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/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | use n | case neg.h.intro
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
β’ β n, β (N : Finset β) (z : β), Late N n β z β s β (N.sum fun n => Complex.abs (f n z)) < e | case h
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
β’ β (N : Finset β) (z : β), Late N n β z β s β (N.sum fun n => Complex.abs (f n z)) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h.intro
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
β’ β n, β (N : Finset β) (z : β), Late N n β z β s β (N.sum fun n => Complex.abs (f n z)) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | intro N z NL zs | case h
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
β’ β (N : Finset β) (z : β), Late N n β z β s β (N.sum fun n => Complex.abs (f n z)) < e | case h
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
N : Finset β
z : β
NL : Late N n
zs : z β s
β’ (N.sum fun n => Complex.abs (f n z)) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
β’ β (N : Finset β) (z : β), Late N n β z β s β (N.sum fun n => Complex.abs (f n z)) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | have a1p : 1 - (a : β) > 0 := by linarith | case h
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
N : Finset β
z : β
NL : Late N n
zs : z β s
β’ (N.sum fun n => Complex.abs (f n z)) < e | case h
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
N : Finset β
z : β
NL : Late N n
zs : z β s
a1p : 1 - a > 0
β’ (N.sum fun n => Complex.abs (f n z)) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
N : Finset β
z : β
NL : Late N n
zs : z β s
β’ (N.sum fun n => Complex.abs (f n z)) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | calc (N.sum fun n β¦ abs (f n z))
_ β€ N.sum fun n β¦ c * a ^ n := Finset.sum_le_sum fun n _ β¦ hf n z zs
_ = c * N.sum fun n β¦ a ^ n := (Finset.mul_sum _ _ _).symm
_ β€ c * (a ^ n * (1 - a)β»ΒΉ) := by bound [late_geometric_bound NL a0 a1]
_ = a ^ n * (c * (1 - a)β»ΒΉ) := by ring
_ β€ t * (c * (1 - a)β»ΒΉ) := by bound
_ = (1 - a) / c * (e / 2) * (c * (1 - a)β»ΒΉ) := rfl
_ = (1 - a) * (1 - a)β»ΒΉ * (c / c) * (e / 2) := by ring
_ = 1 * 1 * (e / 2) := by rw [mul_inv_cancel a1p.ne', div_self c0.ne']
_ = e / 2 := by ring
_ < e := by linarith | case h
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
N : Finset β
z : β
NL : Late N n
zs : z β s
a1p : 1 - a > 0
β’ (N.sum fun n => Complex.abs (f n z)) < e | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
N : Finset β
z : β
NL : Late N n
zs : z β s
a1p : 1 - a > 0
β’ (N.sum fun n => Complex.abs (f n z)) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | bound | f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
β’ t > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
β’ t > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | linarith | f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
N : Finset β
z : β
NL : Late N n
zs : z β s
β’ 1 - a > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
N : Finset β
z : β
NL : Late N n
zs : z β s
β’ 1 - a > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | bound [late_geometric_bound NL a0 a1] | f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
N : Finset β
z : β
NL : Late N n
zs : z β s
a1p : 1 - a > 0
β’ (c * N.sum fun n => a ^ n) β€ c * (a ^ n * (1 - a)β»ΒΉ) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
N : Finset β
z : β
NL : Late N n
zs : z β s
a1p : 1 - a > 0
β’ (c * N.sum fun n => a ^ n) β€ c * (a ^ n * (1 - a)β»ΒΉ)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | ring | f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
N : Finset β
z : β
NL : Late N n
zs : z β s
a1p : 1 - a > 0
β’ c * (a ^ n * (1 - a)β»ΒΉ) = a ^ n * (c * (1 - a)β»ΒΉ) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
N : Finset β
z : β
NL : Late N n
zs : z β s
a1p : 1 - a > 0
β’ c * (a ^ n * (1 - a)β»ΒΉ) = a ^ n * (c * (1 - a)β»ΒΉ)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | bound | f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
N : Finset β
z : β
NL : Late N n
zs : z β s
a1p : 1 - a > 0
β’ a ^ n * (c * (1 - a)β»ΒΉ) β€ t * (c * (1 - a)β»ΒΉ) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
N : Finset β
z : β
NL : Late N n
zs : z β s
a1p : 1 - a > 0
β’ a ^ n * (c * (1 - a)β»ΒΉ) β€ t * (c * (1 - a)β»ΒΉ)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | ring | f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
N : Finset β
z : β
NL : Late N n
zs : z β s
a1p : 1 - a > 0
β’ (1 - a) / c * (e / 2) * (c * (1 - a)β»ΒΉ) = (1 - a) * (1 - a)β»ΒΉ * (c / c) * (e / 2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
N : Finset β
z : β
NL : Late N n
zs : z β s
a1p : 1 - a > 0
β’ (1 - a) / c * (e / 2) * (c * (1 - a)β»ΒΉ) = (1 - a) * (1 - a)β»ΒΉ * (c / c) * (e / 2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | rw [mul_inv_cancel a1p.ne', div_self c0.ne'] | f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
N : Finset β
z : β
NL : Late N n
zs : z β s
a1p : 1 - a > 0
β’ (1 - a) * (1 - a)β»ΒΉ * (c / c) * (e / 2) = 1 * 1 * (e / 2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
N : Finset β
z : β
NL : Late N n
zs : z β s
a1p : 1 - a > 0
β’ (1 - a) * (1 - a)β»ΒΉ * (c / c) * (e / 2) = 1 * 1 * (e / 2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | ring | f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
N : Finset β
z : β
NL : Late N n
zs : z β s
a1p : 1 - a > 0
β’ 1 * 1 * (e / 2) = e / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
N : Finset β
z : β
NL : Late N n
zs : z β s
a1p : 1 - a > 0
β’ 1 * 1 * (e / 2) = e / 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | linarith | f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
N : Finset β
z : β
NL : Late N n
zs : z β s
a1p : 1 - a > 0
β’ e / 2 < e | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
c0 : 0 < c
e : β
ep : e > 0
t : β := (1 - a) / c * (e / 2)
tp : t > 0
n : β
nt : a ^ n < t
N : Finset β
z : β
NL : Late N n
zs : z β s
a1p : 1 - a > 0
β’ e / 2 < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_at | [145, 1] | [154, 36] | set s : Set β := {0} | f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
β’ Summable f | f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
β’ Summable f | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
β’ Summable f
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_at | [145, 1] | [154, 36] | set g : β β β β β := fun n _ β¦ f n | f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
β’ Summable f | f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
g : β β β β β := fun n x => f n
β’ Summable f | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
β’ Summable f
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_at | [145, 1] | [154, 36] | have hg : β n z, z β s β abs (g n z) β€ c * a ^ n := fun n z _ β¦ hf n | f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
g : β β β β β := fun n x => f n
β’ Summable f | f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
g : β β β β β := fun n x => f n
hg : β (n : β), β z β s, Complex.abs (g n z) β€ c * a ^ n
β’ Summable f | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
g : β β β β β := fun n x => f n
β’ Summable f
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_at | [145, 1] | [154, 36] | have u := fast_series_converge_uniformly_on a0 a1 hg | f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
