url stringclasses 147
values | commit stringclasses 147
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https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | set t := Ici (log (log (abs z)) - r) | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | have yt : log (-log p) ∈ t := by
simp only [abs_le, neg_le_sub_iff_le_add, tsub_le_iff_right, add_comm r] at h
simp only [mem_Ici, tsub_le_iff_right, h, t] | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | have lt : log (log (abs z)) ∈ t := by
simp only [mem_Ici, tsub_le_iff_right, le_add_iff_nonneg_right, r0, t] | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | generalize hb : dene (log (log (abs z)) - r) = b | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | have b0 : 0 ≤ b := by rw [←hb]; exact dene_nonneg | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | have bound : ∀ x, x ∈ t → ‖deriv ene x‖ ≤ b := by
intro x m
simp only [Real.dist_eq, mem_Ici, ←hr, t] at m
simp only [deriv_ene, norm_neg, Real.norm_of_nonneg dene_nonneg, ←hb, ←hr]
apply dene_anti (sub_nonneg.mpr (iter_error_le_log_log_abs d z4 cz)) m | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | have m := Convex.norm_image_sub_le_of_norm_deriv_le
(fun x _ ↦ (hasDerivAt_ene x).differentiableAt) bound (convex_Ici _) lt yt | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b
m : ‖ene (-p.log).log - ene (Complex.abs z).log.log‖ ≤ b * ‖(-p.log).log - (Complex.abs z).log.log‖
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | simp only [Real.norm_eq_abs] at m | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b
m : ‖ene (-p.log).log - ene (Complex.abs z).log.log‖ ≤ b * ‖(-p.log).log - (Complex.abs z).log.log‖
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b
m : |ene (-p.log).log - ene (Complex.abs z).log.log| ≤ b * |(-p.log).log - (Complex.abs z).log.log|
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | replace m := le_trans m (mul_le_mul_of_nonneg_left h (by bound)) | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b
m : |ene (-p.log).log - ene (Complex.abs z).log.log| ≤ b * |(-p.log).log - (Complex.abs z).log.log|
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b
m : |ene (-p.log).log - ene (Complex.abs z).log.log| ≤ b * r
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | simp only [ene, Real.exp_log lp0, neg_neg, Real.exp_log p0, Real.exp_log l2, Real.exp_neg,
Real.exp_log z0, inv_eq_one_div] at m | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b
m : |ene (-p.log).log - ene (Complex.abs z).log.log| ≤ b * r
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b
m : |p - 1 / Complex.abs z| ≤ b * r
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | refine le_trans m (le_of_eq ?_) | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b
m : |p - 1 / Complex.abs z| ≤ b * r
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b
m : |p - 1 / Complex.abs z| ≤ b * r
⊢ b * r = potential_error d c z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | simp only [←hr, ←hb, potential_error] | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b
m : |p - 1 / Complex.abs z| ≤ b * r
⊢ b * r = potential_error d c z | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | norm_num | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
⊢ 3 ≤ 4 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | norm_num | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
