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 | bottcher_bound | [513, 1] | [554, 59] | simp only [mem_setOf, map_inv₀] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
⊢ c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
⊢ (Complex.abs c)⁻¹ < (max 16 (Complex.abs c / 2))⁻¹ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | apply inv_lt_inv_of_lt | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
⊢ (Complex.abs c)⁻¹ < (max 16 (Complex.abs c / 2))⁻¹ | case hb
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
⊢ 0 < max 16 (Complex.abs c / 2)
case h
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
⊢ max 16 (Compl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | bound | case hb
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
⊢ 0 < max 16 (Complex.abs c / 2)
case h
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
⊢ max 16 (Compl... | case h
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
⊢ max 16 (Complex.abs c / 2) < Complex.abs c |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | refine max_lt lo (half_lt_self (lt_trans (by norm_num) lo)) | case h
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
⊢ max 16 (Complex.abs c / 2) < Complex.abs c | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | norm_num | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
⊢ 0 < 16 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | norm_num | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
⊢ 2 < 16 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | intro n | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
⊢ ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0 | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
n : ℕ
⊢ (f d c)^[n] ↑c ≠ 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | contrapose mem | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
n : ℕ
⊢ (f d c)^[n] ↑c ≠ 0 | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
n : ℕ
mem : ¬(f d c)^[n] ↑c ≠ 0
⊢ ¬c ∉ multibrot d |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | simp only [not_not] at mem ⊢ | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
n : ℕ
mem : ¬(f d c)^[n] ↑c ≠ 0
⊢ ¬c ∉ multibrot d | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
n : ℕ
mem : (f d c)^[n] ↑c = 0
⊢ c ∈ multibrot d |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | exact multibrot_of_zero mem | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
n : ℕ
mem : (f d c)^[n] ↑c = 0
⊢ c ∈ multibrot d | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | intro n | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
⊢ ∀ (n : ℕ), ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
n : ℕ
⊢ ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | induction' n with n h | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
n : ℕ
⊢ ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹) | case zero
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
⊢ ((f d c)^[0] ↑c)⁻¹ = ↑(g^[0] c⁻¹)
case succ
c✝ : ℂ
d : ℕ
i... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | have cp : c ≠ 0 := Complex.abs.ne_zero_iff.mp (lt_trans (by norm_num) lo).ne' | case zero
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
⊢ ((f d c)^[0] ↑c)⁻¹ = ↑(g^[0] c⁻¹)
case succ
c✝ : ℂ
d : ℕ
i... | case zero
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
cp : c ≠ 0
⊢ ((f d c)^[0] ↑c)⁻¹ = ↑(g^[0] c⁻¹)
case succ
c✝ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | simp only [Function.iterate_zero_apply, inv_coe cp, toComplex_coe] | case zero
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
cp : c ≠ 0
⊢ ((f d c)^[0] ↑c)⁻¹ = ↑(g^[0] c⁻¹)
case succ
c✝ ... | case succ
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
n : ℕ
h : ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
⊢ ((f d c)^[n + 1... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | have e : (f d c)^[n] ↑c = ((g^[n] c⁻¹ : ℂ) : 𝕊)⁻¹ := by rw [← h, inv_inv] | case succ
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
n : ℕ
h : ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
⊢ ((f d c)^[n + 1... | case succ
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
n : ℕ
h : ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
