url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
| input
stringlengths 73
2.09M
|
|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/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:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.open | [244, 1] | [268, 61] | simp only [id, g.zero, s.mem_near 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 | 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
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
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.open | [244, 1] | [268, 61] | have e := start.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
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
⊢ r 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 : 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
e : s.bottcherNear (id c', 0).1 (r (id c', 0).1 (id c', 0).2) = (id c', 0).2
⊢ r 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 : 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
⊢ r c' 0 = a✝
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.open | [244, 1] | [268, 61] | simp only [id, s.bottcherNear_eq_zero m] 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 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
e : s.bottcherNear (id c', 0).1 (r (id c', 0).1 (id c', 0).2) = (id c', 0).2
⊢ r 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 : 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
e : r c' 0 = a✝
⊢ r 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 : 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
e : s.bottcherNear (id c', 0).1 (r (id c', 0).1 (id c', 0).2) = (id c', 0).2
⊢ r c' 0 = a✝
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.open | [244, 1] | [268, 61] | exact 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 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
e : r c' 0 = a✝
⊢ r 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 : 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
e : r c' 0 = a✝
⊢ r c' 0 = a✝
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.open | [244, 1] | [268, 61] | refine eventually_nhdsSet_iff_exists.mpr ⟨a ×ˢ b, ao.prod bo, ?_, ?_⟩ | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
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
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ ({c'} ×ˢ closedBall 0 q), Eqn s n r x | 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
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
⊢ {c'} ×ˢ closedBall 0 q ⊆ a ×ˢ b
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
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
⊢ ∀ x ∈ a ×ˢ b, Eqn s n r 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
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
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ ({c'} ×ˢ closedBall 0 q), Eqn s n r x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.open | [244, 1] | [268, 61] | exact prod_mono (singleton_subset_iff.mpr am') qb | 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
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
⊢ {c'} ×ˢ closedBall 0 q ⊆ a ×ˢ b | 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
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
⊢ {c'} ×ˢ closedBall 0 q ⊆ a ×ˢ b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.open | [244, 1] | [268, 61] | intro x ⟨cm, xm⟩ | 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
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
⊢ ∀ x ∈ a ×ˢ b, Eqn s n r x | 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
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
x : ℂ × ℂ
cm : x.1 ∈ a
xm : x.2 ∈ b
⊢ Eqn s n r x | 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
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
⊢ ∀ x ∈ a ×ˢ b, Eqn s n r x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.open | [244, 1] | [268, 61] | exact sub x ⟨aa _ cm, bb _ xm⟩ | 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
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
x : ℂ × ℂ
cm : x.1 ∈ a
xm : x.2 ∈ b
⊢ Eqn s n r x | 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
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
x : ℂ × ℂ
cm : x.1 ∈ a
xm : x.2 ∈ b
⊢ Eqn s n r x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | GrowOpen.point | [289, 1] | [359, 65] | by_cases za : abs x = 0 | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s (s.np c p) r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ ball 0 p ∧ r' c y = r c y | case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : Complex.abs x = 0
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s (s.np c p) r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ ball 0 p ∧ r' c y = r c y
case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : ¬Complex.abs x = 0
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s (s.np c p) r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ ball 0 p ∧ r' c y = r c y | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s (s.np c p) r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ ball 0 p ∧ r' c y = r c y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | GrowOpen.point | [289, 1] | [359, 65] | replace za := (Ne.symm za).lt_of_le (Complex.abs.nonneg _) | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : ¬Complex.abs x = 0
