url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
| input
stringlengths 73
2.09M
|
|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
Complex.abs_log_one_add_le
|
[14, 1]
|
[65, 80]
|
simp only [ge_iff_le, zero_le_one, uIcc_of_le, mem_Icc] at m
|
case hβ.hz
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
t : β
m : t β uIcc 0 1
β’ |t| * abs z < 1
|
case hβ.hz
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
t : β
m : 0 β€ t β§ t β€ 1
β’ |t| * abs z < 1
|
Please generate a tactic in lean4 to solve the state.
STATE:
case hβ.hz
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
t : β
m : t β uIcc 0 1
β’ |t| * abs z < 1
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
Complex.abs_log_one_add_le
|
[14, 1]
|
[65, 80]
|
apply m1
|
case hβ.hz
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
t : β
m : 0 β€ t β§ t β€ 1
β’ |t| * abs z < 1
|
case hβ.hz.a
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
t : β
m : 0 β€ t β§ t β€ 1
β’ |t| β€ 1
|
Please generate a tactic in lean4 to solve the state.
STATE:
case hβ.hz
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
t : β
m : 0 β€ t β§ t β€ 1
β’ |t| * abs z < 1
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
Complex.abs_log_one_add_le
|
[14, 1]
|
[65, 80]
|
simp only [abs_le]
|
case hβ.hz.a
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
t : β
m : 0 β€ t β§ t β€ 1
β’ |t| β€ 1
|
case hβ.hz.a
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
t : β
m : 0 β€ t β§ t β€ 1
β’ -1 β€ t β§ t β€ 1
|
Please generate a tactic in lean4 to solve the state.
STATE:
case hβ.hz.a
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
t : β
m : 0 β€ t β§ t β€ 1
β’ |t| β€ 1
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
Complex.abs_log_one_add_le
|
[14, 1]
|
[65, 80]
|
exact β¨by linarith, m.2β©
|
case hβ.hz.a
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
t : β
m : 0 β€ t β§ t β€ 1
β’ -1 β€ t β§ t β€ 1
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case hβ.hz.a
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
t : β
m : 0 β€ t β§ t β€ 1
β’ -1 β€ t β§ t β€ 1
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
Complex.abs_log_one_add_le
|
[14, 1]
|
[65, 80]
|
linarith
|
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
t : β
m : 0 β€ t β§ t β€ 1
β’ -1 β€ t
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
t : β
m : 0 β€ t β§ t β€ 1
β’ -1 β€ t
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
Complex.abs_log_one_add_le
|
[14, 1]
|
[65, 80]
|
intro t m
|
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
β’ β t β uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
|
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
t : β
m : t β uIcc 0 1
β’ HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
|
Please generate a tactic in lean4 to solve the state.
STATE:
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
β’ β t β uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
Complex.abs_log_one_add_le
|
[14, 1]
|
[65, 80]
|
simp only [ge_iff_le, zero_le_one, uIcc_of_le, mem_Icc] at m
|
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
t : β
m : t β uIcc 0 1
β’ HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
|
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
t : β
m : 0 β€ t β§ t β€ 1
β’ HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
|
Please generate a tactic in lean4 to solve the state.
STATE:
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
t : β
m : t β uIcc 0 1
β’ HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
Complex.abs_log_one_add_le
|
[14, 1]
|
[65, 80]
|
exact (((hasDerivAt_mul_const _).const_sub _).log ((sub_pos.mpr (m1 _ m.2)).ne')).neg
|
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
t : β
m : 0 β€ t β§ t β€ 1
β’ HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
t : β
m : 0 β€ t β§ t β€ 1
β’ HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
Complex.abs_log_one_add_le
|
[14, 1]
|
[65, 80]
|
apply ContinuousOn.intervalIntegrable_of_Icc zero_le_one
|
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
dr : β t β uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
β’ IntervalIntegrable (fun t => z / (1 + βt * z)) MeasureTheory.volume 0 1
|
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
dr : β t β uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
β’ ContinuousOn (fun t => z / (1 + βt * z)) (Icc 0 1)
|
Please generate a tactic in lean4 to solve the state.
STATE:
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
dr : β t β uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
β’ IntervalIntegrable (fun t => z / (1 + βt * z)) MeasureTheory.volume 0 1
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
Complex.abs_log_one_add_le
|
[14, 1]
|
[65, 80]
|
apply continuousOn_const.div (Continuous.continuousOn (by continuity))
|
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
dr : β t β uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
β’ ContinuousOn (fun t => z / (1 + βt * z)) (Icc 0 1)
|
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
dr : β t β uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
β’ β x β Icc 0 1, 1 + βx * z β 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
dr : β t β uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
β’ ContinuousOn (fun t => z / (1 + βt * z)) (Icc 0 1)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
Complex.abs_log_one_add_le
|
[14, 1]
|
[65, 80]
|
intro t β¨t0,t1β©
|
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
dr : β t β uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
β’ β x β Icc 0 1, 1 + βx * z β 0
|
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
dr : β t β uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
t : β
t0 : 0 β€ t
t1 : t β€ 1
β’ 1 + βt * z β 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
dr : β t β uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
β’ β x β Icc 0 1, 1 + βx * z β 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
Complex.abs_log_one_add_le
|
[14, 1]
|
[65, 80]
|
rw [βComplex.abs.ne_zero_iff]
|
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
dr : β t β uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
t : β
t0 : 0 β€ t
t1 : t β€ 1
β’ 1 + βt * z β 0
|
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
dr : β t β uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
t : β
t0 : 0 β€ t
t1 : t β€ 1
β’ abs (1 + βt * z) β 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
dr : β t β uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
t : β
t0 : 0 β€ t
t1 : t β€ 1
β’ 1 + βt * z β 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
Complex.abs_log_one_add_le
|
[14, 1]
|
[65, 80]
|
apply ne_of_gt
|
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
dr : β t β uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
t : β
t0 : 0 β€ t
t1 : t β€ 1
β’ abs (1 + βt * z) β 0
|
case h
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
dr : β t β uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
t : β
t0 : 0 β€ t
t1 : t β€ 1
β’ 0 < abs (1 + βt * z)
|
Please generate a tactic in lean4 to solve the state.