e : (f d c)^[n] ↑... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | simp only [Function.iterate_succ_apply', e] | case succ
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
n : ℕ
h : ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
e : (f d c)^[n] ↑... | case succ
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
n : ℕ
h : ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
e : (f d c)^[n] ↑... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | generalize hz : g^[n] c⁻¹ = z | case succ
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
n : ℕ
h : ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
e : (f d c)^[n] ↑... | case succ
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
n : ℕ
h : ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
e : (f d c)^[n] ↑... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | simp only [← hg, fl, extChartAt_inf, PartialEquiv.trans_apply, Equiv.toPartialEquiv_apply,
invEquiv_apply, RiemannSphere.inv_inf, coePartialEquiv_symm_apply, toComplex_zero, sub_zero,
Function.comp, add_zero, PartialEquiv.coe_trans_symm, PartialEquiv.symm_symm,
coePartialEquiv_apply, Equiv.toPartialEquiv_symm_app... | case succ
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
n : ℕ
h : ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
e : (f d c)^[n] ↑... | case succ
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
n : ℕ
h : ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
e : (f d c)^[n] ↑... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | rw [coe_toComplex] | case succ
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
n : ℕ
h : ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
e : (f d c)^[n] ↑... | case succ
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
n : ℕ
h : ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
e : (f d c)^[n] ↑... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | simp only [Ne, inv_eq_inf, ← hz, ← h, inv_inv, ← Function.iterate_succ_apply' (f d c)] | case succ
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
n : ℕ
h : ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
e : (f d c)^[n] ↑... | case succ
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
n : ℕ
h : ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
e : (f d c)^[n] ↑... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | apply nz | case succ
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
n : ℕ
h : ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
e : (f d c)^[n] ↑... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | norm_num | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
⊢ 0 < 16 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | rw [← h, inv_inv] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
n : ℕ
h : ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
⊢ (f d c)^[n] ↑c = (↑(g^[n]... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | rw [← hg] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
iter : ∀ (n : ℕ), ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
n : ℕ
right✝ : (c, ... | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
iter : ∀ (n : ℕ), ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
n : ℕ
right✝ : (c, ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | exact bottcherNear_eqn_iter (superNearF d c) ct n | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
iter : ∀ (n : ℕ), ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
n : ℕ
right✝ : (c, ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | apply (pow_left_inj (Complex.abs.nonneg _) (Complex.abs.nonneg _)
(pow_ne_zero n (d_ne_zero d))).mp | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
n : ℕ
right✝ : (c, (f d c)^[n] ↑c) ∈ s.near
eq : bottcher' d c ^ d ^ n ... | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
n : ℕ
right✝ : (c, (f d c)^[n] ↑c) ∈ s.near
eq : bottcher' d c ^ d ^ n ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | simp only [← Complex.abs.map_pow, eq] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
n : ℕ
right✝ : (c, (f d c)^[n] ↑c) ∈ s.near
