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
|
---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
iter_error_le_log_log_abs
|
[245, 1]
|
[259, 83]
|
positivity
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
hl : 1.38 β€ (Complex.abs z).log
hll : 0.32 β€ (Complex.abs z).log.log
β’ 0 β€ 1.38
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
iter_error_le_log_log_abs
|
[245, 1]
|
[259, 83]
|
positivity
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
hl : 1.38 β€ (Complex.abs z).log
hll : 0.32 β€ (Complex.abs z).log.log
β’ 0 β€ Complex.abs z
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_approx
|
[262, 1]
|
[295, 40]
|
set s := superF d
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
β’ |β―.potential c βz - 1 / Complex.abs z| β€ potential_error d c z
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
β’ |s.potential c βz - 1 / Complex.abs z| β€ potential_error d c z
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_approx
|
[262, 1]
|
[295, 40]
|
have z3 : 3 β€ abs z := le_trans (by norm_num) z4
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
β’ |s.potential c βz - 1 / Complex.abs z| β€ potential_error d c z
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
β’ |s.potential c βz - 1 / Complex.abs z| β€ potential_error d c z
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_approx
|
[262, 1]
|
[295, 40]
|
have z0 : 0 < abs z := lt_of_lt_of_le (by norm_num) z3
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
β’ |s.potential c βz - 1 / Complex.abs z| β€ potential_error d c z
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
β’ |s.potential c βz - 1 / Complex.abs z| β€ potential_error d c z
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_approx
|
[262, 1]
|
[295, 40]
|
have l2 : 0 < log (abs z) := Real.log_pos (by linarith)
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
β’ |s.potential c βz - 1 / Complex.abs z| β€ potential_error d c z
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
β’ |s.potential c βz - 1 / Complex.abs z| β€ potential_error d c z
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_approx
|
[262, 1]
|
[295, 40]
|
have h := log_neg_log_potential_approx d z3 cz
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
β’ |s.potential c βz - 1 / Complex.abs z| β€ potential_error d c z
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
h : |(-(β―.potential c βz).log).log - (Complex.abs z).log.log| β€ iter_error d c z
β’ |s.potential c βz - 1 / Complex.abs z| β€ potential_error d c z
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_approx
|
[262, 1]
|
[295, 40]
|
have p0 : 0 < s.potential c z := potential_pos
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
h : |(-(β―.potential c βz).log).log - (Complex.abs z).log.log| β€ iter_error d c z
β’ |s.potential c βz - 1 / Complex.abs z| β€ potential_error d c z
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
h : |(-(β―.potential c βz).log).log - (Complex.abs z).log.log| β€ iter_error d c z
p0 : 0 < s.potential c βz
β’ |s.potential c βz - 1 / Complex.abs z| β€ potential_error d c z
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_approx
|
[262, 1]
|
[295, 40]
|
have lp0 : 0 < -log (s.potential c z) :=
neg_pos.mpr (Real.log_neg p0 (potential_lt_one_of_two_lt (by linarith) cz))
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
h : |(-(β―.potential c βz).log).log - (Complex.abs z).log.log| β€ iter_error d c z
p0 : 0 < s.potential c βz
β’ |s.potential c βz - 1 / Complex.abs z| β€ potential_error d c z
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
h : |(-(β―.potential c βz).log).log - (Complex.abs z).log.log| β€ iter_error d c z
p0 : 0 < s.potential c βz
lp0 : 0 < -(s.potential c βz).log
β’ |s.potential c βz - 1 / Complex.abs z| β€ potential_error d c z
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_approx
|
[262, 1]
|
[295, 40]
|
generalize s.potential c z = p at h p0 lp0
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
h : |(-(β―.potential c βz).log).log - (Complex.abs z).log.log| β€ iter_error d c z
p0 : 0 < s.potential c βz
lp0 : 0 < -(s.potential c βz).log
β’ |s.potential c βz - 1 / Complex.abs z| β€ potential_error d c z
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
h : |(-p.log).log - (Complex.abs z).log.log| β€ iter_error d c z
p0 : 0 < p
lp0 : 0 < -p.log
β’ |p - 1 / Complex.abs z| β€ potential_error d c z
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_approx
|
[262, 1]
|
[295, 40]
|
generalize hr : iter_error d c z = r at h
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
