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/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | exact e.mp (eventually_of_forall fun z e ↦ by rw [e]) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
e : ∀ᶠ (z : ℂ) in atInf, s.bottcher c ↑z = s.bottcherNear c ↑z
⊢ ∀ᶠ (z : ℂ) in atInf, s.bottcher c ↑z * z = s.bottcherNear c ↑z * z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
e : ∀ᶠ (z : ℂ) in atInf, s.bottcher c ↑z = s.bottcherNear c ↑z
⊢ ∀ᶠ (z : ℂ) in atInf, s.bottcher c ↑z * z = s.bottcherNear c ↑z * z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | rw [e] | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
e✝ : ∀ᶠ (z : ℂ) in atInf, s.bottcher c ↑z = s.bottcherNear c ↑z
z : ℂ
e : s.bottcher c ↑z = s.bottcherNear c ↑z
⊢ s.bottcher c ↑z * z = s.bottcherNear c ↑z * z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
e✝ : ∀ᶠ (z : ℂ) in atInf, s.bottcher c ↑z = s.bottcherNear c ↑z
z : ℂ
e : s.bottcher c ↑z = s.bottcherNear c ↑z
⊢ s.bottcher c ↑z * z = s.bottcherNear c ↑z * z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | simp only [one_div, map_inv₀] at zr ⊢ | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
m : ∀ ε > 0, ∃ δ > 0, ∀ {x : ℂ}, Complex.abs x < δ → (Complex.abs x)⁻¹ * Complex.abs (bottcherNear (s.fl c) d x - x) < ε
e : ℝ
ep : e > 0
r : ℝ
rp : r > 0
h : ∀ {x : ℂ}, Complex.abs x < r → (Complex.abs x)⁻¹ * Complex.abs (bottcherNear (s.fl c) d x - x) < e
z : ℂ
zr : 1 / r < Complex.abs z
az0 : Complex.abs z ≠ 0
z0 : z ≠ 0
⊢ Complex.abs z⁻¹ < r | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
m : ∀ ε > 0, ∃ δ > 0, ∀ {x : ℂ}, Complex.abs x < δ → (Complex.abs x)⁻¹ * Complex.abs (bottcherNear (s.fl c) d x - x) < ε
e : ℝ
ep : e > 0
r : ℝ
rp : r > 0
h : ∀ {x : ℂ}, Complex.abs x < r → (Complex.abs x)⁻¹ * Complex.abs (bottcherNear (s.fl c) d x - x) < e
z : ℂ
az0 : Complex.abs z ≠ 0
z0 : z ≠ 0
zr : r⁻¹ < Complex.abs z
⊢ (Complex.abs z)⁻¹ < r | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
m : ∀ ε > 0, ∃ δ > 0, ∀ {x : ℂ}, Complex.abs x < δ → (Complex.abs x)⁻¹ * Complex.abs (bottcherNear (s.fl c) d x - x) < ε
e : ℝ
ep : e > 0
r : ℝ
rp : r > 0
h : ∀ {x : ℂ}, Complex.abs x < r → (Complex.abs x)⁻¹ * Complex.abs (bottcherNear (s.fl c) d x - x) < e
z : ℂ
zr : 1 / r < Complex.abs z
az0 : Complex.abs z ≠ 0
z0 : z ≠ 0
⊢ Complex.abs z⁻¹ < r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | exact inv_lt_of_inv_lt rp zr | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
m : ∀ ε > 0, ∃ δ > 0, ∀ {x : ℂ}, Complex.abs x < δ → (Complex.abs x)⁻¹ * Complex.abs (bottcherNear (s.fl c) d x - x) < ε
e : ℝ
ep : e > 0
r : ℝ
rp : r > 0
h : ∀ {x : ℂ}, Complex.abs x < r → (Complex.abs x)⁻¹ * Complex.abs (bottcherNear (s.fl c) d x - x) < e
z : ℂ
az0 : Complex.abs z ≠ 0
z0 : z ≠ 0
zr : r⁻¹ < Complex.abs z
⊢ (Complex.abs z)⁻¹ < r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
m : ∀ ε > 0, ∃ δ > 0, ∀ {x : ℂ}, Complex.abs x < δ → (Complex.abs x)⁻¹ * Complex.abs (bottcherNear (s.fl c) d x - x) < ε
e : ℝ
ep : e > 0
r : ℝ
rp : r > 0
h : ∀ {x : ℂ}, Complex.abs x < r → (Complex.abs x)⁻¹ * Complex.abs (bottcherNear (s.fl c) d x - x) < e
z : ℂ
az0 : Complex.abs z ≠ 0
z0 : z ≠ 0
zr : r⁻¹ < Complex.abs z
⊢ (Complex.abs z)⁻¹ < r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | potential_tendsto | [744, 1] | [748, 85] | set s := superF d | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
⊢ Tendsto (fun z => ⋯.potential c ↑z * Complex.abs z) atInf (𝓝 1) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
⊢ Tendsto (fun z => s.potential c ↑z * Complex.abs z) atInf (𝓝 1) | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
⊢ Tendsto (fun z => ⋯.potential c ↑z * Complex.abs z) atInf (𝓝 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | potential_tendsto | [744, 1] | [748, 85] | simp only [← s.abs_bottcher, ← Complex.abs.map_mul, ← Complex.abs.map_one] | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
⊢ Tendsto (fun z => s.potential c ↑z * Complex.abs z) atInf (𝓝 1) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
⊢ Tendsto (fun z => Complex.abs (s.bottcher c ↑z * z)) atInf (𝓝 (Complex.abs 1)) | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
⊢ Tendsto (fun z => s.potential c ↑z * Complex.abs z) atInf (𝓝 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | potential_tendsto | [744, 1] | [748, 85] | exact Complex.continuous_abs.continuousAt.tendsto.comp (bottcher_large_approx d c) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
⊢ Tendsto (fun z => Complex.abs (s.bottcher c ↑z * z)) atInf (𝓝 (Complex.abs 1)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
⊢ Tendsto (fun z => Complex.abs (s.bottcher c ↑z * z)) atInf (𝓝 (Complex.abs 1))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.t1Space | [16, 1] | [39, 83] | intro x | A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
⊢ ∀ (x : M), IsClosed {x} | A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x : M
⊢ IsClosed {x} | Please generate a tactic in lean4 to solve the state.
