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stringclasses 147
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stringclasses 147
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stringlengths 7
101
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stringlengths 1
94
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stringlengths 6
10
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stringlengths 6
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stringlengths 1
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2.09M
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2.09M
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|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | norm_num | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ (4 - 1).log / log 4 = log 3 / log 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ (4 - 1).log / log 4 = log 3 / log 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | positivity | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ 0 ≤ log 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ 0 ≤ log 3
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | positivity | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ 0 < log 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ 0 < log 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | norm_num | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ 1.098 / 1.387 ≥ 0.791 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ 1.098 / 1.387 ≥ 0.791
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | positivity | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
⊢ 0 ≤ (x - 1).log / x.log | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
⊢ 0 ≤ (x - 1).log / x.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | linarith | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
⊢ 0 < x - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
⊢ 0 < x - 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | linarith | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
⊢ x - 1 ≤ x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
⊢ x - 1 ≤ x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | positivity | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
⊢ 0 ≤ x.log | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
⊢ 0 ≤ x.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | nlinarith | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ 1 < (x - 1) * x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ 1 < (x - 1) * x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | rw [mul_comm, Real.log_mul (by positivity) (by linarith)] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ ((x - 1) * x).log.log = (x.log + (x - 1).log).log | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ ((x - 1) * x).log.log = (x.log + (x - 1).log).log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | positivity | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ x ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ x ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | linarith | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ x - 1 ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ x - 1 ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | rw [log_add _ _ (by linarith) (by linarith)] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ (x.log + (x - 1).log).log = x.log.log + (1 + (x - 1).log / x.log).log | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ (x.log + (x - 1).log).log = x.log.log + (1 + (x - 1).log / x.log).log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | linarith | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ 0 < x.log | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ 0 < x.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | linarith | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ 0 < x.log + (x - 1).log | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ 0 < x.log + (x - 1).log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | bound [le_log_one_add ll0 ll1] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ x.log.log + (1 + (x - 1).log / x.log).log ≥ x.log.log + log 2 * ((x - 1).log / x.log) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ x.log.log + (1 + (x - 1).log / x.log).log ≥ x.log.log + log 2 * ((x - 1).log / x.log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | bound [lt_log_2] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ x.log.log + log 2 * ((x - 1).log / x.log) ≥ x.log.log + 0.693 * 0.791 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ x.log.log + log 2 * ((x - 1).log / x.log) ≥ x.log.log + 0.693 * 0.791
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | bound | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ x.log.log + 0.693 * 0.791 ≥ x.log.log + 0.548 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ x.log.log + 0.693 * 0.791 ≥ x.log.log + 0.548
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | have d0 : 0 < d := d_pos d | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