h : |(-p.log).log - (Complex.abs z).log.log| β€ iter_error d c z
p0 : 0 < p
lp0 : 0 < -p.log
β’ |p - 1 / Complex.abs z| β€ potential_error d c z
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
β’ |p - 1 / Complex.abs z| β€ potential_error d c z
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_approx
|
[262, 1]
|
[295, 40]
|
have r0 : 0 β€ r := le_trans (abs_nonneg _) h
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
β’ |p - 1 / Complex.abs z| β€ potential_error d c z
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
β’ |p - 1 / Complex.abs z| β€ potential_error d c z
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_approx
|
[262, 1]
|
[295, 40]
|
set t := Ici (log (log (abs z)) - r)
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
β’ |p - 1 / Complex.abs z| β€ potential_error d c z
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
β’ |p - 1 / Complex.abs z| β€ potential_error d c z
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_approx
|
[262, 1]
|
[295, 40]
|
have yt : log (-log p) β t := by
simp only [abs_le, neg_le_sub_iff_le_add, tsub_le_iff_right, add_comm r] at h
simp only [mem_Ici, tsub_le_iff_right, h, t]
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
β’ |p - 1 / Complex.abs z| β€ potential_error d c z
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
β’ |p - 1 / Complex.abs z| β€ potential_error d c z
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_approx
|
[262, 1]
|
[295, 40]
|
have lt : log (log (abs z)) β t := by
simp only [mem_Ici, tsub_le_iff_right, le_add_iff_nonneg_right, r0, t]
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
β’ |p - 1 / Complex.abs z| β€ potential_error d c z
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
β’ |p - 1 / Complex.abs z| β€ potential_error d c z
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_approx
|
[262, 1]
|
[295, 40]
|
generalize hb : dene (log (log (abs z)) - r) = b
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
β’ |p - 1 / Complex.abs z| β€ potential_error d c z
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
β’ |p - 1 / Complex.abs z| β€ potential_error d c z
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_approx
|
[262, 1]
|
[295, 40]
|
have b0 : 0 β€ b := by rw [βhb]; exact dene_nonneg
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
β’ |p - 1 / Complex.abs z| β€ potential_error d c z
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
β’ |p - 1 / Complex.abs z| β€ potential_error d c z
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_approx
|
[262, 1]
|
[295, 40]
|
have bound : β x, x β t β βderiv ene xβ β€ b := by
intro x m
simp only [Real.dist_eq, mem_Ici, βhr, t] at m
simp only [deriv_ene, norm_neg, Real.norm_of_nonneg dene_nonneg, βhb, βhr]
apply dene_anti (sub_nonneg.mpr (iter_error_le_log_log_abs d z4 cz)) m
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
β’ |p - 1 / Complex.abs z| β€ potential_error d c z
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
bound : β x β t, βderiv ene xβ β€ b
β’ |p - 1 / Complex.abs z| β€ potential_error d c z
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_approx
|
[262, 1]
|
[295, 40]
|
have m := Convex.norm_image_sub_le_of_norm_deriv_le
(fun x _ β¦ (hasDerivAt_ene x).differentiableAt) bound (convex_Ici _) lt yt
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
bound : β x β t, βderiv ene xβ β€ b
β’ |p - 1 / Complex.abs z| β€ potential_error d c z
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
bound : β x β t, βderiv ene xβ β€ b
m : βene (-p.log).log - ene (Complex.abs z).log.logβ β€ b * β(-p.log).log - (Complex.abs z).log.logβ
β’ |p - 1 / Complex.abs z| β€ potential_error d c z
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_approx
|
[262, 1]
|
[295, 40]
|
simp only [Real.norm_eq_abs] at m
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
bound : β x β t, βderiv ene xβ β€ b
m : βene (-p.log).log - ene (Complex.abs z).log.logβ β€ b * β(-p.log).log - (Complex.abs z).log.logβ
β’ |p - 1 / Complex.abs z| β€ potential_error d c z
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
bound : β x β t, βderiv ene xβ β€ b
m : |ene (-p.log).log - ene (Complex.abs z).log.log| β€ b * |(-p.log).log - (Complex.abs z).log.log|
β’ |p - 1 / Complex.abs z| β€ potential_error d c z
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_approx
|
[262, 1]
|
[295, 40]
|
replace m := le_trans m (mul_le_mul_of_nonneg_left h (by bound))
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
bound : β x β t, βderiv ene xβ β€ b
