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 |
|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | rw [isClosed_iff_frequently] | case left
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
⊢ IsClosed t | case left
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
⊢ ∀ (x : 𝕊), (∃ᶠ (y : 𝕊) in 𝓝 x, y ∈ t) → x ∈ t |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | intro x e | case left
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
⊢ ∀ (x : 𝕊), (∃ᶠ (y : 𝕊) in 𝓝 x, y ∈ t) → x ∈ t | case left
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
x : 𝕊
e : ∃ᶠ (y : 𝕊) in 𝓝 x, y ∈ t
⊢ x ∈ t |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | by_contra xt | case left
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
x : 𝕊
e : ∃ᶠ (y : 𝕊) in 𝓝 x, y ∈ t
⊢ x ∈ t | case left
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
x : 𝕊
e : ∃ᶠ (y : 𝕊) in 𝓝 x, y ∈ t
xt : x ∉ t
⊢ False |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | have pb : potential d x = abs b := by
apply tendsto_nhds_unique_of_frequently_eq potential_continuous.continuousAt
continuousAt_const
refine e.mp (eventually_of_forall ?_); intro z ⟨_, h⟩; rw [← h.self_of_nhds, abs_bottcher] | case left
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
x : 𝕊
e : ∃ᶠ (y : 𝕊) in 𝓝 x, y ∈ t
xt : x ∉ t
⊢ False | case left
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
x : 𝕊
e : ∃ᶠ (y : 𝕊) in 𝓝 x, y ∈ t
xt : x ∉ t
pb : potential d x = Complex.ab... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | rw [← pb, potential_lt_one] at b1 | case left
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
x : 𝕊
e : ∃ᶠ (y : 𝕊) in 𝓝 x, y ∈ t
xt : x ∉ t
pb : potential d x = Complex.ab... | case left
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
x : 𝕊
b1 : x ∈ multibrotExt d
e : ∃ᶠ (y : 𝕊) in 𝓝 x, y ∈ t
xt : x ∉ t
pb : potential d x = Complex.a... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | have e' : ∃ᶠ y in 𝓝[{x}ᶜ] x, y ∈ t := by
simp only [frequently_nhdsWithin_iff, mem_compl_singleton_iff]
refine e.mp (eventually_of_forall fun z zt ↦ ⟨zt, ?_⟩)
contrapose xt; simp only [not_not] at xt ⊢; rwa [← xt] | case left
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
x : 𝕊
b1 : x ∈ multibrotExt d
e : ∃ᶠ (y : 𝕊) in 𝓝 x, y ∈ t
xt : x ∉ t
pb : potential d x = Complex.a... | case left
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
x : 𝕊
b1 : x ∈ multibrotExt d
e : ∃ᶠ (y : 𝕊) in 𝓝 x, y ∈ t
xt : x ∉ t
pb : potential d x = Complex.a... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | contrapose xt | case left
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
x : 𝕊
b1 : x ∈ multibrotExt d
e : ∃ᶠ (y : 𝕊) in 𝓝 x, y ∈ t
xt : x ∉ t
pb : potential d x = Complex.a... | case left
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
x : 𝕊
b1 : x ∈ multibrotExt d
e : ∃ᶠ (y : 𝕊) in 𝓝 x, y ∈ t
pb : potential d x = Complex.abs b
e' : ∃... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | clear xt | case left
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
x : 𝕊
b1 : x ∈ multibrotExt d
e : ∃ᶠ (y : 𝕊) in 𝓝 x, y ∈ t
pb : potential d x = Complex.abs b
e' : ∃... | case left
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
x : 𝕊
b1 : x ∈ multibrotExt d
e : ∃ᶠ (y : 𝕊) in 𝓝 x, y ∈ t
pb : potential d x = Complex.abs b
e' : ∃... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | simp only [not_not] | case left
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
x : 𝕊
b1 : x ∈ multibrotExt d
e : ∃ᶠ (y : 𝕊) in 𝓝 x, y ∈ t
pb : potential d x = Complex.abs b
e' : ∃... | case left
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
x : 𝕊
b1 : x ∈ multibrotExt d
e : ∃ᶠ (y : 𝕊) in 𝓝 x, y ∈ t
pb : potential d x = Complex.abs b
e' : ∃... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | use b1 | case left
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
x : 𝕊
b1 : x ∈ multibrotExt d
e : ∃ᶠ (y : 𝕊) in 𝓝 x, y ∈ t
pb : potential d x = Complex.abs b
e' : ∃... | case right
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
x : 𝕊
b1 : x ∈ multibrotExt d
e : ∃ᶠ (y : 𝕊) in 𝓝 x, y ∈ t
pb : potential d x = Complex.abs b
e' : ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | cases' HolomorphicAt.eventually_eq_or_eventually_ne (bottcherHolomorphic d _ b1)
holomorphicAt_const with h h | case right
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
x : 𝕊
b1 : x ∈ multibrotExt d
e : ∃ᶠ (y : 𝕊) in 𝓝 x, y ∈ t
pb : potential d x = Complex.abs b
e' : ... | case right.inl
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h✝ : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
x : 𝕊
b1 : x ∈ multibrotExt d
e : ∃ᶠ (y : 𝕊) in 𝓝 x, y ∈ t
pb : potential d x = Complex.abs b
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | use h | case right.inl
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h✝ : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
x : 𝕊
b1 : x ∈ multibrotExt d
e : ∃ᶠ (y : 𝕊) in 𝓝 x, y ∈ t
pb : potential d x = Complex.abs b
... | case right.inr
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h✝ : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
x : 𝕊
b1 : x ∈ multibrotExt d
e : ∃ᶠ (y : 𝕊) in 𝓝 x, y ∈ t
pb : potential d x = Complex.abs b
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | contrapose h | case right.inr
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h✝ : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