g : β β β β β := fun n x => f n
hg : β (n : β), β z β s, Complex.abs (g n z) β€ c * a ^ n
β’ Summable f | f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
g : β β β β β := fun n x => f n
hg : β (n : β), β z β s, Complex.abs (g n z) β€ c * a ^ n
u : HasUniformSum (fun n z => g n z) (tsumOn fun n z => g n z) s
β’ Summable f | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
g : β β β β β := fun n x => f n
hg : β (n : β), β z β s, Complex.abs (g n z) β€ c * a ^ n
β’ Summable f
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_at | [145, 1] | [154, 36] | rw [HasUniformSum] at u | f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
g : β β β β β := fun n x => f n
hg : β (n : β), β z β s, Complex.abs (g n z) β€ c * a ^ n
u : HasUniformSum (fun n z => g n z) (tsumOn fun n z => g n z) s
β’ Summable f | f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
g : β β β β β := fun n x => f n
hg : β (n : β), β z β s, Complex.abs (g n z) β€ c * a ^ n
u : TendstoUniformlyOn (fun N z => N.sum fun n => g n z) (tsumOn fun n z => g n z) atTop s
β’ Summable f | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
g : β β β β β := fun n x => f n
hg : β (n : β), β z β s, Complex.abs (g n z) β€ c * a ^ n
u : HasUniformSum (fun n z => g n z) (tsumOn fun n z => g n z) s
β’ Summable f
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_at | [145, 1] | [154, 36] | rw [tendstoUniformlyOn_singleton_iff_tendsto] at u | f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
g : β β β β β := fun n x => f n
hg : β (n : β), β z β s, Complex.abs (g n z) β€ c * a ^ n
u : TendstoUniformlyOn (fun N z => N.sum fun n => g n z) (tsumOn fun n z => g n z) atTop s
β’ Summable f | f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
g : β β β β β := fun n x => f n
hg : β (n : β), β z β s, Complex.abs (g n z) β€ c * a ^ n
u : Filter.Tendsto (fun n => n.sum fun n => g n 0) atTop (π (tsumOn (fun n z => g n z) 0))
β’ Summable f | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
g : β β β β β := fun n x => f n
hg : β (n : β), β z β s, Complex.abs (g n z) β€ c * a ^ n
u : TendstoUniformlyOn (fun N z => N.sum fun n => g n z) (tsumOn fun n z => g n z) atTop s
β’ Summable f
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_at | [145, 1] | [154, 36] | apply HasSum.summable | f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
g : β β β β β := fun n x => f n
hg : β (n : β), β z β s, Complex.abs (g n z) β€ c * a ^ n
u : Filter.Tendsto (fun n => n.sum fun n => g n 0) atTop (π (tsumOn (fun n z => g n z) 0))
β’ Summable f | case h
f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
g : β β β β β := fun n x => f n
hg : β (n : β), β z β s, Complex.abs (g n z) β€ c * a ^ n
u : Filter.Tendsto (fun n => n.sum fun n => g n 0) atTop (π (tsumOn (fun n z => g n z) 0))
β’ HasSum f ?a
case a
f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
g : β β β β β := fun n x => f n
hg : β (n : β), β z β s, Complex.abs (g n z) β€ c * a ^ n
u : Filter.Tendsto (fun n => n.sum fun n => g n 0) atTop (π (tsumOn (fun n z => g n z) 0))
β’ β | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
g : β β β β β := fun n x => f n
hg : β (n : β), β z β s, Complex.abs (g n z) β€ c * a ^ n
u : Filter.Tendsto (fun n => n.sum fun n => g n 0) atTop (π (tsumOn (fun n z => g n z) 0))
β’ Summable f
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_at | [145, 1] | [154, 36] | assumption | case h
f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
g : β β β β β := fun n x => f n
hg : β (n : β), β z β s, Complex.abs (g n z) β€ c * a ^ n
u : Filter.Tendsto (fun n => n.sum fun n => g n 0) atTop (π (tsumOn (fun n z => g n z) 0))
β’ HasSum f ?a
case a
f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
g : β β β β β := fun n x => f n
hg : β (n : β), β z β s, Complex.abs (g n z) β€ c * a ^ n
u : Filter.Tendsto (fun n => n.sum fun n => g n 0) atTop (π (tsumOn (fun n z => g n z) 0))
β’ β | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
g : β β β β β := fun n x => f n
hg : β (n : β), β z β s, Complex.abs (g n z) β€ c * a ^ n
u : Filter.Tendsto (fun n => n.sum fun n => g n 0) atTop (π (tsumOn (fun n z => g n z) 0))
β’ HasSum f ?a
case a
f : β β β
c a : β
a0 : 0 β€ a
a1 : a < 1
hf : β (n : β), Complex.abs (f n) β€ c * a ^ n
s : Set β := {0}
g : β β β β β := fun n x => f n
hg : β (n : β), β z β s, Complex.abs (g n z) β€ c * a ^ n
u : Filter.Tendsto (fun n => n.sum fun n => g n 0) atTop (π (tsumOn (fun n z => g n z) 0))
β’ β
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge | [157, 1] | [163, 87] | use tsumOn f | f : β β β β β
s : Set β
c a : β
o : IsOpen s
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
β’ β g, AnalyticOn β g s β§ HasSumOn f g s | case h
f : β β β β β
s : Set β
c a : β
o : IsOpen s
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
β’ AnalyticOn β (tsumOn f) s β§ HasSumOn f (tsumOn f) s | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
c a : β
o : IsOpen s
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
β’ β g, AnalyticOn β g s β§ HasSumOn f g s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge | [157, 1] | [163, 87] | constructor | case h
f : β β β β β
s : Set β
c a : β
o : IsOpen s
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
β’ AnalyticOn β (tsumOn f) s β§ HasSumOn f (tsumOn f) s | case h.left
f : β β β β β
s : Set β
c a : β
o : IsOpen s
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
β’ AnalyticOn β (tsumOn f) s
case h.right
f : β β β β β
s : Set β
c a : β
o : IsOpen s
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
β’ HasSumOn f (tsumOn f) s | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β β β
s : Set β
c a : β
o : IsOpen s
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
β’ AnalyticOn β (tsumOn f) s β§ HasSumOn f (tsumOn f) s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge | [157, 1] | [163, 87] | exact uniform_analytic_lim o (fun N β¦ N.analyticOn_sum fun _ _ β¦ h _)
(fast_series_converge_uniformly_on a0 a1 hf) | case h.left
f : β β β β β
s : Set β
c a : β
o : IsOpen s
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
β’ AnalyticOn β (tsumOn f) s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
f : β β β β β
s : Set β
c a : β
o : IsOpen s
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
β’ AnalyticOn β (tsumOn f) s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge | [157, 1] | [163, 87] | exact fun z zs β¦ Summable.hasSum (fast_series_converge_at a0 a1 fun n β¦ hf n z zs) | case h.right
f : β β β β β
s : Set β
c a : β
o : IsOpen s
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
β’ HasSumOn f (tsumOn f) s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
f : β β β β β
s : Set β
c a : β
o : IsOpen s
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z) β€ c * a ^ n
β’ HasSumOn f (tsumOn f) s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_cons | [166, 1] | [174, 32] | rw [HasSum] at h β’ | X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : HasSum f g
β’ HasSum (Stream'.cons a f) (a + g) | X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
β’ Filter.Tendsto (fun s => s.sum fun b => Stream'.cons a f b) atTop (π (a + g)) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : HasSum f g
β’ HasSum (Stream'.cons a f) (a + g)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_cons | [166, 1] | [174, 32] | have ha := Filter.Tendsto.comp (Continuous.tendsto (continuous_add_left a) g) h | X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