eq : bottcher' d c ^ d ^ n ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_tendsto_zero | [557, 1] | [562, 96] | rw [Metric.tendsto_nhds] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ Tendsto (bottcher' d) atInf (𝓝 0) | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ∀ ε > 0, ∀ᶠ (x : ℂ) in atInf, dist (bottcher' d x) 0 < ε |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_tendsto_zero | [557, 1] | [562, 96] | intro r rp | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ∀ ε > 0, ∀ᶠ (x : ℂ) in atInf, dist (bottcher' d x) 0 < ε | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
⊢ ∀ᶠ (x : ℂ) in atInf, dist (bottcher' d x) 0 < r |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_tendsto_zero | [557, 1] | [562, 96] | rw [atInf_basis.eventually_iff] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
⊢ ∀ᶠ (x : ℂ) in atInf, dist (bottcher' d x) 0 < r | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
⊢ ∃ i, True ∧ ∀ ⦃x : ℂ⦄, x ∈ {x | i < ‖x‖} → dist (bottcher' d x) 0 < r |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_tendsto_zero | [557, 1] | [562, 96] | use max 16 (3 / r) | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
⊢ ∃ i, True ∧ ∀ ⦃x : ℂ⦄, x ∈ {x | i < ‖x‖} → dist (bottcher' d x) 0 < r | case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
⊢ True ∧ ∀ ⦃x : ℂ⦄, x ∈ {x | max 16 (3 / r) < ‖x‖} → dist (bottcher' d x) 0 < r |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_tendsto_zero | [557, 1] | [562, 96] | simp only [true_and_iff, mem_setOf, Complex.dist_eq, sub_zero, Complex.norm_eq_abs, max_lt_iff] | case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
⊢ True ∧ ∀ ⦃x : ℂ⦄, x ∈ {x | max 16 (3 / r) < ‖x‖} → dist (bottcher' d x) 0 < r | case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
⊢ ∀ ⦃x : ℂ⦄, 16 < Complex.abs x ∧ 3 / r < Complex.abs x → Complex.abs (bottcher' d x) < r |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_tendsto_zero | [557, 1] | [562, 96] | intro z ⟨lo, rz⟩ | case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
⊢ ∀ ⦃x : ℂ⦄, 16 < Complex.abs x ∧ 3 / r < Complex.abs x → Complex.abs (bottcher' d x) < r | case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
z : ℂ
lo : 16 < Complex.abs z
rz : 3 / r < Complex.abs z
⊢ Complex.abs (bottcher' d z) < r |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_tendsto_zero | [557, 1] | [562, 96] | apply lt_of_le_of_lt (bottcher_bound lo) | case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
z : ℂ
lo : 16 < Complex.abs z
rz : 3 / r < Complex.abs z
⊢ Complex.abs (bottcher' d z) < r | case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
z : ℂ
lo : 16 < Complex.abs z
rz : 3 / r < Complex.abs z
⊢ 3 * Complex.abs z⁻¹ < r |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_tendsto_zero | [557, 1] | [562, 96] | rw [div_lt_iff rp] at rz | case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
z : ℂ
lo : 16 < Complex.abs z
rz : 3 / r < Complex.abs z
⊢ 3 * Complex.abs z⁻¹ < r | case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
z : ℂ
lo : 16 < Complex.abs z
rz : 3 < Complex.abs z * r
⊢ 3 * Complex.abs z⁻¹ < r |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_tendsto_zero | [557, 1] | [562, 96] | rw [map_inv₀, mul_inv_lt_iff (lt_trans (by norm_num) lo)] | case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
z : ℂ
lo : 16 < Complex.abs z
rz : 3 < Complex.abs z * r
⊢ 3 * Complex.abs z⁻¹ < r | case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
z : ℂ
lo : 16 < Complex.abs z
rz : 3 < Complex.abs z * r
⊢ 3 < Complex.abs z * r |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_tendsto_zero | [557, 1] | [562, 96] | exact rz | case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
z : ℂ
lo : 16 < Complex.abs z
rz : 3 < Complex.abs z * r
⊢ 3 < Complex.abs z * r | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_tendsto_zero | [557, 1] | [562, 96] | norm_num | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
z : ℂ
lo : 16 < Complex.abs z
rz : 3 < Complex.abs z * r
⊢ 0 < 16 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_analytic | [565, 1] | [568, 28] | set s := superF d | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ AnalyticOn ℂ (bottcher' d) (multibrot d)ᶜ | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ AnalyticOn ℂ (bottcher' d) (multibrot d)ᶜ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_analytic | [565, 1] | [568, 28] | intro c m | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ AnalyticOn ℂ (bottcher' d) (multibrot d)ᶜ | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