x : 𝕊
b1 : x ∈ multibrotExt d
e : ∃ᶠ (y : 𝕊) in 𝓝 x, y ∈ t
pb : potential d x = Complex.abs b
... | case right.inr
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h✝ : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
x : 𝕊
b1 : x ∈ multibrotExt d
e : ∃ᶠ (y : 𝕊) in 𝓝 x, y ∈ t
pb : potential d x = Complex.abs b
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | simp only [Filter.not_eventually, not_not] at h ⊢ | case right.inr
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h✝ : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
x : 𝕊
b1 : x ∈ multibrotExt d
e : ∃ᶠ (y : 𝕊) in 𝓝 x, y ∈ t
pb : potential d x = Complex.abs b
... | case right.inr
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h✝ : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
x : 𝕊
b1 : x ∈ multibrotExt d
e : ∃ᶠ (y : 𝕊) in 𝓝 x, y ∈ t
pb : potential d x = Complex.abs b
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | exact e'.mp (eventually_of_forall fun y yt ↦ yt.2.self_of_nhds) | case right.inr
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h✝ : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
x : 𝕊
b1 : x ∈ multibrotExt d
e : ∃ᶠ (y : 𝕊) in 𝓝 x, y ∈ t
pb : potential d x = Complex.abs b
... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | apply tendsto_nhds_unique_of_frequently_eq potential_continuous.continuousAt
continuousAt_const | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
x : 𝕊
e : ∃ᶠ (y : 𝕊) in 𝓝 x, y ∈ t
xt : x ∉ t
⊢ potential d x = Complex.abs b | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
x : 𝕊
e : ∃ᶠ (y : 𝕊) in 𝓝 x, y ∈ t
xt : x ∉ t
⊢ ∃ᶠ (x : 𝕊) in 𝓝 x, potential d x = Co... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | refine e.mp (eventually_of_forall ?_) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
x : 𝕊
e : ∃ᶠ (y : 𝕊) in 𝓝 x, y ∈ t
xt : x ∉ t
⊢ ∃ᶠ (x : 𝕊) in 𝓝 x, potential d x = Co... | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
x : 𝕊
e : ∃ᶠ (y : 𝕊) in 𝓝 x, y ∈ t
xt : x ∉ t
⊢ ∀ x ∈ t, potential d x = Complex.abs b |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | intro z ⟨_, h⟩ | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
x : 𝕊
e : ∃ᶠ (y : 𝕊) in 𝓝 x, y ∈ t
xt : x ∉ t
⊢ ∀ x ∈ t, potential d x = Complex.abs b | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h✝ : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
x : 𝕊
e : ∃ᶠ (y : 𝕊) in 𝓝 x, y ∈ t
xt : x ∉ t
z : 𝕊
left✝ : z ∈ multibrotExt d
h : ∀ᶠ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | rw [← h.self_of_nhds, abs_bottcher] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h✝ : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
x : 𝕊
e : ∃ᶠ (y : 𝕊) in 𝓝 x, y ∈ t
xt : x ∉ t
z : 𝕊
left✝ : z ∈ multibrotExt d
h : ∀ᶠ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | simp only [frequently_nhdsWithin_iff, mem_compl_singleton_iff] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
x : 𝕊
b1 : x ∈ multibrotExt d
e : ∃ᶠ (y : 𝕊) in 𝓝 x, y ∈ t
xt : x ∉ t
pb : potential d x = Complex.abs b
⊢ ∃ᶠ ... | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
x : 𝕊
b1 : x ∈ multibrotExt d
e : ∃ᶠ (y : 𝕊) in 𝓝 x, y ∈ t
xt : x ∉ t
pb : potential d x = Complex.abs b
⊢ ∃ᶠ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | refine e.mp (eventually_of_forall fun z zt ↦ ⟨zt, ?_⟩) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
x : 𝕊
b1 : x ∈ multibrotExt d
e : ∃ᶠ (y : 𝕊) in 𝓝 x, y ∈ t
xt : x ∉ t
pb : potential d x = Complex.abs b
⊢ ∃ᶠ ... | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
x : 𝕊
b1 : x ∈ multibrotExt d
e : ∃ᶠ (y : 𝕊) in 𝓝 x, y ∈ t
xt : x ∉ t
pb : potential d x = Complex.abs b
z : �... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | contrapose xt | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
x : 𝕊
b1 : x ∈ multibrotExt d
e : ∃ᶠ (y : 𝕊) in 𝓝 x, y ∈ t
xt : x ∉ t
pb : potential d x = Complex.abs b
z : �... | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
x : 𝕊
b1 : x ∈ multibrotExt d
e : ∃ᶠ (y : 𝕊) in 𝓝 x, y ∈ t
pb : potential d x = Complex.abs b
z : 𝕊
zt : z ∈ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | simp only [not_not] at xt ⊢ | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
x : 𝕊
b1 : x ∈ multibrotExt d
e : ∃ᶠ (y : 𝕊) in 𝓝 x, y ∈ t
pb : potential d x = Complex.abs b
z : 𝕊
zt : z ∈ ... | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
x : 𝕊
b1 : x ∈ multibrotExt d
e : ∃ᶠ (y : 𝕊) in 𝓝 x, y ∈ t
pb : potential d x = Complex.abs b
z : 𝕊
zt : z ∈ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | rwa [← xt] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
x : 𝕊
b1 : x ∈ multibrotExt d
e : ∃ᶠ (y : 𝕊) in 𝓝 x, y ∈ t
pb : potential d x = Complex.abs b
z : 𝕊
zt : z ∈ ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | rw [isOpen_iff_eventually] | case right
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
⊢ IsOpen t | case right
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
⊢ ∀ x ∈ t, ∀ᶠ (y : 𝕊) in 𝓝 x, y ∈ t |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | intro e ⟨m, h⟩ | case right
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
⊢ ∀ x ∈ t, ∀ᶠ (y : 𝕊) in 𝓝 x, y ∈ t | case right
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m✝ : c ∈ multibrotExt d
b : ℂ := bottcher d c
h✝ : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
e : 𝕊