⊢ 0 < 3 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | linarith | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
⊢ 1 < Complex.abs z | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | linarith | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
h : |(-(⋯.potential c ↑z).log).log - (Complex.abs z).log.log| ≤ iter_error d c z
p0 : 0 < s.potential c ↑z
⊢ 2 < Complex.abs z | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | simp only [abs_le, neg_le_sub_iff_le_add, tsub_le_iff_right, add_comm r] at h | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
⊢ (-p.log).log ∈ t | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
h : (Complex.abs z).log.log ≤ (-p.log).log + r ∧ (-p.log).log ≤ (Complex.abs z).log.log + r
⊢ (-p.log).log ∈ t |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | simp only [mem_Ici, tsub_le_iff_right, h, t] | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
h : (Complex.abs z).log.log ≤ (-p.log).log + r ∧ (-p.log).log ≤ (Complex.abs z).log.log + r
⊢ (-p.log).log ∈ t | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | simp only [mem_Ici, tsub_le_iff_right, le_add_iff_nonneg_right, r0, t] | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
⊢ (Complex.abs z).log.log ∈ t | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | rw [←hb] | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
⊢ 0 ≤ b | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
⊢ 0 ≤ dene ((Complex.abs z).log.log - r) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | exact dene_nonneg | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
⊢ 0 ≤ dene ((Complex.abs z).log.log - r) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | intro x m | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
⊢ ∀ x ∈ t, ‖deriv ene x‖ ≤ b | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
x : ℝ
m : x ∈ t
⊢ ‖deriv ene x‖ ≤ b |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | simp only [Real.dist_eq, mem_Ici, ←hr, t] at m | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
x : ℝ
m : x ∈ t
⊢ ‖deriv ene x‖ ≤ b | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
x : ℝ
m : (Complex.abs z).log.log - iter_error d c z ≤ x
⊢ ‖deriv ene x‖ ≤ b |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | simp only [deriv_ene, norm_neg, Real.norm_of_nonneg dene_nonneg, ←hb, ←hr] | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
x : ℝ
m : (Complex.abs z).log.log - iter_error d c z ≤ x
⊢ ‖deriv ene x‖ ≤ b | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
x : ℝ
m : (Complex.abs z).log.log - iter_error d c z ≤ x
⊢ dene x ≤ dene ((Complex.abs z).log.log - iter_error d c z) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | apply dene_anti (sub_nonneg.mpr (iter_error_le_log_log_abs d z4 cz)) m | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
x : ℝ
m : (Complex.abs z).log.log - iter_error d c z ≤ x
⊢ dene x ≤ dene ((Complex.abs z).log.log - iter_error d c z) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | bound | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b
m : |ene (-p.log).log - ene (Complex.abs z).log.log| ≤ b * |(-p.log).log - (Complex.abs z).log.log|
⊢ 0 ≤ b | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.coe_eq_zero | [40, 9] | [41, 37] | simp only [← coe_zero, coe_eq_coe] | z : ℂ