c : ℂ
m : c ∈ (multibrot d)ᶜ
⊢ AnalyticAt ℂ (bottcher' d) c |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_analytic | [565, 1] | [568, 28] | apply HolomorphicAt.analyticAt I I | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
c : ℂ
m : c ∈ (multibrot d)ᶜ
⊢ AnalyticAt ℂ (bottcher' d) c | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
c : ℂ
m : c ∈ (multibrot d)ᶜ
⊢ HolomorphicAt I I (bottcher' d) c |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_analytic | [565, 1] | [568, 28] | exact (s.bottcher_holomorphicOn (c, c) (multibrotPost m)).comp₂_of_eq holomorphicAt_id
(holomorphic_coe _) rfl | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
c : ℂ
m : c ∈ (multibrot d)ᶜ
⊢ HolomorphicAt I I (bottcher' d) c | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherHolomorphic | [571, 1] | [579, 86] | intro c m | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ HolomorphicOn I I (bottcher d) (multibrotExt d) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
⊢ HolomorphicAt I I (bottcher d) c |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherHolomorphic | [571, 1] | [579, 86] | induction c using OnePoint.rec | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
⊢ HolomorphicAt I I (bottcher d) c | case h₁
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
m : ∞ ∈ multibrotExt d
⊢ HolomorphicAt I I (bottcher d) ∞
case h₂
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
x✝ : ℂ
m : ↑x✝ ∈ multibrotExt d
⊢ HolomorphicAt I I (bottcher d) ↑x✝ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherHolomorphic | [571, 1] | [579, 86] | refine holomorphicAt_fill_inf ?_ bottcher_tendsto_zero | case h₁
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
m : ∞ ∈ multibrotExt d
⊢ HolomorphicAt I I (bottcher d) ∞ | case h₁
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
m : ∞ ∈ multibrotExt d
⊢ ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I (bottcher' d) z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherHolomorphic | [571, 1] | [579, 86] | rw [atInf_basis.eventually_iff] | case h₁
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
m : ∞ ∈ multibrotExt d
⊢ ∀ᶠ (z : ℂ) in atInf, HolomorphicAt I I (bottcher' d) z | case h₁
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
m : ∞ ∈ multibrotExt d
⊢ ∃ i, True ∧ ∀ ⦃x : ℂ⦄, x ∈ {x | i < ‖x‖} → HolomorphicAt I I (bottcher' d) x |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherHolomorphic | [571, 1] | [579, 86] | use 2 | case h₁
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
m : ∞ ∈ multibrotExt d
⊢ ∃ i, True ∧ ∀ ⦃x : ℂ⦄, x ∈ {x | i < ‖x‖} → HolomorphicAt I I (bottcher' d) x | case h
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
m : ∞ ∈ multibrotExt d
⊢ True ∧ ∀ ⦃x : ℂ⦄, x ∈ {x | 2 < ‖x‖} → HolomorphicAt I I (bottcher' d) x |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherHolomorphic | [571, 1] | [579, 86] | simp only [true_and_iff, mem_setOf, Complex.norm_eq_abs] | case h
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
m : ∞ ∈ multibrotExt d
⊢ True ∧ ∀ ⦃x : ℂ⦄, x ∈ {x | 2 < ‖x‖} → HolomorphicAt I I (bottcher' d) x | case h
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
m : ∞ ∈ multibrotExt d
⊢ ∀ ⦃x : ℂ⦄, 2 < Complex.abs x → HolomorphicAt I I (bottcher' d) x |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherHolomorphic | [571, 1] | [579, 86] | intro z a | case h
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
m : ∞ ∈ multibrotExt d
⊢ ∀ ⦃x : ℂ⦄, 2 < Complex.abs x → HolomorphicAt I I (bottcher' d) x | case h
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
m : ∞ ∈ multibrotExt d
z : ℂ
a : 2 < Complex.abs z
⊢ HolomorphicAt I I (bottcher' d) z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherHolomorphic | [571, 1] | [579, 86] | exact (bottcher_analytic _ (multibrot_two_lt a)).holomorphicAt I I | case h
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
m : ∞ ∈ multibrotExt d
z : ℂ
a : 2 < Complex.abs z
⊢ HolomorphicAt I I (bottcher' d) z | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherHolomorphic | [571, 1] | [579, 86] | simp only [multibrotExt_coe] at m | case h₂
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