STATE:
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
dr : β t β uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
t : β
t0 : 0 β€ t
t1 : t β€ 1
β’ abs (1 + βt * z) β 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
Complex.abs_log_one_add_le
|
[14, 1]
|
[65, 80]
|
calc abs (1 + t*z)
_ β₯ Complex.abs 1 - abs (t*z) := Complex.abs.le_add _ _
_ = 1 - |t| * abs z := by simp only [map_one, map_mul, Complex.abs_ofReal]
_ > 0 := by refine sub_pos.mpr (m1 _ (abs_le.mpr β¨by linarith, t1β©))
|
case h
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
dr : β t β uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
t : β
t0 : 0 β€ t
t1 : t β€ 1
β’ 0 < abs (1 + βt * z)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
dr : β t β uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
t : β
t0 : 0 β€ t
t1 : t β€ 1
β’ 0 < abs (1 + βt * z)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
Complex.abs_log_one_add_le
|
[14, 1]
|
[65, 80]
|
continuity
|
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
dr : β t β uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
β’ Continuous fun t => 1 + βt * z
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
dr : β t β uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
β’ Continuous fun t => 1 + βt * z
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
Complex.abs_log_one_add_le
|
[14, 1]
|
[65, 80]
|
simp only [map_one, map_mul, Complex.abs_ofReal]
|
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
dr : β t β uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
t : β
t0 : 0 β€ t
t1 : t β€ 1
β’ abs 1 - abs (βt * z) = 1 - |t| * abs z
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
dr : β t β uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
t : β
t0 : 0 β€ t
t1 : t β€ 1
β’ abs 1 - abs (βt * z) = 1 - |t| * abs z
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
Complex.abs_log_one_add_le
|
[14, 1]
|
[65, 80]
|
refine sub_pos.mpr (m1 _ (abs_le.mpr β¨by linarith, t1β©))
|
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
dr : β t β uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
t : β
t0 : 0 β€ t
t1 : t β€ 1
β’ 1 - |t| * abs z > 0
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
dr : β t β uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
t : β
t0 : 0 β€ t
t1 : t β€ 1
β’ 1 - |t| * abs z > 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
Complex.abs_log_one_add_le
|
[14, 1]
|
[65, 80]
|
linarith
|
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
dr : β t β uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
t : β
t0 : 0 β€ t
t1 : t β€ 1
β’ -1 β€ t
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
dr : β t β uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
t : β
t0 : 0 β€ t
t1 : t β€ 1
β’ -1 β€ t
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
Complex.abs_log_one_add_le
|
[14, 1]
|
[65, 80]
|
apply ContinuousOn.intervalIntegrable_of_Icc zero_le_one
|
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
ic : IntervalIntegrable (fun t => z / (1 + βt * z)) MeasureTheory.volume 0 1
dr : β t β uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (abs z / (1 - t * abs z)) t
β’ IntervalIntegrable (fun t => abs z / (1 - t * abs z)) MeasureTheory.volume 0 1
|
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
ic : IntervalIntegrable (fun t => z / (1 + βt * z)) MeasureTheory.volume 0 1
dr : β t β uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (abs z / (1 - t * abs z)) t
β’ ContinuousOn (fun t => abs z / (1 - t * abs z)) (Icc 0 1)
|
Please generate a tactic in lean4 to solve the state.
STATE:
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
ic : IntervalIntegrable (fun t => z / (1 + βt * z)) MeasureTheory.volume 0 1
dr : β t β uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (abs z / (1 - t * abs z)) t
β’ IntervalIntegrable (fun t => abs z / (1 - t * abs z)) MeasureTheory.volume 0 1
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
Complex.abs_log_one_add_le
|
[14, 1]
|
[65, 80]
|
apply continuousOn_const.div (Continuous.continuousOn (by continuity))
|
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
ic : IntervalIntegrable (fun t => z / (1 + βt * z)) MeasureTheory.volume 0 1
dr : β t β uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (abs z / (1 - t * abs z)) t
β’ ContinuousOn (fun t => abs z / (1 - t * abs z)) (Icc 0 1)
|
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
ic : IntervalIntegrable (fun t => z / (1 + βt * z)) MeasureTheory.volume 0 1
dr : β t β uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (abs z / (1 - t * abs z)) t
β’ β x β Icc 0 1, 1 - x * abs z β 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
ic : IntervalIntegrable (fun t => z / (1 + βt * z)) MeasureTheory.volume 0 1
dr : β t β uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (abs z / (1 - t * abs z)) t
β’ ContinuousOn (fun t => abs z / (1 - t * abs z)) (Icc 0 1)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
Complex.abs_log_one_add_le
|
[14, 1]
|
[65, 80]
|
intro t β¨_,t1β©
|
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
ic : IntervalIntegrable (fun t => z / (1 + βt * z)) MeasureTheory.volume 0 1
dr : β t β uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (abs z / (1 - t * abs z)) t
β’ β x β Icc 0 1, 1 - x * abs z β 0
|
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
ic : IntervalIntegrable (fun t => z / (1 + βt * z)) MeasureTheory.volume 0 1
dr : β t β uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (abs z / (1 - t * abs z)) t
t : β
leftβ : 0 β€ t
t1 : t β€ 1
β’ 1 - t * abs z β 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
ic : IntervalIntegrable (fun t => z / (1 + βt * z)) MeasureTheory.volume 0 1
dr : β t β uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (abs z / (1 - t * abs z)) t
