url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
| input
stringlengths 73
2.09M
|
|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | iter_error_le_log_log_abs | [245, 1] | [259, 83] | positivity | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
hl : 1.38 ≤ (Complex.abs z).log
hll : 0.32 ≤ (Complex.abs z).log.log
⊢ 0 ≤ 1.38 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
hl : 1.38 ≤ (Complex.abs z).log
hll : 0.32 ≤ (Complex.abs z).log.log
⊢ 0 ≤ 1.38
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | iter_error_le_log_log_abs | [245, 1] | [259, 83] | positivity | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
hl : 1.38 ≤ (Complex.abs z).log
hll : 0.32 ≤ (Complex.abs z).log.log
⊢ 0 ≤ Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
hl : 1.38 ≤ (Complex.abs z).log
hll : 0.32 ≤ (Complex.abs z).log.log
⊢ 0 ≤ Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | set s := superF d | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ |⋯.potential c ↑z - 1 / Complex.abs z| ≤ potential_error d c z | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
⊢ |s.potential c ↑z - 1 / Complex.abs z| ≤ potential_error d c z | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ |⋯.potential c ↑z - 1 / Complex.abs z| ≤ potential_error d c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | have z3 : 3 ≤ abs z := le_trans (by norm_num) z4 | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
⊢ |s.potential c ↑z - 1 / Complex.abs z| ≤ potential_error d c z | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
⊢ |s.potential c ↑z - 1 / Complex.abs z| ≤ potential_error d c z | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
⊢ |s.potential c ↑z - 1 / Complex.abs z| ≤ potential_error d c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | have z0 : 0 < abs z := lt_of_lt_of_le (by norm_num) z3 | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
⊢ |s.potential c ↑z - 1 / Complex.abs z| ≤ potential_error d c z | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
⊢ |s.potential c ↑z - 1 / Complex.abs z| ≤ potential_error d c z | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
⊢ |s.potential c ↑z - 1 / Complex.abs z| ≤ potential_error d c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | have l2 : 0 < log (abs z) := Real.log_pos (by linarith) | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
⊢ |s.potential c ↑z - 1 / Complex.abs z| ≤ potential_error d c z | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
⊢ |s.potential c ↑z - 1 / Complex.abs z| ≤ potential_error d c z | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
⊢ |s.potential c ↑z - 1 / Complex.abs z| ≤ potential_error d c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | have h := log_neg_log_potential_approx d z3 cz | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
⊢ |s.potential c ↑z - 1 / Complex.abs z| ≤ potential_error d c z | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
h : |(-(⋯.potential c ↑z).log).log - (Complex.abs z).log.log| ≤ iter_error d c z
⊢ |s.potential c ↑z - 1 / Complex.abs z| ≤ potential_error d c z | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
⊢ |s.potential c ↑z - 1 / Complex.abs z| ≤ potential_error d c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | have p0 : 0 < s.potential c z := potential_pos | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
h : |(-(⋯.potential c ↑z).log).log - (Complex.abs z).log.log| ≤ iter_error d c z
⊢ |s.potential c ↑z - 1 / Complex.abs z| ≤ potential_error d c z | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
h : |(-(⋯.potential c ↑z).log).log - (Complex.abs z).log.log| ≤ iter_error d c z
p0 : 0 < s.potential c ↑z
⊢ |s.potential c ↑z - 1 / Complex.abs z| ≤ potential_error d c z | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
h : |(-(⋯.potential c ↑z).log).log - (Complex.abs z).log.log| ≤ iter_error d c z
⊢ |s.potential c ↑z - 1 / Complex.abs z| ≤ potential_error d c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | have lp0 : 0 < -log (s.potential c z) :=
neg_pos.mpr (Real.log_neg p0 (potential_lt_one_of_two_lt (by linarith) cz)) | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
h : |(-(⋯.potential c ↑z).log).log - (Complex.abs z).log.log| ≤ iter_error d c z
p0 : 0 < s.potential c ↑z
⊢ |s.potential c ↑z - 1 / Complex.abs z| ≤ potential_error d c z | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
h : |(-(⋯.potential c ↑z).log).log - (Complex.abs z).log.log| ≤ iter_error d c z
p0 : 0 < s.potential c ↑z
lp0 : 0 < -(s.potential c ↑z).log
⊢ |s.potential c ↑z - 1 / Complex.abs z| ≤ potential_error d c z | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
h : |(-(⋯.potential c ↑z).log).log - (Complex.abs z).log.log| ≤ iter_error d c z
p0 : 0 < s.potential c ↑z
⊢ |s.potential c ↑z - 1 / Complex.abs z| ≤ potential_error d c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | generalize s.potential c z = p at h p0 lp0 | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