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s (s.np c p) r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ ball 0 p ∧ r' c y = r c y | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : 0 < Complex.abs x
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s (s.np c p) r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ ball 0 p ∧ r' c y = r c y | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : ¬Complex.abs x = 0
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s (s.np c p) r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ ball 0 p ∧ r' c y = r c y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | GrowOpen.point | [289, 1] | [359, 65] | set t := ball (0 : ℂ) p | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : 0 < Complex.abs x
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s (s.np c p) r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ ball 0 p ∧ r' c y = r c y | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : 0 < Complex.abs x
t : Set ℂ := ball 0 p
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s (s.np c p) r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : 0 < Complex.abs x
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s (s.np c p) r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ ball 0 p ∧ r' c y = r c y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | GrowOpen.point | [289, 1] | [359, 65] | have xt : x ∈ closure t := by
simp only [closure_ball _ g.pos.ne', mem_closedBall, Complex.dist_eq, sub_zero, ax] | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : 0 < Complex.abs x
t : Set ℂ := ball 0 p
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s (s.np c p) r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : 0 < Complex.abs x
t : Set ℂ := ball 0 p
xt : x ∈ closure t
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s (s.np c p) r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : 0 < Complex.abs x
t : Set ℂ := ball 0 p
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s (s.np c p) r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | GrowOpen.point | [289, 1] | [359, 65] | have ez : ∃ z : S, MapClusterPt z (𝓝[t] x) (r c) :=
@exists_clusterPt_of_compactSpace _ _ _ _
(Filter.map_neBot (hf := mem_closure_iff_nhdsWithin_neBot.mp xt)) | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : 0 < Complex.abs x
t : Set ℂ := ball 0 p
xt : x ∈ closure t
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s (s.np c p) r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : 0 < Complex.abs x
t : Set ℂ := ball 0 p
xt : x ∈ closure t
ez : ∃ z, MapClusterPt z (𝓝[t] x) (r c)
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s (s.np c p) r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : 0 < Complex.abs x
t : Set ℂ := ball 0 p
xt : x ∈ closure t
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s (s.np c p) r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | GrowOpen.point | [289, 1] | [359, 65] | rcases ez with ⟨z, cp⟩ | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : 0 < Complex.abs x
t : Set ℂ := ball 0 p
xt : x ∈ closure t
ez : ∃ z, MapClusterPt z (𝓝[t] x) (r c)
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s (s.np c p) r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y | case neg.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : 0 < Complex.abs x
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
cp : MapClusterPt z (𝓝[t] x) (r c)
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s (s.np c p) r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : 0 < Complex.abs x
t : Set ℂ := ball 0 p
xt : x ∈ closure t
ez : ∃ z, MapClusterPt z (𝓝[t] x) (r c)
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s (s.np c p) r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | GrowOpen.point | [289, 1] | [359, 65] | have pz : s.potential c z = abs x := by
refine eq_of_nhds_neBot (cp.map (Continuous.potential s).along_snd.continuousAt
(Filter.tendsto_map' ?_))
have e : ∀ y, y ∈ t → (s.potential c ∘ r c) y = abs y := by
intro y m; simp only [Function.comp]; exact (g.eqn.self_of_nhdsSet (c, y) ⟨rfl, m⟩).potential
exact tendsto_nhdsWithin_congr (fun t m ↦ (e t m).symm)
Complex.continuous_abs.continuousWithinAt | case neg.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : 0 < Complex.abs x
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
cp : MapClusterPt z (𝓝[t] x) (r c)
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s (s.np c p) r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y | case neg.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : 0 < Complex.abs x
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s (s.np c p) r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : 0 < Complex.abs x
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
cp : MapClusterPt z (𝓝[t] x) (r c)
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s (s.np c p) r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | GrowOpen.point | [289, 1] | [359, 65] | rcases s.nice_np c (lt_of_lt_of_le g.post s.p_le_one) z (_root_.trans (le_of_eq pz) ax)
with ⟨m, nc⟩ | case neg.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : 0 < Complex.abs x
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s (s.np c p) r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y | case neg.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : 0 < Complex.abs x
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
m : (c, (f c)^[s.np c p] z) ∈ s.near
