url stringclasses 147 values | commit stringclasses 147 values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | set t := Ici (log (log (abs z)) - r) | cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
β’ |p - 1 / Complex.abs z| β€ potential_error d c z | cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
β’ |p - 1 / Complex.abs z| β€ potential_error d c z | Please generate a tactic in lean4 to solve the state.
STATE:
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
β’ |p - 1 / Complex.abs z| β€ potential_error d c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | have yt : log (-log p) β 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] | cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
β’ |p - 1 / Complex.abs z| β€ potential_error d c z | cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
β’ |p - 1 / Complex.abs z| β€ potential_error d c z | Please generate a tactic in lean4 to solve the state.
STATE:
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
β’ |p - 1 / Complex.abs z| β€ potential_error d c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | have lt : log (log (abs z)) β t := by
simp only [mem_Ici, tsub_le_iff_right, le_add_iff_nonneg_right, r0, t] | cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
β’ |p - 1 / Complex.abs z| β€ potential_error d c z | cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
β’ |p - 1 / Complex.abs z| β€ potential_error d c z | Please generate a tactic in lean4 to solve the state.
STATE:
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
β’ |p - 1 / Complex.abs z| β€ potential_error d c z
TACTIC:
|
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 | cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
β’ |p - 1 / Complex.abs z| β€ potential_error d c z | cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
β’ |p - 1 / Complex.abs z| β€ potential_error d c z | Please generate a tactic in lean4 to solve the state.
STATE:
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
β’ |p - 1 / Complex.abs z| β€ potential_error d c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | have b0 : 0 β€ b := by rw [βhb]; exact dene_nonneg | cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
β’ |p - 1 / Complex.abs z| β€ potential_error d c z | cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
β’ |p - 1 / Complex.abs z| β€ potential_error d c z | Please generate a tactic in lean4 to solve the state.
STATE:
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
β’ |p - 1 / Complex.abs z| β€ potential_error d c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | have bound : β x, x β t β βderiv ene xβ β€ b := by
intro x m
simp only [Real.dist_eq, mem_Ici, βhr, t] at m
simp only [deriv_ene, norm_neg, Real.norm_of_nonneg dene_nonneg, βhb, βhr]
apply dene_anti (sub_nonneg.mpr (iter_error_le_log_log_abs d z4 cz)) m | cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
β’ |p - 1 / Complex.abs z| β€ potential_error d c z | cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
bound : β x β t, βderiv ene xβ β€ b
β’ |p - 1 / Complex.abs z| β€ potential_error d c z | Please generate a tactic in lean4 to solve the state.
STATE:
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
β’ |p - 1 / Complex.abs z| β€ potential_error d c z
TACTIC:
|
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 _ β¦ (hasDerivAt_ene x).differentiableAt) bound (convex_Ici _) lt yt | cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
bound : β x β t, βderiv ene xβ β€ b
β’ |p - 1 / Complex.abs z| β€ potential_error d c z | cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
bound : β x β t, βderiv ene xβ β€ b
m : βene (-p.log).log - ene (Complex.abs z).log.logβ β€ b * β(-p.log).log - (Complex.abs z).log.logβ
β’ |p - 1 / Complex.abs z| β€ potential_error d c z | Please generate a tactic in lean4 to solve the state.
STATE:
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
bound : β x β t, βderiv ene xβ β€ b
β’ |p - 1 / Complex.abs z| β€ potential_error d c z
TACTIC:
|
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 | cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
bound : β x β t, βderiv ene xβ β€ b
m : βene (-p.log).log - ene (Complex.abs z).log.logβ β€ b * β(-p.log).log - (Complex.abs z).log.logβ
β’ |p - 1 / Complex.abs z| β€ potential_error d c z | cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
bound : β x β t, βderiv ene xβ β€ b
m : |ene (-p.log).log - ene (Complex.abs z).log.log| β€ b * |(-p.log).log - (Complex.abs z).log.log|
β’ |p - 1 / Complex.abs z| β€ potential_error d c z | Please generate a tactic in lean4 to solve the state.
STATE:
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
bound : β x β t, βderiv ene xβ β€ b
m : βene (-p.log).log - ene (Complex.abs z).log.logβ β€ b * β(-p.log).log - (Complex.abs z).log.logβ
β’ |p - 1 / Complex.abs z| β€ potential_error d c z
TACTIC:
|
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)) | cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
bound : β x β t, βderiv ene xβ β€ b
m : |ene (-p.log).log - ene (Complex.abs z).log.log| β€ b * |(-p.log).log - (Complex.abs z).log.log|
β’ |p - 1 / Complex.abs z| β€ potential_error d c z | cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
bound : β x β t, βderiv ene xβ β€ b
m : |ene (-p.log).log - ene (Complex.abs z).log.log| β€ b * r
β’ |p - 1 / Complex.abs z| β€ potential_error d c z | Please generate a tactic in lean4 to solve the state.
STATE:
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
bound : β x β t, βderiv ene xβ β€ b
m : |ene (-p.log).log - ene (Complex.abs z).log.log| β€ b * |(-p.log).log - (Complex.abs z).log.log|
β’ |p - 1 / Complex.abs z| β€ potential_error d c z
TACTIC:
|
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 | cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
bound : β x β t, βderiv ene xβ β€ b
m : |ene (-p.log).log - ene (Complex.abs z).log.log| β€ b * r
β’ |p - 1 / Complex.abs z| β€ potential_error d c z | cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
bound : β x β t, βderiv ene xβ β€ b
m : |p - 1 / Complex.abs z| β€ b * r
β’ |p - 1 / Complex.abs z| β€ potential_error d c z | Please generate a tactic in lean4 to solve the state.