h : |(-(⋯.potential c ↑z).log).log - (Complex.abs z).log.log| ≤ iter_error d c z
p0 : 0 < s.potential c ↑z
lp0 : 0 < -(s.potential c ↑z).log
⊢ |s.potential c ↑z - 1 / Complex.abs z| ≤ potential_error d c z | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
h : |(-p.log).log - (Complex.abs z).log.log| ≤ iter_error d c z
p0 : 0 < p
lp0 : 0 < -p.log
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
h : |(-(⋯.potential c ↑z).log).log - (Complex.abs z).log.log| ≤ iter_error d c z
p0 : 0 < s.potential c ↑z
lp0 : 0 < -(s.potential c ↑z).log
⊢ |s.potential c ↑z - 1 / Complex.abs z| ≤ potential_error d c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | generalize hr : iter_error d c z = r at h | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
h : |(-p.log).log - (Complex.abs z).log.log| ≤ iter_error d c z
p0 : 0 < p
lp0 : 0 < -p.log
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
h : |(-p.log).log - (Complex.abs z).log.log| ≤ iter_error d c z
p0 : 0 < p
lp0 : 0 < -p.log
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | have r0 : 0 ≤ r := le_trans (abs_nonneg _) h | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z
TACTIC:
|
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) | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | have yt : log (-log p) ∈ 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] | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | have lt : log (log (abs z)) ∈ t := by
simp only [mem_Ici, tsub_le_iff_right, le_add_iff_nonneg_right, r0, t] | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | generalize hb : dene (log (log (abs z)) - r) = b | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | have b0 : 0 ≤ b := by rw [←hb]; exact dene_nonneg | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | have bound : ∀ x, x ∈ t → ‖deriv ene x‖ ≤ b := by
intro x m
simp only [Real.dist_eq, mem_Ici, ←hr, t] at m
simp only [deriv_ene, norm_neg, Real.norm_of_nonneg dene_nonneg, ←hb, ←hr]
apply dene_anti (sub_nonneg.mpr (iter_error_le_log_log_abs d z4 cz)) m | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | have m := Convex.norm_image_sub_le_of_norm_deriv_le
(fun x _ ↦ (hasDerivAt_ene x).differentiableAt) bound (convex_Ici _) lt yt | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b
m : ‖ene (-p.log).log - ene (Complex.abs z).log.log‖ ≤ b * ‖(-p.log).log - (Complex.abs z).log.log‖
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | simp only [Real.norm_eq_abs] at m | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b
m : ‖ene (-p.log).log - ene (Complex.abs z).log.log‖ ≤ b * ‖(-p.log).log - (Complex.abs z).log.log‖
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b
m : |ene (-p.log).log - ene (Complex.abs z).log.log| ≤ b * |(-p.log).log - (Complex.abs z).log.log|
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b
m : ‖ene (-p.log).log - ene (Complex.abs z).log.log‖ ≤ b * ‖(-p.log).log - (Complex.abs z).log.log‖
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | replace m := le_trans m (mul_le_mul_of_nonneg_left h (by bound)) | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b
m : |ene (-p.log).log - ene (Complex.abs z).log.log| ≤ b * |(-p.log).log - (Complex.abs z).log.log|
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b
m : |ene (-p.log).log - ene (Complex.abs z).log.log| ≤ b * r
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b
m : |ene (-p.log).log - ene (Complex.abs z).log.log| ≤ b * |(-p.log).log - (Complex.abs z).log.log|
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | simp only [ene, Real.exp_log lp0, neg_neg, Real.exp_log p0, Real.exp_log l2, Real.exp_neg,
Real.exp_log z0, inv_eq_one_div] at m | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b
m : |ene (-p.log).log - ene (Complex.abs z).log.log| ≤ b * r
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b
m : |p - 1 / Complex.abs z| ≤ b * r
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b
m : |ene (-p.log).log - ene (Complex.abs z).log.log| ≤ b * r
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | refine le_trans m (le_of_eq ?_) | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b
m : |p - 1 / Complex.abs z| ≤ b * r
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b
m : |p - 1 / Complex.abs z| ≤ b * r
⊢ b * r = potential_error d c z | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b
m : |p - 1 / Complex.abs z| ≤ b * r
⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | simp only [←hr, ←hb, potential_error] | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b
m : |p - 1 / Complex.abs z| ≤ b * r
⊢ b * r = potential_error d c z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b
m : |p - 1 / Complex.abs z| ≤ b * r
⊢ b * r = potential_error d c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | norm_num | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
⊢ 3 ≤ 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