β’ Filter.Tendsto (fun s => s.sum fun b => Stream'.cons a f b) atTop (π (a + g)) | X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a + b) β fun s => s.sum fun b => f b) atTop (π (a + g))
β’ Filter.Tendsto (fun s => s.sum fun b => Stream'.cons a f b) atTop (π (a + g)) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
β’ Filter.Tendsto (fun s => s.sum fun b => Stream'.cons a f b) atTop (π (a + g))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_cons | [166, 1] | [174, 32] | have s : ((fun z β¦ a + z) β fun N : Finset β β¦ N.sum f) =
(fun N : Finset β β¦ N.sum (Stream'.cons a f)) β push := by
apply funext; intro N; simp; exact push_sum | X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a + b) β fun s => s.sum fun b => f b) atTop (π (a + g))
β’ Filter.Tendsto (fun s => s.sum fun b => Stream'.cons a f b) atTop (π (a + g)) | X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a + b) β fun s => s.sum fun b => f b) atTop (π (a + g))
s : ((fun z => a + z) β fun N => N.sum f) = (fun N => N.sum (Stream'.cons a f)) β push
β’ Filter.Tendsto (fun s => s.sum fun b => Stream'.cons a f b) atTop (π (a + g)) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a + b) β fun s => s.sum fun b => f b) atTop (π (a + g))
β’ Filter.Tendsto (fun s => s.sum fun b => Stream'.cons a f b) atTop (π (a + g))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_cons | [166, 1] | [174, 32] | rw [s] at ha | X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a + b) β fun s => s.sum fun b => f b) atTop (π (a + g))
s : ((fun z => a + z) β fun N => N.sum f) = (fun N => N.sum (Stream'.cons a f)) β push
β’ Filter.Tendsto (fun s => s.sum fun b => Stream'.cons a f b) atTop (π (a + g)) | X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun N => N.sum (Stream'.cons a f)) β push) atTop (π (a + g))
s : ((fun z => a + z) β fun N => N.sum f) = (fun N => N.sum (Stream'.cons a f)) β push
β’ Filter.Tendsto (fun s => s.sum fun b => Stream'.cons a f b) atTop (π (a + g)) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a + b) β fun s => s.sum fun b => f b) atTop (π (a + g))
s : ((fun z => a + z) β fun N => N.sum f) = (fun N => N.sum (Stream'.cons a f)) β push
β’ Filter.Tendsto (fun s => s.sum fun b => Stream'.cons a f b) atTop (π (a + g))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_cons | [166, 1] | [174, 32] | exact tendsto_comp_push.mp ha | X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun N => N.sum (Stream'.cons a f)) β push) atTop (π (a + g))
s : ((fun z => a + z) β fun N => N.sum f) = (fun N => N.sum (Stream'.cons a f)) β push
β’ Filter.Tendsto (fun s => s.sum fun b => Stream'.cons a f b) atTop (π (a + g)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun N => N.sum (Stream'.cons a f)) β push) atTop (π (a + g))
s : ((fun z => a + z) β fun N => N.sum f) = (fun N => N.sum (Stream'.cons a f)) β push
β’ Filter.Tendsto (fun s => s.sum fun b => Stream'.cons a f b) atTop (π (a + g))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_cons | [166, 1] | [174, 32] | apply funext | X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a + b) β fun s => s.sum fun b => f b) atTop (π (a + g))
β’ ((fun z => a + z) β fun N => N.sum f) = (fun N => N.sum (Stream'.cons a f)) β push | case h
X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a + b) β fun s => s.sum fun b => f b) atTop (π (a + g))
β’ β (x : Finset β), ((fun z => a + z) β fun N => N.sum f) x = ((fun N => N.sum (Stream'.cons a f)) β push) x | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a + b) β fun s => s.sum fun b => f b) atTop (π (a + g))
β’ ((fun z => a + z) β fun N => N.sum f) = (fun N => N.sum (Stream'.cons a f)) β push
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_cons | [166, 1] | [174, 32] | intro N | case h
X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a + b) β fun s => s.sum fun b => f b) atTop (π (a + g))
β’ β (x : Finset β), ((fun z => a + z) β fun N => N.sum f) x = ((fun N => N.sum (Stream'.cons a f)) β push) x | case h
X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a + b) β fun s => s.sum fun b => f b) atTop (π (a + g))
N : Finset β
β’ ((fun z => a + z) β fun N => N.sum f) N = ((fun N => N.sum (Stream'.cons a f)) β push) N | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a + b) β fun s => s.sum fun b => f b) atTop (π (a + g))
β’ β (x : Finset β), ((fun z => a + z) β fun N => N.sum f) x = ((fun N => N.sum (Stream'.cons a f)) β push) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_cons | [166, 1] | [174, 32] | simp | case h
X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a + b) β fun s => s.sum fun b => f b) atTop (π (a + g))
N : Finset β
β’ ((fun z => a + z) β fun N => N.sum f) N = ((fun N => N.sum (Stream'.cons a f)) β push) N | case h
X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a + b) β fun s => s.sum fun b => f b) atTop (π (a + g))
N : Finset β
β’ a + N.sum f = (push N).sum (Stream'.cons a f) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a + b) β fun s => s.sum fun b => f b) atTop (π (a + g))
N : Finset β
β’ ((fun z => a + z) β fun N => N.sum f) N = ((fun N => N.sum (Stream'.cons a f)) β push) N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_cons | [166, 1] | [174, 32] | exact push_sum | case h
X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a + b) β fun s => s.sum fun b => f b) atTop (π (a + g))
N : Finset β
β’ a + N.sum f = (push N).sum (Stream'.cons a f) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
instβ : NormedAddCommGroup X
a g : X
f : β β X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a + b) β fun s => s.sum fun b => f b) atTop (π (a + g))
N : Finset β
β’ a + N.sum f = (push N).sum (Stream'.cons a f)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_cons' | [176, 1] | [179, 87] | rcases h with β¨g, hβ© | a : β
f : β β β
h : Summable f
β’ tsum (Stream'.cons a f) = a + tsum f | case intro
a : β
f : β β β
g : β
h : HasSum f g
β’ tsum (Stream'.cons a f) = a + tsum f | Please generate a tactic in lean4 to solve the state.
STATE:
a : β
f : β β β
h : Summable f
β’ tsum (Stream'.cons a f) = a + tsum f
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_cons' | [176, 1] | [179, 87] | rw [HasSum.tsum_eq h] | case intro
a : β
f : β β β
g : β
h : HasSum f g
β’ tsum (Stream'.cons a f) = a + tsum f | case intro
a : β
f : β β β
g : β
h : HasSum f g
β’ tsum (Stream'.cons a f) = a + g | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
a : β
f : β β β
g : β
h : HasSum f g
β’ tsum (Stream'.cons a f) = a + tsum f
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_cons' | [176, 1] | [179, 87] | rw [HasSum.tsum_eq _] | case intro
a : β
f : β β β
g : β
h : HasSum f g
β’ tsum (Stream'.cons a f) = a + g | a : β
f : β β β
g : β
h : HasSum f g
β’ HasSum (fun b => Stream'.cons a f b) (a + g) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
a : β
f : β β β
g : β
h : HasSum f g
β’ tsum (Stream'.cons a f) = a + g
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_cons' | [176, 1] | [179, 87] | exact sum_cons h | a : β
f : β β β
g : β
h : HasSum f g
β’ HasSum (fun b => Stream'.cons a f b) (a + g) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : β
f : β β β
g : β
h : HasSum f g
β’ HasSum (fun b => Stream'.cons a f b) (a + g)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_drop | [181, 1] | [193, 26] | have c := sum_cons (a := -f 0) h | X : Type