m : e ∈ multibrotExt d
h : ∀ᶠ (e : 𝕊) in 𝓝 e, bottcher d e = b
⊢ ∀ᶠ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | apply (isOpen_multibrotExt.eventually_mem m).mp | case right
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m✝ : c ∈ multibrotExt d
b : ℂ := bottcher d c
h✝ : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
e : 𝕊
m : e ∈ multibrotExt d
h : ∀ᶠ (e : 𝕊) in 𝓝 e, bottcher d e = b
⊢ ∀ᶠ ... | case right
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m✝ : c ∈ multibrotExt d
b : ℂ := bottcher d c
h✝ : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
e : 𝕊
m : e ∈ multibrotExt d
h : ∀ᶠ (e : 𝕊) in 𝓝 e, bottcher d e = b
⊢ ∀ᶠ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | apply (eventually_eventually_nhds.mpr h).mp | case right
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m✝ : c ∈ multibrotExt d
b : ℂ := bottcher d c
h✝ : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
e : 𝕊
m : e ∈ multibrotExt d
h : ∀ᶠ (e : 𝕊) in 𝓝 e, bottcher d e = b
⊢ ∀ᶠ ... | case right
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m✝ : c ∈ multibrotExt d
b : ℂ := bottcher d c
h✝ : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
e : 𝕊
m : e ∈ multibrotExt d
h : ∀ᶠ (e : 𝕊) in 𝓝 e, bottcher d e = b
⊢ ∀ᶠ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | exact eventually_of_forall fun f h m ↦ ⟨m, h⟩ | case right
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m✝ : c ∈ multibrotExt d
b : ℂ := bottcher d c
h✝ : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
e : 𝕊
m : e ∈ multibrotExt d
h : ∀ᶠ (e : 𝕊) in 𝓝 e, bottcher d e = b
⊢ ∀ᶠ ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | simp only [tu, mem_univ] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
b : ℂ := bottcher d c
h : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set 𝕊 := {c | c ∈ multibrotExt d ∧ ∀ᶠ (e : 𝕊) in 𝓝 c, bottcher d e = b}
tu : t = univ
⊢ 0 ∈ t | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | set s := superF d | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ bottcher d '' multibrotExt d = ball 0 1 | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ bottcher d '' multibrotExt d = ball 0 1 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | apply subset_antisymm | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ bottcher d '' multibrotExt d = ball 0 1 | case a
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ bottcher d '' multibrotExt d ⊆ ball 0 1
case a
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ ball 0 1 ⊆ bottcher d '' multibrotExt d |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | intro w | case a
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ bottcher d '' multibrotExt d ⊆ ball 0 1 | case a
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
w : ℂ
⊢ w ∈ bottcher d '' multibrotExt d → w ∈ ball 0 1 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | simp only [mem_image] | case a
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
w : ℂ
⊢ w ∈ bottcher d '' multibrotExt d → w ∈ ball 0 1 | case a
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
w : ℂ
⊢ (∃ x ∈ multibrotExt d, bottcher d x = w) → w ∈ ball 0 1 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | intro ⟨c, m, e⟩ | case a
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
w : ℂ
⊢ (∃ x ∈ multibrotExt d, bottcher d x = w) → w ∈ ball 0 1 | case a
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
w : ℂ
c : 𝕊
m : c ∈ multibrotExt d
e : bottcher d c = w
⊢ w ∈ ball 0 1 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | rw [← e] | case a
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
w : ℂ
c : 𝕊
m : c ∈ multibrotExt d
e : bottcher d c = w
⊢ w ∈ ball 0 1 | case a
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
w : ℂ
c : 𝕊
m : c ∈ multibrotExt d
e : bottcher d c = w
⊢ bottcher d c ∈ ball 0 1 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | clear e w | case a
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
w : ℂ
c : 𝕊
m : c ∈ multibrotExt d
e : bottcher d c = w
⊢ bottcher d c ∈ ball 0 1 | case a
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
c : 𝕊
m : c ∈ multibrotExt d
⊢ bottcher d c ∈ ball 0 1 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | induction c using OnePoint.rec | case a
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
c : 𝕊
m : c ∈ multibrotExt d
⊢ bottcher d c ∈ ball 0 1 | case a.h₁
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
m : ∞ ∈ multibrotExt d
⊢ bottcher d ∞ ∈ ball 0 1
case a.h₂
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x✝ : ℂ
m : ↑x✝ ∈ multibrotExt d
⊢ bottcher d ↑x✝ ∈ ball 0 1 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | simp only [bottcher, fill_inf] | case a.h₁
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
m : ∞ ∈ multibrotExt d
⊢ bottcher d ∞ ∈ ball 0 1 | case a.h₁
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
m : ∞ ∈ multibrotExt d
⊢ 0 ∈ ball 0 1 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | exact mem_ball_self one_pos | case a.h₁
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
m : ∞ ∈ multibrotExt d
⊢ 0 ∈ ball 0 1 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | simp only [multibrotExt_coe] at m | case a.h₂
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x✝ : ℂ
m : ↑x✝ ∈ multibrotExt d
⊢ bottcher d ↑x✝ ∈ ball 0 1 | case a.h₂