β’ β x β Icc 0 1, 1 - x * abs z β 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
Complex.abs_log_one_add_le
|
[14, 1]
|
[65, 80]
|
exact ne_of_gt (sub_pos.mpr (m1 _ t1))
|
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
ic : IntervalIntegrable (fun t => z / (1 + βt * z)) MeasureTheory.volume 0 1
dr : β t β uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (abs z / (1 - t * abs z)) t
t : β
leftβ : 0 β€ t
t1 : t β€ 1
β’ 1 - t * abs z β 0
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
ic : IntervalIntegrable (fun t => z / (1 + βt * z)) MeasureTheory.volume 0 1
dr : β t β uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (abs z / (1 - t * abs z)) t
t : β
leftβ : 0 β€ t
t1 : t β€ 1
β’ 1 - t * abs z β 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
Complex.abs_log_one_add_le
|
[14, 1]
|
[65, 80]
|
continuity
|
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
ic : IntervalIntegrable (fun t => z / (1 + βt * z)) MeasureTheory.volume 0 1
dr : β t β uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (abs z / (1 - t * abs z)) t
β’ Continuous fun t => 1 - t * abs z
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
dc : β t β uIcc 0 1, HasDerivAt (fun t => (1 + βt * z).log) (z / (1 + βt * z)) t
ic : IntervalIntegrable (fun t => z / (1 + βt * z)) MeasureTheory.volume 0 1
dr : β t β uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (abs z / (1 - t * abs z)) t
β’ Continuous fun t => 1 - t * abs z
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
Complex.abs_log_one_add_le
|
[14, 1]
|
[65, 80]
|
simp only [map_one, map_mul, Complex.abs_ofReal, _root_.abs_of_nonneg t0]
|
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
ic : IntervalIntegrable (fun t => z / (1 + βt * z)) MeasureTheory.volume 0 1
ir : IntervalIntegrable (fun t => abs z / (1 - t * abs z)) MeasureTheory.volume 0 1
t : β
t0 : 0 β€ t
t1 : t β€ 1
β’ abs 1 - abs (βt * z) = 1 - t * abs z
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
z : β
z1 : abs z < 1
m1 : β t β€ 1, t * abs z < 1
ic : IntervalIntegrable (fun t => z / (1 + βt * z)) MeasureTheory.volume 0 1
ir : IntervalIntegrable (fun t => abs z / (1 - t * abs z)) MeasureTheory.volume 0 1
t : β
t0 : 0 β€ t
t1 : t β€ 1
β’ abs 1 - abs (βt * z) = 1 - t * abs z
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
Real.abs_log_one_add_le
|
[67, 1]
|
[75, 36]
|
have h := Complex.abs_log_one_add_le (z := x) ?_
|
x : β
x1 : |x| < 1
β’ |(1 + x).log| β€ -(1 - |x|).log
|
case refine_2
x : β
x1 : |x| < 1
h : Complex.abs (1 + βx).log β€ -(1 - Complex.abs βx).log
β’ |(1 + x).log| β€ -(1 - |x|).log
case refine_1
x : β
x1 : |x| < 1
β’ Complex.abs βx < 1
|
Please generate a tactic in lean4 to solve the state.
STATE:
x : β
x1 : |x| < 1
β’ |(1 + x).log| β€ -(1 - |x|).log
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
Real.abs_log_one_add_le
|
[67, 1]
|
[75, 36]
|
rw [βComplex.ofReal_one, βComplex.ofReal_add, βComplex.ofReal_log, Complex.abs_ofReal,
Complex.abs_ofReal] at h
|
case refine_2
x : β
x1 : |x| < 1
h : Complex.abs (1 + βx).log β€ -(1 - Complex.abs βx).log
β’ |(1 + x).log| β€ -(1 - |x|).log
|
case refine_2
x : β
x1 : |x| < 1
h : |(1 + x).log| β€ -(1 - |x|).log
β’ |(1 + x).log| β€ -(1 - |x|).log
case refine_2
x : β
x1 : |x| < 1
h : Complex.abs (β(1 + x)).log β€ -(1 - Complex.abs βx).log
β’ 0 β€ 1 + x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
x : β
x1 : |x| < 1
h : Complex.abs (1 + βx).log β€ -(1 - Complex.abs βx).log
β’ |(1 + x).log| β€ -(1 - |x|).log
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
Real.abs_log_one_add_le
|
[67, 1]
|
[75, 36]
|
exact h
|
case refine_2
x : β
x1 : |x| < 1
h : |(1 + x).log| β€ -(1 - |x|).log
β’ |(1 + x).log| β€ -(1 - |x|).log
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
x : β
x1 : |x| < 1
h : |(1 + x).log| β€ -(1 - |x|).log
β’ |(1 + x).log| β€ -(1 - |x|).log
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
Real.abs_log_one_add_le
|
[67, 1]
|
[75, 36]
|
simp only [abs_lt] at x1
|
case refine_2
x : β
x1 : |x| < 1
h : Complex.abs (β(1 + x)).log β€ -(1 - Complex.abs βx).log
β’ 0 β€ 1 + x
|
case refine_2
x : β
h : Complex.abs (β(1 + x)).log β€ -(1 - Complex.abs βx).log
x1 : -1 < x β§ x < 1
β’ 0 β€ 1 + x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
x : β
x1 : |x| < 1
h : Complex.abs (β(1 + x)).log β€ -(1 - Complex.abs βx).log
β’ 0 β€ 1 + x
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
Real.abs_log_one_add_le
|
[67, 1]
|
[75, 36]
|
linarith
|
case refine_2
x : β
h : Complex.abs (β(1 + x)).log β€ -(1 - Complex.abs βx).log
x1 : -1 < x β§ x < 1
β’ 0 β€ 1 + x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
x : β
h : Complex.abs (β(1 + x)).log β€ -(1 - Complex.abs βx).log
x1 : -1 < x β§ x < 1
β’ 0 β€ 1 + x
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
Real.abs_log_one_add_le
|
[67, 1]
|
[75, 36]
|
simpa only [Complex.abs_ofReal]
|
case refine_1
x : β
x1 : |x| < 1
β’ Complex.abs βx < 1
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
x : β
x1 : |x| < 1
β’ Complex.abs βx < 1
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
Real.neg_log_one_sub_mono
|
[77, 1]
|
[80, 59]
|
linarith
|
x y : β
xy : x β€ y
y1 : y < 1
β’ 0 < 1 - y
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
x y : β
xy : x β€ y
y1 : y < 1
β’ 0 < 1 - y
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
Real.neg_log_one_sub_mono
|
[77, 1]
|
[80, 59]
|
linarith
|
x y : β
xy : x β€ y
y1 : y < 1
β’ 1 - y β€ 1 - x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
x y : β
xy : x β€ y
y1 : y < 1
β’ 1 - y β€ 1 - x
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
neg_log_one_sub_le_two
|
[82, 1]
|
[87, 13]
|
apply le_trans (Real.neg_log_one_sub_mono x2 (by linarith)) ?_
|
x : β
x2 : x β€ 1 / 2
β’ -(1 - x).log β€ 2
|
x : β
x2 : x β€ 1 / 2
β’ -(1 - 1 / 2).log β€ 2
|
Please generate a tactic in lean4 to solve the state.