STATE:
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
bound : β x β t, βderiv ene xβ β€ b
m : |ene (-p.log).log - ene (Complex.abs z).log.log| β€ b * r
β’ |p - 1 / Complex.abs z| β€ potential_error d c z
TACTIC:
|
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 ?_) | cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
bound : β x β t, βderiv ene xβ β€ b
m : |p - 1 / Complex.abs z| β€ b * r
β’ |p - 1 / Complex.abs z| β€ potential_error d c z | cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
bound : β x β t, βderiv ene xβ β€ b
m : |p - 1 / Complex.abs z| β€ b * r
β’ b * r = potential_error d c z | Please generate a tactic in lean4 to solve the state.
STATE:
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
bound : β x β t, βderiv ene xβ β€ b
m : |p - 1 / Complex.abs z| β€ b * r
β’ |p - 1 / Complex.abs z| β€ potential_error d c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | simp only [βhr, βhb, potential_error] | cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
bound : β x β t, βderiv ene xβ β€ b
m : |p - 1 / Complex.abs z| β€ b * r
β’ b * r = potential_error d c z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
bound : β x β t, βderiv ene xβ β€ b
m : |p - 1 / Complex.abs z| β€ b * r
β’ b * r = potential_error d c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | norm_num | cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
β’ 3 β€ 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
β’ 3 β€ 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | norm_num | cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
β’ 0 < 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
β’ 0 < 3
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | linarith | cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
β’ 1 < Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
β’ 1 < Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | linarith | cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
h : |(-(β―.potential c βz).log).log - (Complex.abs z).log.log| β€ iter_error d c z
p0 : 0 < s.potential c βz
β’ 2 < Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
h : |(-(β―.potential c βz).log).log - (Complex.abs z).log.log| β€ iter_error d c z
p0 : 0 < s.potential c βz
β’ 2 < Complex.abs z
TACTIC:
|
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 | cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
β’ (-p.log).log β t | cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
h : (Complex.abs z).log.log β€ (-p.log).log + r β§ (-p.log).log β€ (Complex.abs z).log.log + r
β’ (-p.log).log β t | Please generate a tactic in lean4 to solve the state.
STATE:
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
β’ (-p.log).log β t
TACTIC:
|
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] | cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
h : (Complex.abs z).log.log β€ (-p.log).log + r β§ (-p.log).log β€ (Complex.abs z).log.log + r
β’ (-p.log).log β t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
h : (Complex.abs z).log.log β€ (-p.log).log + r β§ (-p.log).log β€ (Complex.abs z).log.log + r
β’ (-p.log).log β t
TACTIC:
|
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] | cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
β’ (Complex.abs z).log.log β t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
β’ (Complex.abs z).log.log β t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | rw [βhb] | cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
β’ 0 β€ b | cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
β’ 0 β€ dene ((Complex.abs z).log.log - r) | Please generate a tactic in lean4 to solve the state.
STATE:
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
β’ 0 β€ b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | exact dene_nonneg | cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
β’ 0 β€ dene ((Complex.abs z).log.log - r) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
β’ 0 β€ dene ((Complex.abs z).log.log - r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | intro x m | cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
β’ β x β t, βderiv ene xβ β€ b | cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
x : β
m : x β t
β’ βderiv ene xβ β€ b | Please generate a tactic in lean4 to solve the state.
STATE:
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
β’ β x β t, βderiv ene xβ β€ b
TACTIC:
|
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, βhr, t] at m | cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
x : β
m : x β t
β’ βderiv ene xβ β€ b | cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
x : β
m : (Complex.abs z).log.log - iter_error d c z β€ x
β’ βderiv ene xβ β€ b | Please generate a tactic in lean4 to solve the state.
STATE:
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
x : β
m : x β t
β’ βderiv ene xβ β€ b
TACTIC:
|
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, βhb, βhr] | cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
x : β
m : (Complex.abs z).log.log - iter_error d c z β€ x
β’ βderiv ene xβ β€ b | cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
x : β
m : (Complex.abs z).log.log - iter_error d c z β€ x
β’ dene x β€ dene ((Complex.abs z).log.log - iter_error d c z) | Please generate a tactic in lean4 to solve the state.
STATE:
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
x : β
m : (Complex.abs z).log.log - iter_error d c z β€ x
β’ βderiv ene xβ β€ b
TACTIC:
|
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 | cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
x : β
m : (Complex.abs z).log.log - iter_error d c z β€ x
β’ dene x β€ dene ((Complex.abs z).log.log - iter_error d c z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
x : β
m : (Complex.abs z).log.log - iter_error d c z β€ x
β’ dene x β€ dene ((Complex.abs z).log.log - iter_error d c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | bound | cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
bound : β x β t, βderiv ene xβ β€ b
m : |ene (-p.log).log - ene (Complex.abs z).log.log| β€ b * |(-p.log).log - (Complex.abs z).log.log|
β’ 0 β€ b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
bound : β x β t, βderiv ene xβ β€ b
m : |ene (-p.log).log - ene (Complex.abs z).log.log| β€ b * |(-p.log).log - (Complex.abs z).log.log|
β’ 0 β€ b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.coe_eq_zero | [40, 9] | [41, 37] | simp only [β coe_zero, coe_eq_coe] | z : β
β’ βz = 0 β z = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z : β
β’ βz = 0 β z = 0
TACTIC:
|
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 β B
f : β β A
i : A
z : π
β’ g (OnePoint.rec i f z) = OnePoint.rec (g i) (g β f) z | case hβ
A : Sort u_1
B : Sort u_2
g : A β B
f : β β A
i : A
β’ g (OnePoint.rec i f β) = OnePoint.rec (g i) (g β f) β
case hβ
A : Sort u_1
B : Sort u_2
g : A β B
f : β β A
i : A
xβ : β
β’ g (OnePoint.rec i f βxβ) = OnePoint.rec (g i) (g β f) βxβ | Please generate a tactic in lean4 to solve the state.