instβ : NormedAddCommGroup X
f : β β X
g : X
h : HasSum f g
β’ HasSum (fun n => f (n + 1)) (g - f 0) | X : Type
instβ : NormedAddCommGroup X
f : β β X
g : X
h : HasSum f g
c : HasSum (Stream'.cons (-f 0) f) (-f 0 + g)
β’ HasSum (fun n => f (n + 1)) (g - f 0) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instβ : NormedAddCommGroup X
f : β β X
g : X
h : HasSum f g
β’ HasSum (fun n => f (n + 1)) (g - f 0)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_drop | [181, 1] | [193, 26] | rw [HasSum] | X : Type
instβ : NormedAddCommGroup X
f : β β X
g : X
h : HasSum f g
c : HasSum (Stream'.cons (-f 0) f) (-f 0 + g)
β’ HasSum (fun n => f (n + 1)) (g - f 0) | X : Type
instβ : NormedAddCommGroup X
f : β β X
g : X
h : HasSum f g
c : HasSum (Stream'.cons (-f 0) f) (-f 0 + g)
β’ Filter.Tendsto (fun s => s.sum fun b => f (b + 1)) atTop (π (g - f 0)) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instβ : NormedAddCommGroup X
f : β β X
g : X
h : HasSum f g
c : HasSum (Stream'.cons (-f 0) f) (-f 0 + g)
β’ HasSum (fun n => f (n + 1)) (g - f 0)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_drop | [181, 1] | [193, 26] | rw [neg_add_eq_sub, HasSum, β tendsto_comp_push, β tendsto_comp_push] at c | X : Type
instβ : NormedAddCommGroup X
f : β β X
g : X
h : HasSum f g
c : HasSum (Stream'.cons (-f 0) f) (-f 0 + g)
β’ Filter.Tendsto (fun s => s.sum fun b => f (b + 1)) atTop (π (g - f 0)) | X : Type
instβ : NormedAddCommGroup X
f : β β X
g : X
h : HasSum f g
c : Filter.Tendsto (((fun s => s.sum fun b => Stream'.cons (-f 0) f b) β push) β push) atTop (π (g - f 0))
β’ Filter.Tendsto (fun s => s.sum fun b => f (b + 1)) atTop (π (g - f 0)) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instβ : NormedAddCommGroup X
f : β β X
g : X
h : HasSum f g
c : HasSum (Stream'.cons (-f 0) f) (-f 0 + g)
β’ Filter.Tendsto (fun s => s.sum fun b => f (b + 1)) atTop (π (g - f 0))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_drop | [181, 1] | [193, 26] | have s : ((fun N : Finset β β¦ N.sum fun n β¦ (Stream'.cons (-f 0) f) n) β push) β push =
fun N : Finset β β¦ N.sum fun n β¦ f (n + 1) := by
clear c h g; apply funext; intro N; simp
nth_rw 2 [β Stream'.eta f]
simp only [βpush_sum, Stream'.head, Stream'.tail, Stream'.get]
abel | X : Type
instβ : NormedAddCommGroup X
f : β β X
g : X
h : HasSum f g
c : Filter.Tendsto (((fun s => s.sum fun b => Stream'.cons (-f 0) f b) β push) β push) atTop (π (g - f 0))
β’ Filter.Tendsto (fun s => s.sum fun b => f (b + 1)) atTop (π (g - f 0)) | X : Type
instβ : NormedAddCommGroup X
f : β β X
g : X
h : HasSum f g
c : Filter.Tendsto (((fun s => s.sum fun b => Stream'.cons (-f 0) f b) β push) β push) atTop (π (g - f 0))
s : ((fun N => N.sum fun n => Stream'.cons (-f 0) f n) β push) β push = fun N => N.sum fun n => f (n + 1)
β’ Filter.Tendsto (fun s => s.sum fun b => f (b + 1)) atTop (π (g - f 0)) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instβ : NormedAddCommGroup X
f : β β X
g : X
h : HasSum f g
c : Filter.Tendsto (((fun s => s.sum fun b => Stream'.cons (-f 0) f b) β push) β push) atTop (π (g - f 0))
β’ Filter.Tendsto (fun s => s.sum fun b => f (b + 1)) atTop (π (g - f 0))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_drop | [181, 1] | [193, 26] | rw [s] at c | X : Type
instβ : NormedAddCommGroup X
f : β β X
g : X
h : HasSum f g
c : Filter.Tendsto (((fun s => s.sum fun b => Stream'.cons (-f 0) f b) β push) β push) atTop (π (g - f 0))
s : ((fun N => N.sum fun n => Stream'.cons (-f 0) f n) β push) β push = fun N => N.sum fun n => f (n + 1)
β’ Filter.Tendsto (fun s => s.sum fun b => f (b + 1)) atTop (π (g - f 0)) | X : Type
instβ : NormedAddCommGroup X
f : β β X
g : X
h : HasSum f g
c : Filter.Tendsto (fun N => N.sum fun n => f (n + 1)) atTop (π (g - f 0))
s : ((fun N => N.sum fun n => Stream'.cons (-f 0) f n) β push) β push = fun N => N.sum fun n => f (n + 1)
β’ Filter.Tendsto (fun s => s.sum fun b => f (b + 1)) atTop (π (g - f 0)) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instβ : NormedAddCommGroup X
f : β β X
g : X
h : HasSum f g
c : Filter.Tendsto (((fun s => s.sum fun b => Stream'.cons (-f 0) f b) β push) β push) atTop (π (g - f 0))
s : ((fun N => N.sum fun n => Stream'.cons (-f 0) f n) β push) β push = fun N => N.sum fun n => f (n + 1)
β’ Filter.Tendsto (fun s => s.sum fun b => f (b + 1)) atTop (π (g - f 0))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_drop | [181, 1] | [193, 26] | assumption | X : Type
instβ : NormedAddCommGroup X
f : β β X
g : X
h : HasSum f g
c : Filter.Tendsto (fun N => N.sum fun n => f (n + 1)) atTop (π (g - f 0))
s : ((fun N => N.sum fun n => Stream'.cons (-f 0) f n) β push) β push = fun N => N.sum fun n => f (n + 1)
β’ Filter.Tendsto (fun s => s.sum fun b => f (b + 1)) atTop (π (g - f 0)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instβ : NormedAddCommGroup X
f : β β X
g : X
h : HasSum f g
c : Filter.Tendsto (fun N => N.sum fun n => f (n + 1)) atTop (π (g - f 0))
s : ((fun N => N.sum fun n => Stream'.cons (-f 0) f n) β push) β push = fun N => N.sum fun n => f (n + 1)
β’ Filter.Tendsto (fun s => s.sum fun b => f (b + 1)) atTop (π (g - f 0))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_drop | [181, 1] | [193, 26] | clear c h g | X : Type
instβ : NormedAddCommGroup X
f : β β X
g : X
h : HasSum f g
c : Filter.Tendsto (((fun s => s.sum fun b => Stream'.cons (-f 0) f b) β push) β push) atTop (π (g - f 0))
β’ ((fun N => N.sum fun n => Stream'.cons (-f 0) f n) β push) β push = fun N => N.sum fun n => f (n + 1) | X : Type
instβ : NormedAddCommGroup X
f : β β X
β’ ((fun N => N.sum fun n => Stream'.cons (-f 0) f n) β push) β push = fun N => N.sum fun n => f (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instβ : NormedAddCommGroup X
f : β β X
g : X
h : HasSum f g
c : Filter.Tendsto (((fun s => s.sum fun b => Stream'.cons (-f 0) f b) β push) β push) atTop (π (g - f 0))
β’ ((fun N => N.sum fun n => Stream'.cons (-f 0) f n) β push) β push = fun N => N.sum fun n => f (n + 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_drop | [181, 1] | [193, 26] | apply funext | X : Type
instβ : NormedAddCommGroup X
f : β β X
β’ ((fun N => N.sum fun n => Stream'.cons (-f 0) f n) β push) β push = fun N => N.sum fun n => f (n + 1) | case h
X : Type
instβ : NormedAddCommGroup X
f : β β X
β’ β (x : Finset β), (((fun N => N.sum fun n => Stream'.cons (-f 0) f n) β push) β push) x = x.sum fun n => f (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instβ : NormedAddCommGroup X
f : β β X
β’ ((fun N => N.sum fun n => Stream'.cons (-f 0) f n) β push) β push = fun N => N.sum fun n => f (n + 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_drop | [181, 1] | [193, 26] | intro N | case h
X : Type
instβ : NormedAddCommGroup X
f : β β X
β’ β (x : Finset β), (((fun N => N.sum fun n => Stream'.cons (-f 0) f n) β push) β push) x = x.sum fun n => f (n + 1) | case h
X : Type
instβ : NormedAddCommGroup X
f : β β X
N : Finset β
β’ (((fun N => N.sum fun n => Stream'.cons (-f 0) f n) β push) β push) N = N.sum fun n => f (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
instβ : NormedAddCommGroup X
f : β β X
β’ β (x : Finset β), (((fun N => N.sum fun n => Stream'.cons (-f 0) f n) β push) β push) x = x.sum fun n => f (n + 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_drop | [181, 1] | [193, 26] | simp | case h
X : Type