STATE:
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
⊢ ∀ (x : M), IsClosed {x}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.t1Space | [16, 1] | [39, 83] | rw [←compl_compl ({x} : Set M)] | A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x : M
⊢ IsClosed {x} | A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x : M
⊢ IsClosed {x}ᶜᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x : M
⊢ IsClosed {x}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.t1Space | [16, 1] | [39, 83] | apply IsOpen.isClosed_compl | A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x : M
⊢ IsClosed {x}ᶜᶜ | case a
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x : M
⊢ IsOpen {x}ᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x : M
⊢ IsClosed {x}ᶜᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.t1Space | [16, 1] | [39, 83] | rw [isOpen_iff_mem_nhds] | case a
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x : M
⊢ IsOpen {x}ᶜ | case a
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x : M
⊢ ∀ x_1 ∈ {x}ᶜ, {x}ᶜ ∈ 𝓝 x_1 | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x : M
⊢ IsOpen {x}ᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.t1Space | [16, 1] | [39, 83] | intro y m | case a
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x : M
⊢ ∀ x_1 ∈ {x}ᶜ, {x}ᶜ ∈ 𝓝 x_1 | case a
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ∈ {x}ᶜ
⊢ {x}ᶜ ∈ 𝓝 y | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x : M
⊢ ∀ x_1 ∈ {x}ᶜ, {x}ᶜ ∈ 𝓝 x_1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.t1Space | [16, 1] | [39, 83] | simp only [mem_compl_singleton_iff] at m | case a
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ∈ {x}ᶜ
⊢ {x}ᶜ ∈ 𝓝 y | case a
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
⊢ {x}ᶜ ∈ 𝓝 y | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ∈ {x}ᶜ
⊢ {x}ᶜ ∈ 𝓝 y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.t1Space | [16, 1] | [39, 83] | simp only [mem_nhds_iff, subset_compl_singleton_iff] | case a
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
⊢ {x}ᶜ ∈ 𝓝 y | case a
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
⊢ ∃ t, x ∉ t ∧ IsOpen t ∧ y ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
⊢ {x}ᶜ ∈ 𝓝 y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.t1Space | [16, 1] | [39, 83] | by_cases xm : x ∈ (chartAt A y).source | case a
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
⊢ ∃ t, x ∉ t ∧ IsOpen t ∧ y ∈ t | case pos
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
⊢ ∃ t, x ∉ t ∧ IsOpen t ∧ y ∈ t
case neg
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∉ (_root_.chartAt A y).source
⊢ ∃ t, x ∉ t ∧ IsOpen t ∧ y ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
⊢ ∃ t, x ∉ t ∧ IsOpen t ∧ y ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.t1Space | [16, 1] | [39, 83] | set t := (chartAt A y).source \ {x} | case pos
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
⊢ ∃ t, x ∉ t ∧ IsOpen t ∧ y ∈ t | case pos
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
⊢ ∃ t, x ∉ t ∧ IsOpen t ∧ y ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
⊢ ∃ t, x ∉ t ∧ IsOpen t ∧ y ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.t1Space | [16, 1] | [39, 83] | have e : t = (chartAt A y).source ∩ chartAt A y ⁻¹'
((chartAt A y).target \ {chartAt A y x}) := by
apply Set.ext; intro z
simp only [mem_diff, mem_singleton_iff, mem_inter_iff, mem_preimage]; constructor
intro ⟨zm, zx⟩; use zm, PartialEquiv.map_source _ zm, (PartialEquiv.injOn _).ne zm xm zx
intro ⟨zm, _, zx⟩; use zm, ((PartialEquiv.injOn _).ne_iff zm xm).mp zx | case pos
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
⊢ ∃ t, x ∉ t ∧ IsOpen t ∧ y ∈ t | case pos
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
e :
t = (_root_.chartAt A y).source ∩ ↑(_root_.chartAt A y) ⁻¹' ((_root_.chartAt A y).target \ {↑(_root_.chartAt A y) x})
⊢ ∃ t, x ∉ t ∧ IsOpen t ∧ y ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
⊢ ∃ t, x ∉ t ∧ IsOpen t ∧ y ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.t1Space | [16, 1] | [39, 83] | use t | case pos
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
e :
t = (_root_.chartAt A y).source ∩ ↑(_root_.chartAt A y) ⁻¹' ((_root_.chartAt A y).target \ {↑(_root_.chartAt A y) x})
⊢ ∃ t, x ∉ t ∧ IsOpen t ∧ y ∈ t | case h
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
e :
t = (_root_.chartAt A y).source ∩ ↑(_root_.chartAt A y) ⁻¹' ((_root_.chartAt A y).target \ {↑(_root_.chartAt A y) x})
⊢ x ∉ t ∧ IsOpen t ∧ y ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
e :
t = (_root_.chartAt A y).source ∩ ↑(_root_.chartAt A y) ⁻¹' ((_root_.chartAt A y).target \ {↑(_root_.chartAt A y) x})
⊢ ∃ t, x ∉ t ∧ IsOpen t ∧ y ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.t1Space | [16, 1] | [39, 83] | apply Set.ext | A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
⊢ t = (_root_.chartAt A y).source ∩ ↑(_root_.chartAt A y) ⁻¹' ((_root_.chartAt A y).target \ {↑(_root_.chartAt A y) x}) | case h
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
⊢ ∀ (x_1 : M),
x_1 ∈ t ↔
x_1 ∈
(_root_.chartAt A y).source ∩
↑(_root_.chartAt A y) ⁻¹' ((_root_.chartAt A y).target \ {↑(_root_.chartAt A y) x}) | Please generate a tactic in lean4 to solve the state.
STATE:
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
⊢ t = (_root_.chartAt A y).source ∩ ↑(_root_.chartAt A y) ⁻¹' ((_root_.chartAt A y).target \ {↑(_root_.chartAt A y) x})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.t1Space | [16, 1] | [39, 83] | intro z | case h
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
⊢ ∀ (x_1 : M),
x_1 ∈ t ↔
x_1 ∈
(_root_.chartAt A y).source ∩
↑(_root_.chartAt A y) ⁻¹' ((_root_.chartAt A y).target \ {↑(_root_.chartAt A y) x}) | case h
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
⊢ z ∈ t ↔
z ∈
(_root_.chartAt A y).source ∩ ↑(_root_.chartAt A y) ⁻¹' ((_root_.chartAt A y).target \ {↑(_root_.chartAt A y) x}) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
⊢ ∀ (x_1 : M),
x_1 ∈ t ↔
x_1 ∈
(_root_.chartAt A y).source ∩
↑(_root_.chartAt A y) ⁻¹' ((_root_.chartAt A y).target \ {↑(_root_.chartAt A y) x})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.t1Space | [16, 1] | [39, 83] | simp only [mem_diff, mem_singleton_iff, mem_inter_iff, mem_preimage] | case h
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
⊢ z ∈ t ↔
z ∈
(_root_.chartAt A y).source ∩ ↑(_root_.chartAt A y) ⁻¹' ((_root_.chartAt A y).target \ {↑(_root_.chartAt A y) x}) | case h
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
⊢ z ∈ t ↔
z ∈ (_root_.chartAt A y).source ∧
↑(_root_.chartAt A y) z ∈ (_root_.chartAt A y).target ∧ ¬↑(_root_.chartAt A y) z = ↑(_root_.chartAt A y) x | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
⊢ z ∈ t ↔
z ∈
(_root_.chartAt A y).source ∩ ↑(_root_.chartAt A y) ⁻¹' ((_root_.chartAt A y).target \ {↑(_root_.chartAt A y) x})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.t1Space | [16, 1] | [39, 83] | constructor | case h
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
⊢ z ∈ t ↔
z ∈ (_root_.chartAt A y).source ∧
↑(_root_.chartAt A y) z ∈ (_root_.chartAt A y).target ∧ ¬↑(_root_.chartAt A y) z = ↑(_root_.chartAt A y) x | case h.mp
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
⊢ z ∈ t →
z ∈ (_root_.chartAt A y).source ∧
↑(_root_.chartAt A y) z ∈ (_root_.chartAt A y).target ∧ ¬↑(_root_.chartAt A y) z = ↑(_root_.chartAt A y) x
case h.mpr
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
⊢ z ∈ (_root_.chartAt A y).source ∧
↑(_root_.chartAt A y) z ∈ (_root_.chartAt A y).target ∧ ¬↑(_root_.chartAt A y) z = ↑(_root_.chartAt A y) x →
z ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
⊢ z ∈ t ↔
z ∈ (_root_.chartAt A y).source ∧
↑(_root_.chartAt A y) z ∈ (_root_.chartAt A y).target ∧ ¬↑(_root_.chartAt A y) z = ↑(_root_.chartAt A y) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.t1Space | [16, 1] | [39, 83] | intro ⟨zm, zx⟩ | case h.mp
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
⊢ z ∈ t →
z ∈ (_root_.chartAt A y).source ∧
↑(_root_.chartAt A y) z ∈ (_root_.chartAt A y).target ∧ ¬↑(_root_.chartAt A y) z = ↑(_root_.chartAt A y) x
case h.mpr
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
⊢ z ∈ (_root_.chartAt A y).source ∧
↑(_root_.chartAt A y) z ∈ (_root_.chartAt A y).target ∧ ¬↑(_root_.chartAt A y) z = ↑(_root_.chartAt A y) x →
z ∈ t | case h.mp
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
zm : z ∈ (_root_.chartAt A y).source
zx : z ∉ {x}
⊢ z ∈ (_root_.chartAt A y).source ∧
↑(_root_.chartAt A y) z ∈ (_root_.chartAt A y).target ∧ ¬↑(_root_.chartAt A y) z = ↑(_root_.chartAt A y) x
case h.mpr
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
⊢ z ∈ (_root_.chartAt A y).source ∧
↑(_root_.chartAt A y) z ∈ (_root_.chartAt A y).target ∧ ¬↑(_root_.chartAt A y) z = ↑(_root_.chartAt A y) x →
z ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
⊢ z ∈ t →
z ∈ (_root_.chartAt A y).source ∧
↑(_root_.chartAt A y) z ∈ (_root_.chartAt A y).target ∧ ¬↑(_root_.chartAt A y) z = ↑(_root_.chartAt A y) x
case h.mpr
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
⊢ z ∈ (_root_.chartAt A y).source ∧
↑(_root_.chartAt A y) z ∈ (_root_.chartAt A y).target ∧ ¬↑(_root_.chartAt A y) z = ↑(_root_.chartAt A y) x →
z ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.t1Space | [16, 1] | [39, 83] | use zm, PartialEquiv.map_source _ zm, (PartialEquiv.injOn _).ne zm xm zx | case h.mp
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
zm : z ∈ (_root_.chartAt A y).source
zx : z ∉ {x}
⊢ z ∈ (_root_.chartAt A y).source ∧
