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.