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x✝ : ℂ
m : x✝ ∉ multibrot d
⊢ bottcher d ↑x✝ ∈ ball 0 1 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | simp only [bottcher, fill_coe, bottcher', mem_ball, Complex.dist_eq, sub_zero] | case a.h₂
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x✝ : ℂ
m : x✝ ∉ multibrot d
⊢ bottcher d ↑x✝ ∈ ball 0 1 | case a.h₂
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x✝ : ℂ
m : x✝ ∉ multibrot d
⊢ Complex.abs (⋯.bottcher x✝ ↑x✝) < 1 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | exact s.bottcher_lt_one (multibrotPost m) | case a.h₂
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x✝ : ℂ
m : x✝ ∉ multibrot d
⊢ Complex.abs (⋯.bottcher x✝ ↑x✝) < 1 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | refine _root_.trans ?_ interior_subset | case a
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ ball 0 1 ⊆ bottcher d '' multibrotExt d | case a
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ ball 0 1 ⊆ interior (bottcher d '' multibrotExt d) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | refine IsPreconnected.relative_clopen (convex_ball _ _).isPreconnected ?_ ?_ ?_ | case a
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ ball 0 1 ⊆ interior (bottcher d '' multibrotExt d) | case a.refine_1
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ (ball 0 1 ∩ bottcher d '' multibrotExt d).Nonempty
case a.refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ ball 0 1 ∩ bottcher d '' multibrotExt d ⊆ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | use 0, mem_ball_self one_pos, ∞ | case a.refine_1
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ (ball 0 1 ∩ bottcher d '' multibrotExt d).Nonempty | case h
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ ∞ ∈ multibrotExt d ∧ bottcher d ∞ = 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | simp only [multibrotExt_inf, bottcher, fill_inf, true_and_iff] | case h
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ ∞ ∈ multibrotExt d ∧ bottcher d ∞ = 0 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | rw [IsOpen.interior_eq] | case a.refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ ball 0 1 ∩ bottcher d '' multibrotExt d ⊆ interior (bottcher d '' multibrotExt d) | case a.refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ ball 0 1 ∩ bottcher d '' multibrotExt d ⊆ bottcher d '' multibrotExt d
case a.refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ IsOpen (bottcher d '' ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | exact inter_subset_right _ _ | case a.refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ ball 0 1 ∩ bottcher d '' multibrotExt d ⊆ bottcher d '' multibrotExt d
case a.refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ IsOpen (bottcher d '' ... | case a.refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ IsOpen (bottcher d '' multibrotExt d) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | rw [isOpen_iff_eventually] | case a.refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ IsOpen (bottcher d '' multibrotExt d) | case a.refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ ∀ x ∈ bottcher d '' multibrotExt d, ∀ᶠ (y : ℂ) in 𝓝 x, y ∈ bottcher d '' multibrotExt d |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | intro z ⟨c, m, e⟩ | case a.refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ ∀ x ∈ bottcher d '' multibrotExt d, ∀ᶠ (y : ℂ) in 𝓝 x, y ∈ bottcher d '' multibrotExt d | case a.refine_2
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
z : ℂ
c : 𝕊
m : c ∈ multibrotExt d
e : bottcher d c = z
⊢ ∀ᶠ (y : ℂ) in 𝓝 z, y ∈ bottcher d '' multibrotExt d |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | rw [← e, (bottcherNontrivial m).nhds_eq_map_nhds, Filter.eventually_map] | case a.refine_2
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
z : ℂ
c : 𝕊
m : c ∈ multibrotExt d
e : bottcher d c = z
⊢ ∀ᶠ (y : ℂ) in 𝓝 z, y ∈ bottcher d '' multibrotExt d | case a.refine_2
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
z : ℂ
c : 𝕊
m : c ∈ multibrotExt d
e : bottcher d c = z
⊢ ∀ᶠ (a : 𝕊) in 𝓝 c, bottcher d a ∈ bottcher d '' multibrotExt d |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | exact
(isOpen_multibrotExt.eventually_mem m).mp (eventually_of_forall fun e m ↦ by use e, m) | case a.refine_2
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
z : ℂ
c : 𝕊
m : c ∈ multibrotExt d
e : bottcher d c = z
⊢ ∀ᶠ (a : 𝕊) in 𝓝 c, bottcher d a ∈ bottcher d '' multibrotExt d | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | use e, m | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
z : ℂ
c : 𝕊
m✝ : c ∈ multibrotExt d
e✝ : bottcher d c = z
e : 𝕊
m : e ∈ multibrotExt d
⊢ bottcher d e ∈ bottcher d '' multibrotExt d | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | intro x ⟨x1, m⟩ | case a.refine_3
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ ball 0 1 ∩ closure (bottcher d '' multibrotExt d) ⊆ bottcher d '' multibrotExt d | case a.refine_3
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
x1 : x ∈ ball 0 1
m : x ∈ closure (bottcher d '' multibrotExt d)
⊢ x ∈ bottcher d '' multibrotExt d |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | simp only [mem_ball, Complex.dist_eq, sub_zero] at x1 | case a.refine_3
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