STATE:
A : Sort u_1
B : Sort u_2
g : A β B
f : β β A
i : A
z : π
β’ g (OnePoint.rec i f z) = OnePoint.rec (g i) (g β f) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.map_rec | [54, 1] | [58, 39] | simp only [rec_inf] | case hβ
A : Sort u_1
B : Sort u_2
g : A β B
f : β β A
i : A
β’ g (OnePoint.rec i f β) = OnePoint.rec (g i) (g β f) β | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hβ
A : Sort u_1
B : Sort u_2
g : A β B
f : β β A
i : A
β’ g (OnePoint.rec i f β) = OnePoint.rec (g i) (g β f) β
TACTIC:
|
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 hβ
A : Sort u_1
B : Sort u_2
g : A β B
f : β β A
i : A
xβ : β
β’ g (OnePoint.rec i f βxβ) = OnePoint.rec (g i) (g β f) βxβ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hβ
A : Sort u_1
B : Sort u_2
g : A β B
f : β β A
i : A
xβ : β
β’ g (OnePoint.rec i f βxβ) = OnePoint.rec (g i) (g β f) βxβ
TACTIC:
|
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 : β
β’ β β βz | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z : β
β’ β β βz
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.inf_ne_zero | [63, 1] | [64, 68] | have e : (0 : π) = ((0 : β) : π) := rfl | β’ β β 0 | e : 0 = β0
β’ β β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
β’ β β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.inf_ne_zero | [63, 1] | [64, 68] | rw [e] | e : 0 = β0
β’ β β 0 | e : 0 = β0
β’ β β β0 | Please generate a tactic in lean4 to solve the state.
STATE:
e : 0 = β0
β’ β β 0
TACTIC:
|
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 = β0
β’ β β β0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
e : 0 = β0
β’ β β β0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.coe_toComplex | [71, 1] | [74, 30] | induction z using OnePoint.rec | z : π
h : z β β
β’ βz.toComplex = z | case hβ
h : β β β
β’ ββ.toComplex = β
case hβ
xβ : β
h : βxβ β β
β’ β(βxβ).toComplex = βxβ | Please generate a tactic in lean4 to solve the state.
STATE:
z : π
h : z β β
β’ βz.toComplex = z
TACTIC:
|
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 hβ
h : β β β
β’ ββ.toComplex = β | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hβ
h : β β β
β’ ββ.toComplex = β
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.coe_toComplex | [71, 1] | [74, 30] | simp only [toComplex_coe] | case hβ
xβ : β
h : βxβ β β
β’ β(βxβ).toComplex = βxβ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hβ
xβ : β
h : βxβ β β
β’ β(βxβ).toComplex = βxβ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.toComplex_zero | [75, 9] | [75, 92] | rw [β coe_zero, toComplex_coe] | β’ OnePoint.toComplex 0 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
β’ OnePoint.toComplex 0 = 0
TACTIC:
|
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 : β
β’ ContinuousAt OnePoint.toComplex βz | z : β
β’ ContinuousAt (fun x => x) z | Please generate a tactic in lean4 to solve the state.
STATE:
z : β
β’ ContinuousAt OnePoint.toComplex βz
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuousAt_toComplex | [76, 1] | [77, 93] | exact continuousAt_id | z : β
β’ ContinuousAt (fun x => x) z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z : β
β’ ContinuousAt (fun x => x) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuousOn_toComplex | [78, 1] | [81, 52] | intro z m | β’ ContinuousOn OnePoint.toComplex {β}αΆ | z : π
m : z β {β}αΆ
β’ ContinuousWithinAt OnePoint.toComplex {β}αΆ z | Please generate a tactic in lean4 to solve the state.
STATE:
β’ ContinuousOn OnePoint.toComplex {β}αΆ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuousOn_toComplex | [78, 1] | [81, 52] | induction z using OnePoint.rec | z : π
m : z β {β}αΆ
β’ ContinuousWithinAt OnePoint.toComplex {β}αΆ z | case hβ
m : β β {β}αΆ
β’ ContinuousWithinAt OnePoint.toComplex {β}αΆ β
case hβ
xβ : β
m : βxβ β {β}αΆ
β’ ContinuousWithinAt OnePoint.toComplex {β}αΆ βxβ | Please generate a tactic in lean4 to solve the state.