STATE:
x : β
x2 : x β€ 1 / 2
β’ -(1 - x).log β€ 2
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
neg_log_one_sub_le_two
|
[82, 1]
|
[87, 13]
|
rw [neg_le, Real.le_log_iff_exp_le]
|
x : β
x2 : x β€ 1 / 2
β’ -(1 - 1 / 2).log β€ 2
|
x : β
x2 : x β€ 1 / 2
β’ (-2).exp β€ 1 - 1 / 2
x : β
x2 : x β€ 1 / 2
β’ 0 < 1 - 1 / 2
|
Please generate a tactic in lean4 to solve the state.
STATE:
x : β
x2 : x β€ 1 / 2
β’ -(1 - 1 / 2).log β€ 2
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
neg_log_one_sub_le_two
|
[82, 1]
|
[87, 13]
|
linarith
|
x : β
x2 : x β€ 1 / 2
β’ 1 / 2 < 1
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
x : β
x2 : x β€ 1 / 2
β’ 1 / 2 < 1
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
neg_log_one_sub_le_two
|
[82, 1]
|
[87, 13]
|
exact (exp_neg_ofNat_lt).le
|
x : β
x2 : x β€ 1 / 2
β’ (-2).exp β€ 1 - 1 / 2
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
x : β
x2 : x β€ 1 / 2
β’ (-2).exp β€ 1 - 1 / 2
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
neg_log_one_sub_le_two
|
[82, 1]
|
[87, 13]
|
norm_num
|
x : β
x2 : x β€ 1 / 2
β’ 0 < 1 - 1 / 2
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
x : β
x2 : x β€ 1 / 2
β’ 0 < 1 - 1 / 2
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
neg_log_one_sub_le_linear
|
[89, 1]
|
[116, 11]
|
rcases le_min_iff.mp xc with β¨x1,xcβ©
|
x c : β
x0 : 0 β€ x
c1 : 1 < c
xc : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
β’ -(1 - x).log β€ c * x
|
case intro
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
x1 : x β€ 1
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
β’ -(1 - x).log β€ c * x
|
Please generate a tactic in lean4 to solve the state.
STATE:
x c : β
x0 : 0 β€ x
c1 : 1 < c
xc : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
β’ -(1 - x).log β€ c * x
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
neg_log_one_sub_le_linear
|
[89, 1]
|
[116, 11]
|
by_cases xz : x = 0
|
case intro
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
x1 : x β€ 1
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
β’ -(1 - x).log β€ c * x
|
case pos
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
x1 : x β€ 1
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : x = 0
β’ -(1 - x).log β€ c * x
case neg
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
x1 : x β€ 1
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
β’ -(1 - x).log β€ c * x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
x1 : x β€ 1
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
β’ -(1 - x).log β€ c * x
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
neg_log_one_sub_le_linear
|
[89, 1]
|
[116, 11]
|
by_cases xe : x = 1
|
case neg
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
x1 : x β€ 1
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
β’ -(1 - x).log β€ c * x
|
case pos
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
x1 : x β€ 1
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : x = 1
β’ -(1 - x).log β€ c * x
case neg
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
x1 : x β€ 1
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
β’ -(1 - x).log β€ c * x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
x1 : x β€ 1
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
β’ -(1 - x).log β€ c * x
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
neg_log_one_sub_le_linear
|
[89, 1]
|
[116, 11]
|
replace x1 := Ne.lt_of_le xe x1
|
case neg
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
x1 : x β€ 1
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
β’ -(1 - x).log β€ c * x
|
case neg
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
β’ -(1 - x).log β€ c * x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
x1 : x β€ 1
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
β’ -(1 - x).log β€ c * x
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
neg_log_one_sub_le_linear
|
[89, 1]
|
[116, 11]
|
have x0' : 0 < x := (Ne.symm xz).lt_of_le x0
|
case neg
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
β’ -(1 - x).log β€ c * x
|
case neg
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
β’ -(1 - x).log β€ c * x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
β’ -(1 - x).log β€ c * x
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
neg_log_one_sub_le_linear
|
[89, 1]
|
[116, 11]
|
have c1p : 0 < (c - 1) * 2 := mul_pos (sub_pos.mpr c1) (by norm_num)
|
case neg
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
β’ -(1 - x).log β€ c * x
|
case neg
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
β’ -(1 - x).log β€ c * x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
β’ -(1 - x).log β€ c * x
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
neg_log_one_sub_le_linear
|
[89, 1]
|
[116, 11]
|
have x1p : 0 < 1 - x := by linarith
|
case neg
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
β’ -(1 - x).log β€ c * x
|
case neg
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
β’ -(1 - x).log β€ c * x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
β’ -(1 - x).log β€ c * x
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
neg_log_one_sub_le_linear
|
[89, 1]
|
[116, 11]
|
have h := Complex.norm_log_one_add_sub_self_le (z := -x)
(by simp only [norm_neg, Complex.norm_eq_abs, Complex.abs_ofReal, abs_of_nonneg x0]; exact x1)
|
case neg
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
β’ -(1 - x).log β€ c * x
|
case neg
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
h : β(1 + -βx).log - -βxβ β€ β-βxβ ^ 2 * (1 - β-βxβ)β»ΒΉ / 2
β’ -(1 - x).log β€ c * x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