x1 : x ∈ ball 0 1
m : x ∈ closure (bottcher d '' multibrotExt d)
⊢ x ∈ bottcher d '' multibrotExt d | case a.refine_3
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
m : x ∈ closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
⊢ x ∈ bottcher d '' multibrotExt d |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | rcases exists_between x1 with ⟨b, xb, b1⟩ | case a.refine_3
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
m : x ∈ closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
⊢ x ∈ bottcher d '' multibrotExt d | case a.refine_3.intro.intro
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
m : x ∈ closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : ℝ
xb : Complex.abs x < b
b1 : b < 1
⊢ x ∈ bottcher d '' multibrotExt d |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | set t := {e | potential d e ≤ b} | case a.refine_3.intro.intro
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
m : x ∈ closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : ℝ
xb : Complex.abs x < b
b1 : b < 1
⊢ x ∈ bottcher d '' multibrotExt d | case a.refine_3.intro.intro
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
m : x ∈ closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : ℝ
xb : Complex.abs x < b
b1 : b < 1
t : Set 𝕊 := {e | potential d e ≤ b}
⊢ x ∈ bottcher d '' multibrotExt d |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | have ct : IsCompact t := (isClosed_le potential_continuous continuous_const).isCompact | case a.refine_3.intro.intro
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
m : x ∈ closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : ℝ
xb : Complex.abs x < b
b1 : b < 1
t : Set 𝕊 := {e | potential d e ≤ b}
⊢ x ∈ bottcher d '' multibrotExt d | case a.refine_3.intro.intro
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
m : x ∈ closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : ℝ
xb : Complex.abs x < b
b1 : b < 1
t : Set 𝕊 := {e | potential d e ≤ b}
ct : IsCompact t
⊢ x ∈ bottcher d '' multib... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | have ts : t ⊆ multibrotExt d := by
intro c m; rw [← potential_lt_one]; exact lt_of_le_of_lt m b1 | case a.refine_3.intro.intro
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
m : x ∈ closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : ℝ
xb : Complex.abs x < b
b1 : b < 1
t : Set 𝕊 := {e | potential d e ≤ b}
ct : IsCompact t
⊢ x ∈ bottcher d '' multib... | case a.refine_3.intro.intro
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
m : x ∈ closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : ℝ
xb : Complex.abs x < b
b1 : b < 1
t : Set 𝕊 := {e | potential d e ≤ b}
ct : IsCompact t
ts : t ⊆ multibrotExt d
⊢ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | have mt : x ∈ closure (bottcher d '' t) := by
rw [mem_closure_iff_frequently] at m ⊢; apply m.mp
have lt : ∀ᶠ y : ℂ in 𝓝 x, abs y < b :=
Complex.continuous_abs.continuousAt.eventually_lt continuousAt_const xb
refine lt.mp (eventually_of_forall fun y lt m ↦ ?_)
rcases m with ⟨c, _, cy⟩; rw [← cy]; rw [← cy,... | case a.refine_3.intro.intro
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
m : x ∈ closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : ℝ
xb : Complex.abs x < b
b1 : b < 1
t : Set 𝕊 := {e | potential d e ≤ b}
ct : IsCompact t
ts : t ⊆ multibrotExt d
⊢ ... | case a.refine_3.intro.intro
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
m : x ∈ closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : ℝ
xb : Complex.abs x < b
b1 : b < 1
t : Set 𝕊 := {e | potential d e ≤ b}
ct : IsCompact t
ts : t ⊆ multibrotExt d
mt... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | apply image_subset _ ts | case a.refine_3.intro.intro
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
m : x ∈ closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : ℝ
xb : Complex.abs x < b
b1 : b < 1
t : Set 𝕊 := {e | potential d e ≤ b}
ct : IsCompact t
ts : t ⊆ multibrotExt d
mt... | case a.refine_3.intro.intro.a
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
m : x ∈ closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : ℝ
xb : Complex.abs x < b
b1 : b < 1
t : Set 𝕊 := {e | potential d e ≤ b}
ct : IsCompact t
ts : t ⊆ multibrotExt d
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | rw [IsClosed.closure_eq] at mt | case a.refine_3.intro.intro.a
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
m : x ∈ closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : ℝ
xb : Complex.abs x < b
b1 : b < 1
t : Set 𝕊 := {e | potential d e ≤ b}
ct : IsCompact t
ts : t ⊆ multibrotExt d
... | case a.refine_3.intro.intro.a
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
m : x ∈ closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : ℝ
xb : Complex.abs x < b
b1 : b < 1
t : Set 𝕊 := {e | potential d e ≤ b}
ct : IsCompact t
ts : t ⊆ multibrotExt d
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | exact mt | case a.refine_3.intro.intro.a
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
m : x ∈ closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : ℝ
xb : Complex.abs x < b
b1 : b < 1
t : Set 𝕊 := {e | potential d e ≤ b}
ct : IsCompact t
ts : t ⊆ multibrotExt d
... | case a.refine_3.intro.intro.a
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
m : x ∈ closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : ℝ