⊢ 3 ≤ 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | norm_num | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
⊢ 0 < 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
⊢ 0 < 3
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | linarith | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
⊢ 1 < Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
⊢ 1 < Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | linarith | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
h : |(-(⋯.potential c ↑z).log).log - (Complex.abs z).log.log| ≤ iter_error d c z
p0 : 0 < s.potential c ↑z
⊢ 2 < Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
h : |(-(⋯.potential c ↑z).log).log - (Complex.abs z).log.log| ≤ iter_error d c z
p0 : 0 < s.potential c ↑z
⊢ 2 < Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | simp only [abs_le, neg_le_sub_iff_le_add, tsub_le_iff_right, add_comm r] at h | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
⊢ (-p.log).log ∈ t | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
h : (Complex.abs z).log.log ≤ (-p.log).log + r ∧ (-p.log).log ≤ (Complex.abs z).log.log + r
⊢ (-p.log).log ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
⊢ (-p.log).log ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | simp only [mem_Ici, tsub_le_iff_right, h, t] | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
h : (Complex.abs z).log.log ≤ (-p.log).log + r ∧ (-p.log).log ≤ (Complex.abs z).log.log + r
⊢ (-p.log).log ∈ t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
h : (Complex.abs z).log.log ≤ (-p.log).log + r ∧ (-p.log).log ≤ (Complex.abs z).log.log + r
⊢ (-p.log).log ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | simp only [mem_Ici, tsub_le_iff_right, le_add_iff_nonneg_right, r0, t] | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
⊢ (Complex.abs z).log.log ∈ t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
⊢ (Complex.abs z).log.log ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | rw [←hb] | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
⊢ 0 ≤ b | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
⊢ 0 ≤ dene ((Complex.abs z).log.log - r) | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
⊢ 0 ≤ b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | exact dene_nonneg | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
⊢ 0 ≤ dene ((Complex.abs z).log.log - r) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
⊢ 0 ≤ dene ((Complex.abs z).log.log - r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | intro x m | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
⊢ ∀ x ∈ t, ‖deriv ene x‖ ≤ b | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
x : ℝ
m : x ∈ t
⊢ ‖deriv ene x‖ ≤ b | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
⊢ ∀ x ∈ t, ‖deriv ene x‖ ≤ b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | simp only [Real.dist_eq, mem_Ici, ←hr, t] at m | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
x : ℝ
m : x ∈ t
⊢ ‖deriv ene x‖ ≤ b | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
x : ℝ
m : (Complex.abs z).log.log - iter_error d c z ≤ x
⊢ ‖deriv ene x‖ ≤ b | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
x : ℝ
m : x ∈ t
⊢ ‖deriv ene x‖ ≤ b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | simp only [deriv_ene, norm_neg, Real.norm_of_nonneg dene_nonneg, ←hb, ←hr] | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
x : ℝ
m : (Complex.abs z).log.log - iter_error d c z ≤ x
⊢ ‖deriv ene x‖ ≤ b | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
x : ℝ
m : (Complex.abs z).log.log - iter_error d c z ≤ x
⊢ dene x ≤ dene ((Complex.abs z).log.log - iter_error d c z) | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
x : ℝ
m : (Complex.abs z).log.log - iter_error d c z ≤ x
⊢ ‖deriv ene x‖ ≤ b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | apply dene_anti (sub_nonneg.mpr (iter_error_le_log_log_abs d z4 cz)) m | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
x : ℝ
m : (Complex.abs z).log.log - iter_error d c z ≤ x
⊢ dene x ≤ dene ((Complex.abs z).log.log - iter_error d c z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
x : ℝ
m : (Complex.abs z).log.log - iter_error d c z ≤ x
⊢ dene x ≤ dene ((Complex.abs z).log.log - iter_error d c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | potential_approx | [262, 1] | [295, 40] | bound | c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b
m : |ene (-p.log).log - ene (Complex.abs z).log.log| ≤ b * |(-p.log).log - (Complex.abs z).log.log|
⊢ 0 ≤ b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
l2 : 0 < (Complex.abs z).log
p : ℝ
p0 : 0 < p
lp0 : 0 < -p.log
r : ℝ
hr : iter_error d c z = r
h : |(-p.log).log - (Complex.abs z).log.log| ≤ r
r0 : 0 ≤ r
t : Set ℝ := Ici ((Complex.abs z).log.log - r)
yt : (-p.log).log ∈ t
lt : (Complex.abs z).log.log ∈ t
b : ℝ
hb : dene ((Complex.abs z).log.log - r) = b
b0 : 0 ≤ b
bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b
m : |ene (-p.log).log - ene (Complex.abs z).log.log| ≤ b * |(-p.log).log - (Complex.abs z).log.log|