x✝ : ℂ
m : ↑x✝ ∈ multibrotExt d
⊢ HolomorphicAt I I (bottcher d) ↑x✝ | case h₂
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
x✝ : ℂ
m : x✝ ∉ multibrot d
⊢ HolomorphicAt I I (bottcher d) ↑x✝ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherHolomorphic | [571, 1] | [579, 86] | exact holomorphicAt_fill_coe ((bottcher_analytic (d := d) _ m).holomorphicAt I I) | case h₂
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
x✝ : ℂ
m : x✝ ∉ multibrot d
⊢ HolomorphicAt I I (bottcher d) ↑x✝ | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | abs_bottcher | [589, 1] | [593, 68] | set s := superF d | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
⊢ Complex.abs (bottcher d c) = potential d c | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
s : Super (f d) d ∞ := superF d
⊢ Complex.abs (bottcher d c) = potential d c |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | abs_bottcher | [589, 1] | [593, 68] | induction c using OnePoint.rec | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
s : Super (f d) d ∞ := superF d
⊢ Complex.abs (bottcher d c) = potential d c | case h₁
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ Complex.abs (bottcher d ∞) = potential d ∞
case h₂
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x✝ : ℂ
⊢ Complex.abs (bottcher d ↑x✝) = potential d ↑x✝ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | abs_bottcher | [589, 1] | [593, 68] | simp only [bottcher, potential, fill_inf, Complex.abs.map_zero] | case h₁
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ Complex.abs (bottcher d ∞) = potential d ∞ | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | abs_bottcher | [589, 1] | [593, 68] | simp only [bottcher, potential, fill_coe] | case h₂
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x✝ : ℂ
⊢ Complex.abs (bottcher d ↑x✝) = potential d ↑x✝ | case h₂
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x✝ : ℂ
⊢ Complex.abs (bottcher' d x✝) = ⋯.potential x✝ ↑x✝ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | abs_bottcher | [589, 1] | [593, 68] | exact s.abs_bottcher | case h₂
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x✝ : ℂ
⊢ Complex.abs (bottcher' d x✝) = ⋯.potential x✝ ↑x✝ | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | potential_continuous | [595, 1] | [603, 49] | set s := superF d | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ Continuous (potential d) | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ Continuous (potential d) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | potential_continuous | [595, 1] | [603, 49] | rw [continuous_iff_continuousAt] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ Continuous (potential d) | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ ∀ (x : 𝕊), ContinuousAt (potential d) x |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | potential_continuous | [595, 1] | [603, 49] | intro c | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ ∀ (x : 𝕊), ContinuousAt (potential d) x | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
c : 𝕊
⊢ ContinuousAt (potential d) c |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | potential_continuous | [595, 1] | [603, 49] | induction c using OnePoint.rec | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
c : 𝕊
⊢ ContinuousAt (potential d) c | case h₁
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ ContinuousAt (potential d) ∞
case h₂
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x✝ : ℂ
⊢ ContinuousAt (potential d) ↑x✝ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | potential_continuous | [595, 1] | [603, 49] | have e : potential d =ᶠ[𝓝 ∞] fun c ↦ abs (bottcher d c) := by
refine eventually_of_forall fun c ↦ ?_; rw [← abs_bottcher] | case h₁
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ ContinuousAt (potential d) ∞ | case h₁
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
e : (𝓝 ∞).EventuallyEq (potential d) fun c => Complex.abs (bottcher d c)
⊢ ContinuousAt (potential d) ∞ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | potential_continuous | [595, 1] | [603, 49] | rw [continuousAt_congr e] | case h₁
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
e : (𝓝 ∞).EventuallyEq (potential d) fun c => Complex.abs (bottcher d c)