STATE:
z : π
m : z β {β}αΆ
β’ ContinuousWithinAt OnePoint.toComplex {β}αΆ z
TACTIC:
|
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 hβ
m : β β {β}αΆ
β’ ContinuousWithinAt OnePoint.toComplex {β}αΆ β | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hβ
m : β β {β}αΆ
β’ ContinuousWithinAt OnePoint.toComplex {β}αΆ β
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuousOn_toComplex | [78, 1] | [81, 52] | exact continuousAt_toComplex.continuousWithinAt | case hβ
xβ : β
m : βxβ β {β}αΆ
β’ ContinuousWithinAt OnePoint.toComplex {β}αΆ βxβ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hβ
xβ : β
m : βxβ β {β}αΆ
β’ ContinuousWithinAt OnePoint.toComplex {β}αΆ βxβ
TACTIC:
|
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] | β’ 0β»ΒΉ = β | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
β’ 0β»ΒΉ = β
TACTIC:
|
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] | β’ ββ»ΒΉ = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
β’ ββ»ΒΉ = 0
TACTIC:
|
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 : β
z0 : z β 0
β’ (βz)β»ΒΉ = βzβ»ΒΉ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z : β
z0 : z β 0
β’ (βz)β»ΒΉ = βzβ»ΒΉ
TACTIC:
|
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 : π
β’ zβ»ΒΉ = β β z = 0 | case hβ
β’ ββ»ΒΉ = β β β = 0
case hβ
xβ : β
β’ (βxβ)β»ΒΉ = β β βxβ = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
z : π
β’ zβ»ΒΉ = β β z = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.inv_eq_inf | [102, 9] | [105, 89] | simp only [inv_inf] | case hβ
β’ ββ»ΒΉ = β β β = 0 | case hβ
β’ 0 = β β β = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hβ
β’ ββ»ΒΉ = β β β = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.inv_eq_inf | [102, 9] | [105, 89] | exact β¨Eq.symm, Eq.symmβ© | case hβ
β’ 0 = β β β = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hβ
β’ 0 = β β β = 0
TACTIC:
|
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 hβ
xβ : β
β’ (βxβ)β»ΒΉ = β β βxβ = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hβ
xβ : β
β’ (βxβ)β»ΒΉ = β β βxβ = 0
TACTIC:
|
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 : π
β’ zβ»ΒΉ = 0 β z = β | case hβ
β’ ββ»ΒΉ = 0 β β = β
case hβ
z : β
β’ (βz)β»ΒΉ = 0 β βz = β | Please generate a tactic in lean4 to solve the state.
STATE:
z : π
β’ zβ»ΒΉ = 0 β z = β
TACTIC:
|
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 hβ
β’ ββ»ΒΉ = 0 β β = β | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hβ
β’ ββ»ΒΉ = 0 β β = β
TACTIC:
|
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 hβ
z : β
β’ (βz)β»ΒΉ = 0 β βz = β | case hβ
z : β
β’ (if βz = 0 then β else βzβ»ΒΉ) = 0 β βz = β | Please generate a tactic in lean4 to solve the state.
STATE:
case hβ
z : β
β’ (βz)β»ΒΉ = 0 β βz = β
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.inv_eq_zero | [106, 9] | [112, 38] | by_cases z0 : (z : π) = 0 | case hβ
z : β
β’ (if βz = 0 then β else βzβ»ΒΉ) = 0 β βz = β | case pos
z : β
z0 : βz = 0
β’ (if βz = 0 then β else βzβ»ΒΉ) = 0 β βz = β
case neg
z : β
z0 : Β¬βz = 0
β’ (if βz = 0 then β else βzβ»ΒΉ) = 0 β βz = β | Please generate a tactic in lean4 to solve the state.
STATE:
case hβ
z : β
β’ (if βz = 0 then β else βzβ»ΒΉ) = 0 β βz = β
TACTIC:
|
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 : β
z0 : βz = 0
β’ (if βz = 0 then β else βzβ»ΒΉ) = 0 β βz = β
case neg
z : β
z0 : Β¬βz = 0
β’ (if βz = 0 then β else βzβ»ΒΉ) = 0 β βz = β | case neg
z : β
z0 : Β¬βz = 0
β’ (if βz = 0 then β else βzβ»ΒΉ) = 0 β βz = β | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
z : β
z0 : βz = 0
β’ (if βz = 0 then β else βzβ»ΒΉ) = 0 β βz = β
case neg
z : β
z0 : Β¬βz = 0
β’ (if βz = 0 then β else βzβ»ΒΉ) = 0 β βz = β
TACTIC:
|
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 : β
z0 : Β¬βz = 0
β’ (if βz = 0 then β else βzβ»ΒΉ) = 0 β βz = β | case neg
z : β
z0 : Β¬βz = 0
β’ Β¬βzβ»ΒΉ = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
z : β
z0 : Β¬βz = 0
β’ (if βz = 0 then β else βzβ»ΒΉ) = 0 β βz = β
TACTIC:
|
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 : β
z0 : Β¬βz = 0
β’ Β¬βzβ»ΒΉ = 0 | case neg
z : β
z0 : Β¬βz = 0
β’ Β¬z = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
z : β
z0 : Β¬βz = 0
β’ Β¬βzβ»ΒΉ = 0
TACTIC:
|
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 : β
z0 : Β¬βz = 0
β’ Β¬z = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
z : β
z0 : Β¬βz = 0
β’ Β¬z = 0
TACTIC:
|
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 : π
β’ zβ»ΒΉ.toComplex = z.toComplexβ»ΒΉ | case hβ
β’ ββ»ΒΉ.toComplex = β.toComplexβ»ΒΉ
case hβ
z : β
β’ (βz)β»ΒΉ.toComplex = (βz).toComplexβ»ΒΉ | Please generate a tactic in lean4 to solve the state.
STATE:
z : π
β’ zβ»ΒΉ.toComplex = z.toComplexβ»ΒΉ
TACTIC:
|
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 hβ
β’ ββ»ΒΉ.toComplex = β.toComplexβ»ΒΉ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hβ
β’ ββ»ΒΉ.toComplex = β.toComplexβ»ΒΉ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.toComplex_inv | [113, 1] | [118, 64] | by_cases z0 : z = 0 | case hβ
z : β
β’ (βz)β»ΒΉ.toComplex = (βz).toComplexβ»ΒΉ | case pos
z : β
z0 : z = 0
β’ (βz)β»ΒΉ.toComplex = (βz).toComplexβ»ΒΉ
case neg
z : β
z0 : Β¬z = 0
β’ (βz)β»ΒΉ.toComplex = (βz).toComplexβ»ΒΉ | Please generate a tactic in lean4 to solve the state.