↑(_root_.chartAt A y) z ∈ (_root_.chartAt A y).target ∧ ¬↑(_root_.chartAt A y) z = ↑(_root_.chartAt A y) x
case h.mpr
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
⊢ z ∈ (_root_.chartAt A y).source ∧
↑(_root_.chartAt A y) z ∈ (_root_.chartAt A y).target ∧ ¬↑(_root_.chartAt A y) z = ↑(_root_.chartAt A y) x →
z ∈ t | case h.mpr
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
⊢ z ∈ (_root_.chartAt A y).source ∧
↑(_root_.chartAt A y) z ∈ (_root_.chartAt A y).target ∧ ¬↑(_root_.chartAt A y) z = ↑(_root_.chartAt A y) x →
z ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
zm : z ∈ (_root_.chartAt A y).source
zx : z ∉ {x}
⊢ z ∈ (_root_.chartAt A y).source ∧
↑(_root_.chartAt A y) z ∈ (_root_.chartAt A y).target ∧ ¬↑(_root_.chartAt A y) z = ↑(_root_.chartAt A y) x
case h.mpr
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
⊢ z ∈ (_root_.chartAt A y).source ∧
↑(_root_.chartAt A y) z ∈ (_root_.chartAt A y).target ∧ ¬↑(_root_.chartAt A y) z = ↑(_root_.chartAt A y) x →
z ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.t1Space | [16, 1] | [39, 83] | intro ⟨zm, _, zx⟩ | case h.mpr
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
⊢ z ∈ (_root_.chartAt A y).source ∧
↑(_root_.chartAt A y) z ∈ (_root_.chartAt A y).target ∧ ¬↑(_root_.chartAt A y) z = ↑(_root_.chartAt A y) x →
z ∈ t | case h.mpr
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
zm : z ∈ (_root_.chartAt A y).source
left✝ : ↑(_root_.chartAt A y) z ∈ (_root_.chartAt A y).target
zx : ¬↑(_root_.chartAt A y) z = ↑(_root_.chartAt A y) x
⊢ z ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
⊢ z ∈ (_root_.chartAt A y).source ∧
↑(_root_.chartAt A y) z ∈ (_root_.chartAt A y).target ∧ ¬↑(_root_.chartAt A y) z = ↑(_root_.chartAt A y) x →
z ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.t1Space | [16, 1] | [39, 83] | use zm, ((PartialEquiv.injOn _).ne_iff zm xm).mp zx | case h.mpr
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
zm : z ∈ (_root_.chartAt A y).source
left✝ : ↑(_root_.chartAt A y) z ∈ (_root_.chartAt A y).target
zx : ¬↑(_root_.chartAt A y) z = ↑(_root_.chartAt A y) x
⊢ z ∈ t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
zm : z ∈ (_root_.chartAt A y).source
left✝ : ↑(_root_.chartAt A y) z ∈ (_root_.chartAt A y).target
zx : ¬↑(_root_.chartAt A y) z = ↑(_root_.chartAt A y) x
⊢ z ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.t1Space | [16, 1] | [39, 83] | use(chartAt A y).source, xm, (chartAt A y).open_source, mem_chart_source A y | case neg
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∉ (_root_.chartAt A y).source
⊢ ∃ t, x ∉ t ∧ IsOpen t ∧ y ∈ t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∉ (_root_.chartAt A y).source
⊢ ∃ t, x ∉ t ∧ IsOpen t ∧ y ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.regularSpace | [43, 1] | [62, 74] | apply RegularSpace.ofExistsMemNhdsIsClosedSubset | A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
⊢ RegularSpace M | case h
A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
⊢ ∀ (x : M), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
⊢ RegularSpace M
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.regularSpace | [43, 1] | [62, 74] | intro x s n | case h
A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
⊢ ∀ (x : M), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s | case h
A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
⊢ ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
⊢ ∀ (x : M), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.regularSpace | [43, 1] | [62, 74] | set t := (chartAt A x).target ∩ (chartAt A x).symm ⁻¹' ((chartAt A x).source ∩ s) | case h
A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
⊢ ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s | case h
A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
t : Set A := (_root_.chartAt A x).target ∩ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s)
⊢ ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
⊢ ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.regularSpace | [43, 1] | [62, 74] | have cn : (chartAt A x).source ∈ 𝓝 x :=
(chartAt A x).open_source.mem_nhds (mem_chart_source A x) | case h
A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
t : Set A := (_root_.chartAt A x).target ∩ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s)
⊢ ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s | case h
A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
t : Set A := (_root_.chartAt A x).target ∩ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s)
cn : (_root_.chartAt A x).source ∈ 𝓝 x
⊢ ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
t : Set A := (_root_.chartAt A x).target ∩ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s)
⊢ ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.regularSpace | [43, 1] | [62, 74] | have tn : t ∈ 𝓝 (chartAt A x x) := by
apply Filter.inter_mem ((chartAt A x).open_target.mem_nhds (mem_chart_target A x))
apply ((chartAt A x).continuousAt_symm (mem_chart_target _ _)).preimage_mem_nhds
rw [(chartAt A x).left_inv (mem_chart_source _ _)]; exact Filter.inter_mem cn n | case h
A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
t : Set A := (_root_.chartAt A x).target ∩ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s)
cn : (_root_.chartAt A x).source ∈ 𝓝 x
⊢ ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s | case h
A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
t : Set A := (_root_.chartAt A x).target ∩ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s)
cn : (_root_.chartAt A x).source ∈ 𝓝 x
tn : t ∈ 𝓝 (↑(_root_.chartAt A x) x)
⊢ ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
t : Set A := (_root_.chartAt A x).target ∩ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s)
cn : (_root_.chartAt A x).source ∈ 𝓝 x
⊢ ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.regularSpace | [43, 1] | [62, 74] | rcases local_compact_nhds tn with ⟨u, un, ut, uc⟩ | case h
A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
t : Set A := (_root_.chartAt A x).target ∩ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s)
cn : (_root_.chartAt A x).source ∈ 𝓝 x
tn : t ∈ 𝓝 (↑(_root_.chartAt A x) x)
⊢ ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s | case h.intro.intro.intro
A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
t : Set A := (_root_.chartAt A x).target ∩ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s)
cn : (_root_.chartAt A x).source ∈ 𝓝 x
tn : t ∈ 𝓝 (↑(_root_.chartAt A x) x)
u : Set A
un : u ∈ 𝓝 (↑(_root_.chartAt A x) x)
ut : u ⊆ t
uc : IsCompact u
⊢ ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
t : Set A := (_root_.chartAt A x).target ∩ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s)
cn : (_root_.chartAt A x).source ∈ 𝓝 x
tn : t ∈ 𝓝 (↑(_root_.chartAt A x) x)
⊢ ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.regularSpace | [43, 1] | [62, 74] | apply Filter.inter_mem ((chartAt A x).open_target.mem_nhds (mem_chart_target A x)) | A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
t : Set A := (_root_.chartAt A x).target ∩ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s)
cn : (_root_.chartAt A x).source ∈ 𝓝 x
⊢ t ∈ 𝓝 (↑(_root_.chartAt A x) x) | A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
t : Set A := (_root_.chartAt A x).target ∩ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s)
cn : (_root_.chartAt A x).source ∈ 𝓝 x
⊢ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s) ∈ 𝓝 (↑(_root_.chartAt A x) x) | Please generate a tactic in lean4 to solve the state.
STATE:
A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
t : Set A := (_root_.chartAt A x).target ∩ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s)
cn : (_root_.chartAt A x).source ∈ 𝓝 x
⊢ t ∈ 𝓝 (↑(_root_.chartAt A x) x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.regularSpace | [43, 1] | [62, 74] | apply ((chartAt A x).continuousAt_symm (mem_chart_target _ _)).preimage_mem_nhds | A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
t : Set A := (_root_.chartAt A x).target ∩ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s)
cn : (_root_.chartAt A x).source ∈ 𝓝 x
⊢ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s) ∈ 𝓝 (↑(_root_.chartAt A x) x) | A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
t : Set A := (_root_.chartAt A x).target ∩ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s)
cn : (_root_.chartAt A x).source ∈ 𝓝 x
⊢ (_root_.chartAt A x).source ∩ s ∈ 𝓝 (↑(_root_.chartAt A x).symm (↑(_root_.chartAt A x) x)) | Please generate a tactic in lean4 to solve the state.
STATE:
A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
t : Set A := (_root_.chartAt A x).target ∩ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s)
cn : (_root_.chartAt A x).source ∈ 𝓝 x
⊢ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s) ∈ 𝓝 (↑(_root_.chartAt A x) x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.regularSpace | [43, 1] | [62, 74] | rw [(chartAt A x).left_inv (mem_chart_source _ _)] | A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
t : Set A := (_root_.chartAt A x).target ∩ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s)
cn : (_root_.chartAt A x).source ∈ 𝓝 x
⊢ (_root_.chartAt A x).source ∩ s ∈ 𝓝 (↑(_root_.chartAt A x).symm (↑(_root_.chartAt A x) x)) | A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
t : Set A := (_root_.chartAt A x).target ∩ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s)
cn : (_root_.chartAt A x).source ∈ 𝓝 x
⊢ (_root_.chartAt A x).source ∩ s ∈ 𝓝 x | Please generate a tactic in lean4 to solve the state.