instβ : NormedAddCommGroup X
f : β β X
N : Finset β
β’ (((fun N => N.sum fun n => Stream'.cons (-f 0) f n) β push) β push) N = N.sum fun n => f (n + 1) | case h
X : Type
instβ : NormedAddCommGroup X
f : β β X
N : Finset β
β’ ((push (push N)).sum fun n => Stream'.cons (-f 0) f n) = N.sum fun n => f (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
instβ : NormedAddCommGroup X
f : β β X
N : Finset β
β’ (((fun N => N.sum fun n => Stream'.cons (-f 0) f n) β push) β push) N = N.sum fun n => f (n + 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_drop | [181, 1] | [193, 26] | nth_rw 2 [β Stream'.eta f] | case h
X : Type
instβ : NormedAddCommGroup X
f : β β X
N : Finset β
β’ ((push (push N)).sum fun n => Stream'.cons (-f 0) f n) = N.sum fun n => f (n + 1) | case h
X : Type
instβ : NormedAddCommGroup X
f : β β X
N : Finset β
β’ ((push (push N)).sum fun n => Stream'.cons (-f 0) (Stream'.cons (Stream'.head f) (Stream'.tail f)) n) =
N.sum fun n => f (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
instβ : NormedAddCommGroup X
f : β β X
N : Finset β
β’ ((push (push N)).sum fun n => Stream'.cons (-f 0) f n) = N.sum fun n => f (n + 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_drop | [181, 1] | [193, 26] | simp only [βpush_sum, Stream'.head, Stream'.tail, Stream'.get] | case h
X : Type
instβ : NormedAddCommGroup X
f : β β X
N : Finset β
β’ ((push (push N)).sum fun n => Stream'.cons (-f 0) (Stream'.cons (Stream'.head f) (Stream'.tail f)) n) =
N.sum fun n => f (n + 1) | case h
X : Type
instβ : NormedAddCommGroup X
f : β β X
N : Finset β
β’ -f 0 + (f 0 + N.sum fun x => f (x + 1)) = N.sum fun x => f (x + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
instβ : NormedAddCommGroup X
f : β β X
N : Finset β
β’ ((push (push N)).sum fun n => Stream'.cons (-f 0) (Stream'.cons (Stream'.head f) (Stream'.tail f)) n) =
N.sum fun n => f (n + 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_drop | [181, 1] | [193, 26] | abel | case h
X : Type
instβ : NormedAddCommGroup X
f : β β X
N : Finset β
β’ -f 0 + (f 0 + N.sum fun x => f (x + 1)) = N.sum fun x => f (x + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
instβ : NormedAddCommGroup X
f : β β X
N : Finset β
β’ -f 0 + (f 0 + N.sum fun x => f (x + 1)) = N.sum fun x => f (x + 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.nonempty_ps | [48, 1] | [49, 73] | simp only [Super.ps, mem_setOf, eq_self_iff_true, true_or_iff] | S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
β’ 1 β s.ps c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
β’ 1 β s.ps c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.compact_ps | [52, 1] | [61, 59] | have pc : Continuous (s.potential c) := (Continuous.potential s).along_snd | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
β’ IsCompact (s.ps c) | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential c)
β’ IsCompact (s.ps c) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
β’ IsCompact (s.ps c)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.compact_ps | [52, 1] | [61, 59] | have c1 : IsCompact {(1 : β)} := isCompact_singleton | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential c)
β’ IsCompact (s.ps c) | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
β’ IsCompact (s.ps c) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential c)
β’ IsCompact (s.ps c)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.compact_ps | [52, 1] | [61, 59] | convert c1.union ((s.isClosed_critical_not_a.snd_preimage c).isCompact.image pc) | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
β’ IsCompact (s.ps c) | case h.e'_3
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
β’ s.ps c = {1} βͺ s.potential c '' {b | (c, b) β {p | Critical (f p.1) p.2 β§ p.2 β a}} | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
β’ IsCompact (s.ps c)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.compact_ps | [52, 1] | [61, 59] | apply Set.ext | case h.e'_3
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
β’ s.ps c = {1} βͺ s.potential c '' {b | (c, b) β {p | Critical (f p.1) p.2 β§ p.2 β a}} | case h.e'_3.h
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
β’ β (x : β), x β s.ps c β x β {1} βͺ s.potential c '' {b | (c, b) β {p | Critical (f p.1) p.2 β§ p.2 β a}} | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
β’ s.ps c = {1} βͺ s.potential c '' {b | (c, b) β {p | Critical (f p.1) p.2 β§ p.2 β a}}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.compact_ps | [52, 1] | [61, 59] | intro p | case h.e'_3.h
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
β’ β (x : β), x β s.ps c β x β {1} βͺ s.potential c '' {b | (c, b) β {p | Critical (f p.1) p.2 β§ p.2 β a}} | case h.e'_3.h
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
p : β
β’ p β s.ps c β p β {1} βͺ s.potential c '' {b | (c, b) β {p | Critical (f p.1) p.2 β§ p.2 β a}} | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
β’ β (x : β), x β s.ps c β x β {1} βͺ s.potential c '' {b | (c, b) β {p | Critical (f p.1) p.2 β§ p.2 β a}}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.compact_ps | [52, 1] | [61, 59] | simp only [mem_setOf, Super.ps, mem_singleton_iff, mem_union, mem_image, Ne, β
s.potential_eq_zero_of_onePreimage c] | case h.e'_3.h
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
p : β
β’ p β s.ps c β p β {1} βͺ s.potential c '' {b | (c, b) β {p | Critical (f p.1) p.2 β§ p.2 β a}} | case h.e'_3.h
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
p : β
β’ (p = 1 β¨ Β¬p = 0 β§ β z, s.potential c z = p β§ Critical (f c) z) β
p = 1 β¨ β x, (Critical (f c) x β§ Β¬s.potential c x = 0) β§ s.potential c x = p | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
p : β
β’ p β s.ps c β p β {1} βͺ s.potential c '' {b | (c, b) β {p | Critical (f p.1) p.2 β§ p.2 β a}}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.compact_ps | [52, 1] | [61, 59] | apply or_congr_right | case h.e'_3.h
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
p : β
β’ (p = 1 β¨ Β¬p = 0 β§ β z, s.potential c z = p β§ Critical (f c) z) β
p = 1 β¨ β x, (Critical (f c) x β§ Β¬s.potential c x = 0) β§ s.potential c x = p | case h.e'_3.h.h
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
p : β
β’ (Β¬p = 0 β§ β z, s.potential c z = p β§ Critical (f c) z) β
β x, (Critical (f c) x β§ Β¬s.potential c x = 0) β§ s.potential c x = p | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
p : β
β’ (p = 1 β¨ Β¬p = 0 β§ β z, s.potential c z = p β§ Critical (f c) z) β
p = 1 β¨ β x, (Critical (f c) x β§ Β¬s.potential c x = 0) β§ s.potential c x = p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.compact_ps | [52, 1] | [61, 59] | constructor | case h.e'_3.h.h
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
p : β
β’ (Β¬p = 0 β§ β z, s.potential c z = p β§ Critical (f c) z) β
β x, (Critical (f c) x β§ Β¬s.potential c x = 0) β§ s.potential c x = p | case h.e'_3.h.h.mp
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
p : β
β’ (Β¬p = 0 β§ β z, s.potential c z = p β§ Critical (f c) z) β
β x, (Critical (f c) x β§ Β¬s.potential c x = 0) β§ s.potential c x = p
case h.e'_3.h.h.mpr
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
p : β
β’ (β x, (Critical (f c) x β§ Β¬s.potential c x = 0) β§ s.potential c x = p) β
Β¬p = 0 β§ β z, s.potential c z = p β§ Critical (f c) z | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.h