xb : Complex.abs x < b
b1 : b < 1
t : Set 𝕊 := {e | potential d e ≤ b}
ct : IsCompact t
ts : t ⊆ multibrotExt d
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | apply IsCompact.isClosed | case a.refine_3.intro.intro.a
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
m : x ∈ closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : ℝ
xb : Complex.abs x < b
b1 : b < 1
t : Set 𝕊 := {e | potential d e ≤ b}
ct : IsCompact t
ts : t ⊆ multibrotExt d
... | case a.refine_3.intro.intro.a.hs
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
m : x ∈ closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : ℝ
xb : Complex.abs x < b
b1 : b < 1
t : Set 𝕊 := {e | potential d e ≤ b}
ct : IsCompact t
ts : t ⊆ multibrotExt... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | apply IsCompact.image_of_continuousOn ct | case a.refine_3.intro.intro.a.hs
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
m : x ∈ closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : ℝ
xb : Complex.abs x < b
b1 : b < 1
t : Set 𝕊 := {e | potential d e ≤ b}
ct : IsCompact t
ts : t ⊆ multibrotExt... | case a.refine_3.intro.intro.a.hs
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
m : x ∈ closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : ℝ
xb : Complex.abs x < b
b1 : b < 1
t : Set 𝕊 := {e | potential d e ≤ b}
ct : IsCompact t
ts : t ⊆ multibrotExt... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | refine ContinuousOn.mono ?_ ts | case a.refine_3.intro.intro.a.hs
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
m : x ∈ closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : ℝ
xb : Complex.abs x < b
b1 : b < 1
t : Set 𝕊 := {e | potential d e ≤ b}
ct : IsCompact t
ts : t ⊆ multibrotExt... | case a.refine_3.intro.intro.a.hs
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
m : x ∈ closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : ℝ
xb : Complex.abs x < b
b1 : b < 1
t : Set 𝕊 := {e | potential d e ≤ b}
ct : IsCompact t
ts : t ⊆ multibrotExt... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | exact (bottcherHolomorphic d).continuousOn | case a.refine_3.intro.intro.a.hs
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
m : x ∈ closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : ℝ
xb : Complex.abs x < b
b1 : b < 1
t : Set 𝕊 := {e | potential d e ≤ b}
ct : IsCompact t
ts : t ⊆ multibrotExt... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | intro c m | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
m : x ∈ closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : ℝ
xb : Complex.abs x < b
b1 : b < 1
t : Set 𝕊 := {e | potential d e ≤ b}
ct : IsCompact t
⊢ t ⊆ multibrotExt d | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
m✝ : x ∈ closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : ℝ
xb : Complex.abs x < b
b1 : b < 1
t : Set 𝕊 := {e | potential d e ≤ b}
ct : IsCompact t
c : 𝕊
m : c ∈ t
⊢ c ∈ multibrotExt d |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | rw [← potential_lt_one] | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
m✝ : x ∈ closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : ℝ
xb : Complex.abs x < b
b1 : b < 1
t : Set 𝕊 := {e | potential d e ≤ b}
ct : IsCompact t
c : 𝕊
m : c ∈ t
⊢ c ∈ multibrotExt d | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
m✝ : x ∈ closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : ℝ
xb : Complex.abs x < b
b1 : b < 1
t : Set 𝕊 := {e | potential d e ≤ b}
ct : IsCompact t
c : 𝕊
m : c ∈ t
⊢ potential d c < 1 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | exact lt_of_le_of_lt m b1 | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
m✝ : x ∈ closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : ℝ
xb : Complex.abs x < b
b1 : b < 1
t : Set 𝕊 := {e | potential d e ≤ b}
ct : IsCompact t
c : 𝕊
m : c ∈ t
⊢ potential d c < 1 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | rw [mem_closure_iff_frequently] at m ⊢ | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
m : x ∈ closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : ℝ
xb : Complex.abs x < b
b1 : b < 1
t : Set 𝕊 := {e | potential d e ≤ b}
ct : IsCompact t
ts : t ⊆ multibrotExt d
⊢ x ∈ closure (bottcher d '' t... | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
m : ∃ᶠ (x : ℂ) in 𝓝 x, x ∈ bottcher d '' multibrotExt d
x1 : Complex.abs x < 1
b : ℝ
xb : Complex.abs x < b
b1 : b < 1
t : Set 𝕊 := {e | potential d e ≤ b}
ct : IsCompact t
ts : t ⊆ multibrotExt d
⊢ ∃ᶠ (x : ℂ) in 𝓝 x... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | apply m.mp | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
m : ∃ᶠ (x : ℂ) in 𝓝 x, x ∈ bottcher d '' multibrotExt d
x1 : Complex.abs x < 1
b : ℝ
xb : Complex.abs x < b
b1 : b < 1
t : Set 𝕊 := {e | potential d e ≤ b}
ct : IsCompact t
ts : t ⊆ multibrotExt d
⊢ ∃ᶠ (x : ℂ) in 𝓝 x... | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
m : ∃ᶠ (x : ℂ) in 𝓝 x, x ∈ bottcher d '' multibrotExt d
x1 : Complex.abs x < 1
b : ℝ
xb : Complex.abs x < b
b1 : b < 1
t : Set 𝕊 := {e | potential d e ≤ b}
ct : IsCompact t
ts : t ⊆ multibrotExt d
⊢ ∀ᶠ (x : ℂ) in 𝓝 x... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | have lt : ∀ᶠ y : ℂ in 𝓝 x, abs y < b :=
Complex.continuous_abs.continuousAt.eventually_lt continuousAt_const xb | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