STATE:
A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
t : Set A := (_root_.chartAt A x).target ∩ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s)
cn : (_root_.chartAt A x).source ∈ 𝓝 x
⊢ (_root_.chartAt A x).source ∩ s ∈ 𝓝 (↑(_root_.chartAt A x).symm (↑(_root_.chartAt A x) x))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.regularSpace | [43, 1] | [62, 74] | exact Filter.inter_mem cn n | A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
t : Set A := (_root_.chartAt A x).target ∩ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s)
cn : (_root_.chartAt A x).source ∈ 𝓝 x
⊢ (_root_.chartAt A x).source ∩ s ∈ 𝓝 x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
t : Set A := (_root_.chartAt A x).target ∩ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s)
cn : (_root_.chartAt A x).source ∈ 𝓝 x
⊢ (_root_.chartAt A x).source ∩ s ∈ 𝓝 x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Mandelbrot.lean | mandelbrot_eq_multibrot | [27, 1] | [31, 6] | ext c | ⊢ mandelbrot = multibrot 2 | case h
c : ℂ
⊢ c ∈ mandelbrot ↔ c ∈ multibrot 2 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ mandelbrot = multibrot 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Mandelbrot.lean | mandelbrot_eq_multibrot | [27, 1] | [31, 6] | simp only [mandelbrot, mem_setOf_eq, multibrot, f_f'_iter, tendsto_inf_iff_tendsto_atInf,
tendsto_atInf_iff_norm_tendsto_atTop, Complex.norm_eq_abs] | case h
c : ℂ
⊢ c ∈ mandelbrot ↔ c ∈ multibrot 2 | case h
c : ℂ
⊢ ¬Tendsto (fun n => Complex.abs ((fun z => z ^ 2 + c)^[n] c)) atTop atTop ↔
¬Tendsto (fun x => Complex.abs ((f' 2 c)^[x] c)) atTop atTop | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c : ℂ
⊢ c ∈ mandelbrot ↔ c ∈ multibrot 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Mandelbrot.lean | mandelbrot_eq_multibrot | [27, 1] | [31, 6] | rfl | case h
c : ℂ
⊢ ¬Tendsto (fun n => Complex.abs ((fun z => z ^ 2 + c)^[n] c)) atTop atTop ↔
¬Tendsto (fun x => Complex.abs ((f' 2 c)^[x] c)) atTop atTop | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c : ℂ
⊢ ¬Tendsto (fun n => Complex.abs ((fun z => z ^ 2 + c)^[n] c)) atTop atTop ↔
¬Tendsto (fun x => Complex.abs ((f' 2 c)^[x] c)) atTop atTop
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Mandelbrot.lean | isConnected_mandelbrot | [34, 1] | [35, 62] | rw [mandelbrot_eq_multibrot] | ⊢ IsConnected mandelbrot | ⊢ IsConnected (multibrot 2) | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ IsConnected mandelbrot
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Mandelbrot.lean | isConnected_mandelbrot | [34, 1] | [35, 62] | exact isConnected_multibrot 2 | ⊢ IsConnected (multibrot 2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ IsConnected (multibrot 2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Mandelbrot.lean | isConnected_compl_mandelbrot | [38, 1] | [39, 68] | rw [mandelbrot_eq_multibrot] | ⊢ IsConnected mandelbrotᶜ | ⊢ IsConnected (multibrot 2)ᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ IsConnected mandelbrotᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Mandelbrot.lean | isConnected_compl_mandelbrot | [38, 1] | [39, 68] | exact isConnected_compl_multibrot 2 | ⊢ IsConnected (multibrot 2)ᶜ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ IsConnected (multibrot 2)ᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.ray_inv | [44, 1] | [49, 35] | rw [← s.ray_bij.image_eq] | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
⊢ ∃ b, HolomorphicOn (I.prod I) I (uncurry b) s.post ∧ ∀ y ∈ s.ext, b y.1 (s.ray y.1 y.2) = y.2 | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
⊢ ∃ b,
HolomorphicOn (I.prod I) I (uncurry b) ((fun y => (y.1, s.ray y.1 y.2)) '' s.ext) ∧
∀ y ∈ s.ext, b y.1 (s.ray y.1 y.2) = y.2 | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
⊢ ∃ b, HolomorphicOn (I.prod I) I (uncurry b) s.post ∧ ∀ y ∈ s.ext, b y.1 (s.ray y.1 y.2) = y.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.ray_inv | [44, 1] | [49, 35] | exact global_complex_inverse_fun_open s.ray_holomorphicOn (fun _ m ↦ s.ray_noncritical m)
s.ray_bij.injOn s.isOpen_ext | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
⊢ ∃ b,
HolomorphicOn (I.prod I) I (uncurry b) ((fun y => (y.1, s.ray y.1 y.2)) '' s.ext) ∧
∀ y ∈ s.ext, b y.1 (s.ray y.1 y.2) = y.2 | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
⊢ ∃ b,
HolomorphicOn (I.prod I) I (uncurry b) ((fun y => (y.1, s.ray y.1 y.2)) '' s.ext) ∧
∀ y ∈ s.ext, b y.1 (s.ray y.1 y.2) = y.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eq_bottcherPost | [67, 1] | [72, 22] | have h : ∃ n, (c, (f c)^[n] z) ∈ s.post := ⟨0, by simpa only [Function.iterate_zero_apply]⟩ | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∈ s.post
⊢ s.bottcher c z = s.bottcherPost c z | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∈ s.post
h : ∃ n, (c, (f c)^[n] z) ∈ s.post
⊢ s.bottcher c z = s.bottcherPost c z | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∈ s.post
⊢ s.bottcher c z = s.bottcherPost c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eq_bottcherPost | [67, 1] | [72, 22] | have h0 := (Nat.find_eq_zero h).mpr m | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∈ s.post
h : ∃ n, (c, (f c)^[n] z) ∈ s.post
⊢ s.bottcher c z = s.bottcherPost c z | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∈ s.post
h : ∃ n, (c, (f c)^[n] z) ∈ s.post
h0 : Nat.find h = 0
⊢ s.bottcher c z = s.bottcherPost c z | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∈ s.post
h : ∃ n, (c, (f c)^[n] z) ∈ s.post
⊢ s.bottcher c z = s.bottcherPost c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eq_bottcherPost | [67, 1] | [72, 22] | simp only [Super.bottcher, h, dif_pos, h0, Function.iterate_zero_apply, pow_zero, inv_one,
Complex.cpow_one] | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∈ s.post
h : ∃ n, (c, (f c)^[n] z) ∈ s.post
h0 : Nat.find h = 0
⊢ s.bottcher c z = s.bottcherPost c z | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∈ s.post
h : ∃ n, (c, (f c)^[n] z) ∈ s.post
h0 : Nat.find h = 0
⊢ s.bottcher c z = s.bottcherPost c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eq_bottcherPost | [67, 1] | [72, 22] | simpa only [Function.iterate_zero_apply] | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∈ s.post
⊢ (c, (f c)^[0] z) ∈ s.post | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∈ s.post
⊢ (c, (f c)^[0] z) ∈ s.post
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_holomorphicOn | [80, 1] | [83, 89] | intro ⟨c, z⟩ m | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
⊢ HolomorphicOn (I.prod I) I (uncurry s.bottcher) s.post | 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✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m : (c, z) ∈ s.post
⊢ HolomorphicAt (I.prod I) I (uncurry s.bottcher) (c, z) | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
⊢ HolomorphicOn (I.prod I) I (uncurry s.bottcher) s.post
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_holomorphicOn | [80, 1] | [83, 89] | apply ((choose_spec s.ray_inv).1 _ m).congr | 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✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m : (c, z) ∈ s.post
⊢ HolomorphicAt (I.prod I) I (uncurry s.bottcher) (c, z) | 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✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m : (c, z) ∈ s.post
⊢ (𝓝 (c, z)).EventuallyEq (uncurry (choose ⋯)) (uncurry s.bottcher) | 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✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m : (c, z) ∈ s.post
⊢ HolomorphicAt (I.prod I) I (uncurry s.bottcher) (c, z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_holomorphicOn | [80, 1] | [83, 89] | exact s.eqOn_bottcher_bottcherPost.symm.eventuallyEq_of_mem (s.isOpen_post.mem_nhds m) | 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✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m : (c, z) ∈ s.post
⊢ (𝓝 (c, z)).EventuallyEq (uncurry (choose ⋯)) (uncurry s.bottcher) | 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✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m : (c, z) ∈ s.post
⊢ (𝓝 (c, z)).EventuallyEq (uncurry (choose ⋯)) (uncurry s.bottcher)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_ray | [86, 1] | [88, 86] | rw [s.bottcher_eq_bottcherPost (s.ray_post m)] | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, x) ∈ s.ext
⊢ s.bottcher c (s.ray c x) = 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, x) ∈ s.ext
⊢ s.bottcherPost c (s.ray c x) = 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, x) ∈ s.ext