⊢ 0 ≤ b
TACTIC:
|
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 : ℕ
A : Finset ℕ
⊢ Late A m ↔ Disjoint A (Finset.range m) | m : ℕ
A : Finset ℕ
⊢ (∀ n ∈ A, m ≤ n) ↔ ∀ a ∈ A, ∀ b < m, ¬a = b | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
A : Finset ℕ
⊢ Late A m ↔ Disjoint A (Finset.range m)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_iff_disjoint_range | [27, 1] | [30, 83] | constructor | m : ℕ
A : Finset ℕ
⊢ (∀ n ∈ A, m ≤ n) ↔ ∀ a ∈ A, ∀ b < m, ¬a = b | case mp
m : ℕ
A : Finset ℕ
⊢ (∀ n ∈ A, m ≤ n) → ∀ a ∈ A, ∀ b < m, ¬a = b
case mpr
m : ℕ
A : Finset ℕ
⊢ (∀ a ∈ A, ∀ b < m, ¬a = b) → ∀ n ∈ A, m ≤ n | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
A : Finset ℕ
⊢ (∀ n ∈ A, m ≤ n) ↔ ∀ a ∈ A, ∀ b < m, ¬a = b
TACTIC:
|
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 : ℕ
A : Finset ℕ
⊢ (∀ n ∈ A, m ≤ n) → ∀ a ∈ A, ∀ b < m, ¬a = b | case mp
m : ℕ
A : Finset ℕ
h : ∀ n ∈ A, m ≤ n
n : ℕ
na : n ∈ A
b : ℕ
bm : b < m
⊢ ¬n = b | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
m : ℕ
A : Finset ℕ
⊢ (∀ n ∈ A, m ≤ n) → ∀ a ∈ A, ∀ b < m, ¬a = b
TACTIC:
|
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 : ℕ
A : Finset ℕ
h : ∀ n ∈ A, m ≤ n
n : ℕ
na : n ∈ A
b : ℕ
bm : b < m
⊢ ¬n = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
m : ℕ
A : Finset ℕ
h : ∀ n ∈ A, m ≤ n
n : ℕ
na : n ∈ A
b : ℕ
bm : b < m
⊢ ¬n = b
TACTIC:
|
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 : ℕ
A : Finset ℕ
⊢ (∀ a ∈ A, ∀ b < m, ¬a = b) → ∀ n ∈ A, m ≤ n | case mpr
m : ℕ
A : Finset ℕ
h : ∀ a ∈ A, ∀ b < m, ¬a = b
n : ℕ
na : n ∈ A
⊢ m ≤ n | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
m : ℕ
A : Finset ℕ
⊢ (∀ a ∈ A, ∀ b < m, ¬a = b) → ∀ n ∈ A, m ≤ n
TACTIC:
|
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 : ℕ
A : Finset ℕ
h : ∀ a ∈ A, ∀ b < m, ¬a = b
n : ℕ
na : n ∈ A
⊢ m ≤ n | case mpr
m : ℕ
A : Finset ℕ
n : ℕ
na : n ∈ A
h : n < m → ¬n = n
⊢ m ≤ n | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
m : ℕ
A : Finset ℕ
h : ∀ a ∈ A, ∀ b < m, ¬a = b
n : ℕ
na : n ∈ A
⊢ m ≤ n
TACTIC:
|
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 : ℕ
A : Finset ℕ
n : ℕ
na : n ∈ A
h : n < m → ¬n = n
⊢ m ≤ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
m : ℕ
A : Finset ℕ
n : ℕ
na : n ∈ A
h : n < m → ¬n = n
⊢ m ≤ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | sdiff_late | [32, 1] | [38, 17] | intro Bm n nAB | m : ℕ
B A : Finset ℕ
⊢ B ≥ Finset.range m → Late (A \ B) m | m : ℕ
B A : Finset ℕ
Bm : B ≥ Finset.range m
n : ℕ
nAB : n ∈ A \ B
⊢ n ≥ m | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
B A : Finset ℕ
⊢ B ≥ Finset.range m → Late (A \ B) m
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | sdiff_late | [32, 1] | [38, 17] | rw [Finset.mem_sdiff] at nAB | m : ℕ
B A : Finset ℕ
Bm : B ≥ Finset.range m
n : ℕ
nAB : n ∈ A \ B
⊢ n ≥ m | m : ℕ
B A : Finset ℕ
Bm : B ≥ Finset.range m
n : ℕ
nAB : n ∈ A ∧ n ∉ B
⊢ n ≥ m | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
B A : Finset ℕ
Bm : B ≥ Finset.range m
n : ℕ
nAB : n ∈ A \ B
⊢ n ≥ m
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | sdiff_late | [32, 1] | [38, 17] | by_contra h | m : ℕ
B A : Finset ℕ
Bm : B ≥ Finset.range m
n : ℕ
nAB : n ∈ A ∧ n ∉ B
⊢ n ≥ m | m : ℕ
B A : Finset ℕ
Bm : B ≥ Finset.range m
n : ℕ
nAB : n ∈ A ∧ n ∉ B
h : ¬n ≥ m
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
B A : Finset ℕ
Bm : B ≥ Finset.range m
n : ℕ
nAB : n ∈ A ∧ n ∉ B
⊢ n ≥ m
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | sdiff_late | [32, 1] | [38, 17] | simp only [not_le] at h | m : ℕ
B A : Finset ℕ
Bm : B ≥ Finset.range m
n : ℕ
nAB : n ∈ A ∧ n ∉ B
h : ¬n ≥ m
⊢ False | m : ℕ
B A : Finset ℕ
Bm : B ≥ Finset.range m
n : ℕ
nAB : n ∈ A ∧ n ∉ B
h : n < m
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
B A : Finset ℕ
Bm : B ≥ Finset.range m
n : ℕ
nAB : n ∈ A ∧ n ∉ B
h : ¬n ≥ m
⊢ False
TACTIC:
|
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 : ℕ
B A : Finset ℕ
Bm : B ≥ Finset.range m
n : ℕ
nAB : n ∈ A ∧ n ∉ B
h : n < m
⊢ False | m : ℕ
B A : Finset ℕ
Bm : B ≥ Finset.range m
n : ℕ
nAB : n ∈ A ∧ n ∉ B
h : n < m
nr : n ∈ Finset.range m
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
B A : Finset ℕ
Bm : B ≥ Finset.range m
n : ℕ
nAB : n ∈ A ∧ n ∉ B
h : n < m
⊢ False
TACTIC:
|
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 : ℕ
B A : Finset ℕ
Bm : B ≥ Finset.range m
n : ℕ
nAB : n ∈ A ∧ n ∉ B
h : n < m
nr : n ∈ Finset.range m
⊢ False | m : ℕ
B A : Finset ℕ
Bm : B ≥ Finset.range m
n : ℕ
nAB : n ∈ A ∧ n ∉ B
h : n < m
nr : n ∈ Finset.range m
nB : n ∈ B
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
B A : Finset ℕ
Bm : B ≥ Finset.range m
n : ℕ
nAB : n ∈ A ∧ n ∉ B
h : n < m
nr : n ∈ Finset.range m
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | sdiff_late | [32, 1] | [38, 17] | exact nAB.2 nB | m : ℕ
B A : Finset ℕ
Bm : B ≥ Finset.range m
n : ℕ
nAB : n ∈ A ∧ n ∉ B
h : n < m
nr : n ∈ Finset.range m
nB : n ∈ B
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