β’ -(1 - x).log β€ c * x
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
neg_log_one_sub_le_linear
|
[89, 1]
|
[116, 11]
|
simp only [Complex.norm_eq_abs, Complex.abs_ofReal, βComplex.ofReal_one, βComplex.ofReal_add,
βComplex.ofReal_log x1p.le, βComplex.ofReal_sub, abs_le, abs_of_nonneg x0, βComplex.ofReal_neg,
βsub_eq_add_neg, abs_neg] at h
|
case neg
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
h : β(1 + -βx).log - -βxβ β€ β-βxβ ^ 2 * (1 - β-βxβ)β»ΒΉ / 2
β’ -(1 - x).log β€ c * x
|
case neg
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
h : -(x ^ 2 * (1 - x)β»ΒΉ / 2) β€ (1 - x).log - -x β§ (1 - x).log - -x β€ x ^ 2 * (1 - x)β»ΒΉ / 2
β’ -(1 - x).log β€ c * x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
h : β(1 + -βx).log - -βxβ β€ β-βxβ ^ 2 * (1 - β-βxβ)β»ΒΉ / 2
β’ -(1 - x).log β€ c * x
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
neg_log_one_sub_le_linear
|
[89, 1]
|
[116, 11]
|
replace h : -Real.log (1 - x) β€ x + x^2 * (1 - x)β»ΒΉ / 2 := by linarith
|
case neg
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
h : -(x ^ 2 * (1 - x)β»ΒΉ / 2) β€ (1 - x).log - -x β§ (1 - x).log - -x β€ x ^ 2 * (1 - x)β»ΒΉ / 2
β’ -(1 - x).log β€ c * x
|
case neg
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
h : -(1 - x).log β€ x + x ^ 2 * (1 - x)β»ΒΉ / 2
β’ -(1 - x).log β€ c * x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
h : -(x ^ 2 * (1 - x)β»ΒΉ / 2) β€ (1 - x).log - -x β§ (1 - x).log - -x β€ x ^ 2 * (1 - x)β»ΒΉ / 2
β’ -(1 - x).log β€ c * x
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
neg_log_one_sub_le_linear
|
[89, 1]
|
[116, 11]
|
apply le_trans h
|
case neg
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
h : -(1 - x).log β€ x + x ^ 2 * (1 - x)β»ΒΉ / 2
β’ -(1 - x).log β€ c * x
|
case neg
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
h : -(1 - x).log β€ x + x ^ 2 * (1 - x)β»ΒΉ / 2
β’ x + x ^ 2 * (1 - x)β»ΒΉ / 2 β€ c * x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
h : -(1 - x).log β€ x + x ^ 2 * (1 - x)β»ΒΉ / 2
β’ -(1 - x).log β€ c * x
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
neg_log_one_sub_le_linear
|
[89, 1]
|
[116, 11]
|
rw [pow_two, mul_assoc, mul_div_assoc, βmul_one_add, mul_comm x _]
|
case neg
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
h : -(1 - x).log β€ x + x ^ 2 * (1 - x)β»ΒΉ / 2
β’ x + x ^ 2 * (1 - x)β»ΒΉ / 2 β€ c * x
|
case neg
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
h : -(1 - x).log β€ x + x ^ 2 * (1 - x)β»ΒΉ / 2
β’ (1 + x * (1 - x)β»ΒΉ / 2) * x β€ c * x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
h : -(1 - x).log β€ x + x ^ 2 * (1 - x)β»ΒΉ / 2
β’ x + x ^ 2 * (1 - x)β»ΒΉ / 2 β€ c * x
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
neg_log_one_sub_le_linear
|
[89, 1]
|
[116, 11]
|
apply mul_le_mul_of_nonneg_right _ x0
|
case neg
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
h : -(1 - x).log β€ x + x ^ 2 * (1 - x)β»ΒΉ / 2
β’ (1 + x * (1 - x)β»ΒΉ / 2) * x β€ c * x
|
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
h : -(1 - x).log β€ x + x ^ 2 * (1 - x)β»ΒΉ / 2
β’ 1 + x * (1 - x)β»ΒΉ / 2 β€ c
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
h : -(1 - x).log β€ x + x ^ 2 * (1 - x)β»ΒΉ / 2
β’ (1 + x * (1 - x)β»ΒΉ / 2) * x β€ c * x
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
neg_log_one_sub_le_linear
|
[89, 1]
|
[116, 11]
|
nth_rw 1 [βinv_inv x]
|
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
h : -(1 - x).log β€ x + x ^ 2 * (1 - x)β»ΒΉ / 2
β’ 1 + x * (1 - x)β»ΒΉ / 2 β€ c
|
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
h : -(1 - x).log β€ x + x ^ 2 * (1 - x)β»ΒΉ / 2
β’ 1 + xβ»ΒΉβ»ΒΉ * (1 - x)β»ΒΉ / 2 β€ c
|
Please generate a tactic in lean4 to solve the state.
STATE:
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
h : -(1 - x).log β€ x + x ^ 2 * (1 - x)β»ΒΉ / 2
β’ 1 + x * (1 - x)β»ΒΉ / 2 β€ c
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
neg_log_one_sub_le_linear
|
[89, 1]
|
[116, 11]
|
rw [βmul_inv, mul_sub, mul_one, inv_mul_cancel xz]
|
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
h : -(1 - x).log β€ x + x ^ 2 * (1 - x)β»ΒΉ / 2
β’ 1 + xβ»ΒΉβ»ΒΉ * (1 - x)β»ΒΉ / 2 β€ c
|
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
h : -(1 - x).log β€ x + x ^ 2 * (1 - x)β»ΒΉ / 2
β’ 1 + (xβ»ΒΉ - 1)β»ΒΉ / 2 β€ c
|
Please generate a tactic in lean4 to solve the state.
STATE:
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
h : -(1 - x).log β€ x + x ^ 2 * (1 - x)β»ΒΉ / 2
β’ 1 + xβ»ΒΉβ»ΒΉ * (1 - x)β»ΒΉ / 2 β€ c
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
neg_log_one_sub_le_linear
|
[89, 1]
|
[116, 11]
|
rw [add_comm, βle_sub_iff_add_le, div_le_iff (by norm_num)]
|
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
h : -(1 - x).log β€ x + x ^ 2 * (1 - x)β»ΒΉ / 2
β’ 1 + (xβ»ΒΉ - 1)β»ΒΉ / 2 β€ c
|
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
h : -(1 - x).log β€ x + x ^ 2 * (1 - x)β»ΒΉ / 2
β’ (xβ»ΒΉ - 1)β»ΒΉ β€ (c - 1) * 2
|
Please generate a tactic in lean4 to solve the state.
STATE:
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
h : -(1 - x).log β€ x + x ^ 2 * (1 - x)β»ΒΉ / 2
β’ 1 + (xβ»ΒΉ - 1)β»ΒΉ / 2 β€ c
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
neg_log_one_sub_le_linear
|
[89, 1]
|
[116, 11]
|
apply inv_le_of_inv_le c1p
|
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
h : -(1 - x).log β€ x + x ^ 2 * (1 - x)β»ΒΉ / 2
β’ (xβ»ΒΉ - 1)β»ΒΉ β€ (c - 1) * 2
|
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
h : -(1 - x).log β€ x + x ^ 2 * (1 - x)β»ΒΉ / 2
β’ ((c - 1) * 2)β»ΒΉ β€ xβ»ΒΉ - 1
|
Please generate a tactic in lean4 to solve the state.