⊢ ContinuousAt (potential d) ∞ | case h₁
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
e : (𝓝 ∞).EventuallyEq (potential d) fun c => Complex.abs (bottcher d c)
⊢ ContinuousAt (fun c => Complex.abs (bottcher d c)) ∞ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | potential_continuous | [595, 1] | [603, 49] | exact Complex.continuous_abs.continuousAt.comp
(bottcherHolomorphic d _ multibrotExt_inf).continuousAt | case h₁
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
e : (𝓝 ∞).EventuallyEq (potential d) fun c => Complex.abs (bottcher d c)
⊢ ContinuousAt (fun c => Complex.abs (bottcher d c)) ∞ | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | potential_continuous | [595, 1] | [603, 49] | refine eventually_of_forall fun c ↦ ?_ | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ (𝓝 ∞).EventuallyEq (potential d) fun c => Complex.abs (bottcher d c) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
c : 𝕊
⊢ potential d c = (fun c => Complex.abs (bottcher d c)) c |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | potential_continuous | [595, 1] | [603, 49] | rw [← abs_bottcher] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
c : 𝕊
⊢ potential d c = (fun c => Complex.abs (bottcher d c)) c | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | potential_continuous | [595, 1] | [603, 49] | exact continuousAt_fill_coe ((Continuous.potential s).comp₂
continuous_id continuous_coe).continuousAt | case h₂
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x✝ : ℂ
⊢ ContinuousAt (potential d) ↑x✝ | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | potential_lt_one | [605, 1] | [616, 48] | set s := superF d | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
⊢ potential d c < 1 ↔ c ∈ multibrotExt d | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
s : Super (f d) d ∞ := superF d
⊢ potential d c < 1 ↔ c ∈ multibrotExt d |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | potential_lt_one | [605, 1] | [616, 48] | induction c using OnePoint.rec | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
s : Super (f d) d ∞ := superF d
⊢ potential d c < 1 ↔ c ∈ multibrotExt d | case h₁
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ potential d ∞ < 1 ↔ ∞ ∈ multibrotExt d
case h₂
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x✝ : ℂ
⊢ potential d ↑x✝ < 1 ↔ ↑x✝ ∈ multibrotExt d |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | potential_lt_one | [605, 1] | [616, 48] | simp only [potential, bottcher, fill_inf, zero_lt_one, multibrotExt_inf] | case h₁
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ potential d ∞ < 1 ↔ ∞ ∈ multibrotExt d | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | potential_lt_one | [605, 1] | [616, 48] | constructor | case h₂
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x✝ : ℂ
⊢ potential d ↑x✝ < 1 ↔ ↑x✝ ∈ multibrotExt d | case h₂.mp
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x✝ : ℂ
⊢ potential d ↑x✝ < 1 → ↑x✝ ∈ multibrotExt d
case h₂.mpr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x✝ : ℂ
⊢ ↑x✝ ∈ multibrotExt d → potential d ↑x✝ < 1 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | potential_lt_one | [605, 1] | [616, 48] | intro h | case h₂.mp
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x✝ : ℂ
⊢ potential d ↑x✝ < 1 → ↑x✝ ∈ multibrotExt d | case h₂.mp
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x✝ : ℂ
h : potential d ↑x✝ < 1
⊢ ↑x✝ ∈ multibrotExt d |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | potential_lt_one | [605, 1] | [616, 48] | contrapose h | case h₂.mp
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x✝ : ℂ
h : potential d ↑x✝ < 1
⊢ ↑x✝ ∈ multibrotExt d | case h₂.mp
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x✝ : ℂ
h : ↑x✝ ∉ multibrotExt d
⊢ ¬potential d ↑x✝ < 1 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | potential_lt_one | [605, 1] | [616, 48] | simp only [not_not, not_lt, multibrot_basin', potential, fill_coe, Super.basin,
mem_setOf, not_exists, multibrotExt_coe] at h ⊢ | case h₂.mp
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x✝ : ℂ
h : ↑x✝ ∉ multibrotExt d
⊢ ¬potential d ↑x✝ < 1 | case h₂.mp
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x✝ : ℂ