nc : ∀ (k : ℕ), s.np c p ≤ k → mfderiv I I (s.bottcherNear c) ((f c)^[k] z) ≠ 0
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s (s.np c p) r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : 0 < Complex.abs x
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s (s.np c p) r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | GrowOpen.point | [289, 1] | [359, 65] | replace nc := nc _ (le_refl _) | case neg.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : 0 < Complex.abs x
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
m : (c, (f c)^[s.np c p] z) ∈ s.near
nc : ∀ (k : ℕ), s.np c p ≤ k → mfderiv I I (s.bottcherNear c) ((f c)^[k] z) ≠ 0
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s (s.np c p) r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y | case neg.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : 0 < Complex.abs x
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
m : (c, (f c)^[s.np c p] z) ∈ s.near
nc : mfderiv I I (s.bottcherNear c) ((f c)^[s.np c p] z) ≠ 0
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s (s.np c p) r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : 0 < Complex.abs x
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
m : (c, (f c)^[s.np c p] z) ∈ s.near
nc : ∀ (k : ℕ), s.np c p ≤ k → mfderiv I I (s.bottcherNear c) ((f c)^[k] z) ≠ 0
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s (s.np c p) r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | GrowOpen.point | [289, 1] | [359, 65] | generalize hn : s.np c p = n | case neg.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : 0 < Complex.abs x
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
m : (c, (f c)^[s.np c p] z) ∈ s.near
nc : mfderiv I I (s.bottcherNear c) ((f c)^[s.np c p] z) ≠ 0
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s (s.np c p) r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y | case neg.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : 0 < Complex.abs x
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
m : (c, (f c)^[s.np c p] z) ∈ s.near
nc : mfderiv I I (s.bottcherNear c) ((f c)^[s.np c p] z) ≠ 0
n : ℕ
hn : s.np c p = n
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : 0 < Complex.abs x
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
m : (c, (f c)^[s.np c p] z) ∈ s.near
nc : mfderiv I I (s.bottcherNear c) ((f c)^[s.np c p] z) ≠ 0
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s (s.np c p) r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | GrowOpen.point | [289, 1] | [359, 65] | rw [hn] at m nc | case neg.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : 0 < Complex.abs x
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
m : (c, (f c)^[s.np c p] z) ∈ s.near
nc : mfderiv I I (s.bottcherNear c) ((f c)^[s.np c p] z) ≠ 0
n : ℕ
hn : s.np c p = n
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y | case neg.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : 0 < Complex.abs x
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
n : ℕ
nc : mfderiv I I (s.bottcherNear c) ((f c)^[n] z) ≠ 0
m : (c, (f c)^[n] z) ∈ s.near
hn : s.np c p = n
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : 0 < Complex.abs x
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
m : (c, (f c)^[s.np c p] z) ∈ s.near
nc : mfderiv I I (s.bottcherNear c) ((f c)^[s.np c p] z) ≠ 0
n : ℕ
hn : s.np c p = n
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | GrowOpen.point | [289, 1] | [359, 65] | generalize hb : s.bottcherNearIter n = b | case neg.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : 0 < Complex.abs x
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
n : ℕ
nc : mfderiv I I (s.bottcherNear c) ((f c)^[n] z) ≠ 0
m : (c, (f c)^[n] z) ∈ s.near
hn : s.np c p = n
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y | case neg.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : 0 < Complex.abs x
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
n : ℕ
nc : mfderiv I I (s.bottcherNear c) ((f c)^[n] z) ≠ 0
m : (c, (f c)^[n] z) ∈ s.near
hn : s.np c p = n
b : ℂ → S → ℂ
hb : s.bottcherNearIter n = b
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : 0 < Complex.abs x
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
n : ℕ
nc : mfderiv I I (s.bottcherNear c) ((f c)^[n] z) ≠ 0
m : (c, (f c)^[n] z) ∈ s.near
hn : s.np c p = n
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | GrowOpen.point | [289, 1] | [359, 65] | have post : Postcritical s c z := lt_of_le_of_lt (_root_.trans (le_of_eq pz) ax) g.post | case neg.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : 0 < Complex.abs x
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
n : ℕ
nc : mfderiv I I (s.bottcherNear c) ((f c)^[n] z) ≠ 0
m : (c, (f c)^[n] z) ∈ s.near
hn : s.np c p = n
b : ℂ → S → ℂ
hb : s.bottcherNearIter n = b
bz : b c z = x ^ d ^ n
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y | case neg.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : 0 < Complex.abs x
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
n : ℕ
nc : mfderiv I I (s.bottcherNear c) ((f c)^[n] z) ≠ 0
m : (c, (f c)^[n] z) ∈ s.near
hn : s.np c p = n
b : ℂ → S → ℂ
hb : s.bottcherNearIter n = b
bz : b c z = x ^ d ^ n
post : Postcritical s c z
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : 0 < Complex.abs x
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
n : ℕ
nc : mfderiv I I (s.bottcherNear c) ((f c)^[n] z) ≠ 0
m : (c, (f c)^[n] z) ∈ s.near