⊢ ↑z = 0 ↔ z = 0 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.map_rec | [54, 1] | [58, 39] | induction z using OnePoint.rec | A : Sort u_1
B : Sort u_2
g : A → B
f : ℂ → A
i : A
z : 𝕊
⊢ g (OnePoint.rec i f z) = OnePoint.rec (g i) (g ∘ f) z | case h₁
A : Sort u_1
B : Sort u_2
g : A → B
f : ℂ → A
i : A
⊢ g (OnePoint.rec i f ∞) = OnePoint.rec (g i) (g ∘ f) ∞
case h₂
A : Sort u_1
B : Sort u_2
g : A → B
f : ℂ → A
i : A
x✝ : ℂ
⊢ g (OnePoint.rec i f ↑x✝) = OnePoint.rec (g i) (g ∘ f) ↑x✝ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.map_rec | [54, 1] | [58, 39] | simp only [rec_inf] | case h₁
A : Sort u_1
B : Sort u_2
g : A → B
f : ℂ → A
i : A
⊢ g (OnePoint.rec i f ∞) = OnePoint.rec (g i) (g ∘ f) ∞ | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.map_rec | [54, 1] | [58, 39] | simp only [rec_coe, Function.comp] | case h₂
A : Sort u_1
B : Sort u_2
g : A → B
f : ℂ → A
i : A
x✝ : ℂ
⊢ g (OnePoint.rec i f ↑x✝) = OnePoint.rec (g i) (g ∘ f) ↑x✝ | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.inf_ne_coe | [61, 1] | [62, 55] | simp only [Ne, OnePoint.infty_ne_coe, not_false_iff] | z : ℂ
⊢ ∞ ≠ ↑z | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.inf_ne_zero | [63, 1] | [64, 68] | have e : (0 : 𝕊) = ((0 : ℂ) : 𝕊) := rfl | ⊢ ∞ ≠ 0 | e : 0 = ↑0
⊢ ∞ ≠ 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.inf_ne_zero | [63, 1] | [64, 68] | rw [e] | e : 0 = ↑0
⊢ ∞ ≠ 0 | e : 0 = ↑0
⊢ ∞ ≠ ↑0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.inf_ne_zero | [63, 1] | [64, 68] | exact inf_ne_coe | e : 0 = ↑0
⊢ ∞ ≠ ↑0 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.coe_toComplex | [71, 1] | [74, 30] | induction z using OnePoint.rec | z : 𝕊
h : z ≠ ∞
⊢ ↑z.toComplex = z | case h₁
h : ∞ ≠ ∞
⊢ ↑∞.toComplex = ∞
case h₂
x✝ : ℂ
h : ↑x✝ ≠ ∞
⊢ ↑(↑x✝).toComplex = ↑x✝ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.coe_toComplex | [71, 1] | [74, 30] | simp only [ne_eq, not_true_eq_false] at h | case h₁
h : ∞ ≠ ∞
⊢ ↑∞.toComplex = ∞ | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.coe_toComplex | [71, 1] | [74, 30] | simp only [toComplex_coe] | case h₂
x✝ : ℂ
h : ↑x✝ ≠ ∞
⊢ ↑(↑x✝).toComplex = ↑x✝ | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.toComplex_zero | [75, 9] | [75, 92] | rw [← coe_zero, toComplex_coe] | ⊢ OnePoint.toComplex 0 = 0 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuousAt_toComplex | [76, 1] | [77, 93] | simp only [OnePoint.continuousAt_coe, Function.comp, toComplex_coe] | z : ℂ
⊢ ContinuousAt OnePoint.toComplex ↑z | z : ℂ
⊢ ContinuousAt (fun x => x) z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuousAt_toComplex | [76, 1] | [77, 93] | exact continuousAt_id | z : ℂ
⊢ ContinuousAt (fun x => x) z | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuousOn_toComplex | [78, 1] | [81, 52] | intro z m | ⊢ ContinuousOn OnePoint.toComplex {∞}ᶜ | z : 𝕊
m : z ∈ {∞}ᶜ
⊢ ContinuousWithinAt OnePoint.toComplex {∞}ᶜ z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuousOn_toComplex | [78, 1] | [81, 52] | induction z using OnePoint.rec | z : 𝕊
m : z ∈ {∞}ᶜ
⊢ ContinuousWithinAt OnePoint.toComplex {∞}ᶜ z | case h₁
m : ∞ ∈ {∞}ᶜ
⊢ ContinuousWithinAt OnePoint.toComplex {∞}ᶜ ∞
case h₂
x✝ : ℂ
m : ↑x✝ ∈ {∞}ᶜ
⊢ ContinuousWithinAt OnePoint.toComplex {∞}ᶜ ↑x✝ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuousOn_toComplex | [78, 1] | [81, 52] | simp only [mem_compl_iff, mem_singleton_iff, not_true] at m | case h₁