⊢ s.bottcher c (s.ray c x) = x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_ray | [86, 1] | [88, 86] | exact (choose_spec s.ray_inv).2 _ m | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, x) ∈ s.ext
⊢ s.bottcherPost c (s.ray c x) = x | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, x) ∈ s.ext
⊢ s.bottcherPost c (s.ray c x) = x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.ray_bottcher | [91, 1] | [93, 65] | rcases s.ray_surj m with ⟨x, m, e⟩ | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∈ s.post
⊢ s.ray c (s.bottcher c z) = z | case 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 x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m✝ : (c, z) ∈ s.post
x : ℂ
m : (c, x) ∈ s.ext
e : s.ray c x = z
⊢ s.ray c (s.bottcher c z) = z | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∈ s.post
⊢ s.ray c (s.bottcher c z) = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.ray_bottcher | [91, 1] | [93, 65] | rw [← e, s.bottcher_ray m] | case 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 x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m✝ : (c, z) ∈ s.post
x : ℂ
m : (c, x) ∈ s.ext
e : s.ray c x = z
⊢ s.ray c (s.bottcher c z) = z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case 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 x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m✝ : (c, z) ∈ s.post
x : ℂ
m : (c, x) ∈ s.ext
e : s.ray c x = z
⊢ s.ray c (s.bottcher c z) = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_ext | [96, 1] | [98, 74] | rcases s.ray_surj m with ⟨x, m, e⟩ | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∈ s.post
⊢ (c, s.bottcher c z) ∈ s.ext | case 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 x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m✝ : (c, z) ∈ s.post
x : ℂ
m : (c, x) ∈ s.ext
e : s.ray c x = z
⊢ (c, s.bottcher c z) ∈ s.ext | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∈ s.post
⊢ (c, s.bottcher c z) ∈ s.ext
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_ext | [96, 1] | [98, 74] | rw [← e, s.bottcher_ray m] | case 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 x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m✝ : (c, z) ∈ s.post
x : ℂ
m : (c, x) ∈ s.ext
e : s.ray c x = z
⊢ (c, s.bottcher c z) ∈ s.ext | case 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 x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m✝ : (c, z) ∈ s.post
x : ℂ
m : (c, x) ∈ s.ext
e : s.ray c x = z
⊢ (c, x) ∈ s.ext | Please generate a tactic in lean4 to solve the state.
STATE:
case 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 x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m✝ : (c, z) ∈ s.post
x : ℂ
m : (c, x) ∈ s.ext
e : s.ray c x = z
⊢ (c, s.bottcher c z) ∈ s.ext
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_ext | [96, 1] | [98, 74] | exact m | case 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 x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m✝ : (c, z) ∈ s.post
x : ℂ
m : (c, x) ∈ s.ext
e : s.ray c x = z
⊢ (c, x) ∈ s.ext | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case 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 x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m✝ : (c, z) ∈ s.post
x : ℂ
m : (c, x) ∈ s.ext
e : s.ray c x = z
⊢ (c, x) ∈ s.ext
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eq_bottcherNear | [101, 1] | [108, 45] | have eq := (s.ray_nontrivial (s.mem_ext c)).nhds_eq_map_nhds | 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✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
⊢ ∀ᶠ (z : S) in 𝓝 a, s.bottcher c z = s.bottcherNear c z | 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✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
eq : 𝓝 (s.ray c 0) = Filter.map (s.ray c) (𝓝 0)
⊢ ∀ᶠ (z : S) in 𝓝 a, s.bottcher c z = s.bottcherNear c z | 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✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
⊢ ∀ᶠ (z : S) in 𝓝 a, s.bottcher c z = s.bottcherNear c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eq_bottcherNear | [101, 1] | [108, 45] | simp only [s.ray_zero] at eq | 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✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
eq : 𝓝 (s.ray c 0) = Filter.map (s.ray c) (𝓝 0)
⊢ ∀ᶠ (z : S) in 𝓝 a, s.bottcher c z = s.bottcherNear c z | 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✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
eq : 𝓝 a = Filter.map (s.ray c) (𝓝 0)
⊢ ∀ᶠ (z : S) in 𝓝 a, s.bottcher c z = s.bottcherNear c z | 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✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
eq : 𝓝 (s.ray c 0) = Filter.map (s.ray c) (𝓝 0)
⊢ ∀ᶠ (z : S) in 𝓝 a, s.bottcher c z = s.bottcherNear c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eq_bottcherNear | [101, 1] | [108, 45] | simp only [eq, Filter.eventually_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✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
eq : 𝓝 a = Filter.map (s.ray c) (𝓝 0)
⊢ ∀ᶠ (z : S) in 𝓝 a, s.bottcher c z = s.bottcherNear c z | 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✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
eq : 𝓝 a = Filter.map (s.ray c) (𝓝 0)
⊢ ∀ᶠ (a_1 : ℂ) in 𝓝 0, s.bottcher c (s.ray c a_1) = s.bottcherNear c (s.ray c a_1) | 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✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
eq : 𝓝 a = Filter.map (s.ray c) (𝓝 0)
⊢ ∀ᶠ (z : S) in 𝓝 a, s.bottcher c z = s.bottcherNear c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eq_bottcherNear | [101, 1] | [108, 45] | apply ((continuousAt_const.prod continuousAt_id).eventually (s.ray_eqn_zero c)).mp | 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✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
eq : 𝓝 a = Filter.map (s.ray c) (𝓝 0)
⊢ ∀ᶠ (a_1 : ℂ) in 𝓝 0, s.bottcher c (s.ray c a_1) = s.bottcherNear c (s.ray c a_1) | 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✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
eq : 𝓝 a = Filter.map (s.ray c) (𝓝 0)
⊢ ∀ᶠ (x : ℂ) in 𝓝 0,
s.bottcherNear (c, id x).1 (s.ray (c, id x).1 (c, id x).2) = (c, id x).2 →
s.bottcher c (s.ray c x) = s.bottcherNear c (s.ray c 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✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
eq : 𝓝 a = Filter.map (s.ray c) (𝓝 0)
⊢ ∀ᶠ (a_1 : ℂ) in 𝓝 0, s.bottcher c (s.ray c a_1) = s.bottcherNear c (s.ray c a_1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eq_bottcherNear | [101, 1] | [108, 45] | refine ((s.isOpen_ext.snd_preimage c).eventually_mem (s.mem_ext c)).mp
(eventually_of_forall fun z m e ↦ ?_) | 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✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
eq : 𝓝 a = Filter.map (s.ray c) (𝓝 0)
⊢ ∀ᶠ (x : ℂ) in 𝓝 0,
s.bottcherNear (c, id x).1 (s.ray (c, id x).1 (c, id x).2) = (c, id x).2 →
s.bottcher c (s.ray c x) = s.bottcherNear c (s.ray c 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✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
eq : 𝓝 a = Filter.map (s.ray c) (𝓝 0)
z : ℂ
m : z ∈ {b | (c, b) ∈ s.ext}
e : s.bottcherNear (c, id z).1 (s.ray (c, id z).1 (c, id z).2) = (c, id z).2
⊢ s.bottcher c (s.ray c z) = s.bottcherNear c (s.ray c z) | 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✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
eq : 𝓝 a = Filter.map (s.ray c) (𝓝 0)
⊢ ∀ᶠ (x : ℂ) in 𝓝 0,
s.bottcherNear (c, id x).1 (s.ray (c, id x).1 (c, id x).2) = (c, id x).2 →
s.bottcher c (s.ray c x) = s.bottcherNear c (s.ray c x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eq_bottcherNear | [101, 1] | [108, 45] | simp only [s.bottcher_ray m] | 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✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
eq : 𝓝 a = Filter.map (s.ray c) (𝓝 0)
z : ℂ
m : z ∈ {b | (c, b) ∈ s.ext}
e : s.bottcherNear (c, id z).1 (s.ray (c, id z).1 (c, id z).2) = (c, id z).2
⊢ s.bottcher c (s.ray c z) = s.bottcherNear c (s.ray c z) | 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✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
eq : 𝓝 a = Filter.map (s.ray c) (𝓝 0)
z : ℂ
m : z ∈ {b | (c, b) ∈ s.ext}
e : s.bottcherNear (c, id z).1 (s.ray (c, id z).1 (c, id z).2) = (c, id z).2
⊢ z = s.bottcherNear c (s.ray c z) | 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✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
eq : 𝓝 a = Filter.map (s.ray c) (𝓝 0)
z : ℂ
m : z ∈ {b | (c, b) ∈ s.ext}
e : s.bottcherNear (c, id z).1 (s.ray (c, id z).1 (c, id z).2) = (c, id z).2