m : |ene (-p.log).log - ene (Complex.abs z).log.log| β€ b * |(-p.log).log - (Complex.abs z).log.log|
β’ |p - 1 / Complex.abs z| β€ potential_error d c z
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
bound : β x β t, βderiv ene xβ β€ b
m : |ene (-p.log).log - ene (Complex.abs z).log.log| β€ b * r
β’ |p - 1 / Complex.abs z| β€ potential_error d c z
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_approx
|
[262, 1]
|
[295, 40]
|
simp only [ene, Real.exp_log lp0, neg_neg, Real.exp_log p0, Real.exp_log l2, Real.exp_neg,
Real.exp_log z0, inv_eq_one_div] at m
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
bound : β x β t, βderiv ene xβ β€ b
m : |ene (-p.log).log - ene (Complex.abs z).log.log| β€ b * r
β’ |p - 1 / Complex.abs z| β€ potential_error d c z
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
bound : β x β t, βderiv ene xβ β€ b
m : |p - 1 / Complex.abs z| β€ b * r
β’ |p - 1 / Complex.abs z| β€ potential_error d c z
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_approx
|
[262, 1]
|
[295, 40]
|
refine le_trans m (le_of_eq ?_)
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
bound : β x β t, βderiv ene xβ β€ b
m : |p - 1 / Complex.abs z| β€ b * r
β’ |p - 1 / Complex.abs z| β€ potential_error d c z
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
bound : β x β t, βderiv ene xβ β€ b
m : |p - 1 / Complex.abs z| β€ b * r
β’ b * r = potential_error d c z
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_approx
|
[262, 1]
|
[295, 40]
|
simp only [βhr, βhb, potential_error]
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
bound : β x β t, βderiv ene xβ β€ b
m : |p - 1 / Complex.abs z| β€ b * r
β’ b * r = potential_error d c z
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_approx
|
[262, 1]
|
[295, 40]
|
norm_num
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
β’ 3 β€ 4
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_approx
|
[262, 1]
|
[295, 40]
|
norm_num
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
β’ 0 < 3
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_approx
|
[262, 1]
|
[295, 40]
|
linarith
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
β’ 1 < Complex.abs z
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_approx
|
[262, 1]
|
[295, 40]
|
linarith
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
h : |(-(β―.potential c βz).log).log - (Complex.abs z).log.log| β€ iter_error d c z
p0 : 0 < s.potential c βz
β’ 2 < Complex.abs z
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_approx
|
[262, 1]
|
[295, 40]
|
simp only [abs_le, neg_le_sub_iff_le_add, tsub_le_iff_right, add_comm r] at h
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
β’ (-p.log).log β t
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
h : (Complex.abs z).log.log β€ (-p.log).log + r β§ (-p.log).log β€ (Complex.abs z).log.log + r
β’ (-p.log).log β t
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_approx
|
[262, 1]
|
[295, 40]
|
simp only [mem_Ici, tsub_le_iff_right, h, t]
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
h : (Complex.abs z).log.log β€ (-p.log).log + r β§ (-p.log).log β€ (Complex.abs z).log.log + r
β’ (-p.log).log β t
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_approx
|
[262, 1]
|
[295, 40]
|
simp only [mem_Ici, tsub_le_iff_right, le_add_iff_nonneg_right, r0, t]
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
β’ (Complex.abs z).log.log β t
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_approx
|
[262, 1]
|
[295, 40]
|
rw [βhb]
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
β’ 0 β€ b
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
β’ 0 β€ dene ((Complex.abs z).log.log - r)
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_approx
|
[262, 1]
|
[295, 40]
|
exact dene_nonneg
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
β’ 0 β€ dene ((Complex.abs z).log.log - r)
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_approx
|
[262, 1]
|
[295, 40]
|
intro x m
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
β’ β x β t, βderiv ene xβ β€ b
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
x : β
m : x β t
β’ βderiv ene xβ β€ b
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_approx
|
[262, 1]
|
[295, 40]
|
simp only [Real.dist_eq, mem_Ici, βhr, t] at m
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
x : β
m : x β t
β’ βderiv ene xβ β€ b
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
x : β
m : (Complex.abs z).log.log - iter_error d c z β€ x
β’ βderiv ene xβ β€ b
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_approx
|