STATE:
case hβ
z : β
β’ (βz)β»ΒΉ.toComplex = (βz).toComplexβ»ΒΉ
TACTIC:
|
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 : β
z0 : z = 0
β’ (βz)β»ΒΉ.toComplex = (βz).toComplexβ»ΒΉ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
z : β
z0 : z = 0
β’ (βz)β»ΒΉ.toComplex = (βz).toComplexβ»ΒΉ
TACTIC:
|
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 : β
z0 : Β¬z = 0
β’ (βz)β»ΒΉ.toComplex = (βz).toComplexβ»ΒΉ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
z : β
z0 : Β¬z = 0
β’ (βz)β»ΒΉ.toComplex = (βz).toComplexβ»ΒΉ
TACTIC:
|
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] | β’ Tendsto (fun z => βz) atInf (π β) | β’ atInf β€ Filter.cocompact β | Please generate a tactic in lean4 to solve the state.
STATE:
β’ Tendsto (fun z => βz) atInf (π β)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.coe_tendsto_inf | [121, 1] | [123, 27] | exact atInf_le_cocompact | β’ atInf β€ Filter.cocompact β | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
β’ atInf β€ Filter.cocompact β
TACTIC:
|
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 : {β}αΆ = range fun z => βz
β’ Tendsto (fun z => βz) atInf (π[β ] β) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
e : {β}αΆ = range fun z => βz
β’ Tendsto (fun z => βz) atInf (π[β ] β)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.coe_tendsto_inf' | [126, 1] | [133, 59] | ext z | β’ {β}αΆ = range fun z => βz | case h
z : π
β’ z β {β}αΆ β z β range fun z => βz | Please generate a tactic in lean4 to solve the state.
STATE:
β’ {β}αΆ = range fun z => βz
TACTIC:
|
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 : π
β’ z β {β}αΆ β z β range fun z => βz | case h.hβ
β’ β β {β}αΆ β β β range fun z => βz
case h.hβ
z : β
β’ βz β {β}αΆ β βz β range fun z => βz | Please generate a tactic in lean4 to solve the state.
STATE:
case h
z : π
β’ z β {β}αΆ β z β range fun z => βz
TACTIC:
|
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.hβ
β’ β β {β}αΆ β β β range fun z => βz | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hβ
β’ β β {β}αΆ β β β range fun z => βz
TACTIC:
|
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.hβ
z : β
β’ βz β {β}αΆ β βz β range fun z => βz | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hβ
z : β
β’ βz β {β}αΆ β βz β range fun z => βz
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_inv | [136, 1] | [166, 63] | rw [continuous_iff_continuousOn_univ] | β’ Continuous fun z => zβ»ΒΉ | β’ ContinuousOn (fun z => zβ»ΒΉ) univ | Please generate a tactic in lean4 to solve the state.
STATE:
β’ Continuous fun z => zβ»ΒΉ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_inv | [136, 1] | [166, 63] | intro z _ | β’ ContinuousOn (fun z => zβ»ΒΉ) univ | z : π
aβ : z β univ
β’ ContinuousWithinAt (fun z => zβ»ΒΉ) univ z | Please generate a tactic in lean4 to solve the state.
STATE:
β’ ContinuousOn (fun z => zβ»ΒΉ) univ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_inv | [136, 1] | [166, 63] | apply ContinuousAt.continuousWithinAt | z : π
aβ : z β univ
β’ ContinuousWithinAt (fun z => zβ»ΒΉ) univ z | case h
z : π
aβ : z β univ
β’ ContinuousAt (fun z => zβ»ΒΉ) z | Please generate a tactic in lean4 to solve the state.
STATE:
z : π
aβ : z β univ
β’ ContinuousWithinAt (fun z => zβ»ΒΉ) univ z
TACTIC:
|
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 : π
aβ : z β univ
β’ ContinuousAt (fun z => zβ»ΒΉ) z | case h.hβ
aβ : β β univ
β’ ContinuousAt (fun z => zβ»ΒΉ) β
case h.hβ
z : β
aβ : βz β univ
β’ ContinuousAt (fun z => zβ»ΒΉ) βz | Please generate a tactic in lean4 to solve the state.