hn : s.np c p = n
b : ℂ → S → ℂ
hb : s.bottcherNearIter n = b
bz : b c z = x ^ d ^ n
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | GrowOpen.point | [289, 1] | [359, 65] | rw [← pz] at za | case neg.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : 0 < Complex.abs x
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
n : ℕ
nc : mfderiv I I (s.bottcherNear c) ((f c)^[n] z) ≠ 0
m : (c, (f c)^[n] z) ∈ s.near
hn : s.np c p = n
b : ℂ → S → ℂ
hb : s.bottcherNearIter n = b
bz : b c z = x ^ d ^ n
post : Postcritical s c z
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y | case neg.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
za : 0 < s.potential c z
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
n : ℕ
nc : mfderiv I I (s.bottcherNear c) ((f c)^[n] z) ≠ 0
m : (c, (f c)^[n] z) ∈ s.near
hn : s.np c p = n
b : ℂ → S → ℂ
hb : s.bottcherNearIter n = b
bz : b c z = x ^ d ^ n
post : Postcritical s c z
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : 0 < Complex.abs x
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
n : ℕ
nc : mfderiv I I (s.bottcherNear c) ((f c)^[n] z) ≠ 0
m : (c, (f c)^[n] z) ∈ s.near
hn : s.np c p = n
b : ℂ → S → ℂ
hb : s.bottcherNearIter n = b
bz : b c z = x ^ d ^ n
post : Postcritical s c z
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | GrowOpen.point | [289, 1] | [359, 65] | have ba := s.bottcherNearIter_holomorphic m | case neg.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
za : 0 < s.potential c z
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
n : ℕ
nc : mfderiv I I (s.bottcherNear c) ((f c)^[n] z) ≠ 0
m : (c, (f c)^[n] z) ∈ s.near
hn : s.np c p = n
b : ℂ → S → ℂ
hb : s.bottcherNearIter n = b
bz : b c z = x ^ d ^ n
post : Postcritical s c z
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y | case neg.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
za : 0 < s.potential c z
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
n : ℕ
nc : mfderiv I I (s.bottcherNear c) ((f c)^[n] z) ≠ 0
m : (c, (f c)^[n] z) ∈ s.near
hn : s.np c p = n
b : ℂ → S → ℂ
hb : s.bottcherNearIter n = b
bz : b c z = x ^ d ^ n
post : Postcritical s c z
ba : HolomorphicAt (I.prod I) I (uncurry (s.bottcherNearIter n)) (c, z)
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
za : 0 < s.potential c z
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
n : ℕ
nc : mfderiv I I (s.bottcherNear c) ((f c)^[n] z) ≠ 0
m : (c, (f c)^[n] z) ∈ s.near
hn : s.np c p = n
b : ℂ → S → ℂ
hb : s.bottcherNearIter n = b
bz : b c z = x ^ d ^ n
post : Postcritical s c z
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | GrowOpen.point | [289, 1] | [359, 65] | replace nc := s.bottcherNearIter_mfderiv_ne_zero nc (post.not_precritical za.ne') | case neg.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
za : 0 < s.potential c z
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
n : ℕ
nc : mfderiv I I (s.bottcherNear c) ((f c)^[n] z) ≠ 0
m : (c, (f c)^[n] z) ∈ s.near
hn : s.np c p = n
b : ℂ → S → ℂ
hb : s.bottcherNearIter n = b
bz : b c z = x ^ d ^ n
post : Postcritical s c z
ba : HolomorphicAt (I.prod I) I (uncurry (s.bottcherNearIter n)) (c, z)
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y | case neg.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
za : 0 < s.potential c z
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
n : ℕ
m : (c, (f c)^[n] z) ∈ s.near
hn : s.np c p = n
b : ℂ → S → ℂ
hb : s.bottcherNearIter n = b
bz : b c z = x ^ d ^ n
post : Postcritical s c z
ba : HolomorphicAt (I.prod I) I (uncurry (s.bottcherNearIter n)) (c, z)
nc : mfderiv I I (s.bottcherNearIter n c) z ≠ 0
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
za : 0 < s.potential c z
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
n : ℕ
nc : mfderiv I I (s.bottcherNear c) ((f c)^[n] z) ≠ 0
m : (c, (f c)^[n] z) ∈ s.near
hn : s.np c p = n
b : ℂ → S → ℂ
hb : s.bottcherNearIter n = b
bz : b c z = x ^ d ^ n
post : Postcritical s c z
ba : HolomorphicAt (I.prod I) I (uncurry (s.bottcherNearIter n)) (c, z)
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | GrowOpen.point | [289, 1] | [359, 65] | rcases complex_inverse_fun ba nc with ⟨i, ia, ib, bi⟩ | case neg.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
za : 0 < s.potential c z
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
n : ℕ
m : (c, (f c)^[n] z) ∈ s.near
hn : s.np c p = n
b : ℂ → S → ℂ
hb : s.bottcherNearIter n = b
bz : b c z = x ^ d ^ n
post : Postcritical s c z
ba : HolomorphicAt (I.prod I) I (uncurry (s.bottcherNearIter n)) (c, z)
nc : mfderiv I I (s.bottcherNearIter n c) z ≠ 0
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y | case neg.intro.intro.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
za : 0 < s.potential c z
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
n : ℕ
m : (c, (f c)^[n] z) ∈ s.near
hn : s.np c p = n
b : ℂ → S → ℂ
hb : s.bottcherNearIter n = b
bz : b c z = x ^ d ^ n
post : Postcritical s c z
ba : HolomorphicAt (I.prod I) I (uncurry (s.bottcherNearIter n)) (c, z)
nc : mfderiv I I (s.bottcherNearIter n c) z ≠ 0
i : ℂ → ℂ → S
ia : HolomorphicAt (I.prod I) I (uncurry i) (c, s.bottcherNearIter n c z)
ib : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), i x.1 (s.bottcherNearIter n x.1 x.2) = x.2
bi : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, s.bottcherNearIter n c z), s.bottcherNearIter n x.1 (i x.1 x.2) = x.2
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