STATE:
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
h : -(1 - x).log β€ x + x ^ 2 * (1 - x)β»ΒΉ / 2
β’ (xβ»ΒΉ - 1)β»ΒΉ β€ (c - 1) * 2
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
neg_log_one_sub_le_linear
|
[89, 1]
|
[116, 11]
|
rw [le_sub_iff_add_le, le_inv (add_pos (inv_pos.mpr c1p) (by norm_num)) x0']
|
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
h : -(1 - x).log β€ x + x ^ 2 * (1 - x)β»ΒΉ / 2
β’ ((c - 1) * 2)β»ΒΉ β€ xβ»ΒΉ - 1
|
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
h : -(1 - x).log β€ x + x ^ 2 * (1 - x)β»ΒΉ / 2
β’ x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
|
Please generate a tactic in lean4 to solve the state.
STATE:
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
h : -(1 - x).log β€ x + x ^ 2 * (1 - x)β»ΒΉ / 2
β’ ((c - 1) * 2)β»ΒΉ β€ xβ»ΒΉ - 1
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
neg_log_one_sub_le_linear
|
[89, 1]
|
[116, 11]
|
exact xc
|
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
h : -(1 - x).log β€ x + x ^ 2 * (1 - x)β»ΒΉ / 2
β’ x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
h : -(1 - x).log β€ x + x ^ 2 * (1 - x)β»ΒΉ / 2
β’ x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
neg_log_one_sub_le_linear
|
[89, 1]
|
[116, 11]
|
simp only [xz, sub_zero, Real.log_one, neg_zero, mul_zero, le_refl]
|
case pos
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
x1 : x β€ 1
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : x = 0
β’ -(1 - x).log β€ c * x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pos
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
x1 : x β€ 1
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : x = 0
β’ -(1 - x).log β€ c * x
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
neg_log_one_sub_le_linear
|
[89, 1]
|
[116, 11]
|
simp only [xe, sub_self, Real.log_zero, neg_zero, mul_one]
|
case pos
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
x1 : x β€ 1
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : x = 1
β’ -(1 - x).log β€ c * x
|
case pos
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
x1 : x β€ 1
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : x = 1
β’ 0 β€ c
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pos
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
x1 : x β€ 1
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : x = 1
β’ -(1 - x).log β€ c * x
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
neg_log_one_sub_le_linear
|
[89, 1]
|
[116, 11]
|
linarith
|
case pos
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
x1 : x β€ 1
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : x = 1
β’ 0 β€ c
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pos
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
x1 : x β€ 1
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : x = 1
β’ 0 β€ c
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
neg_log_one_sub_le_linear
|
[89, 1]
|
[116, 11]
|
norm_num
|
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
β’ 0 < 2
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
β’ 0 < 2
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
neg_log_one_sub_le_linear
|
[89, 1]
|
[116, 11]
|
linarith
|
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
β’ 0 < 1 - x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
β’ 0 < 1 - x
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
neg_log_one_sub_le_linear
|
[89, 1]
|
[116, 11]
|
simp only [norm_neg, Complex.norm_eq_abs, Complex.abs_ofReal, abs_of_nonneg x0]
|
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
β’ β-βxβ < 1
|
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
β’ x < 1
|
Please generate a tactic in lean4 to solve the state.
STATE:
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
β’ β-βxβ < 1
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
neg_log_one_sub_le_linear
|
[89, 1]
|
[116, 11]
|
exact x1
|
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
β’ x < 1
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
β’ x < 1
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
neg_log_one_sub_le_linear
|
[89, 1]
|
[116, 11]
|
linarith
|
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
h : -(x ^ 2 * (1 - x)β»ΒΉ / 2) β€ (1 - x).log - -x β§ (1 - x).log - -x β€ x ^ 2 * (1 - x)β»ΒΉ / 2
β’ -(1 - x).log β€ x + x ^ 2 * (1 - x)β»ΒΉ / 2
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
h : -(x ^ 2 * (1 - x)β»ΒΉ / 2) β€ (1 - x).log - -x β§ (1 - x).log - -x β€ x ^ 2 * (1 - x)β»ΒΉ / 2
β’ -(1 - x).log β€ x + x ^ 2 * (1 - x)β»ΒΉ / 2
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
neg_log_one_sub_le_linear
|
[89, 1]
|
[116, 11]
|
norm_num
|
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
h : -(1 - x).log β€ x + x ^ 2 * (1 - x)β»ΒΉ / 2
β’ 0 < 2
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
h : -(1 - x).log β€ x + x ^ 2 * (1 - x)β»ΒΉ / 2
β’ 0 < 2
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Log1p.lean
|
neg_log_one_sub_le_linear
|
[89, 1]
|
[116, 11]
|
norm_num
|
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
h : -(1 - x).log β€ x + x ^ 2 * (1 - x)β»ΒΉ / 2
β’ 0 < 1
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
x c : β
x0 : 0 β€ x
c1 : 1 < c
xcβ : x β€ min 1 (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xc : x β€ (((c - 1) * 2)β»ΒΉ + 1)β»ΒΉ
xz : Β¬x = 0
xe : Β¬x = 1
x1 : x < 1
x0' : 0 < x
c1p : 0 < (c - 1) * 2
x1p : 0 < 1 - x
h : -(1 - x).log β€ x + x ^ 2 * (1 - x)β»ΒΉ / 2
β’ 0 < 1
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Topology.lean
|
UniformCauchySeqOn.bounded
|
[21, 1]
|
[49, 35]
|
generalize hc : (fun n β¦ Classical.choose ((sc.bddAbove_image (fc n).norm).exists_ge 0)) = c
|
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
u : UniformCauchySeqOn f atTop s
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b
|
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
u : UniformCauchySeqOn f atTop s
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b
|
Please generate a tactic in lean4 to solve the state.
STATE:
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
u : UniformCauchySeqOn f atTop s
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Topology.lean
|
UniformCauchySeqOn.bounded
|
[21, 1]
|
[49, 35]
|
have cs : β n, 0 β€ c n β§ β x, x β s β βf n xβ β€ c n := fun n β¦ by
simpa only [β hc, mem_image, forall_exists_index, and_imp, forall_apply_eq_imp_iffβ] using
Classical.choose_spec ((sc.bddAbove_image (fc n).norm).exists_ge 0)
|
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
u : UniformCauchySeqOn f atTop s
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b
|
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
u : UniformCauchySeqOn f atTop s
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b
|
Please generate a tactic in lean4 to solve the state.
