ∀ x ∈ a' ×ˢ b', Eqn s n r x
a : Set ℂ
aa : ∀ x ∈ a, a' x
ao : IsOpen a
am : c ∈ a
b : Set ℂ
bo : IsOpen b
bp : closedBall 0 p ⊆ b
bb : ∀ x ∈ b, b' x
⊢ ∃ p', p < p' ∧ ∀ᶠ (c' : ℂ) in 𝓝 c, Grow s c' p' n r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.open | [244, 1] | [268, 61] | use q, pq | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
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
p : ℝ
s : Super f d a✝
r : ℂ → ℂ → S
g : Grow s c p n r
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ {c} ×ˢ 𝓝ˢ (closedBall 0 p), Eqn s n r x
a' : Set ℂ
an : a' ∈ 𝓝ˢ {c}
b' : Set ℂ
bn : b' ∈ 𝓝ˢ (closedBall 0 p)
sub : ∀ x ∈ a' ×ˢ b', Eqn s n r x
a : Set ℂ
aa : ∀ x ∈ a, a' x
ao : IsOpen a
am : c ∈ a
b : Set ℂ
bo : IsOpen b
bp : closedBall 0 p ⊆ b
bb : ∀ x ∈ b, b' x
q : ℝ
pq : p < q
qb : closedBall 0 q ⊆ b
⊢ ∃ p', p < p' ∧ ∀ᶠ (c' : ℂ) in 𝓝 c, Grow s c' p' n r | case right
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
p : ℝ
s : Super f d a✝
r : ℂ → ℂ → S
g : Grow s c p n r
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ {c} ×ˢ 𝓝ˢ (closedBall 0 p), Eqn s n r x
a' : Set ℂ
an : a' ∈ 𝓝ˢ {c}
b' : Set ℂ
bn : b' ∈ 𝓝ˢ (closedBall 0 p)
sub : ∀ x ∈ a' ×ˢ b', Eqn s n r x
a : Set ℂ
aa : ∀ x ∈ a, a' x
ao : IsOpen a
am : c ∈ a
b : Set ℂ
bo : IsOpen b
bp : closedBall 0 p ⊆ b
bb : ∀ x ∈ b, b' x
q : ℝ
pq : p < q
qb : closedBall 0 q ⊆ b
⊢ ∀ᶠ (c' : ℂ) in 𝓝 c, Grow s c' q n r | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
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
p : ℝ
s : Super f d a✝
r : ℂ → ℂ → S
g : Grow s c p n r
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ {c} ×ˢ 𝓝ˢ (closedBall 0 p), Eqn s n r x
a' : Set ℂ
an : a' ∈ 𝓝ˢ {c}
b' : Set ℂ
bn : b' ∈ 𝓝ˢ (closedBall 0 p)
sub : ∀ x ∈ a' ×ˢ b', Eqn s n r x
a : Set ℂ
aa : ∀ x ∈ a, a' x
ao : IsOpen a
am : c ∈ a
b : Set ℂ
bo : IsOpen b
bp : closedBall 0 p ⊆ b
bb : ∀ x ∈ b, b' x
q : ℝ
pq : p < q
qb : closedBall 0 q ⊆ b
⊢ ∃ p', p < p' ∧ ∀ᶠ (c' : ℂ) in 𝓝 c, Grow s c' p' n r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.open | [244, 1] | [268, 61] | apply m.mp | case right
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
p : ℝ
s : Super f d a✝
r : ℂ → ℂ → S
g : Grow s c p n r
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ {c} ×ˢ 𝓝ˢ (closedBall 0 p), Eqn s n r x
a' : Set ℂ
an : a' ∈ 𝓝ˢ {c}
b' : Set ℂ
bn : b' ∈ 𝓝ˢ (closedBall 0 p)
sub : ∀ x ∈ a' ×ˢ b', Eqn s n r x
a : Set ℂ
aa : ∀ x ∈ a, a' x
ao : IsOpen a
am : c ∈ a
b : Set ℂ
bo : IsOpen b
bp : closedBall 0 p ⊆ b
bb : ∀ x ∈ b, b' x
q : ℝ
pq : p < q
qb : closedBall 0 q ⊆ b
m : ∀ᶠ (c' : ℂ) in 𝓝 c, (c', r c' 0) ∈ s.near
⊢ ∀ᶠ (c' : ℂ) in 𝓝 c, Grow s c' q n r | case right
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
p : ℝ
s : Super f d a✝
r : ℂ → ℂ → S
g : Grow s c p n r
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ {c} ×ˢ 𝓝ˢ (closedBall 0 p), Eqn s n r x
a' : Set ℂ
an : a' ∈ 𝓝ˢ {c}
b' : Set ℂ
bn : b' ∈ 𝓝ˢ (closedBall 0 p)
sub : ∀ x ∈ a' ×ˢ b', Eqn s n r x
a : Set ℂ
aa : ∀ x ∈ a, a' x
ao : IsOpen a
am : c ∈ a
b : Set ℂ
bo : IsOpen b
bp : closedBall 0 p ⊆ b
bb : ∀ x ∈ b, b' x
q : ℝ
pq : p < q
qb : closedBall 0 q ⊆ b
m : ∀ᶠ (c' : ℂ) in 𝓝 c, (c', r c' 0) ∈ s.near
⊢ ∀ᶠ (x : ℂ) in 𝓝 c, (x, r x 0) ∈ s.near → Grow s x q n r | Please generate a tactic in lean4 to solve the state.