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
p : β
β’ (Β¬p = 0 β§ β z, s.potential c z = p β§ Critical (f c) z) β
β x, (Critical (f c) x β§ Β¬s.potential c x = 0) β§ s.potential c x = p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.compact_ps | [52, 1] | [61, 59] | intro β¨p0, z, e, cβ© | case h.e'_3.h.h.mp
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
p : β
β’ (Β¬p = 0 β§ β z, s.potential c z = p β§ Critical (f c) z) β
β x, (Critical (f c) x β§ Β¬s.potential c x = 0) β§ s.potential c x = p
case h.e'_3.h.h.mpr
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
p : β
β’ (β x, (Critical (f c) x β§ Β¬s.potential c x = 0) β§ s.potential c x = p) β
Β¬p = 0 β§ β z, s.potential c z = p β§ Critical (f c) z | case h.e'_3.h.h.mp
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential cβ)
c1 : IsCompact {1}
p : β
p0 : Β¬p = 0
z : S
e : s.potential cβ z = p
c : Critical (f cβ) z
β’ β x, (Critical (f cβ) x β§ Β¬s.potential cβ x = 0) β§ s.potential cβ x = p
case h.e'_3.h.h.mpr
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
p : β
β’ (β x, (Critical (f c) x β§ Β¬s.potential c x = 0) β§ s.potential c x = p) β
Β¬p = 0 β§ β z, s.potential c z = p β§ Critical (f c) z | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.h.mp
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
p : β
β’ (Β¬p = 0 β§ β z, s.potential c z = p β§ Critical (f c) z) β
β x, (Critical (f c) x β§ Β¬s.potential c x = 0) β§ s.potential c x = p
case h.e'_3.h.h.mpr
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
p : β
β’ (β x, (Critical (f c) x β§ Β¬s.potential c x = 0) β§ s.potential c x = p) β
Β¬p = 0 β§ β z, s.potential c z = p β§ Critical (f c) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.compact_ps | [52, 1] | [61, 59] | rw [β e] at p0 | case h.e'_3.h.h.mp
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential cβ)
c1 : IsCompact {1}
p : β
p0 : Β¬p = 0
z : S
e : s.potential cβ z = p
c : Critical (f cβ) z
β’ β x, (Critical (f cβ) x β§ Β¬s.potential cβ x = 0) β§ s.potential cβ x = p
case h.e'_3.h.h.mpr
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
p : β
β’ (β x, (Critical (f c) x β§ Β¬s.potential c x = 0) β§ s.potential c x = p) β
Β¬p = 0 β§ β z, s.potential c z = p β§ Critical (f c) z | case h.e'_3.h.h.mp
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential cβ)
c1 : IsCompact {1}
p : β
z : S
p0 : Β¬s.potential cβ z = 0
e : s.potential cβ z = p
c : Critical (f cβ) z
β’ β x, (Critical (f cβ) x β§ Β¬s.potential cβ x = 0) β§ s.potential cβ x = p
case h.e'_3.h.h.mpr
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
p : β
β’ (β x, (Critical (f c) x β§ Β¬s.potential c x = 0) β§ s.potential c x = p) β
Β¬p = 0 β§ β z, s.potential c z = p β§ Critical (f c) z | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.h.mp
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential cβ)
c1 : IsCompact {1}
p : β
p0 : Β¬p = 0
z : S
e : s.potential cβ z = p
c : Critical (f cβ) z
β’ β x, (Critical (f cβ) x β§ Β¬s.potential cβ x = 0) β§ s.potential cβ x = p
case h.e'_3.h.h.mpr
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
p : β
β’ (β x, (Critical (f c) x β§ Β¬s.potential c x = 0) β§ s.potential c x = p) β
Β¬p = 0 β§ β z, s.potential c z = p β§ Critical (f c) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.compact_ps | [52, 1] | [61, 59] | exact β¨z, β¨c, p0β©, eβ© | case h.e'_3.h.h.mp
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential cβ)
c1 : IsCompact {1}
p : β
z : S
p0 : Β¬s.potential cβ z = 0
e : s.potential cβ z = p
c : Critical (f cβ) z
β’ β x, (Critical (f cβ) x β§ Β¬s.potential cβ x = 0) β§ s.potential cβ x = p
case h.e'_3.h.h.mpr
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
p : β
β’ (β x, (Critical (f c) x β§ Β¬s.potential c x = 0) β§ s.potential c x = p) β
Β¬p = 0 β§ β z, s.potential c z = p β§ Critical (f c) z | case h.e'_3.h.h.mpr
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
p : β
β’ (β x, (Critical (f c) x β§ Β¬s.potential c x = 0) β§ s.potential c x = p) β
Β¬p = 0 β§ β z, s.potential c z = p β§ Critical (f c) z | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.h.mp
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential cβ)
c1 : IsCompact {1}
p : β
z : S
p0 : Β¬s.potential cβ z = 0
e : s.potential cβ z = p
c : Critical (f cβ) z
β’ β x, (Critical (f cβ) x β§ Β¬s.potential cβ x = 0) β§ s.potential cβ x = p
case h.e'_3.h.h.mpr
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
p : β
β’ (β x, (Critical (f c) x β§ Β¬s.potential c x = 0) β§ s.potential c x = p) β
Β¬p = 0 β§ β z, s.potential c z = p β§ Critical (f c) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.compact_ps | [52, 1] | [61, 59] | intro β¨z, β¨c, p0β©, eβ© | case h.e'_3.h.h.mpr
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
p : β
β’ (β x, (Critical (f c) x β§ Β¬s.potential c x = 0) β§ s.potential c x = p) β
Β¬p = 0 β§ β z, s.potential c z = p β§ Critical (f c) z | case h.e'_3.h.h.mpr
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential cβ)
c1 : IsCompact {1}
p : β
z : S
c : Critical (f cβ) z
p0 : Β¬s.potential cβ z = 0
e : s.potential cβ z = p
β’ Β¬p = 0 β§ β z, s.potential cβ z = p β§ Critical (f cβ) z | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.h.mpr
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
p : β
β’ (β x, (Critical (f c) x β§ Β¬s.potential c x = 0) β§ s.potential c x = p) β
Β¬p = 0 β§ β z, s.potential c z = p β§ Critical (f c) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.compact_ps | [52, 1] | [61, 59] | rw [e] at p0 | case h.e'_3.h.h.mpr
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential cβ)
c1 : IsCompact {1}
p : β
z : S
c : Critical (f cβ) z
p0 : Β¬s.potential cβ z = 0
e : s.potential cβ z = p
β’ Β¬p = 0 β§ β z, s.potential cβ z = p β§ Critical (f cβ) z | case h.e'_3.h.h.mpr
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential cβ)
c1 : IsCompact {1}
p : β
z : S
c : Critical (f cβ) z
p0 : Β¬p = 0
e : s.potential cβ z = p
β’ Β¬p = 0 β§ β z, s.potential cβ z = p β§ Critical (f cβ) z | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.h.mpr
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential cβ)
c1 : IsCompact {1}
p : β
z : S
c : Critical (f cβ) z
p0 : Β¬s.potential cβ z = 0
e : s.potential cβ z = p
β’ Β¬p = 0 β§ β z, s.potential cβ z = p β§ Critical (f cβ) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.compact_ps | [52, 1] | [61, 59] | exact β¨p0, z, e, cβ© | case h.e'_3.h.h.mpr
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential cβ)
c1 : IsCompact {1}
p : β
z : S
c : Critical (f cβ) z
p0 : Β¬p = 0
e : s.potential cβ z = p
β’ Β¬p = 0 β§ β z, s.potential cβ z = p β§ Critical (f cβ) z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.h.mpr
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
pc : Continuous (s.potential cβ)
c1 : IsCompact {1}
p : β
z : S
c : Critical (f cβ) z
p0 : Β¬p = 0
e : s.potential cβ z = p
β’ Β¬p = 0 β§ β z, s.potential cβ z = p β§ Critical (f cβ) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.ps_pos | [64, 1] | [66, 44] | cases' m with m m | S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
c : β
p : β
m : p β s.ps c
β’ 0 < p | case inl
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
c : β
p : β
m : p = 1
β’ 0 < p
case inr
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
c : β
p : β
m : p β 0 β§ β z, s.potential c z = p β§ Critical (f c) z
β’ 0 < p | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