m : ∞ ∈ {∞}ᶜ
⊢ ContinuousWithinAt OnePoint.toComplex {∞}ᶜ ∞ | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuousOn_toComplex | [78, 1] | [81, 52] | exact continuousAt_toComplex.continuousWithinAt | case h₂
x✝ : ℂ
m : ↑x✝ ∈ {∞}ᶜ
⊢ ContinuousWithinAt OnePoint.toComplex {∞}ᶜ ↑x✝ | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.inv_zero' | [97, 9] | [97, 100] | simp only [inv_def, inv, eq_self_iff_true, if_true] | ⊢ 0⁻¹ = ∞ | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.inv_inf | [98, 9] | [98, 85] | simp [inv_def, inv, inf_ne_zero] | ⊢ ∞⁻¹ = 0 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.inv_coe | [100, 1] | [101, 90] | simp only [inv_def, inv, z0, WithTop.coe_eq_zero, toComplex_coe, if_false, coe_eq_zero] | z : ℂ
z0 : z ≠ 0
⊢ (↑z)⁻¹ = ↑z⁻¹ | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.inv_eq_inf | [102, 9] | [105, 89] | induction z using OnePoint.rec | z : 𝕊
⊢ z⁻¹ = ∞ ↔ z = 0 | case h₁
⊢ ∞⁻¹ = ∞ ↔ ∞ = 0
case h₂
x✝ : ℂ
⊢ (↑x✝)⁻¹ = ∞ ↔ ↑x✝ = 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.inv_eq_inf | [102, 9] | [105, 89] | simp only [inv_inf] | case h₁
⊢ ∞⁻¹ = ∞ ↔ ∞ = 0 | case h₁
⊢ 0 = ∞ ↔ ∞ = 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.inv_eq_inf | [102, 9] | [105, 89] | exact ⟨Eq.symm, Eq.symm⟩ | case h₁
⊢ 0 = ∞ ↔ ∞ = 0 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.inv_eq_inf | [102, 9] | [105, 89] | simp only [inv_def, inv, not_not, imp_false, ite_eq_left_iff, OnePoint.coe_ne_infty] | case h₂
x✝ : ℂ
⊢ (↑x✝)⁻¹ = ∞ ↔ ↑x✝ = 0 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.inv_eq_zero | [106, 9] | [112, 38] | induction' z using OnePoint.rec with z | z : 𝕊
⊢ z⁻¹ = 0 ↔ z = ∞ | case h₁
⊢ ∞⁻¹ = 0 ↔ ∞ = ∞
case h₂
z : ℂ
⊢ (↑z)⁻¹ = 0 ↔ ↑z = ∞ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.inv_eq_zero | [106, 9] | [112, 38] | simp only [inv_inf, eq_self_iff_true] | case h₁
⊢ ∞⁻¹ = 0 ↔ ∞ = ∞ | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.inv_eq_zero | [106, 9] | [112, 38] | simp only [inv_def, inv, toComplex_coe] | case h₂
z : ℂ
⊢ (↑z)⁻¹ = 0 ↔ ↑z = ∞ | case h₂
z : ℂ
⊢ (if ↑z = 0 then ∞ else ↑z⁻¹) = 0 ↔ ↑z = ∞ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.inv_eq_zero | [106, 9] | [112, 38] | by_cases z0 : (z : 𝕊) = 0 | case h₂
z : ℂ
⊢ (if ↑z = 0 then ∞ else ↑z⁻¹) = 0 ↔ ↑z = ∞ | case pos
z : ℂ
z0 : ↑z = 0
⊢ (if ↑z = 0 then ∞ else ↑z⁻¹) = 0 ↔ ↑z = ∞
case neg
z : ℂ
z0 : ¬↑z = 0
⊢ (if ↑z = 0 then ∞ else ↑z⁻¹) = 0 ↔ ↑z = ∞ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.inv_eq_zero | [106, 9] | [112, 38] | simp only [if_pos, z0, inf_ne_zero, inf_ne_zero.symm] | case pos
z : ℂ
z0 : ↑z = 0
⊢ (if ↑z = 0 then ∞ else ↑z⁻¹) = 0 ↔ ↑z = ∞
case neg
z : ℂ
z0 : ¬↑z = 0
⊢ (if ↑z = 0 then ∞ else ↑z⁻¹) = 0 ↔ ↑z = ∞ | case neg
z : ℂ
z0 : ¬↑z = 0
⊢ (if ↑z = 0 then ∞ else ↑z⁻¹) = 0 ↔ ↑z = ∞ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.inv_eq_zero | [106, 9] | [112, 38] | simp only [if_neg z0, coe_ne_inf, iff_false_iff] | case neg
z : ℂ
z0 : ¬↑z = 0
⊢ (if ↑z = 0 then ∞ else ↑z⁻¹) = 0 ↔ ↑z = ∞ | case neg
z : ℂ
z0 : ¬↑z = 0
⊢ ¬↑z⁻¹ = 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.inv_eq_zero | [106, 9] | [112, 38] | rw [coe_eq_zero, _root_.inv_eq_zero] | case neg
z : ℂ
z0 : ¬↑z = 0
⊢ ¬↑z⁻¹ = 0 | case neg
z : ℂ
z0 : ¬↑z = 0