STATE:
case right
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
p : ℝ
s : Super f d a✝
r : ℂ → ℂ → S
g : Grow s c p n r
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ {c} ×ˢ 𝓝ˢ (closedBall 0 p), Eqn s n r x
a' : Set ℂ
an : a' ∈ 𝓝ˢ {c}
b' : Set ℂ
bn : b' ∈ 𝓝ˢ (closedBall 0 p)
sub : ∀ x ∈ a' ×ˢ b', Eqn s n r x
a : Set ℂ
aa : ∀ x ∈ a, a' x
ao : IsOpen a
am : c ∈ a
b : Set ℂ
bo : IsOpen b
bp : closedBall 0 p ⊆ b
bb : ∀ x ∈ b, b' x
q : ℝ
pq : p < q
qb : closedBall 0 q ⊆ b
m : ∀ᶠ (c' : ℂ) in 𝓝 c, (c', r c' 0) ∈ s.near
⊢ ∀ᶠ (c' : ℂ) in 𝓝 c, Grow s c' q n r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.open | [244, 1] | [268, 61] | apply ((continuousAt_id.prod continuousAt_const).eventually g.start.eventually_nhds).mp | case right
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
p : ℝ
s : Super f d a✝
r : ℂ → ℂ → S
g : Grow s c p n r
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ {c} ×ˢ 𝓝ˢ (closedBall 0 p), Eqn s n r x
a' : Set ℂ
an : a' ∈ 𝓝ˢ {c}
b' : Set ℂ
bn : b' ∈ 𝓝ˢ (closedBall 0 p)
sub : ∀ x ∈ a' ×ˢ b', Eqn s n r x
a : Set ℂ
aa : ∀ x ∈ a, a' x
ao : IsOpen a
am : c ∈ a
b : Set ℂ
bo : IsOpen b
bp : closedBall 0 p ⊆ b
bb : ∀ x ∈ b, b' x
q : ℝ
pq : p < q
qb : closedBall 0 q ⊆ b
m : ∀ᶠ (c' : ℂ) in 𝓝 c, (c', r c' 0) ∈ s.near
⊢ ∀ᶠ (x : ℂ) in 𝓝 c, (x, r x 0) ∈ s.near → Grow s x q n r | case right
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
p : ℝ
s : Super f d a✝
r : ℂ → ℂ → S
g : Grow s c p n r
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ {c} ×ˢ 𝓝ˢ (closedBall 0 p), Eqn s n r x
a' : Set ℂ
an : a' ∈ 𝓝ˢ {c}
b' : Set ℂ
bn : b' ∈ 𝓝ˢ (closedBall 0 p)
sub : ∀ x ∈ a' ×ˢ b', Eqn s n r x
a : Set ℂ
aa : ∀ x ∈ a, a' x
ao : IsOpen a
am : c ∈ a
b : Set ℂ
bo : IsOpen b
bp : closedBall 0 p ⊆ b
bb : ∀ x ∈ b, b' x
q : ℝ
pq : p < q
qb : closedBall 0 q ⊆ b
m : ∀ᶠ (c' : ℂ) in 𝓝 c, (c', r c' 0) ∈ s.near
⊢ ∀ᶠ (x : ℂ) in 𝓝 c,
(∀ᶠ (x : ℂ × ℂ) in 𝓝 (id x, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2) → (x, r x 0) ∈ s.near → Grow s x q n r | Please generate a tactic in lean4 to solve the state.