[262, 1]
|
[295, 40]
|
simp only [deriv_ene, norm_neg, Real.norm_of_nonneg dene_nonneg, βhb, βhr]
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
x : β
m : (Complex.abs z).log.log - iter_error d c z β€ x
β’ βderiv ene xβ β€ b
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
x : β
m : (Complex.abs z).log.log - iter_error d c z β€ x
β’ dene x β€ dene ((Complex.abs z).log.log - iter_error d c z)
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_approx
|
[262, 1]
|
[295, 40]
|
apply dene_anti (sub_nonneg.mpr (iter_error_le_log_log_abs d z4 cz)) m
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
x : β
m : (Complex.abs z).log.log - iter_error d c z β€ x
β’ dene x β€ dene ((Complex.abs z).log.log - iter_error d c z)
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_approx
|
[262, 1]
|
[295, 40]
|
bound
|
cβ zβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c z : β
z4 : 4 β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 β€ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : β
p0 : 0 < p
lp0 : 0 < -p.log
r : β
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| β€ r
r0 : 0 β€ r
t : Set β := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log β t
lt : (Complex.abs z).log.log β t
b : β
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 β€ b
bound : β x β t, βderiv ene xβ β€ b
m : |ene (-p.log).log - ene (Complex.abs z).log.log| β€ b * |(-p.log).log - (Complex.abs z).log.log|
β’ 0 β€ b
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
late_iff_disjoint_range
|
[27, 1]
|
[30, 83]
|
simp only [Late, ge_iff_le, Finset.disjoint_iff_ne, Finset.mem_range, ne_eq]
|
m : β
A : Finset β
β’ Late A m β Disjoint A (Finset.range m)
|
m : β
A : Finset β
β’ (β n β A, m β€ n) β β a β A, β b < m, Β¬a = b
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
late_iff_disjoint_range
|
[27, 1]
|
[30, 83]
|
constructor
|
m : β
A : Finset β
β’ (β n β A, m β€ n) β β a β A, β b < m, Β¬a = b
|
case mp
m : β
A : Finset β
β’ (β n β A, m β€ n) β β a β A, β b < m, Β¬a = b
case mpr
m : β
A : Finset β
β’ (β a β A, β b < m, Β¬a = b) β β n β A, m β€ n
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
late_iff_disjoint_range
|
[27, 1]
|
[30, 83]
|
intro h n na b bm
|
case mp
m : β
A : Finset β
β’ (β n β A, m β€ n) β β a β A, β b < m, Β¬a = b
|
case mp
m : β
A : Finset β
h : β n β A, m β€ n
n : β
na : n β A
b : β
bm : b < m
β’ Β¬n = b
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
late_iff_disjoint_range
|
[27, 1]
|
[30, 83]
|
linarith [h _ na]
|
case mp
m : β
A : Finset β
h : β n β A, m β€ n
n : β
na : n β A
b : β
bm : b < m
β’ Β¬n = b
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
late_iff_disjoint_range
|
[27, 1]
|
[30, 83]
|
intro h n na
|
case mpr
m : β
A : Finset β
β’ (β a β A, β b < m, Β¬a = b) β β n β A, m β€ n
|
case mpr
m : β
A : Finset β
h : β a β A, β b < m, Β¬a = b
n : β
na : n β A
β’ m β€ n
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
late_iff_disjoint_range
|
[27, 1]
|
[30, 83]
|
specialize h n na n
|
case mpr
m : β
A : Finset β
h : β a β A, β b < m, Β¬a = b
n : β
na : n β A
β’ m β€ n
|
case mpr
m : β
A : Finset β
n : β
na : n β A
h : n < m β Β¬n = n
β’ m β€ n
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
late_iff_disjoint_range
|
[27, 1]
|
[30, 83]
|
simpa [not_true, imp_false, not_lt] using h
|
case mpr
m : β
A : Finset β
n : β
na : n β A
h : n < m β Β¬n = n
β’ m β€ n
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
sdiff_late
|
[32, 1]
|
[38, 17]
|
intro Bm n nAB
|
m : β
B A : Finset β
β’ B β₯ Finset.range m β Late (A \ B) m
|
m : β
B A : Finset β
Bm : B β₯ Finset.range m
n : β
nAB : n β A \ B
β’ n β₯ m
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
sdiff_late
|
[32, 1]
|
[38, 17]
|
rw [Finset.mem_sdiff] at nAB
|
m : β
B A : Finset β
Bm : B β₯ Finset.range m
n : β
nAB : n β A \ B
β’ n β₯ m
|
m : β
B A : Finset β
Bm : B β₯ Finset.range m
n : β
nAB : n β A β§ n β B
β’ n β₯ m
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
sdiff_late
|
[32, 1]
|
[38, 17]
|
by_contra h
|
m : β
B A : Finset β
Bm : B β₯ Finset.range m
n : β
nAB : n β A β§ n β B
β’ n β₯ m
|
m : β
B A : Finset β
Bm : B β₯ Finset.range m
n : β
nAB : n β A β§ n β B
h : Β¬n β₯ m
β’ False
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
sdiff_late
|
[32, 1]
|
[38, 17]
|
simp only [not_le] at h
|
m : β
B A : Finset β
Bm : B β₯ Finset.range m
n : β
nAB : n β A β§ n β B
h : Β¬n β₯ m
β’ False
|
m : β
B A : Finset β
Bm : B β₯ Finset.range m
n : β
nAB : n β A β§ n β B
h : n < m
β’ False
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