⊢ ¬z = 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.inv_eq_zero | [106, 9] | [112, 38] | simpa only [coe_eq_zero] using z0 | case neg
z : ℂ
z0 : ¬↑z = 0
⊢ ¬z = 0 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.toComplex_inv | [113, 1] | [118, 64] | induction' z using OnePoint.rec with z | z : 𝕊
⊢ z⁻¹.toComplex = z.toComplex⁻¹ | case h₁
⊢ ∞⁻¹.toComplex = ∞.toComplex⁻¹
case h₂
z : ℂ
⊢ (↑z)⁻¹.toComplex = (↑z).toComplex⁻¹ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.toComplex_inv | [113, 1] | [118, 64] | simp only [inv_inf, toComplex_zero, toComplex_inf, inv_zero', inv_zero, eq_self_iff_true] | case h₁
⊢ ∞⁻¹.toComplex = ∞.toComplex⁻¹ | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.toComplex_inv | [113, 1] | [118, 64] | by_cases z0 : z = 0 | case h₂
z : ℂ
⊢ (↑z)⁻¹.toComplex = (↑z).toComplex⁻¹ | case pos
z : ℂ
z0 : z = 0
⊢ (↑z)⁻¹.toComplex = (↑z).toComplex⁻¹
case neg
z : ℂ
z0 : ¬z = 0
⊢ (↑z)⁻¹.toComplex = (↑z).toComplex⁻¹ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.toComplex_inv | [113, 1] | [118, 64] | simp only [z0, coe_zero, inv_zero', toComplex_inf, toComplex_zero, inv_zero] | case pos
z : ℂ
z0 : z = 0
⊢ (↑z)⁻¹.toComplex = (↑z).toComplex⁻¹ | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.toComplex_inv | [113, 1] | [118, 64] | simp only [z0, inv_coe, Ne, not_false_iff, toComplex_coe] | case neg
z : ℂ
z0 : ¬z = 0
⊢ (↑z)⁻¹.toComplex = (↑z).toComplex⁻¹ | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.coe_tendsto_inf | [121, 1] | [123, 27] | rw [Filter.tendsto_iff_comap, OnePoint.comap_coe_nhds_infty, Filter.coclosedCompact_eq_cocompact] | ⊢ Tendsto (fun z => ↑z) atInf (𝓝 ∞) | ⊢ atInf ≤ Filter.cocompact ℂ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.coe_tendsto_inf | [121, 1] | [123, 27] | exact atInf_le_cocompact | ⊢ atInf ≤ Filter.cocompact ℂ | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.coe_tendsto_inf' | [126, 1] | [133, 59] | simp only [e, tendsto_nhdsWithin_range, coe_tendsto_inf] | e : {∞}ᶜ = range fun z => ↑z
⊢ Tendsto (fun z => ↑z) atInf (𝓝[≠] ∞) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.coe_tendsto_inf' | [126, 1] | [133, 59] | ext z | ⊢ {∞}ᶜ = range fun z => ↑z | case h
z : 𝕊
⊢ z ∈ {∞}ᶜ ↔ z ∈ range fun z => ↑z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.coe_tendsto_inf' | [126, 1] | [133, 59] | induction' z using OnePoint.rec with z | case h
z : 𝕊
⊢ z ∈ {∞}ᶜ ↔ z ∈ range fun z => ↑z | case h.h₁
⊢ ∞ ∈ {∞}ᶜ ↔ ∞ ∈ range fun z => ↑z
case h.h₂
z : ℂ
⊢ ↑z ∈ {∞}ᶜ ↔ ↑z ∈ range fun z => ↑z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.coe_tendsto_inf' | [126, 1] | [133, 59] | simp only [mem_compl_iff, mem_singleton_iff, not_true, mem_range, OnePoint.coe_ne_infty,
exists_false] | case h.h₁
⊢ ∞ ∈ {∞}ᶜ ↔ ∞ ∈ range fun z => ↑z | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.coe_tendsto_inf' | [126, 1] | [133, 59] | simp only [mem_compl_iff, mem_singleton_iff, OnePoint.coe_ne_infty, not_false_eq_true,
mem_range, coe_eq_coe, exists_eq] | case h.h₂
z : ℂ
⊢ ↑z ∈ {∞}ᶜ ↔ ↑z ∈ range fun z => ↑z | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_inv | [136, 1] | [166, 63] | rw [continuous_iff_continuousOn_univ] | ⊢ Continuous fun z => z⁻¹ | ⊢ ContinuousOn (fun z => z⁻¹) univ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_inv | [136, 1] | [166, 63] | intro z _ | ⊢ ContinuousOn (fun z => z⁻¹) univ | z : 𝕊
a✝ : z ∈ univ