⊢ Postcritical ⋯ c ↑z | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
⊢ Postcritical ⋯ c ↑z | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
⊢ Postcritical ⋯ c ↑z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | have lcz : log (log (abs c)) ≤ log (log (abs z)) := log_log_mono (by linarith) cz | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
⊢ Postcritical ⋯ c ↑z | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
⊢ Postcritical ⋯ c ↑z | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
⊢ Postcritical ⋯ c ↑z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | simp only [Postcritical, multibrot_p] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
⊢ Postcritical ⋯ c ↑z | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
⊢ ⋯.potential c ↑z < ⋯.potential c 0 | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
⊢ Postcritical ⋯ c ↑z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | set s := superF d | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
⊢ ⋯.potential c ↑z < ⋯.potential c 0 | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
⊢ s.potential c ↑z < s.potential c 0 | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
⊢ ⋯.potential c ↑z < ⋯.potential c 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | rw [← Real.pow_rpow_inv_natCast s.potential_nonneg d0.ne', ←
Real.pow_rpow_inv_natCast (s.potential_nonneg : 0 ≤ s.potential c 0) d0.ne'] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
⊢ s.potential c ↑z < s.potential c 0 | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
⊢ (s.potential c ↑z ^ d) ^ (↑d)⁻¹ < (s.potential c 0 ^ d) ^ (↑d)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
⊢ s.potential c ↑z < s.potential c 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | simp only [← s.potential_eqn] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
⊢ (s.potential c ↑z ^ d) ^ (↑d)⁻¹ < (s.potential c 0 ^ d) ^ (↑d)⁻¹ | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
⊢ s.potential c (f d c ↑z) ^ (↑d)⁻¹ < s.potential c (f d c 0) ^ (↑d)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
⊢ (s.potential c ↑z ^ d) ^ (↑d)⁻¹ < (s.potential c 0 ^ d) ^ (↑d)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | refine Real.rpow_lt_rpow s.potential_nonneg ?_ (by bound) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
⊢ s.potential c (f d c ↑z) ^ (↑d)⁻¹ < s.potential c (f d c 0) ^ (↑d)⁻¹ | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
⊢ s.potential c (f d c ↑z) < s.potential c (f d c 0) | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
⊢ s.potential c (f d c ↑z) ^ (↑d)⁻¹ < s.potential c (f d c 0) ^ (↑d)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | generalize hw : f' d c z = w | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
⊢ s.potential c (f d c ↑z) < s.potential c (f d c 0) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
⊢ s.potential c (f d c ↑z) < s.potential c (f d c 0) | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
⊢ s.potential c (f d c ↑z) < s.potential c (f d c 0)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | have e : f d c z = w := by rw [f, lift_coe', hw] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
⊢ s.potential c (f d c ↑z) < s.potential c (f d c 0) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
e : f d c ↑z = ↑w
⊢ s.potential c (f d c ↑z) < s.potential c (f d c 0) | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
⊢ s.potential c (f d c ↑z) < s.potential c (f d c 0)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | simp only [f_0, e] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
e : f d c ↑z = ↑w
⊢ s.potential c (f d c ↑z) < s.potential c (f d c 0) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
e : f d c ↑z = ↑w
⊢ s.potential c ↑w < s.potential c ↑c | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
e : f d c ↑z = ↑w
⊢ s.potential c (f d c ↑z) < s.potential c (f d c 0)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | clear e | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
e : f d c ↑z = ↑w
⊢ s.potential c ↑w < s.potential c ↑c | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
⊢ s.potential c ↑w < s.potential c ↑c | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
e : f d c ↑z = ↑w
⊢ s.potential c ↑w < s.potential c ↑c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | have zw : abs z ≤ abs w := by rw [←hw]; exact le_self_iter d (by linarith) cz 1 | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
⊢ s.potential c ↑w < s.potential c ↑c | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
⊢ s.potential c ↑w < s.potential c ↑c | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
⊢ s.potential c ↑w < s.potential c ↑c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | have cw : abs c ≤ abs w := le_trans cz zw | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
⊢ s.potential c ↑w < s.potential c ↑c | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