sdiff_late
|
[32, 1]
|
[38, 17]
|
have nr := Finset.mem_range.mpr h
|
m : β
B A : Finset β
Bm : B β₯ Finset.range m
n : β
nAB : n β A β§ n β B
h : n < m
β’ False
|
m : β
B A : Finset β
Bm : B β₯ Finset.range m
n : β
nAB : n β A β§ n β B
h : n < m
nr : n β Finset.range m
β’ False
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
sdiff_late
|
[32, 1]
|
[38, 17]
|
have nB := Finset.mem_of_subset Bm nr
|
m : β
B A : Finset β
Bm : B β₯ Finset.range m
n : β
nAB : n β A β§ n β B
h : n < m
nr : n β Finset.range m
β’ False
|
m : β
B A : Finset β
Bm : B β₯ Finset.range m
n : β
nAB : n β A β§ n β B
h : n < m
nr : n β Finset.range m
nB : n β B
β’ False
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
sdiff_late
|
[32, 1]
|
[38, 17]
|
exact nAB.2 nB
|
m : β
B A : Finset β
Bm : B β₯ Finset.range m
n : β
nAB : n β A β§ n β B
h : n < m
nr : n β Finset.range m
nB : n β B
β’ False
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
partial_geometric_bound
|
[41, 1]
|
[44, 57]
|
intro n _
|
a : β
N : Finset β
a0 : 0 β€ a
a1 : a < 1
β’ β n β N, 0 β€ a ^ n
|
a : β
N : Finset β
a0 : 0 β€ a
a1 : a < 1
n : β
aβ : n β N
β’ 0 β€ a ^ n
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
partial_geometric_bound
|
[41, 1]
|
[44, 57]
|
bound
|
a : β
N : Finset β
a0 : 0 β€ a
a1 : a < 1
n : β
aβ : n β N
β’ 0 β€ a ^ n
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
partial_scaled_geometric_bound
|
[46, 1]
|
[49, 42]
|
rw [βFinset.mul_sum]
|
a : β
c : ββ₯0
N : Finset β
a0 : 0 β€ a
a1 : a < 1
β’ (N.sum fun n => βc * a ^ n) β€ βc * (1 - a)β»ΒΉ
|
a : β
c : ββ₯0
N : Finset β
a0 : 0 β€ a
a1 : a < 1
β’ (βc * N.sum fun i => a ^ i) β€ βc * (1 - a)β»ΒΉ
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
partial_scaled_geometric_bound
|
[46, 1]
|
[49, 42]
|
bound [partial_geometric_bound N a0 a1]
|
a : β
c : ββ₯0
N : Finset β
a0 : 0 β€ a
a1 : a < 1
β’ (βc * N.sum fun i => a ^ i) β€ βc * (1 - a)β»ΒΉ
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
late_series_sum
|
[52, 1]
|
[66, 44]
|
set Ns := Finset.image (fun n β¦ n - m) N
|
m : β
N : Finset β
h : Late N m
f : β β β
β’ N.sum f = (Finset.image (fun n => n - m) N).sum fun n => f (n + m)
|
m : β
N : Finset β
h : Late N m
f : β β β
Ns : Finset β := Finset.image (fun n => n - m) N
β’ N.sum f = Ns.sum fun n => f (n + m)
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
late_series_sum
|
[52, 1]
|
[66, 44]
|
rw [NNs]
|
m : β
N : Finset β
h : Late N m
f : β β β
Ns : Finset β := Finset.image (fun n => n - m) N
NNs : N = Finset.image (fun n => n + m) Ns
β’ N.sum f = Ns.sum fun n => f (n + m)
|
m : β
N : Finset β
h : Late N m
f : β β β
Ns : Finset β := Finset.image (fun n => n - m) N
NNs : N = Finset.image (fun n => n + m) Ns
β’ (Finset.image (fun n => n + m) Ns).sum f = Ns.sum fun n => f (n + m)
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
late_series_sum
|
[52, 1]
|
[66, 44]
|
apply Finset.sum_image
|
m : β
N : Finset β
h : Late N m
f : β β β
Ns : Finset β := Finset.image (fun n => n - m) N
NNs : N = Finset.image (fun n => n + m) Ns
β’ (Finset.image (fun n => n + m) Ns).sum f = Ns.sum fun n => f (n + m)
|
case a
m : β
N : Finset β
h : Late N m
f : β β β
Ns : Finset β := Finset.image (fun n => n - m) N
NNs : N = Finset.image (fun n => n + m) Ns
β’ β x β Ns, β y β Ns, x + m = y + m β x = y
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
late_series_sum
|
[52, 1]
|
[66, 44]
|
intro a _ b _
|
case a
m : β
N : Finset β
h : Late N m
f : β β β
Ns : Finset β := Finset.image (fun n => n - m) N
NNs : N = Finset.image (fun n => n + m) Ns
β’ β x β Ns, β y β Ns, x + m = y + m β x = y
|
case a
m : β
N : Finset β
h : Late N m
f : β β β
Ns : Finset β := Finset.image (fun n => n - m) N
NNs : N = Finset.image (fun n => n + m) Ns
a : β
aβΒΉ : a β Ns
b : β
aβ : b β Ns
β’ a + m = b + m β a = b
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
late_series_sum
|
[52, 1]
|
[66, 44]
|
exact Nat.add_right_cancel
|
case a
m : β
N : Finset β
h : Late N m
f : β β β
Ns : Finset β := Finset.image (fun n => n - m) N
NNs : N = Finset.image (fun n => n + m) Ns
a : β
aβΒΉ : a β Ns
b : β
aβ : b β Ns
β’ a + m = b + m β a = b
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
late_series_sum
|
[52, 1]
|
[66, 44]
|
apply Finset.ext
|
m : β
N : Finset β
h : Late N m
f : β β β
Ns : Finset β := Finset.image (fun n => n - m) N
β’ N = Finset.image (fun n => n + m) Ns
|
case a
m : β
N : Finset β
h : Late N m
f : β β β
Ns : Finset β := Finset.image (fun n => n - m) N
β’ β (a : β), a β N β a β Finset.image (fun n => n + m) Ns
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