⊢ ContinuousWithinAt (fun z => z⁻¹) univ z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_inv | [136, 1] | [166, 63] | apply ContinuousAt.continuousWithinAt | z : 𝕊
a✝ : z ∈ univ
⊢ ContinuousWithinAt (fun z => z⁻¹) univ z | case h
z : 𝕊
a✝ : z ∈ univ
⊢ ContinuousAt (fun z => z⁻¹) z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_inv | [136, 1] | [166, 63] | induction' z using OnePoint.rec with z | case h
z : 𝕊
a✝ : z ∈ univ
⊢ ContinuousAt (fun z => z⁻¹) z | case h.h₁
a✝ : ∞ ∈ univ
⊢ ContinuousAt (fun z => z⁻¹) ∞
case h.h₂
z : ℂ
a✝ : ↑z ∈ univ
⊢ ContinuousAt (fun z => z⁻¹) ↑z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_inv | [136, 1] | [166, 63] | simp only [OnePoint.continuousAt_infty', Function.comp, Filter.coclosedCompact_eq_cocompact,
inv_inf, ← atInf_eq_cocompact] | case h.h₁
a✝ : ∞ ∈ univ
⊢ ContinuousAt (fun z => z⁻¹) ∞ | case h.h₁
a✝ : ∞ ∈ univ
⊢ Tendsto (fun x => (↑x)⁻¹) atInf (𝓝 0) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_inv | [136, 1] | [166, 63] | have e : ∀ᶠ z : ℂ in atInf, ↑z⁻¹ = (↑z : 𝕊)⁻¹ := by
refine (eventually_atInf 0).mp (eventually_of_forall fun z z0 ↦ ?_)
simp only [gt_iff_lt, Complex.norm_eq_abs, AbsoluteValue.pos_iff] at z0; rw [inv_coe z0] | case h.h₁
a✝ : ∞ ∈ univ
⊢ Tendsto (fun x => (↑x)⁻¹) atInf (𝓝 0) | case h.h₁
a✝ : ∞ ∈ univ
e : ∀ᶠ (z : ℂ) in atInf, ↑z⁻¹ = (↑z)⁻¹
⊢ Tendsto (fun x => (↑x)⁻¹) atInf (𝓝 0) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_inv | [136, 1] | [166, 63] | apply Filter.Tendsto.congr' e | case h.h₁
a✝ : ∞ ∈ univ
e : ∀ᶠ (z : ℂ) in atInf, ↑z⁻¹ = (↑z)⁻¹
⊢ Tendsto (fun x => (↑x)⁻¹) atInf (𝓝 0) | case h.h₁
a✝ : ∞ ∈ univ
e : ∀ᶠ (z : ℂ) in atInf, ↑z⁻¹ = (↑z)⁻¹
⊢ Tendsto (fun x => ↑x⁻¹) atInf (𝓝 0) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_inv | [136, 1] | [166, 63] | exact Filter.Tendsto.comp continuous_coe.continuousAt inv_tendsto_atInf' | case h.h₁
a✝ : ∞ ∈ univ
e : ∀ᶠ (z : ℂ) in atInf, ↑z⁻¹ = (↑z)⁻¹
⊢ Tendsto (fun x => ↑x⁻¹) atInf (𝓝 0) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_inv | [136, 1] | [166, 63] | refine (eventually_atInf 0).mp (eventually_of_forall fun z z0 ↦ ?_) | a✝ : ∞ ∈ univ
⊢ ∀ᶠ (z : ℂ) in atInf, ↑z⁻¹ = (↑z)⁻¹ | a✝ : ∞ ∈ univ
z : ℂ
z0 : ‖z‖ > 0
⊢ ↑z⁻¹ = (↑z)⁻¹ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_inv | [136, 1] | [166, 63] | simp only [gt_iff_lt, Complex.norm_eq_abs, AbsoluteValue.pos_iff] at z0 | a✝ : ∞ ∈ univ
z : ℂ
z0 : ‖z‖ > 0
⊢ ↑z⁻¹ = (↑z)⁻¹ | a✝ : ∞ ∈ univ
z : ℂ
z0 : z ≠ 0
⊢ ↑z⁻¹ = (↑z)⁻¹ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_inv | [136, 1] | [166, 63] | rw [inv_coe z0] | a✝ : ∞ ∈ univ
z : ℂ
z0 : z ≠ 0
⊢ ↑z⁻¹ = (↑z)⁻¹ | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_inv | [136, 1] | [166, 63] | simp only [OnePoint.continuousAt_coe, Function.comp, inv_def, inv, WithTop.coe_eq_zero,
toComplex_coe] | case h.h₂
z : ℂ
a✝ : ↑z ∈ univ
⊢ ContinuousAt (fun z => z⁻¹) ↑z | case h.h₂
z : ℂ
a✝ : ↑z ∈ univ
⊢ ContinuousAt (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_inv | [136, 1] | [166, 63] | by_cases z0 : z = 0 | case h.h₂
z : ℂ
a✝ : ↑z ∈ univ
⊢ ContinuousAt (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) z | case pos
z : ℂ
a✝ : ↑z ∈ univ
z0 : z = 0
⊢ ContinuousAt (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) z
case neg
z : ℂ
a✝ : ↑z ∈ univ
z0 : ¬z = 0
⊢ ContinuousAt (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_inv | [136, 1] | [166, 63] | simp only [z0, ContinuousAt, OnePoint.nhds_infty_eq, eq_self_iff_true, if_true,