STATE:
case right
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
p : ℝ
s : Super f d a✝
r : ℂ → ℂ → S
g : Grow s c p n r
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ {c} ×ˢ 𝓝ˢ (closedBall 0 p), Eqn s n r x
a' : Set ℂ
an : a' ∈ 𝓝ˢ {c}
b' : Set ℂ
bn : b' ∈ 𝓝ˢ (closedBall 0 p)
sub : ∀ x ∈ a' ×ˢ b', Eqn s n r x
a : Set ℂ
aa : ∀ x ∈ a, a' x
ao : IsOpen a
am : c ∈ a
b : Set ℂ
bo : IsOpen b
bp : closedBall 0 p ⊆ b
bb : ∀ x ∈ b, b' x
q : ℝ
pq : p < q
qb : closedBall 0 q ⊆ b
m : ∀ᶠ (c' : ℂ) in 𝓝 c, (c', r c' 0) ∈ s.near
⊢ ∀ᶠ (x : ℂ) in 𝓝 c, (x, r x 0) ∈ s.near → Grow s x q n r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.open | [244, 1] | [268, 61] | refine eventually_nhds_iff.mpr ⟨a, ?_, ao, am⟩ | case right
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
p : ℝ
s : Super f d a✝
r : ℂ → ℂ → S
g : Grow s c p n r
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ {c} ×ˢ 𝓝ˢ (closedBall 0 p), Eqn s n r x
a' : Set ℂ
an : a' ∈ 𝓝ˢ {c}
b' : Set ℂ
bn : b' ∈ 𝓝ˢ (closedBall 0 p)
sub : ∀ x ∈ a' ×ˢ b', Eqn s n r x
a : Set ℂ
aa : ∀ x ∈ a, a' x
ao : IsOpen a
am : c ∈ a
b : Set ℂ
bo : IsOpen b
bp : closedBall 0 p ⊆ b
bb : ∀ x ∈ b, b' x
q : ℝ
pq : p < q
qb : closedBall 0 q ⊆ b
m : ∀ᶠ (c' : ℂ) in 𝓝 c, (c', r c' 0) ∈ s.near
⊢ ∀ᶠ (x : ℂ) in 𝓝 c,
(∀ᶠ (x : ℂ × ℂ) in 𝓝 (id x, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2) → (x, r x 0) ∈ s.near → Grow s x q n r | case right
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
p : ℝ
s : Super f d a✝
r : ℂ → ℂ → S
g : Grow s c p n r
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ {c} ×ˢ 𝓝ˢ (closedBall 0 p), Eqn s n r x
a' : Set ℂ
an : a' ∈ 𝓝ˢ {c}
b' : Set ℂ
bn : b' ∈ 𝓝ˢ (closedBall 0 p)
sub : ∀ x ∈ a' ×ˢ b', Eqn s n r x
a : Set ℂ
aa : ∀ x ∈ a, a' x
ao : IsOpen a
am : c ∈ a
b : Set ℂ
bo : IsOpen b
bp : closedBall 0 p ⊆ b
bb : ∀ x ∈ b, b' x
q : ℝ
pq : p < q
qb : closedBall 0 q ⊆ b
m : ∀ᶠ (c' : ℂ) in 𝓝 c, (c', r c' 0) ∈ s.near
⊢ ∀ x ∈ a, (∀ᶠ (x : ℂ × ℂ) in 𝓝 (id x, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2) → (x, r x 0) ∈ s.near → Grow s x q n r | Please generate a tactic in lean4 to solve the state.