late_series_sum
|
[52, 1]
|
[66, 44]
|
intro k
|
case a
m : β
N : Finset β
h : Late N m
f : β β β
Ns : Finset β := Finset.image (fun n => n - m) N
β’ β (a : β), a β N β a β Finset.image (fun n => n + m) Ns
|
case a
m : β
N : Finset β
h : Late N m
f : β β β
Ns : Finset β := Finset.image (fun n => n - m) N
k : β
β’ k β N β k β Finset.image (fun n => n + m) Ns
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
late_series_sum
|
[52, 1]
|
[66, 44]
|
rw [Finset.image_image, Finset.mem_image]
|
case a
m : β
N : Finset β
h : Late N m
f : β β β
Ns : Finset β := Finset.image (fun n => n - m) N
k : β
β’ k β N β k β Finset.image (fun n => n + m) Ns
|
case a
m : β
N : Finset β
h : Late N m
f : β β β
Ns : Finset β := Finset.image (fun n => n - m) N
k : β
β’ k β N β β a β N, ((fun n => n + m) β fun n => n - m) a = k
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
late_series_sum
|
[52, 1]
|
[66, 44]
|
simp
|
case a
m : β
N : Finset β
h : Late N m
f : β β β
Ns : Finset β := Finset.image (fun n => n - m) N
k : β
β’ k β N β β a β N, ((fun n => n + m) β fun n => n - m) a = k
|
case a
m : β
N : Finset β
h : Late N m
f : β β β
Ns : Finset β := Finset.image (fun n => n - m) N
k : β
β’ k β N β β a β N, a - m + m = k
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
late_series_sum
|
[52, 1]
|
[66, 44]
|
apply Iff.intro
|
case a
m : β
N : Finset β
h : Late N m
f : β β β
Ns : Finset β := Finset.image (fun n => n - m) N
k : β
β’ k β N β β a β N, a - m + m = k
|
case a.mp
m : β
N : Finset β
h : Late N m
f : β β β
Ns : Finset β := Finset.image (fun n => n - m) N
k : β
β’ k β N β β a β N, a - m + m = k
case a.mpr
m : β
N : Finset β
h : Late N m
f : β β β
Ns : Finset β := Finset.image (fun n => n - m) N
k : β
β’ (β a β N, a - m + m = k) β k β N
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
late_series_sum
|
[52, 1]
|
[66, 44]
|
intro kN
|
case a.mp
m : β
N : Finset β
h : Late N m
f : β β β
Ns : Finset β := Finset.image (fun n => n - m) N
k : β
β’ k β N β β a β N, a - m + m = k
|
case a.mp
m : β
N : Finset β
h : Late N m
f : β β β
Ns : Finset β := Finset.image (fun n => n - m) N
k : β
kN : k β N
β’ β a β N, a - m + m = k
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
late_series_sum
|
[52, 1]
|
[66, 44]
|
exists k
|
case a.mp
m : β
N : Finset β
h : Late N m
f : β β β
Ns : Finset β := Finset.image (fun n => n - m) N
k : β
kN : k β N
β’ β a β N, a - m + m = k
|
case a.mp
m : β
N : Finset β
h : Late N m
f : β β β
Ns : Finset β := Finset.image (fun n => n - m) N
k : β
kN : k β N
β’ k β N β§ k - m + m = k
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
late_series_sum
|
[52, 1]
|
[66, 44]
|
apply And.intro
|
case a.mp
m : β
N : Finset β
h : Late N m
f : β β β
Ns : Finset β := Finset.image (fun n => n - m) N
k : β
kN : k β N
β’ k β N β§ k - m + m = k
|
case a.mp.left
m : β
N : Finset β
h : Late N m
f : β β β
Ns : Finset β := Finset.image (fun n => n - m) N
k : β
kN : k β N
β’ k β N
case a.mp.right
m : β
N : Finset β
h : Late N m
f : β β β
Ns : Finset β := Finset.image (fun n => n - m) N
k : β
kN : k β N
β’ k - m + m = k
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
late_series_sum
|
[52, 1]
|
[66, 44]
|
assumption
|
case a.mp.left
m : β
N : Finset β
h : Late N m
f : β β β
Ns : Finset β := Finset.image (fun n => n - m) N
k : β
kN : k β N
β’ k β N
case a.mp.right
m : β
N : Finset β
h : Late N m
f : β β β
Ns : Finset β := Finset.image (fun n => n - m) N
k : β
kN : k β N
β’ k - m + m = k
|
case a.mp.right
m : β
N : Finset β
h : Late N m
f : β β β
Ns : Finset β := Finset.image (fun n => n - m) N
k : β
kN : k β N
β’ k - m + m = k
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
late_series_sum
|
[52, 1]
|
[66, 44]
|
exact Nat.sub_add_cancel (h k kN)
|
case a.mp.right
m : β
N : Finset β
h : Late N m
f : β β β
Ns : Finset β := Finset.image (fun n => n - m) N
k : β
kN : k β N
β’ k - m + m = k
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
late_series_sum
|
[52, 1]
|
[66, 44]
|
intro ha
|
case a.mpr
m : β
N : Finset β
h : Late N m
f : β β β
Ns : Finset β := Finset.image (fun n => n - m) N
k : β
β’ (β a β N, a - m + m = k) β k β N
|
case a.mpr
m : β
N : Finset β
h : Late N m
f : β β β
Ns : Finset β := Finset.image (fun n => n - m) N
k : β
ha : β a β N, a - m + m = k
β’ k β N
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
late_series_sum
|
[52, 1]
|
[66, 44]
|
rcases ha with β¨a, aN, akβ©
|
case a.mpr
m : β
N : Finset β
h : Late N m
f : β β β
Ns : Finset β := Finset.image (fun n => n - m) N
k : β
ha : β a β N, a - m + m = k
β’ k β N