m : ∃ᶠ (x : ℂ) in 𝓝 x, x ∈ bottcher d '' multibrotExt d
x1 : Complex.abs x < 1
b : ℝ
xb : Complex.abs x < b
b1 : b < 1
t : Set 𝕊 := {e | potential d e ≤ b}
ct : IsCompact t
ts : t ⊆ multibrotExt d
⊢ ∀ᶠ (x : ℂ) in 𝓝 x... | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
m : ∃ᶠ (x : ℂ) in 𝓝 x, x ∈ bottcher d '' multibrotExt d
x1 : Complex.abs x < 1
b : ℝ
xb : Complex.abs x < b
b1 : b < 1
t : Set 𝕊 := {e | potential d e ≤ b}
ct : IsCompact t
ts : t ⊆ multibrotExt d
lt : ∀ᶠ (y : ℂ) in �... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | refine lt.mp (eventually_of_forall fun y lt m ↦ ?_) | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
m : ∃ᶠ (x : ℂ) in 𝓝 x, x ∈ bottcher d '' multibrotExt d
x1 : Complex.abs x < 1
b : ℝ
xb : Complex.abs x < b
b1 : b < 1
t : Set 𝕊 := {e | potential d e ≤ b}
ct : IsCompact t
ts : t ⊆ multibrotExt d
lt : ∀ᶠ (y : ℂ) in �... | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
m✝ : ∃ᶠ (x : ℂ) in 𝓝 x, x ∈ bottcher d '' multibrotExt d
x1 : Complex.abs x < 1
b : ℝ
xb : Complex.abs x < b
b1 : b < 1
t : Set 𝕊 := {e | potential d e ≤ b}
ct : IsCompact t
ts : t ⊆ multibrotExt d
lt✝ : ∀ᶠ (y : ℂ) in... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | rcases m with ⟨c, _, cy⟩ | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
m✝ : ∃ᶠ (x : ℂ) in 𝓝 x, x ∈ bottcher d '' multibrotExt d
x1 : Complex.abs x < 1
b : ℝ
xb : Complex.abs x < b
b1 : b < 1
t : Set 𝕊 := {e | potential d e ≤ b}
ct : IsCompact t
ts : t ⊆ multibrotExt d
lt✝ : ∀ᶠ (y : ℂ) in... | case intro.intro
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
m : ∃ᶠ (x : ℂ) in 𝓝 x, x ∈ bottcher d '' multibrotExt d
x1 : Complex.abs x < 1
b : ℝ
xb : Complex.abs x < b
b1 : b < 1
t : Set 𝕊 := {e | potential d e ≤ b}
ct : IsCompact t
ts : t ⊆ multibrotExt d
lt... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | rw [← cy] | case intro.intro
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
m : ∃ᶠ (x : ℂ) in 𝓝 x, x ∈ bottcher d '' multibrotExt d
x1 : Complex.abs x < 1
b : ℝ
xb : Complex.abs x < b
b1 : b < 1
t : Set 𝕊 := {e | potential d e ≤ b}
ct : IsCompact t
ts : t ⊆ multibrotExt d
lt... | case intro.intro
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
m : ∃ᶠ (x : ℂ) in 𝓝 x, x ∈ bottcher d '' multibrotExt d
x1 : Complex.abs x < 1
b : ℝ
xb : Complex.abs x < b
b1 : b < 1
t : Set 𝕊 := {e | potential d e ≤ b}
ct : IsCompact t
ts : t ⊆ multibrotExt d
lt... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | rw [← cy, abs_bottcher] at lt | case intro.intro
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
m : ∃ᶠ (x : ℂ) in 𝓝 x, x ∈ bottcher d '' multibrotExt d
x1 : Complex.abs x < 1
b : ℝ
xb : Complex.abs x < b
b1 : b < 1
t : Set 𝕊 := {e | potential d e ≤ b}
ct : IsCompact t
ts : t ⊆ multibrotExt d
lt... | case intro.intro
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
m : ∃ᶠ (x : ℂ) in 𝓝 x, x ∈ bottcher d '' multibrotExt d
x1 : Complex.abs x < 1
b : ℝ
xb : Complex.abs x < b
b1 : b < 1
t : Set 𝕊 := {e | potential d e ≤ b}
ct : IsCompact t
ts : t ⊆ multibrotExt d
lt... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | exact ⟨c, lt.le, rfl⟩ | case intro.intro
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x : ℂ
m : ∃ᶠ (x : ℂ) in 𝓝 x, x ∈ bottcher d '' multibrotExt d
x1 : Complex.abs x < 1
b : ℝ
xb : Complex.abs x < b
b1 : b < 1
t : Set 𝕊 := {e | potential d e ≤ b}
ct : IsCompact t
ts : t ⊆ multibrotExt d
lt... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | set s := superF d | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
⊢ Tendsto (fun z => ⋯.bottcher c ↑z * 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.bottcher c ↑z * z) atInf (𝓝 1) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | have e : ∀ᶠ z : ℂ in atInf, s.bottcher c z * z = s.bottcherNear c z * z := by
suffices e : ∀ᶠ z : ℂ in atInf, s.bottcher c z = s.bottcherNear c z by
exact e.mp (eventually_of_forall fun z e ↦ by rw [e])
refine coe_tendsto_inf.eventually (p := fun z ↦ s.bottcher c z = s.bottcherNear c z) ?_
apply s.bottcher_eq... | 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.bottcher c ↑z * 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
e : ∀ᶠ (z : ℂ) in atInf, s.bottcher c ↑z * z = s.bottcherNear c ↑z * z
⊢ Tendsto (fun z => s.bottcher c ↑z * z) atInf (𝓝 1) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | rw [Filter.tendsto_congr' 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 * z = s.bottcherNear c ↑z * z
⊢ Tendsto (fun z => s.bottcher c ↑z * 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
e : ∀ᶠ (z : ℂ) in atInf, s.bottcher c ↑z * z = s.bottcherNear c ↑z * z
⊢ Tendsto (fun x => s.bottcherNear c ↑x * x) atInf (𝓝 1) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | clear 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 * z = s.bottcherNear c ↑z * z
⊢ Tendsto (fun x => s.bottcherNear c ↑x * x) 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 x => s.bottcherNear c ↑x * x) atInf (𝓝 1) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | have m := bottcherNear_monic (s.superNearC.s (mem_univ 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 x => s.bottcherNear c ↑x * x) 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
m : HasDerivAt (bottcherNear (s.fl c) d) 1 0