STATE:
case right
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
p : ℝ
s : Super f d a✝
r : ℂ → ℂ → S
g : Grow s c p n r
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ {c} ×ˢ 𝓝ˢ (closedBall 0 p), Eqn s n r x
a' : Set ℂ
an : a' ∈ 𝓝ˢ {c}
b' : Set ℂ
bn : b' ∈ 𝓝ˢ (closedBall 0 p)
sub : ∀ x ∈ a' ×ˢ b', Eqn s n r x
a : Set ℂ
aa : ∀ x ∈ a, a' x
ao : IsOpen a
am : c ∈ a
b : Set ℂ
bo : IsOpen b
bp : closedBall 0 p ⊆ b
bb : ∀ x ∈ b, b' x
q : ℝ
pq : p < q
qb : closedBall 0 q ⊆ b
m : ∀ᶠ (c' : ℂ) in 𝓝 c, (c', r c' 0) ∈ s.near
⊢ ∀ᶠ (x : ℂ) in 𝓝 c,
(∀ᶠ (x : ℂ × ℂ) in 𝓝 (id x, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2) → (x, r x 0) ∈ s.near → Grow s x q n r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.open | [244, 1] | [268, 61] | intro c' am' start m | case right
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
p : ℝ
s : Super f d a✝
r : ℂ → ℂ → S
g : Grow s c p n r
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ {c} ×ˢ 𝓝ˢ (closedBall 0 p), Eqn s n r x
a' : Set ℂ
an : a' ∈ 𝓝ˢ {c}
b' : Set ℂ
bn : b' ∈ 𝓝ˢ (closedBall 0 p)
sub : ∀ x ∈ a' ×ˢ b', Eqn s n r x
a : Set ℂ
aa : ∀ x ∈ a, a' x
ao : IsOpen a
am : c ∈ a
b : Set ℂ
bo : IsOpen b
bp : closedBall 0 p ⊆ b
bb : ∀ x ∈ b, b' x
q : ℝ
pq : p < q
qb : closedBall 0 q ⊆ b
m : ∀ᶠ (c' : ℂ) in 𝓝 c, (c', r c' 0) ∈ s.near
⊢ ∀ x ∈ a, (∀ᶠ (x : ℂ × ℂ) in 𝓝 (id x, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2) → (x, r x 0) ∈ s.near → Grow s x q n r | case right
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
p : ℝ
s : Super f d a✝
r : ℂ → ℂ → S
g : Grow s c p n r
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ {c} ×ˢ 𝓝ˢ (closedBall 0 p), Eqn s n r x
a' : Set ℂ
an : a' ∈ 𝓝ˢ {c}
b' : Set ℂ
bn : b' ∈ 𝓝ˢ (closedBall 0 p)
sub : ∀ x ∈ a' ×ˢ b', Eqn s n r x
a : Set ℂ
aa : ∀ x ∈ a, a' x
ao : IsOpen a
am : c ∈ a
b : Set ℂ
bo : IsOpen b
bp : closedBall 0 p ⊆ b
bb : ∀ x ∈ b, b' x
q : ℝ
pq : p < q
qb : closedBall 0 q ⊆ b
m✝ : ∀ᶠ (c' : ℂ) in 𝓝 c, (c', r c' 0) ∈ s.near
c' : ℂ
am' : c' ∈ a
start : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (id c', 0), s.bottcherNear x.1 (r x.1 x.2) = x.2
m : (c', r c' 0) ∈ s.near
⊢ Grow s c' q n r | Please generate a tactic in lean4 to solve the state.
STATE:
case right
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
p : ℝ
s : Super f d a✝
r : ℂ → ℂ → S
g : Grow s c p n r
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ {c} ×ˢ 𝓝ˢ (closedBall 0 p), Eqn s n r x
a' : Set ℂ
an : a' ∈ 𝓝ˢ {c}
b' : Set ℂ
bn : b' ∈ 𝓝ˢ (closedBall 0 p)
sub : ∀ x ∈ a' ×ˢ b', Eqn s n r x
a : Set ℂ
aa : ∀ x ∈ a, a' x
ao : IsOpen a
am : c ∈ a
b : Set ℂ
bo : IsOpen b
bp : closedBall 0 p ⊆ b
bb : ∀ x ∈ b, b' x
q : ℝ
pq : p < q
qb : closedBall 0 q ⊆ b
m : ∀ᶠ (c' : ℂ) in 𝓝 c, (c', r c' 0) ∈ s.near
⊢ ∀ x ∈ a, (∀ᶠ (x : ℂ × ℂ) in 𝓝 (id x, 0), s.bottcherNear x.1 (r x.1 x.2) = x.2) → (x, r x 0) ∈ s.near → Grow s x q n r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.open | [244, 1] | [268, 61] | refine (continuousAt_id.prod ?_).eventually_mem (s.isOpen_near.mem_nhds ?_) | 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
p : ℝ
s : Super f d a✝
r : ℂ → ℂ → S
g : Grow s c p n r
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ {c} ×ˢ 𝓝ˢ (closedBall 0 p), Eqn s n r x