|
case a.mpr.intro.intro
m : β
N : Finset β
h : Late N m
f : β β β
Ns : Finset β := Finset.image (fun n => n - m) N
k a : β
aN : a β N
ak : a - m + m = k
β’ k β N
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
late_series_sum
|
[52, 1]
|
[66, 44]
|
rw [Nat.sub_add_cancel (h a aN)] at ak
|
case a.mpr.intro.intro
m : β
N : Finset β
h : Late N m
f : β β β
Ns : Finset β := Finset.image (fun n => n - m) N
k a : β
aN : a β N
ak : a - m + m = k
β’ k β N
|
case a.mpr.intro.intro
m : β
N : Finset β
h : Late N m
f : β β β
Ns : Finset β := Finset.image (fun n => n - m) N
k a : β
aN : a β N
ak : a = k
β’ k β N
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
late_series_sum
|
[52, 1]
|
[66, 44]
|
rw [β ak]
|
case a.mpr.intro.intro
m : β
N : Finset β
h : Late N m
f : β β β
Ns : Finset β := Finset.image (fun n => n - m) N
k a : β
aN : a β N
ak : a = k
β’ k β N
|
case a.mpr.intro.intro
m : β
N : Finset β
h : Late N m
f : β β β
Ns : Finset β := Finset.image (fun n => n - m) N
k a : β
aN : a β N
ak : a = k
β’ a β N
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
late_series_sum
|
[52, 1]
|
[66, 44]
|
assumption
|
case a.mpr.intro.intro
m : β
N : Finset β
h : Late N m
f : β β β
Ns : Finset β := Finset.image (fun n => n - m) N
k a : β
aN : a β N
ak : a = k
β’ a β N
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
late_series_sum'
|
[69, 1]
|
[72, 28]
|
exists Finset.image (fun n β¦ n - m) N
|
m : β
N : Finset β
h : Late N m
f : β β β
β’ β M, N.sum f = M.sum fun n => f (n + m)
|
m : β
N : Finset β
h : Late N m
f : β β β
β’ N.sum f = (Finset.image (fun n => n - m) N).sum fun n => f (n + m)
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
late_series_sum'
|
[69, 1]
|
[72, 28]
|
exact late_series_sum h f
|
m : β
N : Finset β
h : Late N m
f : β β β
β’ N.sum f = (Finset.image (fun n => n - m) N).sum fun n => f (n + m)
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
late_geometric_bound
|
[74, 1]
|
[83, 35]
|
rcases late_series_sum' h (fun n β¦ a^n) with β¨M,Lβ©
|
m : β
a : β
N : Finset β
h : Late N m
a0 : 0 β€ a
a1 : a < 1
β’ (N.sum fun n => a ^ n) β€ a ^ m * (1 - a)β»ΒΉ
|
case intro
m : β
a : β
N : Finset β
h : Late N m
a0 : 0 β€ a
a1 : a < 1
M : Finset β
L : (N.sum fun n => a ^ n) = M.sum fun n => a ^ (n + m)
β’ (N.sum fun n => a ^ n) β€ a ^ m * (1 - a)β»ΒΉ
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
late_geometric_bound
|
[74, 1]
|
[83, 35]
|
rw [L]
|
case intro
m : β
a : β
N : Finset β
h : Late N m
a0 : 0 β€ a
a1 : a < 1
M : Finset β
L : (N.sum fun n => a ^ n) = M.sum fun n => a ^ (n + m)
β’ (N.sum fun n => a ^ n) β€ a ^ m * (1 - a)β»ΒΉ
|
case intro
m : β
a : β
N : Finset β
h : Late N m
a0 : 0 β€ a
a1 : a < 1
M : Finset β
L : (N.sum fun n => a ^ n) = M.sum fun n => a ^ (n + m)
β’ (M.sum fun n => a ^ (n + m)) β€ a ^ m * (1 - a)β»ΒΉ
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
late_geometric_bound
|
[74, 1]
|
[83, 35]
|
clear L
|
case intro
m : β
a : β
N : Finset β
h : Late N m
a0 : 0 β€ a
a1 : a < 1
M : Finset β
L : (N.sum fun n => a ^ n) = M.sum fun n => a ^ (n + m)
β’ (M.sum fun n => a ^ (n + m)) β€ a ^ m * (1 - a)β»ΒΉ
|
case intro
m : β
a : β
N : Finset β
h : Late N m
a0 : 0 β€ a
a1 : a < 1
M : Finset β
β’ (M.sum fun n => a ^ (n + m)) β€ a ^ m * (1 - a)β»ΒΉ
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
late_geometric_bound
|
[74, 1]
|
[83, 35]
|
have pa : (fun n β¦ a^(n + m)) = (fun n β¦ a^n * a^m) := by apply funext; intro n; rw [pow_add]
|
case intro
m : β
a : β
N : Finset β
h : Late N m
a0 : 0 β€ a
a1 : a < 1
M : Finset β
β’ (M.sum fun n => a ^ (n + m)) β€ a ^ m * (1 - a)β»ΒΉ
|
case intro
m : β
a : β
N : Finset β
h : Late N m
a0 : 0 β€ a
a1 : a < 1
M : Finset β
pa : (fun n => a ^ (n + m)) = fun n => a ^ n * a ^ m
β’ (M.sum fun n => a ^ (n + m)) β€ a ^ m * (1 - a)β»ΒΉ
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
late_geometric_bound
|
[74, 1]
|
[83, 35]
|
calc
M.sum (fun n β¦ a^(n + m)) = M.sum (fun n β¦ a^n * a^m) := by rw [ pa ]
_ = M.sum (fun n β¦ a^n) * a^m := (Finset.sum_mul _ _ _).symm
_ β€ (1 - a)β»ΒΉ * a^m := by bound [partial_geometric_bound M a0 a1]
_ = a^m * (1 - a)β»ΒΉ := by ring
|
case intro
m : β
a : β
N : Finset β
h : Late N m
a0 : 0 β€ a
a1 : a < 1
M : Finset β
pa : (fun n => a ^ (n + m)) = fun n => a ^ n * a ^ m
β’ (M.sum fun n => a ^ (n + m)) β€ a ^ m * (1 - a)β»ΒΉ
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
late_geometric_bound
|
[74, 1]
|
[83, 35]
|
apply funext
|
m : β
a : β
N : Finset β
h : Late N m
a0 : 0 β€ a
a1 : a < 1
M : Finset β
β’ (fun n => a ^ (n + m)) = fun n => a ^ n * a ^ m