⊢ s.potential c ↑w < s.potential c ↑c | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
⊢ s.potential c ↑w < s.potential c ↑c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | have pc1 : s.potential c c < 1 := potential_lt_one_of_two_lt (by linarith) (le_refl _) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
⊢ s.potential c ↑w < s.potential c ↑c | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
⊢ s.potential c ↑w < s.potential c ↑c | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
⊢ s.potential c ↑w < s.potential c ↑c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | have pw1 : s.potential c w < 1 := potential_lt_one_of_two_lt (by linarith) (by linarith) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
⊢ s.potential c ↑w < s.potential c ↑c | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
⊢ s.potential c ↑w < s.potential c ↑c | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
⊢ s.potential c ↑w < s.potential c ↑c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | rw [←log_neg_log_strict_anti potential_pos potential_pos pw1 pc1] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
⊢ s.potential c ↑w < s.potential c ↑c | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
⊢ (-(⋯.potential c ↑c).log).log < (-(⋯.potential c ↑w).log).log | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
⊢ s.potential c ↑w < s.potential c ↑c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | refine lt_of_lt_of_le ?_ (le_sub_iff_add_le.mp (abs_le.mp
(log_neg_log_potential_approx d (by linarith) cw)).1) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
⊢ (-(⋯.potential c ↑c).log).log < (-(⋯.potential c ↑w).log).log | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
⊢ (-(⋯.potential c ↑c).log).log < -iter_error d c w + (Complex.abs w).log.log | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
⊢ (-(⋯.potential c ↑c).log).log < (-(⋯.potential c ↑w).log).log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | refine lt_of_le_of_lt (sub_le_iff_le_add.mp (abs_le.mp
(log_neg_log_potential_approx d (by linarith) (le_refl _))).2) ?_ | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
⊢ (-(⋯.potential c ↑c).log).log < -iter_error d c w + (Complex.abs w).log.log | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
⊢ iter_error d c c + (Complex.abs c).log.log < -iter_error d c w + (Complex.abs w).log.log | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
⊢ (-(⋯.potential c ↑c).log).log < -iter_error d c w + (Complex.abs w).log.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | have lzw : log (log (abs z)) + 0.548 ≤ log (log (abs w)) := by
rw [←hw]; exact log_log_iter (by linarith) cz | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
⊢ iter_error d c c + (Complex.abs c).log.log < -iter_error d c w + (Complex.abs w).log.log | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z).log.log + 0.548 ≤ (Complex.abs w).log.log
⊢ iter_error d c c + (Complex.abs c).log.log < -iter_error d c w + (Complex.abs w).log.log | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
⊢ iter_error d c c + (Complex.abs c).log.log < -iter_error d c w + (Complex.abs w).log.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | have ie : ∀ z : ℂ, 4 ≤ abs z → abs c ≤ abs z → iter_error d c z ≤ 0.15 := by
intro z z4 cz
refine le_trans (iter_error_le_of_z4 d z4 cz) ?_
calc 0.8095 / (abs z * log (abs z))
_ ≤ 0.8095 / (4 * log 4) := by bound
_ ≤ 0.8095 / (4 * 1.386) := by bound [lt_log_4]
_ ≤ 0.15 := by norm_num | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z).log.log + 0.548 ≤ (Complex.abs w).log.log
⊢ iter_error d c c + (Complex.abs c).log.log < -iter_error d c w + (Complex.abs w).log.log | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z).log.log + 0.548 ≤ (Complex.abs w).log.log
ie : ∀ (z : ℂ), 4 ≤ Complex.abs z → Complex.abs c ≤ Complex.abs z → iter_error d c z ≤ 0.15
⊢ iter_error d c c + (Complex.abs c).log.log < -iter_error d c w + (Complex.abs w).log.log | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z).log.log + 0.548 ≤ (Complex.abs w).log.log
⊢ iter_error d c c + (Complex.abs c).log.log < -iter_error d c w + (Complex.abs w).log.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | have iec := ie c c4 (le_refl _) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z).log.log + 0.548 ≤ (Complex.abs w).log.log
ie : ∀ (z : ℂ), 4 ≤ Complex.abs z → Complex.abs c ≤ Complex.abs z → iter_error d c z ≤ 0.15
⊢ iter_error d c c + (Complex.abs c).log.log < -iter_error d c w + (Complex.abs w).log.log | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z).log.log + 0.548 ≤ (Complex.abs w).log.log
ie : ∀ (z : ℂ), 4 ≤ Complex.abs z → Complex.abs c ≤ Complex.abs z → iter_error d c z ≤ 0.15
iec : iter_error d c c ≤ 0.15
⊢ iter_error d c c + (Complex.abs c).log.log < -iter_error d c w + (Complex.abs w).log.log | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z).log.log + 0.548 ≤ (Complex.abs w).log.log