⊢ s.bottcher c (s.ray c z) = s.bottcherNear c (s.ray c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eq_bottcherNear | [101, 1] | [108, 45] | exact e.symm | 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✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
eq : 𝓝 a = Filter.map (s.ray c) (𝓝 0)
z : ℂ
m : z ∈ {b | (c, b) ∈ s.ext}
e : s.bottcherNear (c, id z).1 (s.ray (c, id z).1 (c, id z).2) = (c, id z).2
⊢ z = s.bottcherNear c (s.ray c z) | 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✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
eq : 𝓝 a = Filter.map (s.ray c) (𝓝 0)
z : ℂ
m : z ∈ {b | (c, b) ∈ s.ext}
e : s.bottcherNear (c, id z).1 (s.ray (c, id z).1 (c, id z).2) = (c, id z).2
⊢ z = s.bottcherNear c (s.ray c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.post_connected | [149, 1] | [151, 66] | have e : s.post = s.homeomorph '' s.ext := s.homeomorph.image_source_eq_target.symm | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
⊢ IsConnected s.post | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
e : s.post = ↑s.homeomorph '' s.ext
⊢ IsConnected s.post | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
⊢ IsConnected s.post
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.post_connected | [149, 1] | [151, 66] | rw [e] | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
e : s.post = ↑s.homeomorph '' s.ext
⊢ IsConnected s.post | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
e : s.post = ↑s.homeomorph '' s.ext
⊢ IsConnected (↑s.homeomorph '' s.ext) | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
e : s.post = ↑s.homeomorph '' s.ext
⊢ IsConnected s.post
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.post_connected | [149, 1] | [151, 66] | exact s.ext_connected.image _ s.homeomorph.continuousOn | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
e : s.post = ↑s.homeomorph '' s.ext
⊢ IsConnected (↑s.homeomorph '' s.ext) | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
e : s.post = ↑s.homeomorph '' s.ext
⊢ IsConnected (↑s.homeomorph '' s.ext)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.post_slice_connected | [154, 1] | [158, 85] | have e : {z | (c, z) ∈ s.post} = s.homeomorphSlice c '' {x | (c, x) ∈ s.ext} :=
(s.homeomorphSlice c).image_source_eq_target.symm | 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✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
⊢ IsConnected {z | (c, z) ∈ s.post} | 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✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
e : {z | (c, z) ∈ s.post} = ↑(s.homeomorphSlice c) '' {x | (c, x) ∈ s.ext}
⊢ IsConnected {z | (c, z) ∈ s.post} | 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✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
⊢ IsConnected {z | (c, z) ∈ s.post}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.post_slice_connected | [154, 1] | [158, 85] | rw [e] | 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✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
e : {z | (c, z) ∈ s.post} = ↑(s.homeomorphSlice c) '' {x | (c, x) ∈ s.ext}
⊢ IsConnected {z | (c, z) ∈ s.post} | 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✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
e : {z | (c, z) ∈ s.post} = ↑(s.homeomorphSlice c) '' {x | (c, x) ∈ s.ext}
⊢ IsConnected (↑(s.homeomorphSlice c) '' {x | (c, x) ∈ s.ext}) | 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✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
e : {z | (c, z) ∈ s.post} = ↑(s.homeomorphSlice c) '' {x | (c, x) ∈ s.ext}
⊢ IsConnected {z | (c, z) ∈ s.post}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.post_slice_connected | [154, 1] | [158, 85] | exact (s.ext_slice_connected c).image _ (s.homeomorphSlice c).continuousOn | 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✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
e : {z | (c, z) ∈ s.post} = ↑(s.homeomorphSlice c) '' {x | (c, x) ∈ s.ext}
⊢ IsConnected (↑(s.homeomorphSlice c) '' {x | (c, x) ∈ s.ext}) | 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✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
e : {z | (c, z) ∈ s.post} = ↑(s.homeomorphSlice c) '' {x | (c, x) ∈ s.ext}
⊢ IsConnected (↑(s.homeomorphSlice c) '' {x | (c, x) ∈ s.ext})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_not_basin | [161, 1] | [167, 63] | have p : ¬∃ n, (c, (f c)^[n] z) ∈ s.post := by
contrapose m; simp only [not_not] at m ⊢; rcases m with ⟨n, m⟩
rcases s.post_basin m with ⟨k, m⟩
simp only [← Function.iterate_add_apply] at m; exact ⟨k + n, m⟩ | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∉ s.basin
⊢ s.bottcher c z = 1 | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∉ s.basin
p : ¬∃ n, (c, (f c)^[n] z) ∈ s.post
⊢ s.bottcher c z = 1 | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∉ s.basin
⊢ s.bottcher c z = 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_not_basin | [161, 1] | [167, 63] | simp only [Super.bottcher, p] | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∉ s.basin
p : ¬∃ n, (c, (f c)^[n] z) ∈ s.post
⊢ s.bottcher c z = 1 | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∉ s.basin
p : ¬∃ n, (c, (f c)^[n] z) ∈ s.post
⊢ (if h : False then (fun w => w ^ (↑d)⁻¹)^[Nat.find ⋯] (s.bottcherPost c ((f c)^[Nat.find ⋯] z)) else 1) = 1 | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∉ s.basin
p : ¬∃ n, (c, (f c)^[n] z) ∈ s.post
⊢ s.bottcher c z = 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_not_basin | [161, 1] | [167, 63] | rw [dif_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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∉ s.basin
p : ¬∃ n, (c, (f c)^[n] z) ∈ s.post
⊢ (if h : False then (fun w => w ^ (↑d)⁻¹)^[Nat.find ⋯] (s.bottcherPost c ((f c)^[Nat.find ⋯] z)) else 1) = 1 | case hnc
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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∉ s.basin
p : ¬∃ n, (c, (f c)^[n] z) ∈ s.post
⊢ ¬False | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∉ s.basin
p : ¬∃ n, (c, (f c)^[n] z) ∈ s.post
⊢ (if h : False then (fun w => w ^ (↑d)⁻¹)^[Nat.find ⋯] (s.bottcherPost c ((f c)^[Nat.find ⋯] z)) else 1) = 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_not_basin | [161, 1] | [167, 63] | exact not_false | case hnc
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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∉ s.basin
p : ¬∃ n, (c, (f c)^[n] z) ∈ s.post
⊢ ¬False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hnc
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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∉ s.basin
p : ¬∃ n, (c, (f c)^[n] z) ∈ s.post
⊢ ¬False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_not_basin | [161, 1] | [167, 63] | contrapose m | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∉ s.basin
⊢ ¬∃ n, (c, (f c)^[n] z) ∈ s.post | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : ¬¬∃ n, (c, (f c)^[n] z) ∈ s.post
⊢ ¬(c, z) ∉ s.basin | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∉ s.basin
⊢ ¬∃ n, (c, (f c)^[n] z) ∈ s.post
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_not_basin | [161, 1] | [167, 63] | simp only [not_not] at m ⊢ | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : ¬¬∃ n, (c, (f c)^[n] z) ∈ s.post
⊢ ¬(c, z) ∉ s.basin | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : ∃ n, (c, (f c)^[n] z) ∈ s.post
⊢ (c, z) ∈ s.basin | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : ¬¬∃ n, (c, (f c)^[n] z) ∈ s.post
⊢ ¬(c, z) ∉ s.basin
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_not_basin | [161, 1] | [167, 63] | rcases m with ⟨n, m⟩ | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : ∃ n, (c, (f c)^[n] z) ∈ s.post
⊢ (c, z) ∈ s.basin | case 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 x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
n : ℕ
m : (c, (f c)^[n] z) ∈ s.post
⊢ (c, z) ∈ s.basin | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : ∃ n, (c, (f c)^[n] z) ∈ s.post
⊢ (c, z) ∈ s.basin
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_not_basin | [161, 1] | [167, 63] | rcases s.post_basin m with ⟨k, m⟩ | case 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 x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
n : ℕ
m : (c, (f c)^[n] z) ∈ s.post
⊢ (c, z) ∈ s.basin | case 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 x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
n : ℕ
m✝ : (c, (f c)^[n] z) ∈ s.post
k : ℕ
m : ((c, (f c)^[n] z).1, (f (c, (f c)^[n] z).1)^[k] (c, (f c)^[n] z).2) ∈ s.near
⊢ (c, z) ∈ s.basin | Please generate a tactic in lean4 to solve the state.