|
case h
m : β
a : β
N : Finset β
h : Late N m
a0 : 0 β€ a
a1 : a < 1
M : Finset β
β’ β (x : β), a ^ (x + m) = a ^ x * a ^ m
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
late_geometric_bound
|
[74, 1]
|
[83, 35]
|
intro n
|
case h
m : β
a : β
N : Finset β
h : Late N m
a0 : 0 β€ a
a1 : a < 1
M : Finset β
β’ β (x : β), a ^ (x + m) = a ^ x * a ^ m
|
case h
m : β
a : β
N : Finset β
h : Late N m
a0 : 0 β€ a
a1 : a < 1
M : Finset β
n : β
β’ a ^ (n + m) = a ^ n * a ^ m
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
late_geometric_bound
|
[74, 1]
|
[83, 35]
|
rw [pow_add]
|
case h
m : β
a : β
N : Finset β
h : Late N m
a0 : 0 β€ a
a1 : a < 1
M : Finset β
n : β
β’ a ^ (n + m) = a ^ n * a ^ m
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
late_geometric_bound
|
[74, 1]
|
[83, 35]
|
rw [ pa ]
|
m : β
a : β
N : Finset β
h : Late N m
a0 : 0 β€ a
a1 : a < 1
M : Finset β
pa : (fun n => a ^ (n + m)) = fun n => a ^ n * a ^ m
β’ (M.sum fun n => a ^ (n + m)) = M.sum fun n => a ^ n * a ^ m
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
late_geometric_bound
|
[74, 1]
|
[83, 35]
|
bound [partial_geometric_bound M a0 a1]
|
m : β
a : β
N : Finset β
h : Late N m
a0 : 0 β€ a
a1 : a < 1
M : Finset β
pa : (fun n => a ^ (n + m)) = fun n => a ^ n * a ^ m
β’ (M.sum fun n => a ^ n) * a ^ m β€ (1 - a)β»ΒΉ * a ^ m
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
late_geometric_bound
|
[74, 1]
|
[83, 35]
|
ring
|
m : β
a : β
N : Finset β
h : Late N m
a0 : 0 β€ a
a1 : a < 1
M : Finset β
pa : (fun n => a ^ (n + m)) = fun n => a ^ n * a ^ m
β’ (1 - a)β»ΒΉ * a ^ m = a ^ m * (1 - a)β»ΒΉ
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
finset_partition
|
[85, 1]
|
[95, 21]
|
apply Finset.ext
|
A B : Finset β
β’ A = A \ B βͺ A β© B
|
case a
A B : Finset β
β’ β (a : β), a β A β a β A \ B βͺ A β© B
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
finset_partition
|
[85, 1]
|
[95, 21]
|
simp only [Finset.mem_union, Finset.mem_sdiff, Finset.mem_inter]
|
case a
A B : Finset β
β’ β (a : β), a β A β a β A \ B βͺ A β© B
|
case a
A B : Finset β
β’ β (a : β), a β A β a β A β§ a β B β¨ a β A β§ a β B
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
finset_partition
|
[85, 1]
|
[95, 21]
|
intro x
|
case a
A B : Finset β
β’ β (a : β), a β A β a β A β§ a β B β¨ a β A β§ a β B
|
case a
A B : Finset β
x : β
β’ x β A β x β A β§ x β B β¨ x β A β§ x β B
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
finset_partition
|
[85, 1]
|
[95, 21]
|
constructor
|
case a
A B : Finset β
x : β
β’ x β A β x β A β§ x β B β¨ x β A β§ x β B
|
case a.mp
A B : Finset β
x : β
β’ x β A β x β A β§ x β B β¨ x β A β§ x β B
case a.mpr
A B : Finset β
x : β
β’ x β A β§ x β B β¨ x β A β§ x β B β x β A
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
finset_partition
|
[85, 1]
|
[95, 21]
|
intro a
|
case a.mp
A B : Finset β
x : β
β’ x β A β x β A β§ x β B β¨ x β A β§ x β B
|
case a.mp
A B : Finset β
x : β
a : x β A
β’ x β A β§ x β B β¨ x β A β§ x β B
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
finset_partition
|
[85, 1]
|
[95, 21]
|
by_cases b : x β B
|
case a.mp
A B : Finset β
x : β
a : x β A
β’ x β A β§ x β B β¨ x β A β§ x β B
|
case pos
A B : Finset β
x : β
a : x β A
b : x β B
β’ x β A β§ x β B β¨ x β A β§ x β B
case neg
A B : Finset β
x : β
a : x β A
b : x β B
β’ x β A β§ x β B β¨ x β A β§ x β B
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
finset_partition
|
[85, 1]
|
[95, 21]
|
right
|
case pos
A B : Finset β
x : β
a : x β A
b : x β B
β’ x β A β§ x β B β¨ x β A β§ x β B
|
case pos.h
A B : Finset β
x : β
a : x β A
b : x β B
β’ x β A β§ x β B
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
finset_partition
|
[85, 1]
|
[95, 21]
|
use a,b
|
case pos.h
A B : Finset β
x : β
a : x β A
b : x β B
β’ x β A β§ x β B
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
finset_partition
|
[85, 1]
|
[95, 21]
|
left
|
case neg
A B : Finset β
x : β
a : x β A
b : x β B
β’ x β A β§ x β B β¨ x β A β§ x β B
|
case neg.h
A B : Finset β
x : β
a : x β A
b : x β B
β’ x β A β§ x β B
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
finset_partition
|
[85, 1]
|
[95, 21]
|
use a,b
|
case neg.h
A B : Finset β
x : β
a : x β A
b : x β B
β’ x β A β§ x β B
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Misc/Bounds.lean
|
finset_partition
|
[85, 1]
|
[95, 21]
|
intro h
|
case a.mpr
A B : Finset β
x : β
β’ x β A β§ x β B β¨ x β A β§ x β B β x β A
|
case a.mpr
A B : Finset β
x : β
h : x β A β§ x β B β¨ x β A β§ x β B
β’ x β A
|
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