STATE:
case h
z : π
aβ : z β univ
β’ ContinuousAt (fun z => zβ»ΒΉ) z
TACTIC:
|
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, β atInf_eq_cocompact] | case h.hβ
aβ : β β univ
β’ ContinuousAt (fun z => zβ»ΒΉ) β | case h.hβ
aβ : β β univ
β’ Tendsto (fun x => (βx)β»ΒΉ) atInf (π 0) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hβ
aβ : β β univ
β’ ContinuousAt (fun z => zβ»ΒΉ) β
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_inv | [136, 1] | [166, 63] | have e : βαΆ z : β in atInf, βzβ»ΒΉ = (βz : π)β»ΒΉ := by
refine (eventually_atInf 0).mp (eventually_of_forall fun z z0 β¦ ?_)
simp only [gt_iff_lt, Complex.norm_eq_abs, AbsoluteValue.pos_iff] at z0; rw [inv_coe z0] | case h.hβ
aβ : β β univ
β’ Tendsto (fun x => (βx)β»ΒΉ) atInf (π 0) | case h.hβ
aβ : β β univ
e : βαΆ (z : β) in atInf, βzβ»ΒΉ = (βz)β»ΒΉ
β’ Tendsto (fun x => (βx)β»ΒΉ) atInf (π 0) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hβ
aβ : β β univ
β’ Tendsto (fun x => (βx)β»ΒΉ) atInf (π 0)
TACTIC:
|
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.hβ
aβ : β β univ
e : βαΆ (z : β) in atInf, βzβ»ΒΉ = (βz)β»ΒΉ
β’ Tendsto (fun x => (βx)β»ΒΉ) atInf (π 0) | case h.hβ
aβ : β β univ
e : βαΆ (z : β) in atInf, βzβ»ΒΉ = (βz)β»ΒΉ
β’ Tendsto (fun x => βxβ»ΒΉ) atInf (π 0) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hβ
aβ : β β univ
e : βαΆ (z : β) in atInf, βzβ»ΒΉ = (βz)β»ΒΉ
β’ Tendsto (fun x => (βx)β»ΒΉ) atInf (π 0)
TACTIC:
|
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.hβ
aβ : β β univ
e : βαΆ (z : β) in atInf, βzβ»ΒΉ = (βz)β»ΒΉ
β’ Tendsto (fun x => βxβ»ΒΉ) atInf (π 0) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hβ
aβ : β β univ
e : βαΆ (z : β) in atInf, βzβ»ΒΉ = (βz)β»ΒΉ
β’ Tendsto (fun x => βxβ»ΒΉ) atInf (π 0)
TACTIC:
|
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 β¦ ?_) | aβ : β β univ
β’ βαΆ (z : β) in atInf, βzβ»ΒΉ = (βz)β»ΒΉ | aβ : β β univ
z : β
z0 : βzβ > 0
β’ βzβ»ΒΉ = (βz)β»ΒΉ | Please generate a tactic in lean4 to solve the state.
STATE:
aβ : β β univ
β’ βαΆ (z : β) in atInf, βzβ»ΒΉ = (βz)β»ΒΉ
TACTIC:
|
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 | aβ : β β univ
z : β
z0 : βzβ > 0
β’ βzβ»ΒΉ = (βz)β»ΒΉ | aβ : β β univ
z : β
z0 : z β 0
β’ βzβ»ΒΉ = (βz)β»ΒΉ | Please generate a tactic in lean4 to solve the state.
STATE:
aβ : β β univ
z : β
z0 : βzβ > 0
β’ βzβ»ΒΉ = (βz)β»ΒΉ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_inv | [136, 1] | [166, 63] | rw [inv_coe z0] | aβ : β β univ
z : β
z0 : z β 0
β’ βzβ»ΒΉ = (βz)β»ΒΉ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
aβ : β β univ
z : β
z0 : z β 0
β’ βzβ»ΒΉ = (βz)β»ΒΉ
TACTIC:
|
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.hβ
z : β
aβ : βz β univ
β’ ContinuousAt (fun z => zβ»ΒΉ) βz | case h.hβ
z : β
aβ : βz β univ
β’ ContinuousAt (fun x => if βx = 0 then β else βxβ»ΒΉ) z | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hβ
z : β
aβ : βz β univ
β’ ContinuousAt (fun z => zβ»ΒΉ) βz
TACTIC:
|
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.hβ
z : β
aβ : βz β univ
β’ ContinuousAt (fun x => if βx = 0 then β else βxβ»ΒΉ) z | case pos
z : β
aβ : βz β univ
z0 : z = 0
β’ ContinuousAt (fun x => if βx = 0 then β else βxβ»ΒΉ) z
case neg
z : β
aβ : βz β univ
z0 : Β¬z = 0
β’ ContinuousAt (fun x => if βx = 0 then β else βxβ»ΒΉ) z | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hβ
z : β
aβ : βz β univ
β’ ContinuousAt (fun x => if βx = 0 then β else βxβ»ΒΉ) z
TACTIC:
|
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 : β
aβ : βz β univ
z0 : z = 0
β’ ContinuousAt (fun x => if βx = 0 then β else βxβ»ΒΉ) z | case pos
z : β
aβ : βz β univ
z0 : z = 0
β’ Tendsto (fun x => if βx = 0 then β else βxβ»ΒΉ) (π 0) (π (if β0 = 0 then β else β0β»ΒΉ)) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
z : β
aβ : βz β univ
z0 : z = 0
β’ ContinuousAt (fun x => if βx = 0 then β else βxβ»ΒΉ) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_inv | [136, 1] | [166, 63] | simp only [β nhdsWithin_compl_singleton_sup_pure, Filter.tendsto_sup] | case pos
z : β
aβ : βz β univ