B A : Finset ℕ
Bm : B ≥ Finset.range m
n : ℕ
nAB : n ∈ A ∧ n ∉ B
h : n < m
nr : n ∈ Finset.range m
nB : n ∈ B
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | partial_geometric_bound | [41, 1] | [44, 57] | intro n _ | a : ℝ
N : Finset ℕ
a0 : 0 ≤ a
a1 : a < 1
⊢ ∀ n ∉ N, 0 ≤ a ^ n | a : ℝ
N : Finset ℕ
a0 : 0 ≤ a
a1 : a < 1
n : ℕ
a✝ : n ∉ N
⊢ 0 ≤ a ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℝ
N : Finset ℕ
a0 : 0 ≤ a
a1 : a < 1
⊢ ∀ n ∉ N, 0 ≤ a ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | partial_geometric_bound | [41, 1] | [44, 57] | bound | a : ℝ
N : Finset ℕ
a0 : 0 ≤ a
a1 : a < 1
n : ℕ
a✝ : n ∉ N
⊢ 0 ≤ a ^ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℝ
N : Finset ℕ
a0 : 0 ≤ a
a1 : a < 1
n : ℕ
a✝ : n ∉ N
⊢ 0 ≤ a ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | partial_scaled_geometric_bound | [46, 1] | [49, 42] | rw [←Finset.mul_sum] | a : ℝ
c : ℝ≥0
N : Finset ℕ
a0 : 0 ≤ a
a1 : a < 1
⊢ (N.sum fun n => ↑c * a ^ n) ≤ ↑c * (1 - a)⁻¹ | a : ℝ
c : ℝ≥0
N : Finset ℕ
a0 : 0 ≤ a
a1 : a < 1
⊢ (↑c * N.sum fun i => a ^ i) ≤ ↑c * (1 - a)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℝ
c : ℝ≥0
N : Finset ℕ
a0 : 0 ≤ a
a1 : a < 1
⊢ (N.sum fun n => ↑c * a ^ n) ≤ ↑c * (1 - a)⁻¹
TACTIC:
|
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 : ℝ
c : ℝ≥0
N : Finset ℕ
a0 : 0 ≤ a
a1 : a < 1
⊢ (↑c * N.sum fun i => a ^ i) ≤ ↑c * (1 - a)⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℝ
c : ℝ≥0
N : Finset ℕ
a0 : 0 ≤ a
a1 : a < 1
⊢ (↑c * N.sum fun i => a ^ i) ≤ ↑c * (1 - a)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_series_sum | [52, 1] | [66, 44] | set Ns := Finset.image (fun n ↦ n - m) N | m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
⊢ N.sum f = (Finset.image (fun n => n - m) N).sum fun n => f (n + m) | m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
⊢ N.sum f = Ns.sum fun n => f (n + m) | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
⊢ N.sum f = (Finset.image (fun n => n - m) N).sum fun n => f (n + m)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_series_sum | [52, 1] | [66, 44] | rw [NNs] | m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
NNs : N = Finset.image (fun n => n + m) Ns
⊢ N.sum f = Ns.sum fun n => f (n + m) | m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
NNs : N = Finset.image (fun n => n + m) Ns
⊢ (Finset.image (fun n => n + m) Ns).sum f = Ns.sum fun n => f (n + m) | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
NNs : N = Finset.image (fun n => n + m) Ns
⊢ N.sum f = Ns.sum fun n => f (n + m)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_series_sum | [52, 1] | [66, 44] | apply Finset.sum_image | m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
NNs : N = Finset.image (fun n => n + m) Ns
⊢ (Finset.image (fun n => n + m) Ns).sum f = Ns.sum fun n => f (n + m) | case a
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
NNs : N = Finset.image (fun n => n + m) Ns
⊢ ∀ x ∈ Ns, ∀ y ∈ Ns, x + m = y + m → x = y | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
NNs : N = Finset.image (fun n => n + m) Ns
⊢ (Finset.image (fun n => n + m) Ns).sum f = Ns.sum fun n => f (n + m)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_series_sum | [52, 1] | [66, 44] | intro a _ b _ | case a
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
NNs : N = Finset.image (fun n => n + m) Ns
⊢ ∀ x ∈ Ns, ∀ y ∈ Ns, x + m = y + m → x = y | case a
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
NNs : N = Finset.image (fun n => n + m) Ns
a : ℕ
a✝¹ : a ∈ Ns
b : ℕ
a✝ : b ∈ Ns
⊢ a + m = b + m → a = b | Please generate a tactic in lean4 to solve the state.
STATE:
case a
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
NNs : N = Finset.image (fun n => n + m) Ns
⊢ ∀ x ∈ Ns, ∀ y ∈ Ns, x + m = y + m → x = y
TACTIC:
|
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 : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
NNs : N = Finset.image (fun n => n + m) Ns
a : ℕ
a✝¹ : a ∈ Ns
b : ℕ
a✝ : b ∈ Ns
⊢ a + m = b + m → a = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
NNs : N = Finset.image (fun n => n + m) Ns
a : ℕ
a✝¹ : a ∈ Ns
b : ℕ
a✝ : b ∈ Ns
⊢ a + m = b + m → a = b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_series_sum | [52, 1] | [66, 44] | apply Finset.ext | m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
⊢ N = Finset.image (fun n => n + m) Ns | case a
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
⊢ ∀ (a : ℕ), a ∈ N ↔ a ∈ Finset.image (fun n => n + m) Ns | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
⊢ N = Finset.image (fun n => n + m) Ns
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_series_sum | [52, 1] | [66, 44] | intro k | case a
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
⊢ ∀ (a : ℕ), a ∈ N ↔ a ∈ Finset.image (fun n => n + m) Ns | case a
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
⊢ k ∈ N ↔ k ∈ Finset.image (fun n => n + m) Ns | Please generate a tactic in lean4 to solve the state.