⊢ Tendsto (fun x => s.bottcherNear c ↑x * x) atInf (𝓝 1) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | simp only [hasDerivAt_iff_tendsto, sub_zero, bottcherNear_zero, smul_eq_mul, mul_one,
Metric.tendsto_nhds_nhds, Real.dist_eq, Complex.norm_eq_abs, Complex.dist_eq, abs_mul,
abs_of_nonneg (Complex.abs.nonneg _), abs_inv] at m | 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 : HasDerivAt (bottcherNear (s.fl c) d) 1 0
⊢ Tendsto (fun x => s.bottcherNear c ↑x * x) 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
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
⊢ Tendsto (fun x => s.bottcherNear c ↑x * x) atInf (𝓝 1) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | simp only [Metric.tendsto_nhds, atInf_basis.eventually_iff, true_and_iff, mem_setOf,
Complex.dist_eq, Complex.norm_eq_abs] | 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
⊢ Tendsto (fun x => s.bottcherNear c ↑x * x) 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
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
⊢ ∀ ε > 0, ∃ i, ∀ ⦃x : ℂ⦄, i < Complex.abs x → Complex.abs (s.bottcherNear c ↑x * x - 1) < ε |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | intro e ep | 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
⊢ ∀ ε > 0, ∃ i, ∀ ⦃x : ℂ⦄, i < Complex.abs x → Complex.abs (s.bottcherNear c ↑x * x - 1) < ε | 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
⊢ ∃ i, ∀ ⦃x : ℂ⦄, i < Complex.abs x → Complex.abs (s.bottcherNear c ↑x * x - 1) ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | rcases m e ep with ⟨r, rp, h⟩ | 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
⊢ ∃ i, ∀ ⦃x : ℂ⦄, i < Complex.abs x → Complex.abs (s.bottcherNear c ↑x * x - 1) ... | case intro.intro
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.ab... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | use 1 / r | case intro.intro
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.ab... | case h
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 * C... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | intro z zr | case h
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 * C... | case h
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 * C... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | have az0 : abs z ≠ 0 := (lt_trans (one_div_pos.mpr rp) zr).ne' | case h
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 * C... | case h
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 * C... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | have z0 : z ≠ 0 := Complex.abs.ne_zero_iff.mp az0 | case h
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 * C... | case h
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 * C... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | have zir : abs (z⁻¹) < r := by
simp only [one_div, map_inv₀] at zr ⊢; exact inv_lt_of_inv_lt rp zr | case h
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 * C... | case h
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 * C... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | specialize @h z⁻¹ zir | case h
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 * C... | case h
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
z : ℂ
zr : 1 / r < Complex.abs z
az0 : Complex.abs z ≠ 0... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | simp only [map_inv₀, inv_inv, ← Complex.abs.map_mul, sub_mul, inv_mul_cancel z0,
mul_comm z _] at h | case h
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
z : ℂ
zr : 1 / r < Complex.abs z
az0 : Complex.abs z ≠ 0... | case h
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
z : ℂ
zr : 1 / r < Complex.abs z
az0 : Complex.abs z ≠ 0... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | simp only [Super.bottcherNear, extChartAt_inf, PartialEquiv.trans_apply,
coePartialEquiv_symm_apply, Equiv.toPartialEquiv_apply, invEquiv_apply, RiemannSphere.inv_inf,
toComplex_zero, sub_zero, inv_coe z0, toComplex_coe] | case h
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
z : ℂ
zr : 1 / r < Complex.abs z
az0 : Complex.abs z ≠ 0... | case h
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
z : ℂ
zr : 1 / r < Complex.abs z
az0 : Complex.abs z ≠ 0... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | exact h | case h
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
z : ℂ
zr : 1 / r < Complex.abs z
az0 : Complex.abs z ≠ 0... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | suffices e : ∀ᶠ z : ℂ in atInf, s.bottcher c z = s.bottcherNear c z by
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
⊢ ∀ᶠ (z : ℂ) in atInf, s.bottcher c ↑z * z = s.bottcherNear c ↑z * z | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
⊢ ∀ᶠ (z : ℂ) in atInf, s.bottcher c ↑z = s.bottcherNear c ↑z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | refine coe_tendsto_inf.eventually (p := fun z ↦ s.bottcher c z = s.bottcherNear c z) ?_ | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
⊢ ∀ᶠ (z : ℂ) in atInf, s.bottcher c ↑z = s.bottcherNear c ↑z | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
⊢ ∀ᶠ (y : 𝕊) in 𝓝 ∞, (fun z => s.bottcher c z = s.bottcherNear c z) y |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | apply s.bottcher_eq_bottcherNear | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
⊢ ∀ᶠ (y : 𝕊) in 𝓝 ∞, (fun z => s.bottcher c z = s.bottcherNear c z) y | no goals |
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