Filter.coclosedCompact_eq_cocompact] | case pos
z : ℂ
a✝ : ↑z ∈ univ
z0 : z = 0
⊢ ContinuousAt (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) z | case pos
z : ℂ
a✝ : ↑z ∈ univ
z0 : z = 0
⊢ Tendsto (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) (𝓝 0) (𝓝 (if ↑0 = 0 then ∞ else ↑0⁻¹)) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_inv | [136, 1] | [166, 63] | simp only [← nhdsWithin_compl_singleton_sup_pure, Filter.tendsto_sup] | case pos
z : ℂ
a✝ : ↑z ∈ univ
z0 : z = 0
⊢ Tendsto (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) (𝓝 0) (𝓝 (if ↑0 = 0 then ∞ else ↑0⁻¹)) | case pos
z : ℂ
a✝ : ↑z ∈ univ
z0 : z = 0
⊢ Tendsto (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) (𝓝[≠] 0)
((𝓝[≠] if ↑0 = 0 then ∞ else ↑0⁻¹) ⊔ pure (if ↑0 = 0 then ∞ else ↑0⁻¹)) ∧
Tendsto (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) (pure 0)
((𝓝[≠] if ↑0 = 0 then ∞ else ↑0⁻¹) ⊔ pure (if ↑0 = 0 then ∞ else ↑0⁻¹)) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_inv | [136, 1] | [166, 63] | constructor | case pos
z : ℂ
a✝ : ↑z ∈ univ
z0 : z = 0
⊢ Tendsto (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) (𝓝[≠] 0)
((𝓝[≠] if ↑0 = 0 then ∞ else ↑0⁻¹) ⊔ pure (if ↑0 = 0 then ∞ else ↑0⁻¹)) ∧
Tendsto (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) (pure 0)
((𝓝[≠] if ↑0 = 0 then ∞ else ↑0⁻¹) ⊔ pure (if ↑0 = 0 then ∞ else ↑0⁻¹)) | case pos.left
z : ℂ
a✝ : ↑z ∈ univ
z0 : z = 0
⊢ Tendsto (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) (𝓝[≠] 0)
((𝓝[≠] if ↑0 = 0 then ∞ else ↑0⁻¹) ⊔ pure (if ↑0 = 0 then ∞ else ↑0⁻¹))
case pos.right
z : ℂ
a✝ : ↑z ∈ univ
z0 : z = 0
⊢ Tendsto (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) (pure 0)
((𝓝[≠] if ↑0 = 0 then ∞ else ↑0⁻¹) ⊔ pure (if ↑0 = 0 then ∞ else ↑0⁻¹)) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_inv | [136, 1] | [166, 63] | refine Filter.Tendsto.mono_right ?_ le_sup_left | case pos.left
z : ℂ
a✝ : ↑z ∈ univ
z0 : z = 0
⊢ Tendsto (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) (𝓝[≠] 0)
((𝓝[≠] if ↑0 = 0 then ∞ else ↑0⁻¹) ⊔ pure (if ↑0 = 0 then ∞ else ↑0⁻¹)) | case pos.left
z : ℂ
a✝ : ↑z ∈ univ
z0 : z = 0
⊢ Tendsto (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) (𝓝[≠] 0) (𝓝[≠] if ↑0 = 0 then ∞ else ↑0⁻¹) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_inv | [136, 1] | [166, 63] | apply tendsto_nhdsWithin_congr (f := fun z : ℂ ↦ (↑z⁻¹ : 𝕊)) | case pos.left
z : ℂ
a✝ : ↑z ∈ univ
z0 : z = 0
⊢ Tendsto (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) (𝓝[≠] 0) (𝓝[≠] if ↑0 = 0 then ∞ else ↑0⁻¹) | case pos.left.hfg
z : ℂ
a✝ : ↑z ∈ univ
z0 : z = 0
⊢ ∀ x ∈ {0}ᶜ, ↑x⁻¹ = if ↑x = 0 then ∞ else ↑x⁻¹
case pos.left.hf
z : ℂ
a✝ : ↑z ∈ univ
z0 : z = 0
⊢ Tendsto (fun z => ↑z⁻¹) (𝓝[≠] 0) (𝓝[≠] if ↑0 = 0 then ∞ else ↑0⁻¹) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_inv | [136, 1] | [166, 63] | intro z m | case pos.left.hfg
z : ℂ
a✝ : ↑z ∈ univ
z0 : z = 0
⊢ ∀ x ∈ {0}ᶜ, ↑x⁻¹ = if ↑x = 0 then ∞ else ↑x⁻¹ | case pos.left.hfg
z✝ : ℂ
a✝ : ↑z✝ ∈ univ
z0 : z✝ = 0
z : ℂ
m : z ∈ {0}ᶜ
⊢ ↑z⁻¹ = if ↑z = 0 then ∞ else ↑z⁻¹ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_inv | [136, 1] | [166, 63] | rw [mem_compl_singleton_iff] at m | case pos.left.hfg
z✝ : ℂ
a✝ : ↑z✝ ∈ univ
z0 : z✝ = 0
z : ℂ
m : z ∈ {0}ᶜ
⊢ ↑z⁻¹ = if ↑z = 0 then ∞ else ↑z⁻¹ | case pos.left.hfg