za : 0 < s.potential c z
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
n : ℕ
m : (c, (f c)^[n] z) ∈ s.near
hn : s.np c p = n
b : ℂ → S → ℂ
hb : s.bottcherNearIter n = b
bz : b c z = x ^ d ^ n
post : Postcritical s c z
ba : HolomorphicAt (I.prod I) I (uncurry (s.bottcherNearIter n)) (c, z)
nc : mfderiv I I (s.bottcherNearIter n c) z ≠ 0
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | GrowOpen.point | [289, 1] | [359, 65] | simp only [hb, bz] at ia bi ib | case neg.intro.intro.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
za : 0 < s.potential c z
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
n : ℕ
m : (c, (f c)^[n] z) ∈ s.near
hn : s.np c p = n
b : ℂ → S → ℂ
hb : s.bottcherNearIter n = b
bz : b c z = x ^ d ^ n
post : Postcritical s c z
ba : HolomorphicAt (I.prod I) I (uncurry (s.bottcherNearIter n)) (c, z)
nc : mfderiv I I (s.bottcherNearIter n c) z ≠ 0
i : ℂ → ℂ → S
ia : HolomorphicAt (I.prod I) I (uncurry i) (c, s.bottcherNearIter n c z)
ib : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), i x.1 (s.bottcherNearIter n x.1 x.2) = x.2
bi : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, s.bottcherNearIter n c z), s.bottcherNearIter n x.1 (i x.1 x.2) = x.2
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y | case neg.intro.intro.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
za : 0 < s.potential c z
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
n : ℕ
m : (c, (f c)^[n] z) ∈ s.near
hn : s.np c p = n
b : ℂ → S → ℂ
hb : s.bottcherNearIter n = b
bz : b c z = x ^ d ^ n
post : Postcritical s c z
ba : HolomorphicAt (I.prod I) I (uncurry (s.bottcherNearIter n)) (c, z)
nc : mfderiv I I (s.bottcherNearIter n c) z ≠ 0
i : ℂ → ℂ → S
ia : HolomorphicAt (I.prod I) I (uncurry i) (c, x ^ d ^ n)
bi : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, x ^ d ^ n), b x.1 (i x.1 x.2) = x.2
ib : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), i x.1 (b x.1 x.2) = x.2
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
za : 0 < s.potential c z
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
n : ℕ
m : (c, (f c)^[n] z) ∈ s.near
hn : s.np c p = n
b : ℂ → S → ℂ
hb : s.bottcherNearIter n = b
bz : b c z = x ^ d ^ n
post : Postcritical s c z
ba : HolomorphicAt (I.prod I) I (uncurry (s.bottcherNearIter n)) (c, z)
nc : mfderiv I I (s.bottcherNearIter n c) z ≠ 0
i : ℂ → ℂ → S
ia : HolomorphicAt (I.prod I) I (uncurry i) (c, s.bottcherNearIter n c z)
ib : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), i x.1 (s.bottcherNearIter n x.1 x.2) = x.2
bi : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, s.bottcherNearIter n c z), s.bottcherNearIter n x.1 (i x.1 x.2) = x.2
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | GrowOpen.point | [289, 1] | [359, 65] | have pt : Tendsto (fun p : ℂ × ℂ ↦ (p.1, p.2 ^ d ^ n)) (𝓝 (c, x)) (𝓝 (c, x ^ d ^ n)) :=
continuousAt_fst.prod (continuousAt_snd.pow _) | case neg.intro.intro.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
za : 0 < s.potential c z
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
n : ℕ
m : (c, (f c)^[n] z) ∈ s.near
hn : s.np c p = n
b : ℂ → S → ℂ
hb : s.bottcherNearIter n = b
bz : b c z = x ^ d ^ n
post : Postcritical s c z
ba : HolomorphicAt (I.prod I) I (uncurry (s.bottcherNearIter n)) (c, z)
nc : mfderiv I I (s.bottcherNearIter n c) z ≠ 0
i : ℂ → ℂ → S
ia : HolomorphicAt (I.prod I) I (uncurry i) (c, x ^ d ^ n)
bi : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, x ^ d ^ n), b x.1 (i x.1 x.2) = x.2
ib : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), i x.1 (b x.1 x.2) = x.2
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y | case neg.intro.intro.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
za : 0 < s.potential c z
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
n : ℕ
m : (c, (f c)^[n] z) ∈ s.near
hn : s.np c p = n
b : ℂ → S → ℂ
hb : s.bottcherNearIter n = b
bz : b c z = x ^ d ^ n
post : Postcritical s c z
ba : HolomorphicAt (I.prod I) I (uncurry (s.bottcherNearIter n)) (c, z)
nc : mfderiv I I (s.bottcherNearIter n c) z ≠ 0
i : ℂ → ℂ → S
ia : HolomorphicAt (I.prod I) I (uncurry i) (c, x ^ d ^ n)
bi : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, x ^ d ^ n), b x.1 (i x.1 x.2) = x.2
ib : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), i x.1 (b x.1 x.2) = x.2
pt : Tendsto (fun p => (p.1, p.2 ^ d ^ n)) (𝓝 (c, x)) (𝓝 (c, x ^ d ^ n))
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
za : 0 < s.potential c z
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
n : ℕ
m : (c, (f c)^[n] z) ∈ s.near
hn : s.np c p = n
b : ℂ → S → ℂ
hb : s.bottcherNearIter n = b
bz : b c z = x ^ d ^ n
post : Postcritical s c z
ba : HolomorphicAt (I.prod I) I (uncurry (s.bottcherNearIter n)) (c, z)
nc : mfderiv I I (s.bottcherNearIter n c) z ≠ 0
i : ℂ → ℂ → S
ia : HolomorphicAt (I.prod I) I (uncurry i) (c, x ^ d ^ n)
bi : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, x ^ d ^ n), b x.1 (i x.1 x.2) = x.2
ib : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), i x.1 (b x.1 x.2) = x.2
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | GrowOpen.point | [289, 1] | [359, 65] | have ian : HolomorphicAt II I (uncurry fun e y : ℂ ↦ i e (y ^ d ^ n)) (c, x) :=
ia.comp₂_of_eq holomorphicAt_fst holomorphicAt_snd.pow rfl | case neg.intro.intro.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
za : 0 < s.potential c z
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