h : ∀ (x : ℕ), (x✝, (f d x✝)^[x] ↑x✝) ∉ ⋯.near
⊢ 1 ≤ ⋯.potential x✝ ↑x✝ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | potential_lt_one | [605, 1] | [616, 48] | rw [s.potential_eq_one] | case h₂.mp
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x✝ : ℂ
h : ∀ (x : ℕ), (x✝, (f d x✝)^[x] ↑x✝) ∉ ⋯.near
⊢ 1 ≤ ⋯.potential x✝ ↑x✝ | case h₂.mp
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x✝ : ℂ
h : ∀ (x : ℕ), (x✝, (f d x✝)^[x] ↑x✝) ∉ ⋯.near
⊢ ∀ (n : ℕ), (x✝, (f d x✝)^[n] ↑x✝) ∉ s.near |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | potential_lt_one | [605, 1] | [616, 48] | exact h | case h₂.mp
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x✝ : ℂ
h : ∀ (x : ℕ), (x✝, (f d x✝)^[x] ↑x✝) ∉ ⋯.near
⊢ ∀ (n : ℕ), (x✝, (f d x✝)^[n] ↑x✝) ∉ s.near | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | potential_lt_one | [605, 1] | [616, 48] | intro m | case h₂.mpr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x✝ : ℂ
⊢ ↑x✝ ∈ multibrotExt d → potential d ↑x✝ < 1 | case h₂.mpr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x✝ : ℂ
m : ↑x✝ ∈ multibrotExt d
⊢ potential d ↑x✝ < 1 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | potential_lt_one | [605, 1] | [616, 48] | rw [← abs_bottcher] | case h₂.mpr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x✝ : ℂ
m : ↑x✝ ∈ multibrotExt d
⊢ potential d ↑x✝ < 1 | case h₂.mpr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x✝ : ℂ
m : ↑x✝ ∈ multibrotExt d
⊢ Complex.abs (bottcher d ↑x✝) < 1 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | potential_lt_one | [605, 1] | [616, 48] | simp only [bottcher, fill_coe] | case h₂.mpr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x✝ : ℂ
m : ↑x✝ ∈ multibrotExt d
⊢ Complex.abs (bottcher d ↑x✝) < 1 | case h₂.mpr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x✝ : ℂ
m : ↑x✝ ∈ multibrotExt d
⊢ Complex.abs (bottcher' d x✝) < 1 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | potential_lt_one | [605, 1] | [616, 48] | simp only [multibrotExt_coe] at m | case h₂.mpr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x✝ : ℂ
m : ↑x✝ ∈ multibrotExt d
⊢ Complex.abs (bottcher' d x✝) < 1 | case h₂.mpr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x✝ : ℂ
m : x✝ ∉ multibrot d
⊢ Complex.abs (bottcher' d x✝) < 1 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | potential_lt_one | [605, 1] | [616, 48] | exact s.bottcher_lt_one (multibrotPost m) | case h₂.mpr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
x✝ : ℂ
m : x✝ ∉ multibrot d
⊢ Complex.abs (bottcher' d x✝) < 1 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | potential_nonneg | [618, 1] | [621, 71] | induction c using OnePoint.rec | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
⊢ 0 ≤ potential d c | case h₁
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ 0 ≤ potential d ∞
case h₂
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x✝ : ℂ
⊢ 0 ≤ potential d ↑x✝ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | potential_nonneg | [618, 1] | [621, 71] | simp only [potential, fill_inf, le_refl] | case h₁
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ 0 ≤ potential d ∞ | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | potential_nonneg | [618, 1] | [621, 71] | simp only [potential, fill_coe] | case h₂
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x✝ : ℂ
⊢ 0 ≤ potential d ↑x✝ | case h₂
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x✝ : ℂ
⊢ 0 ≤ ⋯.potential x✝ ↑x✝ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | potential_nonneg | [618, 1] | [621, 71] | exact (superF d).potential_nonneg | case h₂
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x✝ : ℂ
⊢ 0 ≤ ⋯.potential x✝ ↑x✝ | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | potential_eq_zero | [623, 1] | [626, 81] | induction c using OnePoint.rec | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
⊢ potential d c = 0 ↔ c = ∞ | case h₁
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ potential d ∞ = 0 ↔ ∞ = ∞
case h₂
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x✝ : ℂ
⊢ potential d ↑x✝ = 0 ↔ ↑x✝ = ∞ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | potential_eq_zero | [623, 1] | [626, 81] | simp only [potential, fill_inf, eq_self_iff_true] | case h₁