z0 : z = 0
β’ Tendsto (fun x => if βx = 0 then β else βxβ»ΒΉ) (π 0) (π (if β0 = 0 then β else β0β»ΒΉ)) | case pos
z : β
aβ : βz β univ
z0 : z = 0
β’ Tendsto (fun x => if βx = 0 then β else βxβ»ΒΉ) (π[β ] 0)
((π[β ] if β0 = 0 then β else β0β»ΒΉ) β pure (if β0 = 0 then β else β0β»ΒΉ)) β§
Tendsto (fun x => if βx = 0 then β else βxβ»ΒΉ) (pure 0)
((π[β ] if β0 = 0 then β else β0β»ΒΉ) β pure (if β0 = 0 then β else β0β»ΒΉ)) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
z : β
aβ : βz β univ
z0 : z = 0
β’ Tendsto (fun x => if βx = 0 then β else βxβ»ΒΉ) (π 0) (π (if β0 = 0 then β else β0β»ΒΉ))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_inv | [136, 1] | [166, 63] | constructor | case pos
z : β
aβ : βz β univ
z0 : z = 0
β’ Tendsto (fun x => if βx = 0 then β else βxβ»ΒΉ) (π[β ] 0)
((π[β ] if β0 = 0 then β else β0β»ΒΉ) β pure (if β0 = 0 then β else β0β»ΒΉ)) β§
Tendsto (fun x => if βx = 0 then β else βxβ»ΒΉ) (pure 0)
((π[β ] if β0 = 0 then β else β0β»ΒΉ) β pure (if β0 = 0 then β else β0β»ΒΉ)) | case pos.left
z : β
aβ : βz β univ
z0 : z = 0
β’ Tendsto (fun x => if βx = 0 then β else βxβ»ΒΉ) (π[β ] 0)
((π[β ] if β0 = 0 then β else β0β»ΒΉ) β pure (if β0 = 0 then β else β0β»ΒΉ))
case pos.right
z : β
aβ : βz β univ
z0 : z = 0
β’ Tendsto (fun x => if βx = 0 then β else βxβ»ΒΉ) (pure 0)
((π[β ] if β0 = 0 then β else β0β»ΒΉ) β pure (if β0 = 0 then β else β0β»ΒΉ)) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
z : β
aβ : βz β univ
z0 : z = 0
β’ Tendsto (fun x => if βx = 0 then β else βxβ»ΒΉ) (π[β ] 0)
((π[β ] if β0 = 0 then β else β0β»ΒΉ) β pure (if β0 = 0 then β else β0β»ΒΉ)) β§
Tendsto (fun x => if βx = 0 then β else βxβ»ΒΉ) (pure 0)
((π[β ] if β0 = 0 then β else β0β»ΒΉ) β pure (if β0 = 0 then β else β0β»ΒΉ))
TACTIC:
|
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 : β
aβ : βz β univ
z0 : z = 0
β’ Tendsto (fun x => if βx = 0 then β else βxβ»ΒΉ) (π[β ] 0)
((π[β ] if β0 = 0 then β else β0β»ΒΉ) β pure (if β0 = 0 then β else β0β»ΒΉ)) | case pos.left
z : β
aβ : βz β univ
z0 : z = 0
β’ Tendsto (fun x => if βx = 0 then β else βxβ»ΒΉ) (π[β ] 0) (π[β ] if β0 = 0 then β else β0β»ΒΉ) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.left
z : β
aβ : βz β univ
z0 : z = 0
β’ Tendsto (fun x => if βx = 0 then β else βxβ»ΒΉ) (π[β ] 0)
((π[β ] if β0 = 0 then β else β0β»ΒΉ) β pure (if β0 = 0 then β else β0β»ΒΉ))
TACTIC:
|
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 : β β¦ (βzβ»ΒΉ : π)) | case pos.left
z : β
aβ : βz β univ
z0 : z = 0
β’ Tendsto (fun x => if βx = 0 then β else βxβ»ΒΉ) (π[β ] 0) (π[β ] if β0 = 0 then β else β0β»ΒΉ) | case pos.left.hfg
z : β
aβ : βz β univ
z0 : z = 0
β’ β x β {0}αΆ, βxβ»ΒΉ = if βx = 0 then β else βxβ»ΒΉ
case pos.left.hf
z : β
aβ : βz β univ
z0 : z = 0
β’ Tendsto (fun z => βzβ»ΒΉ) (π[β ] 0) (π[β ] if β0 = 0 then β else β0β»ΒΉ) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.left
z : β
aβ : βz β univ
z0 : z = 0
β’ Tendsto (fun x => if βx = 0 then β else βxβ»ΒΉ) (π[β ] 0) (π[β ] if β0 = 0 then β else β0β»ΒΉ)
TACTIC:
|
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 : β
aβ : βz β univ
z0 : z = 0
β’ β x β {0}αΆ, βxβ»ΒΉ = if βx = 0 then β else βxβ»ΒΉ | case pos.left.hfg
zβ : β
aβ : βzβ β univ
z0 : zβ = 0
z : β
m : z β {0}αΆ
β’ βzβ»ΒΉ = if βz = 0 then β else βzβ»ΒΉ | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.left.hfg
z : β
aβ : βz β univ
z0 : z = 0
β’ β x β {0}αΆ, βxβ»ΒΉ = if βx = 0 then β else βxβ»ΒΉ
TACTIC:
|
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
zβ : β
aβ : βzβ β univ
z0 : zβ = 0
z : β
m : z β {0}αΆ
β’ βzβ»ΒΉ = if βz = 0 then β else βzβ»ΒΉ | case pos.left.hfg
zβ : β
aβ : βzβ β univ
z0 : zβ = 0
z : β
m : z β 0
β’ βzβ»ΒΉ = if βz = 0 then β else βzβ»ΒΉ | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.left.hfg
zβ : β
aβ : βzβ β univ
z0 : zβ = 0
z : β
m : z β {0}αΆ
β’ βzβ»ΒΉ = if βz = 0 then β else βzβ»ΒΉ
TACTIC:
|
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
zβ : β
aβ : βzβ β univ
z0 : zβ = 0
z : β
m : z β 0
β’ βzβ»ΒΉ = if βz = 0 then β else βzβ»ΒΉ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.left.hfg