STATE:
case 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 x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
n : ℕ
m : (c, (f c)^[n] z) ∈ s.post
⊢ (c, z) ∈ s.basin
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_not_basin | [161, 1] | [167, 63] | simp only [← Function.iterate_add_apply] at m | case 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 x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
n : ℕ
m✝ : (c, (f c)^[n] z) ∈ s.post
k : ℕ
m : ((c, (f c)^[n] z).1, (f (c, (f c)^[n] z).1)^[k] (c, (f c)^[n] z).2) ∈ s.near
⊢ (c, z) ∈ s.basin | case 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 x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
n : ℕ
m✝ : (c, (f c)^[n] z) ∈ s.post
k : ℕ
m : (c, (f c)^[k + n] z) ∈ s.near
⊢ (c, z) ∈ s.basin | Please generate a tactic in lean4 to solve the state.
STATE:
case 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 x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
n : ℕ
m✝ : (c, (f c)^[n] z) ∈ s.post
k : ℕ
m : ((c, (f c)^[n] z).1, (f (c, (f c)^[n] z).1)^[k] (c, (f c)^[n] z).2) ∈ s.near
⊢ (c, z) ∈ s.basin
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_not_basin | [161, 1] | [167, 63] | exact ⟨k + n, m⟩ | case 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 x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
n : ℕ
m✝ : (c, (f c)^[n] z) ∈ s.post
k : ℕ
m : (c, (f c)^[k + n] z) ∈ s.near
⊢ (c, z) ∈ s.basin | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case 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 x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
n : ℕ
m✝ : (c, (f c)^[n] z) ∈ s.post
k : ℕ
m : (c, (f c)^[k + n] z) ∈ s.near
⊢ (c, z) ∈ s.basin
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | have h0 : ∀ {c z}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d := by
intro c z m
suffices e : ∀ᶠ w in 𝓝 a, s.bottcher c (f c w) = s.bottcher c w ^ d by
refine (HolomorphicOn.eq_of_locally_eq ?_ (fun z m ↦
(s.bottcher_holomorphicOn (c, z) m).along_snd.pow) (s.post_slice_connected c).isPreconnected
⟨a, s.post_a c, e⟩).self_of_nhdsSet m
intro z m
exact (s.bottcher_holomorphicOn _ (s.stays_post m)).along_snd.comp (s.fa _).along_snd
have e := s.bottcher_eq_bottcherNear c
have fc := (s.fa (c, a)).along_snd.continuousAt; simp only [ContinuousAt, s.f0] at fc
apply e.mp; apply (fc.eventually e).mp
apply ((s.isOpen_near.snd_preimage c).eventually_mem (s.mem_near c)).mp
refine eventually_of_forall fun w m e0 e1 ↦ ?_; simp only at m e0 e1
simp only [e0, e1]; exact s.bottcherNear_eqn m | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
⊢ s.bottcher c (f c z) = s.bottcher c z ^ d | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
⊢ s.bottcher c (f c z) = s.bottcher c z ^ d | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
⊢ s.bottcher c (f c z) = s.bottcher c z ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | by_cases p : (c, z) ∈ s.post | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
⊢ s.bottcher c (f c z) = s.bottcher c z ^ d | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∈ s.post
⊢ s.bottcher c (f c z) = s.bottcher c z ^ d
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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
⊢ s.bottcher c (f c z) = s.bottcher c z ^ d | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
⊢ s.bottcher c (f c z) = s.bottcher c z ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | simp only [h0 p] | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∈ s.post
⊢ s.bottcher c (f c z) = s.bottcher c z ^ d
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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
⊢ s.bottcher c (f c z) = s.bottcher c z ^ d | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
⊢ s.bottcher c (f c z) = s.bottcher c z ^ d | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∈ s.post
⊢ s.bottcher c (f c z) = s.bottcher c z ^ d
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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
⊢ s.bottcher c (f c z) = s.bottcher c z ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | by_cases m : (c, z) ∈ s.basin | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
⊢ s.bottcher c (f c z) = s.bottcher c z ^ d | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
⊢ s.bottcher c (f c z) = s.bottcher c z ^ d
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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∉ s.basin
⊢ s.bottcher c (f c z) = s.bottcher c z ^ d | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
⊢ s.bottcher c (f c z) = s.bottcher c z ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | have m1 : (c, f c z) ∉ s.basin := by
contrapose m; simp only [not_not] at m ⊢
rcases m with ⟨n, m⟩; use n + 1; simp only at m ⊢; rwa [Function.iterate_succ_apply] | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∉ s.basin
⊢ s.bottcher c (f c z) = s.bottcher c z ^ d | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∉ s.basin
m1 : (c, f c z) ∉ s.basin
⊢ s.bottcher c (f c z) = s.bottcher c z ^ d | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∉ s.basin
⊢ s.bottcher c (f c z) = s.bottcher c z ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | simp only [s.bottcher_not_basin m, s.bottcher_not_basin m1, one_pow] | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∉ s.basin
m1 : (c, f c z) ∉ s.basin
⊢ s.bottcher c (f c z) = s.bottcher c z ^ d | no goals | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∉ s.basin
m1 : (c, f c z) ∉ s.basin
⊢ s.bottcher c (f c z) = s.bottcher c z ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | intro c z m | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
⊢ ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d | 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✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m : (c, z) ∈ s.post
⊢ s.bottcher c (f c z) = s.bottcher c z ^ d | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
⊢ ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | suffices e : ∀ᶠ w in 𝓝 a, s.bottcher c (f c w) = s.bottcher c w ^ d by
refine (HolomorphicOn.eq_of_locally_eq ?_ (fun z m ↦
(s.bottcher_holomorphicOn (c, z) m).along_snd.pow) (s.post_slice_connected c).isPreconnected
⟨a, s.post_a c, e⟩).self_of_nhdsSet m
intro z m
exact (s.bottcher_holomorphicOn _ (s.stays_post m)).along_snd.comp (s.fa _).along_snd | 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✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m : (c, z) ∈ s.post
⊢ s.bottcher c (f c z) = s.bottcher c z ^ d | 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✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m : (c, z) ∈ s.post
⊢ ∀ᶠ (w : S) in 𝓝 a, s.bottcher c (f c w) = s.bottcher c w ^ d | 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✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m : (c, z) ∈ s.post
⊢ s.bottcher c (f c z) = s.bottcher c z ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | have e := s.bottcher_eq_bottcherNear c | 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✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m : (c, z) ∈ s.post
⊢ ∀ᶠ (w : S) in 𝓝 a, s.bottcher c (f c w) = s.bottcher c w ^ d | 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✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m : (c, z) ∈ s.post
e : ∀ᶠ (z : S) in 𝓝 a, s.bottcher c z = s.bottcherNear c z
⊢ ∀ᶠ (w : S) in 𝓝 a, s.bottcher c (f c w) = s.bottcher c w ^ d | 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✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m : (c, z) ∈ s.post
⊢ ∀ᶠ (w : S) in 𝓝 a, s.bottcher c (f c w) = s.bottcher c w ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | have fc := (s.fa (c, a)).along_snd.continuousAt | 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✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m : (c, z) ∈ s.post
e : ∀ᶠ (z : S) in 𝓝 a, s.bottcher c z = s.bottcherNear c z
⊢ ∀ᶠ (w : S) in 𝓝 a, s.bottcher c (f c w) = s.bottcher c w ^ d | 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✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m : (c, z) ∈ s.post
e : ∀ᶠ (z : S) in 𝓝 a, s.bottcher c z = s.bottcherNear c z
fc : ContinuousAt (fun y => f c y) a