a' : Set ℂ
an : a' ∈ 𝓝ˢ {c}
b' : Set ℂ
bn : b' ∈ 𝓝ˢ (closedBall 0 p)
sub : ∀ x ∈ a' ×ˢ b', Eqn s n r x
a : Set ℂ
aa : ∀ x ∈ a, a' x
ao : IsOpen a
am : c ∈ a
b : Set ℂ
bo : IsOpen b
bp : closedBall 0 p ⊆ b
bb : ∀ x ∈ b, b' x
q : ℝ
pq : p < q
qb : closedBall 0 q ⊆ b
⊢ ∀ᶠ (c' : ℂ) in 𝓝 c, (c', r c' 0) ∈ s.near | case refine_1
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
p : ℝ
s : Super f d a✝
r : ℂ → ℂ → S
g : Grow s c p n r
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ {c} ×ˢ 𝓝ˢ (closedBall 0 p), Eqn s n r x
a' : Set ℂ
an : a' ∈ 𝓝ˢ {c}
b' : Set ℂ
bn : b' ∈ 𝓝ˢ (closedBall 0 p)
sub : ∀ x ∈ a' ×ˢ b', Eqn s n r x
a : Set ℂ
aa : ∀ x ∈ a, a' x
ao : IsOpen a
am : c ∈ a
b : Set ℂ
bo : IsOpen b
bp : closedBall 0 p ⊆ b
bb : ∀ x ∈ b, b' x
q : ℝ
pq : p < q
qb : closedBall 0 q ⊆ b
⊢ ContinuousAt (fun c' => r c' 0) c
case refine_2
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
p : ℝ
s : Super f d a✝
r : ℂ → ℂ → S
g : Grow s c p n r
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ {c} ×ˢ 𝓝ˢ (closedBall 0 p), Eqn s n r x
a' : Set ℂ
an : a' ∈ 𝓝ˢ {c}
b' : Set ℂ
bn : b' ∈ 𝓝ˢ (closedBall 0 p)
sub : ∀ x ∈ a' ×ˢ b', Eqn s n r x
a : Set ℂ
aa : ∀ x ∈ a, a' x
ao : IsOpen a
am : c ∈ a
b : Set ℂ
bo : IsOpen b
bp : closedBall 0 p ⊆ b
bb : ∀ x ∈ b, b' x
q : ℝ
pq : p < q
qb : closedBall 0 q ⊆ b
⊢ (id c, r c 0) ∈ s.near | 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
p : ℝ
s : Super f d a✝
r : ℂ → ℂ → S
g : Grow s c p n r
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ {c} ×ˢ 𝓝ˢ (closedBall 0 p), Eqn s n r x
a' : Set ℂ
an : a' ∈ 𝓝ˢ {c}
b' : Set ℂ
bn : b' ∈ 𝓝ˢ (closedBall 0 p)
sub : ∀ x ∈ a' ×ˢ b', Eqn s n r x
a : Set ℂ
aa : ∀ x ∈ a, a' x
ao : IsOpen a
am : c ∈ a
b : Set ℂ
bo : IsOpen b
bp : closedBall 0 p ⊆ b
bb : ∀ x ∈ b, b' x
q : ℝ
pq : p < q
qb : closedBall 0 q ⊆ b
⊢ ∀ᶠ (c' : ℂ) in 𝓝 c, (c', r c' 0) ∈ s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.open | [244, 1] | [268, 61] | exact (g.eqn.filter_mono (nhds_le_nhdsSet (mem_domain c
g.nonneg))).self_of_nhds.holo.along_fst.continuousAt | case refine_1
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
p : ℝ
s : Super f d a✝
r : ℂ → ℂ → S
g : Grow s c p n r
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ {c} ×ˢ 𝓝ˢ (closedBall 0 p), Eqn s n r x
a' : Set ℂ
an : a' ∈ 𝓝ˢ {c}
b' : Set ℂ
bn : b' ∈ 𝓝ˢ (closedBall 0 p)
sub : ∀ x ∈ a' ×ˢ b', Eqn s n r x
a : Set ℂ
aa : ∀ x ∈ a, a' x
ao : IsOpen a
am : c ∈ a
b : Set ℂ
bo : IsOpen b
bp : closedBall 0 p ⊆ b
bb : ∀ x ∈ b, b' x
q : ℝ
pq : p < q
qb : closedBall 0 q ⊆ b
⊢ ContinuousAt (fun c' => r c' 0) c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
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
p : ℝ
s : Super f d a✝
r : ℂ → ℂ → S
g : Grow s c p n r
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ {c} ×ˢ 𝓝ˢ (closedBall 0 p), Eqn s n r x
a' : Set ℂ
an : a' ∈ 𝓝ˢ {c}
b' : Set ℂ
bn : b' ∈ 𝓝ˢ (closedBall 0 p)
sub : ∀ x ∈ a' ×ˢ b', Eqn s n r x
a : Set ℂ
aa : ∀ x ∈ a, a' x
ao : IsOpen a
am : c ∈ a
b : Set ℂ
bo : IsOpen b
bp : closedBall 0 p ⊆ b
bb : ∀ x ∈ b, b' x
q : ℝ
pq : p < q
qb : closedBall 0 q ⊆ b
⊢ ContinuousAt (fun c' => r c' 0) c
TACTIC:
|
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