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ potential d ∞ = 0 ↔ ∞ = ∞ | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | potential_eq_zero | [623, 1] | [626, 81] | simp only [potential, fill_coe, (superF d).potential_eq_zero_of_onePreimage] | case h₂
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x✝ : ℂ
⊢ potential d ↑x✝ = 0 ↔ ↑x✝ = ∞ | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | by_cases h : ∃ᶠ e in 𝓝 c, bottcher d e ≠ bottcher d c | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
⊢ NontrivialHolomorphicAt (bottcher d) c | case pos
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
h : ∃ᶠ (e : 𝕊) in 𝓝 c, bottcher d e ≠ bottcher d c
⊢ NontrivialHolomorphicAt (bottcher d) c
case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
h : ¬∃ᶠ (e : 𝕊) in 𝓝 c, bottcher d e ≠ bottcher d c
⊢ NontrivialHolomorphicAt... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | exact
{ holomorphicAt := bottcherHolomorphic d _ m
nonconst := h } | case pos
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
h : ∃ᶠ (e : 𝕊) in 𝓝 c, bottcher d e ≠ bottcher d c
⊢ NontrivialHolomorphicAt (bottcher d) c
case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
h : ¬∃ᶠ (e : 𝕊) in 𝓝 c, bottcher d e ≠ bottcher d c
⊢ NontrivialHolomorphicAt... | case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
h : ¬∃ᶠ (e : 𝕊) in 𝓝 c, bottcher d e ≠ bottcher d c
⊢ NontrivialHolomorphicAt (bottcher d) c |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | exfalso | case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
h : ¬∃ᶠ (e : 𝕊) in 𝓝 c, bottcher d e ≠ bottcher d c
⊢ NontrivialHolomorphicAt (bottcher d) c | case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
h : ¬∃ᶠ (e : 𝕊) in 𝓝 c, bottcher d e ≠ bottcher d c
⊢ False |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | simp only [Filter.not_frequently, not_not] at h | case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
h : ¬∃ᶠ (e : 𝕊) in 𝓝 c, bottcher d e ≠ bottcher d c
⊢ False | case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
h : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = bottcher d c
⊢ False |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | set b := bottcher d c | case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multibrotExt d
h : ∀ᶠ (x : 𝕊) in 𝓝 c, bottcher d x = bottcher d c
⊢ False | case neg
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
⊢ False |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | have b1 : abs b < 1 := by simp only [abs_bottcher, potential_lt_one, m, b] | case neg
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
⊢ False | case neg
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
⊢ False |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | set t := {c | c ∈ multibrotExt d ∧ ∀ᶠ e in 𝓝 c, bottcher d e = b} | case neg
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
⊢ False | case neg
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}
⊢ False |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | have m0 : (0 : 𝕊) ∈ multibrotExt d :=
haveI m : (0 : 𝕊) ∈ t := by simp only [tu, mem_univ]
m.1 | case neg
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
⊢ False | case neg
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
m0 : 0 ∈ multibrotExt d
⊢ False |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | simp only [← coe_zero, multibrotExt_coe, multibrot_zero, not_true] at m0 | case neg
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
m0 : 0 ∈ multibrotExt d
⊢ False | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | simp only [abs_bottcher, potential_lt_one, m, 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
⊢ Complex.abs b < 1 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | refine IsClopen.eq_univ ?_ ⟨c, m, 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}
⊢ t = 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}
⊢ IsClopen t |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | constructor | 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}
⊢ IsClopen 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}
⊢ IsClosed t
case right
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : 𝕊
m : c ∈ multib... |
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