⊢ ∀ᶠ (w : S) in 𝓝 a, s.bottcher c (f c w) = s.bottcher c w ^ d | 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✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m : (c, z) ∈ s.post
e : ∀ᶠ (z : S) in 𝓝 a, s.bottcher c z = s.bottcherNear c z
⊢ ∀ᶠ (w : S) in 𝓝 a, s.bottcher c (f c w) = s.bottcher c w ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | simp only [ContinuousAt, s.f0] at fc | 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✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m : (c, z) ∈ s.post
e : ∀ᶠ (z : S) in 𝓝 a, s.bottcher c z = s.bottcherNear c z
fc : ContinuousAt (fun y => f c y) a
⊢ ∀ᶠ (w : S) in 𝓝 a, s.bottcher c (f c w) = s.bottcher c w ^ d | 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✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m : (c, z) ∈ s.post
e : ∀ᶠ (z : S) in 𝓝 a, s.bottcher c z = s.bottcherNear c z
fc : Tendsto (fun y => f c y) (𝓝 a) (𝓝 a)
⊢ ∀ᶠ (w : S) in 𝓝 a, s.bottcher c (f c w) = s.bottcher c w ^ d | 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✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m : (c, z) ∈ s.post
e : ∀ᶠ (z : S) in 𝓝 a, s.bottcher c z = s.bottcherNear c z
fc : ContinuousAt (fun y => f c y) a
⊢ ∀ᶠ (w : S) in 𝓝 a, s.bottcher c (f c w) = s.bottcher c w ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | apply e.mp | 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✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m : (c, z) ∈ s.post
e : ∀ᶠ (z : S) in 𝓝 a, s.bottcher c z = s.bottcherNear c z
fc : Tendsto (fun y => f c y) (𝓝 a) (𝓝 a)
⊢ ∀ᶠ (w : S) in 𝓝 a, s.bottcher c (f c w) = s.bottcher c w ^ d | 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✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m : (c, z) ∈ s.post
e : ∀ᶠ (z : S) in 𝓝 a, s.bottcher c z = s.bottcherNear c z
fc : Tendsto (fun y => f c y) (𝓝 a) (𝓝 a)
⊢ ∀ᶠ (x : S) in 𝓝 a, s.bottcher c x = s.bottcherNear c x → s.bottcher c (f c x) = s.bottcher c x ^ d | 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✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m : (c, z) ∈ s.post
e : ∀ᶠ (z : S) in 𝓝 a, s.bottcher c z = s.bottcherNear c z
fc : Tendsto (fun y => f c y) (𝓝 a) (𝓝 a)
⊢ ∀ᶠ (w : S) in 𝓝 a, s.bottcher c (f c w) = s.bottcher c w ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | apply (fc.eventually e).mp | 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✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m : (c, z) ∈ s.post
e : ∀ᶠ (z : S) in 𝓝 a, s.bottcher c z = s.bottcherNear c z
fc : Tendsto (fun y => f c y) (𝓝 a) (𝓝 a)
⊢ ∀ᶠ (x : S) in 𝓝 a, s.bottcher c x = s.bottcherNear c x → s.bottcher c (f c x) = s.bottcher c x ^ d | 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✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m : (c, z) ∈ s.post
e : ∀ᶠ (z : S) in 𝓝 a, s.bottcher c z = s.bottcherNear c z
fc : Tendsto (fun y => f c y) (𝓝 a) (𝓝 a)
⊢ ∀ᶠ (x : S) in 𝓝 a,
s.bottcher c (f c x) = s.bottcherNear c (f c x) →
s.bottcher c x = s.bottcherNear c x → s.bottcher c (f c x) = s.bottcher c x ^ d | 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✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m : (c, z) ∈ s.post
e : ∀ᶠ (z : S) in 𝓝 a, s.bottcher c z = s.bottcherNear c z
fc : Tendsto (fun y => f c y) (𝓝 a) (𝓝 a)
⊢ ∀ᶠ (x : S) in 𝓝 a, s.bottcher c x = s.bottcherNear c x → s.bottcher c (f c x) = s.bottcher c x ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | apply ((s.isOpen_near.snd_preimage c).eventually_mem (s.mem_near c)).mp | 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✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m : (c, z) ∈ s.post
e : ∀ᶠ (z : S) in 𝓝 a, s.bottcher c z = s.bottcherNear c z
fc : Tendsto (fun y => f c y) (𝓝 a) (𝓝 a)
⊢ ∀ᶠ (x : S) in 𝓝 a,
s.bottcher c (f c x) = s.bottcherNear c (f c x) →
s.bottcher c x = s.bottcherNear c x → s.bottcher c (f c x) = s.bottcher c x ^ d | 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✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m : (c, z) ∈ s.post
e : ∀ᶠ (z : S) in 𝓝 a, s.bottcher c z = s.bottcherNear c z
fc : Tendsto (fun y => f c y) (𝓝 a) (𝓝 a)
⊢ ∀ᶠ (x : S) in 𝓝 a,
x ∈ {b | (c, b) ∈ s.near} →
s.bottcher c (f c x) = s.bottcherNear c (f c x) →
s.bottcher c x = s.bottcherNear c x → s.bottcher c (f c x) = s.bottcher c x ^ d | 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✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m : (c, z) ∈ s.post
e : ∀ᶠ (z : S) in 𝓝 a, s.bottcher c z = s.bottcherNear c z
fc : Tendsto (fun y => f c y) (𝓝 a) (𝓝 a)
⊢ ∀ᶠ (x : S) in 𝓝 a,
s.bottcher c (f c x) = s.bottcherNear c (f c x) →
s.bottcher c x = s.bottcherNear c x → s.bottcher c (f c x) = s.bottcher c x ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | refine eventually_of_forall fun w m e0 e1 ↦ ?_ | 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✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m : (c, z) ∈ s.post
e : ∀ᶠ (z : S) in 𝓝 a, s.bottcher c z = s.bottcherNear c z
fc : Tendsto (fun y => f c y) (𝓝 a) (𝓝 a)
⊢ ∀ᶠ (x : S) in 𝓝 a,
x ∈ {b | (c, b) ∈ s.near} →
s.bottcher c (f c x) = s.bottcherNear c (f c x) →
s.bottcher c x = s.bottcherNear c x → s.bottcher c (f c x) = s.bottcher c x ^ d | 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✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m✝ : (c, z) ∈ s.post
e : ∀ᶠ (z : S) in 𝓝 a, s.bottcher c z = s.bottcherNear c z
fc : Tendsto (fun y => f c y) (𝓝 a) (𝓝 a)
w : S
m : w ∈ {b | (c, b) ∈ s.near}
e0 : s.bottcher c (f c w) = s.bottcherNear c (f c w)
e1 : s.bottcher c w = s.bottcherNear c w
⊢ s.bottcher c (f c w) = s.bottcher c w ^ d | 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✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m : (c, z) ∈ s.post
e : ∀ᶠ (z : S) in 𝓝 a, s.bottcher c z = s.bottcherNear c z
fc : Tendsto (fun y => f c y) (𝓝 a) (𝓝 a)
⊢ ∀ᶠ (x : S) in 𝓝 a,
x ∈ {b | (c, b) ∈ s.near} →
s.bottcher c (f c x) = s.bottcherNear c (f c x) →
s.bottcher c x = s.bottcherNear c x → s.bottcher c (f c x) = s.bottcher c x ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | simp only [e0, e1] | 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✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m✝ : (c, z) ∈ s.post
e : ∀ᶠ (z : S) in 𝓝 a, s.bottcher c z = s.bottcherNear c z
fc : Tendsto (fun y => f c y) (𝓝 a) (𝓝 a)
w : S
m : w ∈ {b | (c, b) ∈ s.near}
e0 : s.bottcher c (f c w) = s.bottcherNear c (f c w)
e1 : s.bottcher c w = s.bottcherNear c w
⊢ s.bottcher c (f c w) = s.bottcher c w ^ d | 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✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m✝ : (c, z) ∈ s.post
e : ∀ᶠ (z : S) in 𝓝 a, s.bottcher c z = s.bottcherNear c z
fc : Tendsto (fun y => f c y) (𝓝 a) (𝓝 a)
w : S
m : w ∈ {b | (c, b) ∈ s.near}
e0 : s.bottcher c (f c w) = s.bottcherNear c (f c w)
e1 : s.bottcher c w = s.bottcherNear c w
⊢ s.bottcherNear c (f c w) = s.bottcherNear c w ^ d | 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✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m✝ : (c, z) ∈ s.post
e : ∀ᶠ (z : S) in 𝓝 a, s.bottcher c z = s.bottcherNear c z
fc : Tendsto (fun y => f c y) (𝓝 a) (𝓝 a)
w : S
m : w ∈ {b | (c, b) ∈ s.near}
e0 : s.bottcher c (f c w) = s.bottcherNear c (f c w)
e1 : s.bottcher c w = s.bottcherNear c w
⊢ s.bottcher c (f c w) = s.bottcher c w ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | exact s.bottcherNear_eqn m | 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✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m✝ : (c, z) ∈ s.post
e : ∀ᶠ (z : S) in 𝓝 a, s.bottcher c z = s.bottcherNear c z
fc : Tendsto (fun y => f c y) (𝓝 a) (𝓝 a)
w : S
m : w ∈ {b | (c, b) ∈ s.near}
e0 : s.bottcher c (f c w) = s.bottcherNear c (f c w)
e1 : s.bottcher c w = s.bottcherNear c w
⊢ s.bottcherNear c (f c w) = s.bottcherNear c w ^ d | 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✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m✝ : (c, z) ∈ s.post
e : ∀ᶠ (z : S) in 𝓝 a, s.bottcher c z = s.bottcherNear c z
fc : Tendsto (fun y => f c y) (𝓝 a) (𝓝 a)
w : S
m : w ∈ {b | (c, b) ∈ s.near}
e0 : s.bottcher c (f c w) = s.bottcherNear c (f c w)
e1 : s.bottcher c w = s.bottcherNear c w
⊢ s.bottcherNear c (f c w) = s.bottcherNear c w ^ d
TACTIC:
|
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