c : β
p : β
m : p β s.ps c
β’ 0 < p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.ps_pos | [64, 1] | [66, 44] | simp only [m, zero_lt_one] | case inl
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
c : β
p : β
m : p = 1
β’ 0 < p
case inr
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
c : β
p : β
m : p β 0 β§ β z, s.potential c z = p β§ Critical (f c) z
β’ 0 < p | case inr
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
c : β
p : β
m : p β 0 β§ β z, s.potential c z = p β§ Critical (f c) z
β’ 0 < p | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
c : β
p : β
m : p = 1
β’ 0 < p
case inr
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
c : β
p : β
m : p β 0 β§ β z, s.potential c z = p β§ Critical (f c) z
β’ 0 < p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.ps_pos | [64, 1] | [66, 44] | rcases m with β¨p0, z, e, cβ© | case inr
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
c : β
p : β
m : p β 0 β§ β z, s.potential c z = p β§ Critical (f c) z
β’ 0 < p | case inr.intro.intro.intro
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβΒΉ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
cβ : β
p : β
p0 : p β 0
z : S
e : s.potential cβ z = p
c : Critical (f cβ) z
β’ 0 < p | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
c : β
p : β
m : p β 0 β§ β z, s.potential c z = p β§ Critical (f c) z
β’ 0 < p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.ps_pos | [64, 1] | [66, 44] | rw [β e] at p0 β’ | case inr.intro.intro.intro
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβΒΉ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
cβ : β
p : β
p0 : p β 0
z : S
e : s.potential cβ z = p
c : Critical (f cβ) z
β’ 0 < p | case inr.intro.intro.intro
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβΒΉ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
cβ : β
p : β
z : S
p0 : s.potential cβ z β 0
e : s.potential cβ z = p
c : Critical (f cβ) z
β’ 0 < s.potential cβ z | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.intro.intro
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβΒΉ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
cβ : β
p : β
p0 : p β 0
z : S
e : s.potential cβ z = p
c : Critical (f cβ) z
β’ 0 < p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.ps_pos | [64, 1] | [66, 44] | exact p0.symm.lt_of_le s.potential_nonneg | case inr.intro.intro.intro
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβΒΉ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
cβ : β
p : β
z : S
p0 : s.potential cβ z β 0
e : s.potential cβ z = p
c : Critical (f cβ) z
β’ 0 < s.potential cβ z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.intro.intro
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβΒΉ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
cβ : β
p : β
z : S
p0 : s.potential cβ z β 0
e : s.potential cβ z = p
c : Critical (f cβ) z
β’ 0 < s.potential cβ z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.mem_ps | [73, 1] | [74, 95] | rw [β s.compact_ps.isClosed.closure_eq] | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
c : β
instβ : OnePreimage s
β’ s.p c β s.ps c | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
c : β
instβ : OnePreimage s
β’ s.p c β closure (s.ps c) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
c : β
instβ : OnePreimage s
β’ s.p c β s.ps c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.mem_ps | [73, 1] | [74, 95] | exact csInf_mem_closure s.nonempty_ps s.bddBelow_ps | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
c : β
instβ : OnePreimage s
β’ s.p c β closure (s.ps c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
c : β
instβ : OnePreimage s
β’ s.p c β closure (s.ps c)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | intro c p h | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
β’ LowerSemicontinuous s.p | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : p < s.p c
β’ βαΆ (x' : β) in π c, p < s.p x' | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
β’ LowerSemicontinuous s.p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | contrapose h | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : p < s.p c
β’ βαΆ (x' : β) in π c, p < s.p x' | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : Β¬βαΆ (x' : β) in π c, p < s.p x'
β’ Β¬p < s.p c | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : p < s.p c
β’ βαΆ (x' : β) in π c, p < s.p x'
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | simp only [not_lt, Filter.not_eventually] at h β’ | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : Β¬βαΆ (x' : β) in π c, p < s.p x'
β’ Β¬p < s.p c | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
β’ s.p c β€ p | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : Β¬βαΆ (x' : β) in π c, p < s.p x'
β’ Β¬p < s.p c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | apply le_of_forall_lt' | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
β’ s.p c β€ p | case H
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
β’ β (c_1 : β), p < c_1 β s.p c < c_1 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
β’ s.p c β€ p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | intro q' pq' | case H
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
β’ β (c_1 : β), p < c_1 β s.p c < c_1 | case H
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q' : β
pq' : p < q'
β’ s.p c < q' | Please generate a tactic in lean4 to solve the state.
STATE:
case H
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
β’ β (c_1 : β), p < c_1 β s.p c < c_1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | rcases exists_between pq' with β¨q, pq, qqβ© | case H
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q' : β
pq' : p < q'
β’ s.p c < q' | case H.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q' : β
pq' : p < q'
q : β
pq : p < q
qq : q < q'
β’ s.p c < q' | Please generate a tactic in lean4 to solve the state.
STATE:
case H
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q' : β
pq' : p < q'
β’ s.p c < q'
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | refine lt_of_le_of_lt ?_ qq | case H.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q' : β
pq' : p < q'
q : β
pq : p < q
qq : q < q'
β’ s.p c < q' | case H.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q' : β
pq' : p < q'
q : β
pq : p < q
qq : q < q'
β’ s.p c β€ q | Please generate a tactic in lean4 to solve the state.
STATE:
case H.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q' : β
pq' : p < q'
q : β
pq : p < q
qq : q < q'
β’ s.p c < q'
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | clear qq pq' q' | case H.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q' : β
pq' : p < q'
q : β
pq : p < q
qq : q < q'
β’ s.p c β€ q | case H.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
β’ s.p c β€ q | Please generate a tactic in lean4 to solve the state.
STATE:
case H.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q' : β
pq' : p < q'
q : β
pq : p < q
qq : q < q'
β’ s.p c β€ q
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | by_cases q1 : 1 β€ q | case H.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
β’ s.p c β€ q | case pos
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : 1 β€ q
β’ s.p c β€ q
case neg
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : Β¬1 β€ q
β’ s.p c β€ q | Please generate a tactic in lean4 to solve the state.