STATE:
case a
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
⊢ ∀ (a : ℕ), a ∈ N ↔ a ∈ Finset.image (fun n => n + m) Ns
TACTIC:
|
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 : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
⊢ k ∈ N ↔ k ∈ Finset.image (fun n => n + m) Ns | case a
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
⊢ k ∈ N ↔ ∃ a ∈ N, ((fun n => n + m) ∘ fun n => n - m) a = k | Please generate a tactic in lean4 to solve the state.
STATE:
case a
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
⊢ k ∈ N ↔ k ∈ Finset.image (fun n => n + m) Ns
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_series_sum | [52, 1] | [66, 44] | simp | case a
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
⊢ k ∈ N ↔ ∃ a ∈ N, ((fun n => n + m) ∘ fun n => n - m) a = k | case a
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
⊢ k ∈ N ↔ ∃ a ∈ N, a - m + m = k | Please generate a tactic in lean4 to solve the state.
STATE:
case a
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
⊢ k ∈ N ↔ ∃ a ∈ N, ((fun n => n + m) ∘ fun n => n - m) a = k
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_series_sum | [52, 1] | [66, 44] | apply Iff.intro | case a
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
⊢ k ∈ N ↔ ∃ a ∈ N, a - m + m = k | case a.mp
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
⊢ k ∈ N → ∃ a ∈ N, a - m + m = k
case a.mpr
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
⊢ (∃ a ∈ N, a - m + m = k) → k ∈ N | Please generate a tactic in lean4 to solve the state.
STATE:
case a
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
⊢ k ∈ N ↔ ∃ a ∈ N, a - m + m = k
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_series_sum | [52, 1] | [66, 44] | intro kN | case a.mp
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
⊢ k ∈ N → ∃ a ∈ N, a - m + m = k | case a.mp
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
kN : k ∈ N
⊢ ∃ a ∈ N, a - m + m = k | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
⊢ k ∈ N → ∃ a ∈ N, a - m + m = k
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_series_sum | [52, 1] | [66, 44] | exists k | case a.mp
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
kN : k ∈ N
⊢ ∃ a ∈ N, a - m + m = k | case a.mp
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
kN : k ∈ N
⊢ k ∈ N ∧ k - m + m = k | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
kN : k ∈ N
⊢ ∃ a ∈ N, a - m + m = k
TACTIC:
|
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 : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
kN : k ∈ N
⊢ k ∈ N ∧ k - m + m = k | case a.mp.left
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
kN : k ∈ N
⊢ k ∈ N
case a.mp.right
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
kN : k ∈ N
⊢ k - m + m = k | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
kN : k ∈ N
⊢ k ∈ N ∧ k - m + m = k
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_series_sum | [52, 1] | [66, 44] | assumption | case a.mp.left
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
kN : k ∈ N
⊢ k ∈ N
case a.mp.right
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
kN : k ∈ N
⊢ k - m + m = k | case a.mp.right
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
kN : k ∈ N
⊢ k - m + m = k | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp.left
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
kN : k ∈ N
⊢ k ∈ N
case a.mp.right
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
kN : k ∈ N
⊢ k - m + m = k
TACTIC:
|
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 : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
kN : k ∈ N
⊢ k - m + m = k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp.right
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
kN : k ∈ N
⊢ k - m + m = k
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_series_sum | [52, 1] | [66, 44] | intro ha | case a.mpr
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
⊢ (∃ a ∈ N, a - m + m = k) → k ∈ N | case a.mpr
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
ha : ∃ a ∈ N, a - m + m = k
⊢ k ∈ N | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
⊢ (∃ a ∈ N, a - m + m = k) → k ∈ N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_series_sum | [52, 1] | [66, 44] | rcases ha with ⟨a, aN, ak⟩ | case a.mpr
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
ha : ∃ a ∈ N, a - m + m = k
⊢ k ∈ N | case a.mpr.intro.intro
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k a : ℕ
aN : a ∈ N
ak : a - m + m = k
⊢ k ∈ N | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
ha : ∃ a ∈ N, a - m + m = k
⊢ k ∈ N
TACTIC:
|
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 : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k a : ℕ
aN : a ∈ N
ak : a - m + m = k
⊢ k ∈ N | case a.mpr.intro.intro
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k a : ℕ
aN : a ∈ N
ak : a = k
⊢ k ∈ N | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr.intro.intro
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k a : ℕ
aN : a ∈ N
ak : a - m + m = k
⊢ k ∈ N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_series_sum | [52, 1] | [66, 44] | rw [← ak] | case a.mpr.intro.intro