ie : ∀ (z : ℂ), 4 ≤ Complex.abs z → Complex.abs c ≤ Complex.abs z → iter_error d c z ≤ 0.15
⊢ iter_error d c c + (Complex.abs c).log.log < -iter_error d c w + (Complex.abs w).log.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | have iew := ie w (by linarith) cw | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z).log.log + 0.548 ≤ (Complex.abs w).log.log
ie : ∀ (z : ℂ), 4 ≤ Complex.abs z → Complex.abs c ≤ Complex.abs z → iter_error d c z ≤ 0.15
iec : iter_error d c c ≤ 0.15
⊢ iter_error d c c + (Complex.abs c).log.log < -iter_error d c w + (Complex.abs w).log.log | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z).log.log + 0.548 ≤ (Complex.abs w).log.log
ie : ∀ (z : ℂ), 4 ≤ Complex.abs z → Complex.abs c ≤ Complex.abs z → iter_error d c z ≤ 0.15
iec : iter_error d c c ≤ 0.15
iew : iter_error d c w ≤ 0.15
⊢ iter_error d c c + (Complex.abs c).log.log < -iter_error d c w + (Complex.abs w).log.log | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z).log.log + 0.548 ≤ (Complex.abs w).log.log
ie : ∀ (z : ℂ), 4 ≤ Complex.abs z → Complex.abs c ≤ Complex.abs z → iter_error d c z ≤ 0.15
iec : iter_error d c c ≤ 0.15
⊢ iter_error d c c + (Complex.abs c).log.log < -iter_error d c w + (Complex.abs w).log.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | refine lt_of_lt_of_le (lt_of_le_of_lt (add_le_add iec lcz) ?_) (add_le_add (neg_le_neg iew) lzw) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z).log.log + 0.548 ≤ (Complex.abs w).log.log
ie : ∀ (z : ℂ), 4 ≤ Complex.abs z → Complex.abs c ≤ Complex.abs z → iter_error d c z ≤ 0.15
iec : iter_error d c c ≤ 0.15
iew : iter_error d c w ≤ 0.15
⊢ iter_error d c c + (Complex.abs c).log.log < -iter_error d c w + (Complex.abs w).log.log | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z).log.log + 0.548 ≤ (Complex.abs w).log.log
ie : ∀ (z : ℂ), 4 ≤ Complex.abs z → Complex.abs c ≤ Complex.abs z → iter_error d c z ≤ 0.15
iec : iter_error d c c ≤ 0.15
iew : iter_error d c w ≤ 0.15
⊢ 0.15 + (Complex.abs z).log.log < -0.15 + ((Complex.abs z).log.log + 0.548) | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z).log.log + 0.548 ≤ (Complex.abs w).log.log
ie : ∀ (z : ℂ), 4 ≤ Complex.abs z → Complex.abs c ≤ Complex.abs z → iter_error d c z ≤ 0.15
iec : iter_error d c c ≤ 0.15
iew : iter_error d c w ≤ 0.15
⊢ iter_error d c c + (Complex.abs c).log.log < -iter_error d c w + (Complex.abs w).log.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | ring_nf | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z).log.log + 0.548 ≤ (Complex.abs w).log.log
ie : ∀ (z : ℂ), 4 ≤ Complex.abs z → Complex.abs c ≤ Complex.abs z → iter_error d c z ≤ 0.15
iec : iter_error d c c ≤ 0.15
iew : iter_error d c w ≤ 0.15
⊢ 0.15 + (Complex.abs z).log.log < -0.15 + ((Complex.abs z).log.log + 0.548) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z).log.log + 0.548 ≤ (Complex.abs w).log.log
ie : ∀ (z : ℂ), 4 ≤ Complex.abs z → Complex.abs c ≤ Complex.abs z → iter_error d c z ≤ 0.15
iec : iter_error d c c ≤ 0.15
iew : iter_error d c w ≤ 0.15
⊢ 3 / 20 + (Complex.abs z).log.log < 199 / 500 + (Complex.abs z).log.log | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z).log.log + 0.548 ≤ (Complex.abs w).log.log
ie : ∀ (z : ℂ), 4 ≤ Complex.abs z → Complex.abs c ≤ Complex.abs z → iter_error d c z ≤ 0.15
iec : iter_error d c c ≤ 0.15
iew : iter_error d c w ≤ 0.15
⊢ 0.15 + (Complex.abs z).log.log < -0.15 + ((Complex.abs z).log.log + 0.548)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | simp only [add_lt_add_iff_right] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z).log.log + 0.548 ≤ (Complex.abs w).log.log
ie : ∀ (z : ℂ), 4 ≤ Complex.abs z → Complex.abs c ≤ Complex.abs z → iter_error d c z ≤ 0.15
iec : iter_error d c c ≤ 0.15
iew : iter_error d c w ≤ 0.15
⊢ 3 / 20 + (Complex.abs z).log.log < 199 / 500 + (Complex.abs z).log.log | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z).log.log + 0.548 ≤ (Complex.abs w).log.log
ie : ∀ (z : ℂ), 4 ≤ Complex.abs z → Complex.abs c ≤ Complex.abs z → iter_error d c z ≤ 0.15
iec : iter_error d c c ≤ 0.15
iew : iter_error d c w ≤ 0.15
⊢ 3 / 20 < 199 / 500 | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z).log.log + 0.548 ≤ (Complex.abs w).log.log
ie : ∀ (z : ℂ), 4 ≤ Complex.abs z → Complex.abs c ≤ Complex.abs z → iter_error d c z ≤ 0.15
iec : iter_error d c c ≤ 0.15
iew : iter_error d c w ≤ 0.15
⊢ 3 / 20 + (Complex.abs z).log.log < 199 / 500 + (Complex.abs z).log.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | norm_num | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z).log.log + 0.548 ≤ (Complex.abs w).log.log
ie : ∀ (z : ℂ), 4 ≤ Complex.abs z → Complex.abs c ≤ Complex.abs z → iter_error d c z ≤ 0.15
iec : iter_error d c c ≤ 0.15
iew : iter_error d c w ≤ 0.15
⊢ 3 / 20 < 199 / 500 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z).log.log + 0.548 ≤ (Complex.abs w).log.log
ie : ∀ (z : ℂ), 4 ≤ Complex.abs z → Complex.abs c ≤ Complex.abs z → iter_error d c z ≤ 0.15
iec : iter_error d c c ≤ 0.15