STATE:
case H.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
β’ s.p c β€ q
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | exact _root_.trans s.p_le_one q1 | case pos
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : 1 β€ q
β’ s.p c β€ q
case neg
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : Β¬1 β€ q
β’ s.p c β€ q | case neg
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : Β¬1 β€ q
β’ s.p c β€ q | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : 1 β€ q
β’ s.p c β€ q
case neg
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : Β¬1 β€ q
β’ s.p c β€ q
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | simp only [not_le] at q1 | case neg
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : Β¬1 β€ q
β’ s.p c β€ q | case neg
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
β’ s.p c β€ q | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : Β¬1 β€ q
β’ s.p c β€ q
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | set t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a} | case neg
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
β’ s.p c β€ q | case neg
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
β’ s.p c β€ q | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
β’ s.p c β€ q
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | have ct : IsClosed t :=
(isClosed_le (Continuous.potential s) continuous_const).inter s.isClosed_critical_not_a | case neg
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
β’ s.p c β€ q | case neg
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
β’ s.p c β€ q | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
β’ s.p c β€ q
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | set u := Prod.fst '' t | case neg
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
β’ s.p c β€ q | case neg
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
β’ s.p c β€ q | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
β’ s.p c β€ q
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | have cu : IsClosed u := isClosedMap_fst_of_compactSpace _ ct | case neg
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
β’ s.p c β€ q | case neg
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
β’ s.p c β€ q | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
β’ s.p c β€ q
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | suffices m : c β u by
rcases(mem_image _ _ _).mp m with β¨β¨c', zβ©, β¨zp, zc, zaβ©, ccβ©
simp only at cc za zc zp; simp only [cc] at za zc zp; clear cc c'
simp only [Ne, β s.potential_eq_zero_of_onePreimage c] at za
refine _root_.trans (csInf_le s.bddBelow_ps ?_) zp; right; use za, z, rfl, zc | case neg
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
β’ s.p c β€ q | case neg
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
β’ c β u | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
β’ s.p c β€ q
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | refine Filter.Frequently.mem_of_closed ?_ cu | case neg
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
β’ c β u | case neg
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
β’ βαΆ (x : β) in π c, x β u | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
β’ c β u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | refine h.mp (eventually_of_forall fun e h β¦ ?_) | case neg
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
β’ βαΆ (x : β) in π c, x β u | case neg
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
hβ : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
e : β
h : s.p e β€ p
β’ e β u | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
β’ βαΆ (x : β) in π c, x β u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | rcases exists_lt_of_csInf_lt s.nonempty_ps (lt_of_le_of_lt h pq) with β¨r, m, rqβ© | case neg
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
hβ : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
e : β
h : s.p e β€ p
β’ e β u | case neg.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
hβ : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
e : β
h : s.p e β€ p
r : β
m : r β s.ps e
rq : r < q
β’ e β u | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
hβ : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
e : β
h : s.p e β€ p
β’ e β u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | cases' m with m m | case neg.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
hβ : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
e : β
h : s.p e β€ p
r : β
m : r β s.ps e
rq : r < q
β’ e β u | case neg.intro.intro.inl
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
hβ : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
e : β
h : s.p e β€ p
r : β
rq : r < q
m : r = 1
β’ e β u
case neg.intro.intro.inr
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
hβ : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
e : β
h : s.p e β€ p
r : β
rq : r < q
m : r β 0 β§ β z, s.potential e z = r β§ Critical (f e) z
β’ e β u | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
hβ : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
e : β
h : s.p e β€ p
r : β
m : r β s.ps e
rq : r < q
β’ e β u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | linarith | case neg.intro.intro.inl
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
hβ : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
e : β
h : s.p e β€ p
r : β
rq : r < q
m : r = 1
β’ e β u
case neg.intro.intro.inr
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
hβ : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
e : β
h : s.p e β€ p
r : β
rq : r < q
m : r β 0 β§ β z, s.potential e z = r β§ Critical (f e) z
β’ e β u | case neg.intro.intro.inr
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
hβ : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
e : β
h : s.p e β€ p
r : β
rq : r < q
m : r β 0 β§ β z, s.potential e z = r β§ Critical (f e) z
β’ e β u | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.inl
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
hβ : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
e : β
h : s.p e β€ p
r : β
rq : r < q
m : r = 1
β’ e β u
case neg.intro.intro.inr
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
hβ : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
e : β
h : s.p e β€ p
r : β
rq : r < q
m : r β 0 β§ β z, s.potential e z = r β§ Critical (f e) z
β’ e β u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | rcases m with β¨r0, z, zr, zcβ© | case neg.intro.intro.inr
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
hβ : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
e : β
h : s.p e β€ p
r : β
rq : r < q
m : r β 0 β§ β z, s.potential e z = r β§ Critical (f e) z
β’ e β u | case neg.intro.intro.inr.intro.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
hβ : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
e : β
h : s.p e β€ p
r : β
rq : r < q
r0 : r β 0
z : S
zr : s.potential e z = r
zc : Critical (f e) z
β’ e β u | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.inr
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
hβ : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
e : β
h : s.p e β€ p
r : β
rq : r < q
m : r β 0 β§ β z, s.potential e z = r β§ Critical (f e) z
β’ e β u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | rw [β zr, Ne, s.potential_eq_zero_of_onePreimage] at r0 | case neg.intro.intro.inr.intro.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
hβ : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
e : β
h : s.p e β€ p
r : β
rq : r < q
r0 : r β 0
z : S
zr : s.potential e z = r
zc : Critical (f e) z
β’ e β u | case neg.intro.intro.inr.intro.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
hβ : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
e : β
h : s.p e β€ p
r : β
rq : r < q
z : S
r0 : Β¬z = a
zr : s.potential e z = r
zc : Critical (f e) z
β’ e β u | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.inr.intro.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
hβ : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
e : β
h : s.p e β€ p
r : β
rq : r < q
r0 : r β 0
z : S
zr : s.potential e z = r
zc : Critical (f e) z
β’ e β u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | rw [mem_image] | case neg.intro.intro.inr.intro.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
hβ : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
e : β
h : s.p e β€ p
r : β
rq : r < q
z : S
r0 : Β¬z = a
zr : s.potential e z = r
zc : Critical (f e) z
β’ e β u | case neg.intro.intro.inr.intro.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
hβ : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
e : β
h : s.p e β€ p
r : β
rq : r < q
z : S
r0 : Β¬z = a
zr : s.potential e z = r
zc : Critical (f e) z
β’ β x β t, x.1 = e | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.inr.intro.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
hβ : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
e : β
h : s.p e β€ p
r : β
rq : r < q
z : S
r0 : Β¬z = a
zr : s.potential e z = r
zc : Critical (f e) z
β’ e β u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | refine β¨(e, z), β¨?_, zc, r0β©, rflβ© | case neg.intro.intro.inr.intro.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
hβ : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
e : β
h : s.p e β€ p
r : β
rq : r < q
z : S
r0 : Β¬z = a
zr : s.potential e z = r
zc : Critical (f e) z
β’ β x β t, x.1 = e | case neg.intro.intro.inr.intro.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
hβ : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
e : β
h : s.p e β€ p
r : β
rq : r < q
z : S
r0 : Β¬z = a
zr : s.potential e z = r
zc : Critical (f e) z
β’ s.potential (e, z).1 (e, z).2 β€ q | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.inr.intro.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
hβ : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
e : β
h : s.p e β€ p
r : β
rq : r < q
z : S
r0 : Β¬z = a
zr : s.potential e z = r
zc : Critical (f e) z
β’ β x β t, x.1 = e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | simp only [zr] | case neg.intro.intro.inr.intro.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
hβ : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
e : β
h : s.p e β€ p
r : β
rq : r < q
z : S
r0 : Β¬z = a
zr : s.potential e z = r
zc : Critical (f e) z
β’ s.potential (e, z).1 (e, z).2 β€ q | case neg.intro.intro.inr.intro.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
hβ : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
e : β
h : s.p e β€ p
r : β
rq : r < q
z : S
r0 : Β¬z = a
zr : s.potential e z = r
zc : Critical (f e) z
β’ r β€ q | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.inr.intro.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
hβ : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
e : β
h : s.p e β€ p
r : β
rq : r < q
z : S
r0 : Β¬z = a
zr : s.potential e z = r
zc : Critical (f e) z
β’ s.potential (e, z).1 (e, z).2 β€ q
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | exact rq.le | case neg.intro.intro.inr.intro.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
hβ : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
e : β
h : s.p e β€ p
r : β
rq : r < q
z : S
r0 : Β¬z = a
zr : s.potential e z = r
zc : Critical (f e) z
β’ r β€ q | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.inr.intro.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
hβ : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
e : β
h : s.p e β€ p
r : β
rq : r < q
z : S
r0 : Β¬z = a
zr : s.potential e z = r
zc : Critical (f e) z
β’ r β€ q
TACTIC:
|
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