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k a : ℕ
aN : a ∈ N
ak : a = k
⊢ k ∈ N | case a.mpr.intro.intro
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k a : ℕ
aN : a ∈ N
ak : a = k
⊢ a ∈ N | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr.intro.intro
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k a : ℕ
aN : a ∈ N
ak : a = k
⊢ k ∈ N
TACTIC:
|
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 : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k a : ℕ
aN : a ∈ N
ak : a = k
⊢ a ∈ N | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr.intro.intro
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k a : ℕ
aN : a ∈ N
ak : a = k
⊢ a ∈ N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_series_sum' | [69, 1] | [72, 28] | exists Finset.image (fun n ↦ n - m) N | m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
⊢ ∃ M, N.sum f = M.sum fun n => f (n + m) | m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
⊢ N.sum f = (Finset.image (fun n => n - m) N).sum fun n => f (n + m) | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
⊢ ∃ M, N.sum f = M.sum fun n => f (n + m)
TACTIC:
|
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 : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
⊢ N.sum f = (Finset.image (fun n => n - m) N).sum fun n => f (n + m) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
⊢ N.sum f = (Finset.image (fun n => n - m) N).sum fun n => f (n + m)
TACTIC:
|
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 ↦ a^n) with ⟨M,L⟩ | m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
⊢ (N.sum fun n => a ^ n) ≤ a ^ m * (1 - a)⁻¹ | case intro
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
L : (N.sum fun n => a ^ n) = M.sum fun n => a ^ (n + m)
⊢ (N.sum fun n => a ^ n) ≤ a ^ m * (1 - a)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
⊢ (N.sum fun n => a ^ n) ≤ a ^ m * (1 - a)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_geometric_bound | [74, 1] | [83, 35] | rw [L] | case intro
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
L : (N.sum fun n => a ^ n) = M.sum fun n => a ^ (n + m)
⊢ (N.sum fun n => a ^ n) ≤ a ^ m * (1 - a)⁻¹ | case intro
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
L : (N.sum fun n => a ^ n) = M.sum fun n => a ^ (n + m)
⊢ (M.sum fun n => a ^ (n + m)) ≤ a ^ m * (1 - a)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
L : (N.sum fun n => a ^ n) = M.sum fun n => a ^ (n + m)
⊢ (N.sum fun n => a ^ n) ≤ a ^ m * (1 - a)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_geometric_bound | [74, 1] | [83, 35] | clear L | case intro
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
L : (N.sum fun n => a ^ n) = M.sum fun n => a ^ (n + m)
⊢ (M.sum fun n => a ^ (n + m)) ≤ a ^ m * (1 - a)⁻¹ | case intro
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
⊢ (M.sum fun n => a ^ (n + m)) ≤ a ^ m * (1 - a)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
L : (N.sum fun n => a ^ n) = M.sum fun n => a ^ (n + m)
⊢ (M.sum fun n => a ^ (n + m)) ≤ a ^ m * (1 - a)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_geometric_bound | [74, 1] | [83, 35] | have pa : (fun n ↦ a^(n + m)) = (fun n ↦ a^n * a^m) := by apply funext; intro n; rw [pow_add] | case intro
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
⊢ (M.sum fun n => a ^ (n + m)) ≤ a ^ m * (1 - a)⁻¹ | case intro
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
pa : (fun n => a ^ (n + m)) = fun n => a ^ n * a ^ m
⊢ (M.sum fun n => a ^ (n + m)) ≤ a ^ m * (1 - a)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
⊢ (M.sum fun n => a ^ (n + m)) ≤ a ^ m * (1 - a)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_geometric_bound | [74, 1] | [83, 35] | calc
M.sum (fun n ↦ a^(n + m)) = M.sum (fun n ↦ a^n * a^m) := by rw [ pa ]
_ = M.sum (fun n ↦ a^n) * a^m := (Finset.sum_mul _ _ _).symm
_ ≤ (1 - a)⁻¹ * a^m := by bound [partial_geometric_bound M a0 a1]
_ = a^m * (1 - a)⁻¹ := by ring | case intro
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
pa : (fun n => a ^ (n + m)) = fun n => a ^ n * a ^ m
⊢ (M.sum fun n => a ^ (n + m)) ≤ a ^ m * (1 - a)⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
pa : (fun n => a ^ (n + m)) = fun n => a ^ n * a ^ m
⊢ (M.sum fun n => a ^ (n + m)) ≤ a ^ m * (1 - a)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_geometric_bound | [74, 1] | [83, 35] | apply funext | m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
⊢ (fun n => a ^ (n + m)) = fun n => a ^ n * a ^ m | case h
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
⊢ ∀ (x : ℕ), a ^ (x + m) = a ^ x * a ^ m | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
⊢ (fun n => a ^ (n + m)) = fun n => a ^ n * a ^ m
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_geometric_bound | [74, 1] | [83, 35] | intro n | case h
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
⊢ ∀ (x : ℕ), a ^ (x + m) = a ^ x * a ^ m | case h
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
n : ℕ
⊢ a ^ (n + m) = a ^ n * a ^ m | Please generate a tactic in lean4 to solve the state.