iew : iter_error d c w ≤ 0.15
⊢ 3 / 20 < 199 / 500
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | linarith | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
⊢ 1 < Complex.abs c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
⊢ 1 < Complex.abs c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | bound | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
⊢ 0 < (↑d)⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
⊢ 0 < (↑d)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | rw [f, lift_coe', hw] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
⊢ f d c ↑z = ↑w | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
⊢ f d c ↑z = ↑w
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | rw [←hw] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
⊢ Complex.abs z ≤ Complex.abs w | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
⊢ Complex.abs z ≤ Complex.abs (f' d c z) | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
⊢ Complex.abs z ≤ Complex.abs w
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | exact le_self_iter d (by linarith) cz 1 | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
⊢ Complex.abs z ≤ Complex.abs (f' d c z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
⊢ Complex.abs z ≤ Complex.abs (f' d c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | linarith | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
⊢ 3 ≤ Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
⊢ 3 ≤ Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | linarith | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
⊢ 2 < Complex.abs c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
⊢ 2 < Complex.abs c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | linarith | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
⊢ 2 < Complex.abs w | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
⊢ 2 < Complex.abs w
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | linarith | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
⊢ Complex.abs c ≤ Complex.abs w | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
⊢ Complex.abs c ≤ Complex.abs w
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | linarith | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
⊢ 3 ≤ Complex.abs w | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
⊢ 3 ≤ Complex.abs w
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | linarith | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
⊢ 3 ≤ Complex.abs c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
⊢ 3 ≤ Complex.abs c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | rw [←hw] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
⊢ (Complex.abs z).log.log + 0.548 ≤ (Complex.abs w).log.log | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
⊢ (Complex.abs z).log.log + 0.548 ≤ (Complex.abs (f' d c z)).log.log | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
⊢ (Complex.abs z).log.log + 0.548 ≤ (Complex.abs w).log.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | exact log_log_iter (by linarith) cz | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
⊢ (Complex.abs z).log.log + 0.548 ≤ (Complex.abs (f' d c z)).log.log | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
⊢ (Complex.abs z).log.log + 0.548 ≤ (Complex.abs (f' d c z)).log.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | linarith | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
⊢ 4 ≤ Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
⊢ 4 ≤ Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | intro z z4 cz | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z).log.log + 0.548 ≤ (Complex.abs w).log.log
⊢ ∀ (z : ℂ), 4 ≤ Complex.abs z → Complex.abs c ≤ Complex.abs z → iter_error d c z ≤ 0.15 | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c4 : 4 ≤ Complex.abs c
cz✝ : Complex.abs c ≤ Complex.abs z✝
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z✝).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z✝ = w
zw : Complex.abs z✝ ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z✝).log.log + 0.548 ≤ (Complex.abs w).log.log
z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ iter_error d c z ≤ 0.15 | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z).log.log + 0.548 ≤ (Complex.abs w).log.log
⊢ ∀ (z : ℂ), 4 ≤ Complex.abs z → Complex.abs c ≤ Complex.abs z → iter_error d c z ≤ 0.15
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | refine le_trans (iter_error_le_of_z4 d z4 cz) ?_ | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c4 : 4 ≤ Complex.abs c
cz✝ : Complex.abs c ≤ Complex.abs z✝
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z✝).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z✝ = w
zw : Complex.abs z✝ ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z✝).log.log + 0.548 ≤ (Complex.abs w).log.log
z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ iter_error d c z ≤ 0.15 | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c4 : 4 ≤ Complex.abs c
cz✝ : Complex.abs c ≤ Complex.abs z✝