n : ℕ
m : (c, (f c)^[n] z) ∈ s.near
hn : s.np c p = n
b : ℂ → S → ℂ
hb : s.bottcherNearIter n = b
bz : b c z = x ^ d ^ n
post : Postcritical s c z
ba : HolomorphicAt (I.prod I) I (uncurry (s.bottcherNearIter n)) (c, z)
nc : mfderiv I I (s.bottcherNearIter n c) z ≠ 0
i : ℂ → ℂ → S
ia : HolomorphicAt (I.prod I) I (uncurry i) (c, x ^ d ^ n)
bi : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, x ^ d ^ n), b x.1 (i x.1 x.2) = x.2
ib : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), i x.1 (b x.1 x.2) = x.2
pt : Tendsto (fun p => (p.1, p.2 ^ d ^ n)) (𝓝 (c, x)) (𝓝 (c, x ^ d ^ n))
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y | case neg.intro.intro.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
za : 0 < s.potential c z
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
n : ℕ
m : (c, (f c)^[n] z) ∈ s.near
hn : s.np c p = n
b : ℂ → S → ℂ
hb : s.bottcherNearIter n = b
bz : b c z = x ^ d ^ n
post : Postcritical s c z
ba : HolomorphicAt (I.prod I) I (uncurry (s.bottcherNearIter n)) (c, z)
nc : mfderiv I I (s.bottcherNearIter n c) z ≠ 0
i : ℂ → ℂ → S
ia : HolomorphicAt (I.prod I) I (uncurry i) (c, x ^ d ^ n)
bi : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, x ^ d ^ n), b x.1 (i x.1 x.2) = x.2
ib : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), i x.1 (b x.1 x.2) = x.2
pt : Tendsto (fun p => (p.1, p.2 ^ d ^ n)) (𝓝 (c, x)) (𝓝 (c, x ^ d ^ n))
ian : HolomorphicAt (I.prod I) I (uncurry fun e y => i e (y ^ d ^ n)) (c, x)
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
za : 0 < s.potential c z
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
n : ℕ
m : (c, (f c)^[n] z) ∈ s.near
hn : s.np c p = n
b : ℂ → S → ℂ
hb : s.bottcherNearIter n = b
bz : b c z = x ^ d ^ n
post : Postcritical s c z
ba : HolomorphicAt (I.prod I) I (uncurry (s.bottcherNearIter n)) (c, z)
nc : mfderiv I I (s.bottcherNearIter n c) z ≠ 0
i : ℂ → ℂ → S
ia : HolomorphicAt (I.prod I) I (uncurry i) (c, x ^ d ^ n)
bi : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, x ^ d ^ n), b x.1 (i x.1 x.2) = x.2
ib : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), i x.1 (b x.1 x.2) = x.2
pt : Tendsto (fun p => (p.1, p.2 ^ d ^ n)) (𝓝 (c, x)) (𝓝 (c, x ^ d ^ n))
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | GrowOpen.point | [289, 1] | [359, 65] | use fun e y ↦ i e (y ^ d ^ n) | case neg.intro.intro.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
za : 0 < s.potential c z
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
n : ℕ
m : (c, (f c)^[n] z) ∈ s.near
hn : s.np c p = n
b : ℂ → S → ℂ
hb : s.bottcherNearIter n = b
bz : b c z = x ^ d ^ n
post : Postcritical s c z
ba : HolomorphicAt (I.prod I) I (uncurry (s.bottcherNearIter n)) (c, z)
nc : mfderiv I I (s.bottcherNearIter n c) z ≠ 0
i : ℂ → ℂ → S
ia : HolomorphicAt (I.prod I) I (uncurry i) (c, x ^ d ^ n)
bi : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, x ^ d ^ n), b x.1 (i x.1 x.2) = x.2
ib : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), i x.1 (b x.1 x.2) = x.2
pt : Tendsto (fun p => (p.1, p.2 ^ d ^ n)) (𝓝 (c, x)) (𝓝 (c, x ^ d ^ n))
ian : HolomorphicAt (I.prod I) I (uncurry fun e y => i e (y ^ d ^ n)) (c, x)
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y | case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
za : 0 < s.potential c z
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
n : ℕ
m : (c, (f c)^[n] z) ∈ s.near
hn : s.np c p = n
b : ℂ → S → ℂ
hb : s.bottcherNearIter n = b
bz : b c z = x ^ d ^ n
post : Postcritical s c z
ba : HolomorphicAt (I.prod I) I (uncurry (s.bottcherNearIter n)) (c, z)
nc : mfderiv I I (s.bottcherNearIter n c) z ≠ 0
i : ℂ → ℂ → S
ia : HolomorphicAt (I.prod I) I (uncurry i) (c, x ^ d ^ n)
bi : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, x ^ d ^ n), b x.1 (i x.1 x.2) = x.2
ib : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), i x.1 (b x.1 x.2) = x.2
pt : Tendsto (fun p => (p.1, p.2 ^ d ^ n)) (𝓝 (c, x)) (𝓝 (c, x ^ d ^ n))
ian : HolomorphicAt (I.prod I) I (uncurry fun e y => i e (y ^ d ^ n)) (c, x)
⊢ (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n (fun e y => i e (y ^ d ^ n)) y) ∧
∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ i c (y ^ d ^ n) = r c y | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
za : 0 < s.potential c z
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
n : ℕ
m : (c, (f c)^[n] z) ∈ s.near
hn : s.np c p = n
b : ℂ → S → ℂ
hb : s.bottcherNearIter n = b
bz : b c z = x ^ d ^ n
post : Postcritical s c z
ba : HolomorphicAt (I.prod I) I (uncurry (s.bottcherNearIter n)) (c, z)
nc : mfderiv I I (s.bottcherNearIter n c) z ≠ 0
i : ℂ → ℂ → S
ia : HolomorphicAt (I.prod I) I (uncurry i) (c, x ^ d ^ n)
bi : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, x ^ d ^ n), b x.1 (i x.1 x.2) = x.2
ib : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), i x.1 (b x.1 x.2) = x.2
pt : Tendsto (fun p => (p.1, p.2 ^ d ^ n)) (𝓝 (c, x)) (𝓝 (c, x ^ d ^ n))
ian : HolomorphicAt (I.prod I) I (uncurry fun e y => i e (y ^ d ^ n)) (c, x)
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ r' c y = r c y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | GrowOpen.point | [289, 1] | [359, 65] | constructor | case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
za : 0 < s.potential c z
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
n : ℕ
m : (c, (f c)^[n] z) ∈ s.near
hn : s.np c p = n
b : ℂ → S → ℂ
hb : s.bottcherNearIter n = b