STATE:
case h
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
⊢ ∀ (x : ℕ), a ^ (x + m) = a ^ x * a ^ m
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_geometric_bound | [74, 1] | [83, 35] | rw [pow_add] | case h
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
n : ℕ
⊢ a ^ (n + m) = a ^ n * a ^ m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
n : ℕ
⊢ a ^ (n + m) = a ^ n * a ^ m
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_geometric_bound | [74, 1] | [83, 35] | rw [ pa ] | m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
pa : (fun n => a ^ (n + m)) = fun n => a ^ n * a ^ m
⊢ (M.sum fun n => a ^ (n + m)) = M.sum fun n => a ^ n * a ^ m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
pa : (fun n => a ^ (n + m)) = fun n => a ^ n * a ^ m
⊢ (M.sum fun n => a ^ (n + m)) = M.sum fun n => a ^ n * a ^ m
TACTIC:
|
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 : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
pa : (fun n => a ^ (n + m)) = fun n => a ^ n * a ^ m
⊢ (M.sum fun n => a ^ n) * a ^ m ≤ (1 - a)⁻¹ * a ^ m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
pa : (fun n => a ^ (n + m)) = fun n => a ^ n * a ^ m
⊢ (M.sum fun n => a ^ n) * a ^ m ≤ (1 - a)⁻¹ * a ^ m
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_geometric_bound | [74, 1] | [83, 35] | ring | m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
pa : (fun n => a ^ (n + m)) = fun n => a ^ n * a ^ m
⊢ (1 - a)⁻¹ * a ^ m = a ^ m * (1 - a)⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
pa : (fun n => a ^ (n + m)) = fun n => a ^ n * a ^ m
⊢ (1 - a)⁻¹ * a ^ m = a ^ m * (1 - a)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | finset_partition | [85, 1] | [95, 21] | apply Finset.ext | A B : Finset ℕ
⊢ A = A \ B ∪ A ∩ B | case a
A B : Finset ℕ
⊢ ∀ (a : ℕ), a ∈ A ↔ a ∈ A \ B ∪ A ∩ B | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Finset ℕ
⊢ A = A \ B ∪ A ∩ B
TACTIC:
|
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 ℕ
⊢ ∀ (a : ℕ), a ∈ A ↔ a ∈ A \ B ∪ A ∩ B | case a
A B : Finset ℕ
⊢ ∀ (a : ℕ), a ∈ A ↔ a ∈ A ∧ a ∉ B ∨ a ∈ A ∧ a ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A B : Finset ℕ
⊢ ∀ (a : ℕ), a ∈ A ↔ a ∈ A \ B ∪ A ∩ B
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | finset_partition | [85, 1] | [95, 21] | intro x | case a
A B : Finset ℕ
⊢ ∀ (a : ℕ), a ∈ A ↔ a ∈ A ∧ a ∉ B ∨ a ∈ A ∧ a ∈ B | case a
A B : Finset ℕ
x : ℕ
⊢ x ∈ A ↔ x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A B : Finset ℕ
⊢ ∀ (a : ℕ), a ∈ A ↔ a ∈ A ∧ a ∉ B ∨ a ∈ A ∧ a ∈ B
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | finset_partition | [85, 1] | [95, 21] | constructor | case a
A B : Finset ℕ
x : ℕ
⊢ x ∈ A ↔ x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B | case a.mp
A B : Finset ℕ
x : ℕ
⊢ x ∈ A → x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B
case a.mpr
A B : Finset ℕ
x : ℕ
⊢ x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B → x ∈ A | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A B : Finset ℕ
x : ℕ
⊢ x ∈ A ↔ x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B
TACTIC:
|
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 ℕ
x : ℕ
⊢ x ∈ A → x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B | case a.mp
A B : Finset ℕ
x : ℕ
a : x ∈ A
⊢ x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp
A B : Finset ℕ
x : ℕ
⊢ x ∈ A → x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | finset_partition | [85, 1] | [95, 21] | by_cases b : x ∈ B | case a.mp
A B : Finset ℕ
x : ℕ
a : x ∈ A
⊢ x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B | case pos
A B : Finset ℕ
x : ℕ
a : x ∈ A
b : x ∈ B
⊢ x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B
case neg
A B : Finset ℕ
x : ℕ
a : x ∈ A
b : x ∉ B
⊢ x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp
A B : Finset ℕ
x : ℕ
a : x ∈ A
⊢ x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | finset_partition | [85, 1] | [95, 21] | right | case pos
A B : Finset ℕ
x : ℕ
a : x ∈ A
b : x ∈ B
⊢ x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B | case pos.h
A B : Finset ℕ
x : ℕ
a : x ∈ A
b : x ∈ B
⊢ x ∈ A ∧ x ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A B : Finset ℕ
x : ℕ
a : x ∈ A
b : x ∈ B
⊢ x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B
TACTIC:
|
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 ℕ
x : ℕ
a : x ∈ A
b : x ∈ B
⊢ x ∈ A ∧ x ∈ B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.h
A B : Finset ℕ
x : ℕ
a : x ∈ A
b : x ∈ B
⊢ x ∈ A ∧ x ∈ B
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | finset_partition | [85, 1] | [95, 21] | left | case neg
A B : Finset ℕ
x : ℕ
a : x ∈ A
b : x ∉ B
⊢ x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B | case neg.h
A B : Finset ℕ
x : ℕ
a : x ∈ A
b : x ∉ B
⊢ x ∈ A ∧ x ∉ B | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A B : Finset ℕ
x : ℕ
a : x ∈ A
b : x ∉ B
⊢ x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B
TACTIC:
|
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 ℕ
x : ℕ
a : x ∈ A
b : x ∉ B
⊢ x ∈ A ∧ x ∉ B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h
A B : Finset ℕ
x : ℕ
a : x ∈ A
b : x ∉ B
⊢ x ∈ A ∧ x ∉ B
TACTIC:
|
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 ℕ
x : ℕ
⊢ x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B → x ∈ A | case a.mpr
A B : Finset ℕ
x : ℕ
h : x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B
⊢ x ∈ A | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr
A B : Finset ℕ
x : ℕ
⊢ x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B → x ∈ A
TACTIC:
|
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