z✝ : ℂ
a✝ : ↑z✝ ∈ univ
z0 : z✝ = 0
z : ℂ
m : z ≠ 0
⊢ ↑z⁻¹ = if ↑z = 0 then ∞ else ↑z⁻¹ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_inv | [136, 1] | [166, 63] | simp only [coe_eq_zero, m, ite_false] | case pos.left.hfg
z✝ : ℂ
a✝ : ↑z✝ ∈ univ
z0 : z✝ = 0
z : ℂ
m : z ≠ 0
⊢ ↑z⁻¹ = if ↑z = 0 then ∞ else ↑z⁻¹ | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_inv | [136, 1] | [166, 63] | simp only [coe_zero, ite_true] | case pos.left.hf
z : ℂ
a✝ : ↑z ∈ univ
z0 : z = 0
⊢ Tendsto (fun z => ↑z⁻¹) (𝓝[≠] 0) (𝓝[≠] if ↑0 = 0 then ∞ else ↑0⁻¹) | case pos.left.hf
z : ℂ
a✝ : ↑z ∈ univ
z0 : z = 0
⊢ Tendsto (fun z => ↑z⁻¹) (𝓝[≠] 0) (𝓝[≠] ∞) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_inv | [136, 1] | [166, 63] | apply coe_tendsto_inf'.comp | case pos.left.hf
z : ℂ
a✝ : ↑z ∈ univ
z0 : z = 0
⊢ Tendsto (fun z => ↑z⁻¹) (𝓝[≠] 0) (𝓝[≠] ∞) | case pos.left.hf
z : ℂ
a✝ : ↑z ∈ univ
z0 : z = 0
⊢ Tendsto (fun z => z⁻¹) (𝓝[≠] 0) atInf |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_inv | [136, 1] | [166, 63] | rw [← @tendsto_atInf_iff_tendsto_nhds_zero ℂ ℂ _ _ fun z : ℂ ↦ z] | case pos.left.hf
z : ℂ
a✝ : ↑z ∈ univ
z0 : z = 0
⊢ Tendsto (fun z => z⁻¹) (𝓝[≠] 0) atInf | case pos.left.hf
z : ℂ
a✝ : ↑z ∈ univ
z0 : z = 0
⊢ Tendsto (fun z => z) atInf atInf |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_inv | [136, 1] | [166, 63] | exact Filter.tendsto_id | case pos.left.hf
z : ℂ
a✝ : ↑z ∈ univ
z0 : z = 0
⊢ Tendsto (fun z => z) atInf atInf | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_inv | [136, 1] | [166, 63] | refine Filter.Tendsto.mono_right ?_ le_sup_right | case pos.right
z : ℂ
a✝ : ↑z ∈ univ
z0 : z = 0
⊢ Tendsto (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) (pure 0)
((𝓝[≠] if ↑0 = 0 then ∞ else ↑0⁻¹) ⊔ pure (if ↑0 = 0 then ∞ else ↑0⁻¹)) | case pos.right
z : ℂ
a✝ : ↑z ∈ univ
z0 : z = 0
⊢ Tendsto (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) (pure 0) (pure (if ↑0 = 0 then ∞ else ↑0⁻¹)) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_inv | [136, 1] | [166, 63] | simp only [Filter.pure_zero, Filter.tendsto_pure, ite_eq_left_iff, Filter.eventually_zero,
eq_self_iff_true, not_true, IsEmpty.forall_iff] | case pos.right
z : ℂ
a✝ : ↑z ∈ univ
z0 : z = 0
⊢ Tendsto (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) (pure 0) (pure (if ↑0 = 0 then ∞ else ↑0⁻¹)) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_inv | [136, 1] | [166, 63] | have e : ∀ᶠ w : ℂ in 𝓝 z, (if w = 0 then ∞ else ↑w⁻¹ : 𝕊) = ↑w⁻¹ := by
refine (continuousAt_id.eventually_ne z0).mp (eventually_of_forall fun w w0 ↦ ?_)
simp only [Ne, id_eq] at w0; simp only [w0, if_false] | case neg
z : ℂ
a✝ : ↑z ∈ univ
z0 : ¬z = 0
⊢ ContinuousAt (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) z | case neg
z : ℂ
a✝ : ↑z ∈ univ
z0 : ¬z = 0
e : ∀ᶠ (w : ℂ) in 𝓝 z, (if w = 0 then ∞ else ↑w⁻¹) = ↑w⁻¹
⊢ ContinuousAt (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_inv | [136, 1] | [166, 63] | simp only [coe_eq_zero, continuousAt_congr e] | case neg
z : ℂ
a✝ : ↑z ∈ univ
z0 : ¬z = 0
e : ∀ᶠ (w : ℂ) in 𝓝 z, (if w = 0 then ∞ else ↑w⁻¹) = ↑w⁻¹
⊢ ContinuousAt (fun x => if ↑x = 0 then ∞ else ↑x⁻¹) z | case neg
z : ℂ
a✝ : ↑z ∈ univ
z0 : ¬z = 0
e : ∀ᶠ (w : ℂ) in 𝓝 z, (if w = 0 then ∞ else ↑w⁻¹) = ↑w⁻¹
⊢ ContinuousAt (fun x => ↑x⁻¹) z |
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