bz : b c z = x ^ d ^ n
post : Postcritical s c z
ba : HolomorphicAt (I.prod I) I (uncurry (s.bottcherNearIter n)) (c, z)
nc : mfderiv I I (s.bottcherNearIter n c) z ≠ 0
i : ℂ → ℂ → S
ia : HolomorphicAt (I.prod I) I (uncurry i) (c, x ^ d ^ n)
bi : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, x ^ d ^ n), b x.1 (i x.1 x.2) = x.2
ib : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), i x.1 (b x.1 x.2) = x.2
pt : Tendsto (fun p => (p.1, p.2 ^ d ^ n)) (𝓝 (c, x)) (𝓝 (c, x ^ d ^ n))
ian : HolomorphicAt (I.prod I) I (uncurry fun e y => i e (y ^ d ^ n)) (c, x)
⊢ (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n (fun e y => i e (y ^ d ^ n)) y) ∧
∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ i c (y ^ d ^ n) = r c y | case h.left
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
za : 0 < s.potential c z
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
n : ℕ
m : (c, (f c)^[n] z) ∈ s.near
hn : s.np c p = n
b : ℂ → S → ℂ
hb : s.bottcherNearIter n = b
bz : b c z = x ^ d ^ n
post : Postcritical s c z
ba : HolomorphicAt (I.prod I) I (uncurry (s.bottcherNearIter n)) (c, z)
nc : mfderiv I I (s.bottcherNearIter n c) z ≠ 0
i : ℂ → ℂ → S
ia : HolomorphicAt (I.prod I) I (uncurry i) (c, x ^ d ^ n)
bi : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, x ^ d ^ n), b x.1 (i x.1 x.2) = x.2
ib : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), i x.1 (b x.1 x.2) = x.2
pt : Tendsto (fun p => (p.1, p.2 ^ d ^ n)) (𝓝 (c, x)) (𝓝 (c, x ^ d ^ n))
ian : HolomorphicAt (I.prod I) I (uncurry fun e y => i e (y ^ d ^ n)) (c, x)
⊢ ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n (fun e y => i e (y ^ d ^ n)) y
case h.right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
za : 0 < s.potential c z
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
n : ℕ
m : (c, (f c)^[n] z) ∈ s.near
hn : s.np c p = n
b : ℂ → S → ℂ
hb : s.bottcherNearIter n = b
bz : b c z = x ^ d ^ n
post : Postcritical s c z
ba : HolomorphicAt (I.prod I) I (uncurry (s.bottcherNearIter n)) (c, z)
nc : mfderiv I I (s.bottcherNearIter n c) z ≠ 0
i : ℂ → ℂ → S
ia : HolomorphicAt (I.prod I) I (uncurry i) (c, x ^ d ^ n)
bi : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, x ^ d ^ n), b x.1 (i x.1 x.2) = x.2
ib : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), i x.1 (b x.1 x.2) = x.2
pt : Tendsto (fun p => (p.1, p.2 ^ d ^ n)) (𝓝 (c, x)) (𝓝 (c, x ^ d ^ n))
ian : HolomorphicAt (I.prod I) I (uncurry fun e y => i e (y ^ d ^ n)) (c, x)
⊢ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ i c (y ^ d ^ n) = r c y | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
za : 0 < s.potential c z
cp : MapClusterPt z (𝓝[t] x) (r c)
pz : s.potential c z = Complex.abs x
n : ℕ
m : (c, (f c)^[n] z) ∈ s.near
hn : s.np c p = n
b : ℂ → S → ℂ
hb : s.bottcherNearIter n = b
bz : b c z = x ^ d ^ n
post : Postcritical s c z
ba : HolomorphicAt (I.prod I) I (uncurry (s.bottcherNearIter n)) (c, z)
nc : mfderiv I I (s.bottcherNearIter n c) z ≠ 0
i : ℂ → ℂ → S
ia : HolomorphicAt (I.prod I) I (uncurry i) (c, x ^ d ^ n)
bi : ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, x ^ d ^ n), b x.1 (i x.1 x.2) = x.2
ib : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), i x.1 (b x.1 x.2) = x.2
pt : Tendsto (fun p => (p.1, p.2 ^ d ^ n)) (𝓝 (c, x)) (𝓝 (c, x ^ d ^ n))
ian : HolomorphicAt (I.prod I) I (uncurry fun e y => i e (y ^ d ^ n)) (c, x)
⊢ (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n (fun e y => i e (y ^ d ^ n)) y) ∧
∃ᶠ (y : ℂ) in 𝓝 x, y ∈ t ∧ i c (y ^ d ^ n) = r c y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | GrowOpen.point | [289, 1] | [359, 65] | use r | case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : Complex.abs x = 0
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s (s.np c p) r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ ball 0 p ∧ r' c y = r c y | case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : Complex.abs x = 0
⊢ (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s (s.np c p) r y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ ball 0 p ∧ r c y = r c y | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : Complex.abs x = 0
⊢ ∃ r', (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s (s.np c p) r' y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ ball 0 p ∧ r' c y = r c y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | GrowOpen.point | [289, 1] | [359, 65] | simp only [Complex.abs.eq_zero] at za | case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : Complex.abs x = 0
⊢ (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s (s.np c p) r y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ ball 0 p ∧ r c y = r c y | case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : x = 0
⊢ (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s (s.np c p) r y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ ball 0 p ∧ r c y = r c y | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : Complex.abs x = 0
⊢ (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s (s.np c p) r y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ ball 0 p ∧ r c y = r c y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | GrowOpen.point | [289, 1] | [359, 65] | simp only [za, eq_self_iff_true, and_true_iff] | case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : x = 0