STATE:
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
u : UniformCauchySeqOn f atTop s
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Topology.lean
|
UniformCauchySeqOn.bounded
|
[21, 1]
|
[49, 35]
|
rw [Metric.uniformCauchySeqOn_iff] at u
|
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
u : UniformCauchySeqOn f atTop s
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b
|
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
u : β Ξ΅ > 0, β N, β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < Ξ΅
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b
|
Please generate a tactic in lean4 to solve the state.
STATE:
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
u : UniformCauchySeqOn f atTop s
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Topology.lean
|
UniformCauchySeqOn.bounded
|
[21, 1]
|
[49, 35]
|
rcases u 1 (by norm_num) with β¨N, Hβ©
|
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
u : β Ξ΅ > 0, β N, β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < Ξ΅
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b
|
case intro
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
u : β Ξ΅ > 0, β N, β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < Ξ΅
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b
|
Please generate a tactic in lean4 to solve the state.
STATE:
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
u : β Ξ΅ > 0, β N, β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < Ξ΅
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Topology.lean
|
UniformCauchySeqOn.bounded
|
[21, 1]
|
[49, 35]
|
clear u
|
case intro
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
u : β Ξ΅ > 0, β N, β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < Ξ΅
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b
|
case intro
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
u : β Ξ΅ > 0, β N, β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < Ξ΅
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Topology.lean
|
UniformCauchySeqOn.bounded
|
[21, 1]
|
[49, 35]
|
generalize hbs : Finset.image c (Finset.range (N + 1)) = bs
|
case intro
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b
|
case intro
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Topology.lean
|
UniformCauchySeqOn.bounded
|
[21, 1]
|
[49, 35]
|
have c0 : c 0 β bs := by simp [β hbs]; exists 0; simp
|
case intro
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b
|
case intro
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Topology.lean
|
UniformCauchySeqOn.bounded
|
[21, 1]
|
[49, 35]
|
generalize hb : 1 + bs.max' β¨_, c0β© = b
|
case intro
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b
|
case intro
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Topology.lean
|
UniformCauchySeqOn.bounded
|
[21, 1]
|
[49, 35]
|
exists b
|
case intro
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b
|
case intro
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
β’ 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Topology.lean
|
UniformCauchySeqOn.bounded
|
[21, 1]
|
[49, 35]
|
constructor
|
case intro
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
β’ 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b
|
case intro.left
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
β’ 0 β€ b
case intro.right
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
β’ β (n : β), β x β s, βf n xβ β€ b
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
β’ 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Topology.lean
|
UniformCauchySeqOn.bounded
|
[21, 1]
|
[49, 35]
|
simpa only [β hc, mem_image, forall_exists_index, and_imp, forall_apply_eq_imp_iffβ] using
Classical.choose_spec ((sc.bddAbove_image (fc n).norm).exists_ge 0)
|
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
u : UniformCauchySeqOn f atTop s
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
n : β
β’ 0 β€ c n β§ β x β s, βf n xβ β€ c n
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
u : UniformCauchySeqOn f atTop s
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
n : β
β’ 0 β€ c n β§ β x β s, βf n xβ β€ c n
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Topology.lean
|
UniformCauchySeqOn.bounded
|
[21, 1]
|
[49, 35]
|
norm_num
|
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
u : β Ξ΅ > 0, β N, β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < Ξ΅
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
β’ 1 > 0
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
u : β Ξ΅ > 0, β N, β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < Ξ΅
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
β’ 1 > 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Topology.lean
|
UniformCauchySeqOn.bounded
|
[21, 1]
|
[49, 35]
|
simp [β hbs]
|
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
β’ c 0 β bs
|
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
β’ β a < N + 1, c a = c 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
β’ c 0 β bs
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Topology.lean
|
UniformCauchySeqOn.bounded
|
[21, 1]
|
[49, 35]
|
exists 0
|
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
β’ β a < N + 1, c a = c 0
|
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
β’ 0 < N + 1 β§ c 0 = c 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
β’ β a < N + 1, c a = c 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Topology.lean
|
UniformCauchySeqOn.bounded
|
[21, 1]
|
[49, 35]
|
simp
|
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
β’ 0 < N + 1 β§ c 0 = c 0
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
β’ 0 < N + 1 β§ c 0 = c 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Topology.lean
|
UniformCauchySeqOn.bounded
|
[21, 1]
|
[49, 35]
|
rw [β hb]
|
case intro.left
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
β’ 0 β€ b
|
case intro.left
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
β’ 0 β€ 1 + bs.max' β―
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro.left
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
β’ 0 β€ b
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Topology.lean
|
UniformCauchySeqOn.bounded
|
[21, 1]
|
[49, 35]
|
exact add_nonneg (by norm_num) (_root_.trans (cs 0).1 (Finset.le_max' _ _ c0))
|
case intro.left
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
β’ 0 β€ 1 + bs.max' β―
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro.left
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
β’ 0 β€ 1 + bs.max' β―
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Topology.lean
|
UniformCauchySeqOn.bounded
|
[21, 1]
|
[49, 35]
|
norm_num
|
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
β’ 0 β€ 1
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
β’ 0 β€ 1
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Topology.lean
|
UniformCauchySeqOn.bounded
|
[21, 1]
|
[49, 35]
|
intro n x xs
|
case intro.right
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
β’ β (n : β), β x β s, βf n xβ β€ b
|
case intro.right
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
β’ βf n xβ β€ b
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro.right
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
β’ β (n : β), β x β s, βf n xβ β€ b
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Topology.lean
|
UniformCauchySeqOn.bounded
|
[21, 1]
|
[49, 35]
|
by_cases nN : n β€ N
|
case intro.right
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
β’ βf n xβ β€ b
|
case pos
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : n β€ N
β’ βf n xβ β€ b
case neg
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : Β¬n β€ N
β’ βf n xβ β€ b
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro.right
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
β’ βf n xβ β€ b
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Topology.lean
|
UniformCauchySeqOn.bounded
|
[21, 1]
|
[49, 35]
|
have cn : c n β bs := by simp [β hbs]; exists n; simp [Nat.lt_add_one_iff.mpr nN]
|
case pos
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : n β€ N
β’ βf n xβ β€ b
|
case pos
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : n β€ N
cn : c n β bs
β’ βf n xβ β€ b
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pos
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : n β€ N
β’ βf n xβ β€ b
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Topology.lean
|
UniformCauchySeqOn.bounded
|
[21, 1]
|
[49, 35]
|
exact _root_.trans ((cs n).2 x xs) (_root_.trans (Finset.le_max' _ _ cn)
(by simp only [le_add_iff_nonneg_left, zero_le_one, β hb]))
|
case pos
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : n β€ N
cn : c n β bs
β’ βf n xβ β€ b
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pos
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : n β€ N
cn : c n β bs
β’ βf n xβ β€ b
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Topology.lean
|
UniformCauchySeqOn.bounded
|
[21, 1]
|
[49, 35]
|
simp [β hbs]
|
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : n β€ N
β’ c n β bs
|
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : n β€ N
β’ β a < N + 1, c a = c n
|
Please generate a tactic in lean4 to solve the state.