zβ : β
aβ : βzβ β univ
z0 : zβ = 0
z : β
m : z β 0
β’ βzβ»ΒΉ = if βz = 0 then β else βzβ»ΒΉ
TACTIC:
|
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 : β
aβ : βz β univ
z0 : z = 0
β’ Tendsto (fun z => βzβ»ΒΉ) (π[β ] 0) (π[β ] if β0 = 0 then β else β0β»ΒΉ) | case pos.left.hf
z : β
aβ : βz β univ
z0 : z = 0
β’ Tendsto (fun z => βzβ»ΒΉ) (π[β ] 0) (π[β ] β) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.left.hf
z : β
aβ : βz β univ
z0 : z = 0
β’ Tendsto (fun z => βzβ»ΒΉ) (π[β ] 0) (π[β ] if β0 = 0 then β else β0β»ΒΉ)
TACTIC:
|
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 : β
aβ : βz β univ
z0 : z = 0
β’ Tendsto (fun z => βzβ»ΒΉ) (π[β ] 0) (π[β ] β) | case pos.left.hf
z : β
aβ : βz β univ
z0 : z = 0
β’ Tendsto (fun z => zβ»ΒΉ) (π[β ] 0) atInf | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.left.hf
z : β
aβ : βz β univ
z0 : z = 0
β’ Tendsto (fun z => βzβ»ΒΉ) (π[β ] 0) (π[β ] β)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_inv | [136, 1] | [166, 63] | rw [β @tendsto_atInf_iff_tendsto_nhds_zero β β _ _ fun z : β β¦ z] | case pos.left.hf
z : β
aβ : βz β univ
z0 : z = 0
β’ Tendsto (fun z => zβ»ΒΉ) (π[β ] 0) atInf | case pos.left.hf
z : β
aβ : βz β univ
z0 : z = 0
β’ Tendsto (fun z => z) atInf atInf | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.left.hf
z : β
aβ : βz β univ
z0 : z = 0
β’ Tendsto (fun z => zβ»ΒΉ) (π[β ] 0) atInf
TACTIC:
|
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 : β
aβ : βz β univ
z0 : z = 0
β’ Tendsto (fun z => z) atInf atInf | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.left.hf
z : β
aβ : βz β univ
z0 : z = 0
β’ Tendsto (fun z => z) atInf atInf
TACTIC:
|
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 : β
aβ : βz β univ
z0 : z = 0
β’ Tendsto (fun x => if βx = 0 then β else βxβ»ΒΉ) (pure 0)
((π[β ] if β0 = 0 then β else β0β»ΒΉ) β pure (if β0 = 0 then β else β0β»ΒΉ)) | case pos.right
z : β
aβ : βz β univ
z0 : z = 0
β’ Tendsto (fun x => if βx = 0 then β else βxβ»ΒΉ) (pure 0) (pure (if β0 = 0 then β else β0β»ΒΉ)) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.right
z : β
aβ : βz β univ
z0 : z = 0
β’ Tendsto (fun x => if βx = 0 then β else βxβ»ΒΉ) (pure 0)
((π[β ] if β0 = 0 then β else β0β»ΒΉ) β pure (if β0 = 0 then β else β0β»ΒΉ))
TACTIC:
|
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 : β
aβ : βz β univ
z0 : z = 0
β’ Tendsto (fun x => if βx = 0 then β else βxβ»ΒΉ) (pure 0) (pure (if β0 = 0 then β else β0β»ΒΉ)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.right
z : β
aβ : βz β univ
z0 : z = 0
β’ Tendsto (fun x => if βx = 0 then β else βxβ»ΒΉ) (pure 0) (pure (if β0 = 0 then β else β0β»ΒΉ))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuous_inv | [136, 1] | [166, 63] | have e : βαΆ w : β in π z, (if w = 0 then β else βwβ»ΒΉ : π) = βwβ»ΒΉ := by
refine (continuousAt_id.eventually_ne z0).mp (eventually_of_forall fun w w0 β¦ ?_)
simp only [Ne, id_eq] at w0; simp only [w0, if_false] | case neg
z : β
aβ : βz β univ
z0 : Β¬z = 0
β’ ContinuousAt (fun x => if βx = 0 then β else βxβ»ΒΉ) z | case neg
z : β
aβ : βz β univ
z0 : Β¬z = 0
e : βαΆ (w : β) in π z, (if w = 0 then β else βwβ»ΒΉ) = βwβ»ΒΉ
β’ ContinuousAt (fun x => if βx = 0 then β else βxβ»ΒΉ) z | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
z : β
aβ : βz β univ
z0 : Β¬z = 0
β’ ContinuousAt (fun x => if βx = 0 then β else βxβ»ΒΉ) z
TACTIC:
|
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 : β
aβ : βz β univ
z0 : Β¬z = 0
e : βαΆ (w : β) in π z, (if w = 0 then β else βwβ»ΒΉ) = βwβ»ΒΉ
β’ ContinuousAt (fun x => if βx = 0 then β else βxβ»ΒΉ) z | case neg
z : β
aβ : βz β univ
z0 : Β¬z = 0
e : βαΆ (w : β) in π z, (if w = 0 then β else βwβ»ΒΉ) = βwβ»ΒΉ
β’ ContinuousAt (fun x => βxβ»ΒΉ) z | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
z : β
aβ : βz β univ
z0 : Β¬z = 0
e : βαΆ (w : β) in π z, (if w = 0 then β else βwβ»ΒΉ) = βwβ»ΒΉ
β’ ContinuousAt (fun x => if βx = 0 then β else βxβ»ΒΉ) z
TACTIC:
|
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