⊢ (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s (s.np c p) r y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ ball 0 p ∧ r c y = r c y | case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : x = 0
⊢ (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, 0), Eqn s (s.np c p) r y) ∧ ∃ᶠ (y : ℂ) in 𝓝 0, y ∈ ball 0 p | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : x = 0
⊢ (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s (s.np c p) r y) ∧ ∃ᶠ (y : ℂ) in 𝓝 x, y ∈ ball 0 p ∧ r c y = r c y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | GrowOpen.point | [289, 1] | [359, 65] | constructor | case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : x = 0
⊢ (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, 0), Eqn s (s.np c p) r y) ∧ ∃ᶠ (y : ℂ) in 𝓝 0, y ∈ ball 0 p | case h.left
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : x = 0
⊢ ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, 0), Eqn s (s.np c p) r y
case h.right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : x = 0
⊢ ∃ᶠ (y : ℂ) in 𝓝 0, y ∈ ball 0 p | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : x = 0
⊢ (∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, 0), Eqn s (s.np c p) r y) ∧ ∃ᶠ (y : ℂ) in 𝓝 0, y ∈ ball 0 p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | GrowOpen.point | [289, 1] | [359, 65] | refine g.eqn.filter_mono (nhds_le_nhdsSet ?_) | case h.left
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : x = 0
⊢ ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, 0), Eqn s (s.np c p) r y
case h.right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : x = 0
⊢ ∃ᶠ (y : ℂ) in 𝓝 0, y ∈ ball 0 p | case h.left
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : x = 0
⊢ (c, 0) ∈ {c} ×ˢ ball 0 p
case h.right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : x = 0
⊢ ∃ᶠ (y : ℂ) in 𝓝 0, y ∈ ball 0 p | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : x = 0
⊢ ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, 0), Eqn s (s.np c p) r y
case h.right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : x = 0
⊢ ∃ᶠ (y : ℂ) in 𝓝 0, y ∈ ball 0 p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | GrowOpen.point | [289, 1] | [359, 65] | exact mk_mem_prod rfl (mem_ball_self g.pos) | case h.left
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : x = 0
⊢ (c, 0) ∈ {c} ×ˢ ball 0 p
case h.right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : x = 0
⊢ ∃ᶠ (y : ℂ) in 𝓝 0, y ∈ ball 0 p | case h.right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : x = 0
⊢ ∃ᶠ (y : ℂ) in 𝓝 0, y ∈ ball 0 p | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : x = 0
⊢ (c, 0) ∈ {c} ×ˢ ball 0 p
case h.right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : x = 0
⊢ ∃ᶠ (y : ℂ) in 𝓝 0, y ∈ ball 0 p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | GrowOpen.point | [289, 1] | [359, 65] | exact (isOpen_ball.eventually_mem (mem_ball_self g.pos)).frequently | case h.right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : x = 0
⊢ ∃ᶠ (y : ℂ) in 𝓝 0, y ∈ ball 0 p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : x = 0
⊢ ∃ᶠ (y : ℂ) in 𝓝 0, y ∈ ball 0 p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | GrowOpen.point | [289, 1] | [359, 65] | simp only [closure_ball _ g.pos.ne', mem_closedBall, Complex.dist_eq, sub_zero, ax] | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : 0 < Complex.abs x
t : Set ℂ := ball 0 p
⊢ x ∈ closure t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : 0 < Complex.abs x
t : Set ℂ := ball 0 p
⊢ x ∈ closure t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | GrowOpen.point | [289, 1] | [359, 65] | refine eq_of_nhds_neBot (cp.map (Continuous.potential s).along_snd.continuousAt
(Filter.tendsto_map' ?_)) | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : 0 < Complex.abs x
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
cp : MapClusterPt z (𝓝[t] x) (r c)
⊢ s.potential c z = Complex.abs x | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : 0 < Complex.abs x
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
cp : MapClusterPt z (𝓝[t] x) (r c)
⊢ Tendsto (s.potential c ∘ r c) (𝓝[t] x) (𝓝 (Complex.abs x)) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : 0 < Complex.abs x
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
cp : MapClusterPt z (𝓝[t] x) (r c)
⊢ s.potential c z = Complex.abs x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | GrowOpen.point | [289, 1] | [359, 65] | have e : ∀ y, y ∈ t → (s.potential c ∘ r c) y = abs y := by
intro y m; simp only [Function.comp]; exact (g.eqn.self_of_nhdsSet (c, y) ⟨rfl, m⟩).potential | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : 0 < Complex.abs x
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
cp : MapClusterPt z (𝓝[t] x) (r c)
⊢ Tendsto (s.potential c ∘ r c) (𝓝[t] x) (𝓝 (Complex.abs x)) | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : 0 < Complex.abs x
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
cp : MapClusterPt z (𝓝[t] x) (r c)
e : ∀ y ∈ t, (s.potential c ∘ r c) y = Complex.abs y
⊢ Tendsto (s.potential c ∘ r c) (𝓝[t] x) (𝓝 (Complex.abs x)) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : 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 : GrowOpen s c p r
inst✝ : OnePreimage s
x : ℂ
ax : Complex.abs x ≤ p
za : 0 < Complex.abs x
t : Set ℂ := ball 0 p
xt : x ∈ closure t
z : S
cp : MapClusterPt z (𝓝[t] x) (r c)
⊢ Tendsto (s.potential c ∘ r c) (𝓝[t] x) (𝓝 (Complex.abs x))
TACTIC:
|
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