STATE:
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : n β€ N
β’ c n β bs
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Topology.lean
|
UniformCauchySeqOn.bounded
|
[21, 1]
|
[49, 35]
|
exists n
|
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : n β€ N
β’ β a < N + 1, c a = c n
|
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : n β€ N
β’ n < N + 1 β§ c n = c n
|
Please generate a tactic in lean4 to solve the state.
STATE:
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : n β€ N
β’ β a < N + 1, c a = c n
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Topology.lean
|
UniformCauchySeqOn.bounded
|
[21, 1]
|
[49, 35]
|
simp [Nat.lt_add_one_iff.mpr nN]
|
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : n β€ N
β’ n < N + 1 β§ c n = c n
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : n β€ N
β’ n < N + 1 β§ c n = c n
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Topology.lean
|
UniformCauchySeqOn.bounded
|
[21, 1]
|
[49, 35]
|
simp only [le_add_iff_nonneg_left, zero_le_one, β hb]
|
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : n β€ N
cn : c n β bs
β’ bs.max' β― β€ b
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : n β€ N
cn : c n β bs
β’ bs.max' β― β€ b
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Topology.lean
|
UniformCauchySeqOn.bounded
|
[21, 1]
|
[49, 35]
|
simp at nN
|
case neg
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : Β¬n β€ N
β’ βf n xβ β€ b
|
case neg
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : N < n
β’ βf n xβ β€ b
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : Β¬n β€ N
β’ βf n xβ β€ b
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Topology.lean
|
UniformCauchySeqOn.bounded
|
[21, 1]
|
[49, 35]
|
specialize H N le_rfl n nN.le x xs
|
case neg
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : N < n
β’ βf n xβ β€ b
|
case neg
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : N < n
H : dist (f N x) (f n x) < 1
β’ βf n xβ β€ b
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : N < n
β’ βf n xβ β€ b
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Topology.lean
|
UniformCauchySeqOn.bounded
|
[21, 1]
|
[49, 35]
|
have cN : c N β bs := by simp [β hbs]; exists N; simp
|
case neg
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : N < n
H : dist (f N x) (f n x) < 1
β’ βf n xβ β€ b
|
case neg
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : N < n
H : dist (f N x) (f n x) < 1
cN : c N β bs
β’ βf n xβ β€ b
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : N < n
H : dist (f N x) (f n x) < 1
β’ βf n xβ β€ b
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Topology.lean
|
UniformCauchySeqOn.bounded
|
[21, 1]
|
[49, 35]
|
have bN := _root_.trans ((cs N).2 x xs) (Finset.le_max' _ _ cN)
|
case neg
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : N < n
H : dist (f N x) (f n x) < 1
cN : c N β bs
β’ βf n xβ β€ b
|
case neg
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : N < n
H : dist (f N x) (f n x) < 1
cN : c N β bs
bN : βf N xβ β€ bs.max' β―
β’ βf n xβ β€ b
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : N < n
H : dist (f N x) (f n x) < 1
cN : c N β bs
β’ βf n xβ β€ b
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Topology.lean
|
UniformCauchySeqOn.bounded
|
[21, 1]
|
[49, 35]
|
rw [dist_eq_norm] at H
|
case neg
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : N < n
H : dist (f N x) (f n x) < 1
cN : c N β bs
bN : βf N xβ β€ bs.max' β―
β’ βf n xβ β€ b
|
case neg
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : N < n
H : βf N x - f n xβ < 1
cN : c N β bs
bN : βf N xβ β€ bs.max' β―
β’ βf n xβ β€ b
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : N < n
H : dist (f N x) (f n x) < 1
cN : c N β bs
bN : βf N xβ β€ bs.max' β―
β’ βf n xβ β€ b
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Topology.lean
|
UniformCauchySeqOn.bounded
|
[21, 1]
|
[49, 35]
|
calc βf n xβ = βf N x - (f N x - f n x)β := by rw [sub_sub_cancel]
_ β€ βf N xβ + βf N x - f n xβ := norm_sub_le _ _
_ β€ bs.max' _ + 1 := add_le_add bN H.le
_ = 1 + bs.max' _ := by ring
_ = b := by simp only [hb]
|
case neg
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : N < n
H : βf N x - f n xβ < 1
cN : c N β bs
bN : βf N xβ β€ bs.max' β―
β’ βf n xβ β€ b
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : N < n
H : βf N x - f n xβ < 1
cN : c N β bs
bN : βf N xβ β€ bs.max' β―
β’ βf n xβ β€ b
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Topology.lean
|
UniformCauchySeqOn.bounded
|
[21, 1]
|
[49, 35]
|
simp [β hbs]
|
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : N < n
H : dist (f N x) (f n x) < 1
β’ c N β bs
|
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : N < n
H : dist (f N x) (f n x) < 1
β’ β a < N + 1, c a = c N
|
Please generate a tactic in lean4 to solve the state.
STATE:
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : N < n
H : dist (f N x) (f n x) < 1
β’ c N β bs
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Topology.lean
|
UniformCauchySeqOn.bounded
|
[21, 1]
|
[49, 35]
|
exists N
|
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : N < n
H : dist (f N x) (f n x) < 1
β’ β a < N + 1, c a = c N
|
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : N < n
H : dist (f N x) (f n x) < 1
β’ N < N + 1 β§ c N = c N
|
Please generate a tactic in lean4 to solve the state.
STATE:
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : N < n
H : dist (f N x) (f n x) < 1
β’ β a < N + 1, c a = c N
TACTIC:
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.