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z✝).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z✝ = w
zw : Complex.abs z✝ ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z✝).log.log + 0.548 ≤ (Complex.abs w).log.log
z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 0.8095 / (Complex.abs z * (Complex.abs z).log) ≤ 0.15 | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c4 : 4 ≤ Complex.abs c
cz✝ : Complex.abs c ≤ Complex.abs z✝
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z✝).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z✝ = w
zw : Complex.abs z✝ ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z✝).log.log + 0.548 ≤ (Complex.abs w).log.log
z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ iter_error d c z ≤ 0.15
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | calc 0.8095 / (abs z * log (abs z))
_ ≤ 0.8095 / (4 * log 4) := by bound
_ ≤ 0.8095 / (4 * 1.386) := by bound [lt_log_4]
_ ≤ 0.15 := by norm_num | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c4 : 4 ≤ Complex.abs c
cz✝ : Complex.abs c ≤ Complex.abs z✝
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z✝).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z✝ = w
zw : Complex.abs z✝ ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z✝).log.log + 0.548 ≤ (Complex.abs w).log.log
z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 0.8095 / (Complex.abs z * (Complex.abs z).log) ≤ 0.15 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c4 : 4 ≤ Complex.abs c
cz✝ : Complex.abs c ≤ Complex.abs z✝
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z✝).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z✝ = w
zw : Complex.abs z✝ ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z✝).log.log + 0.548 ≤ (Complex.abs w).log.log
z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 0.8095 / (Complex.abs z * (Complex.abs z).log) ≤ 0.15
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | bound | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c4 : 4 ≤ Complex.abs c
cz✝ : Complex.abs c ≤ Complex.abs z✝
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z✝).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z✝ = w
zw : Complex.abs z✝ ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z✝).log.log + 0.548 ≤ (Complex.abs w).log.log
z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 0.8095 / (Complex.abs z * (Complex.abs z).log) ≤ 0.8095 / (4 * log 4) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c4 : 4 ≤ Complex.abs c
cz✝ : Complex.abs c ≤ Complex.abs z✝
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z✝).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z✝ = w
zw : Complex.abs z✝ ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z✝).log.log + 0.548 ≤ (Complex.abs w).log.log
z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 0.8095 / (Complex.abs z * (Complex.abs z).log) ≤ 0.8095 / (4 * log 4)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | bound [lt_log_4] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c4 : 4 ≤ Complex.abs c
cz✝ : Complex.abs c ≤ Complex.abs z✝
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z✝).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z✝ = w
zw : Complex.abs z✝ ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z✝).log.log + 0.548 ≤ (Complex.abs w).log.log
z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 0.8095 / (4 * log 4) ≤ 0.8095 / (4 * 1.386) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c4 : 4 ≤ Complex.abs c
cz✝ : Complex.abs c ≤ Complex.abs z✝
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z✝).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z✝ = w
zw : Complex.abs z✝ ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z✝).log.log + 0.548 ≤ (Complex.abs w).log.log
z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 0.8095 / (4 * log 4) ≤ 0.8095 / (4 * 1.386)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | norm_num | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c4 : 4 ≤ Complex.abs c
cz✝ : Complex.abs c ≤ Complex.abs z✝
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z✝).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z✝ = w
zw : Complex.abs z✝ ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z✝).log.log + 0.548 ≤ (Complex.abs w).log.log
z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 0.8095 / (4 * 1.386) ≤ 0.15 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c4 : 4 ≤ Complex.abs c
cz✝ : Complex.abs c ≤ Complex.abs z✝
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z✝).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z✝ = w
zw : Complex.abs z✝ ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z✝).log.log + 0.548 ≤ (Complex.abs w).log.log
z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 0.8095 / (4 * 1.386) ≤ 0.15
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | linarith | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z).log.log + 0.548 ≤ (Complex.abs w).log.log
ie : ∀ (z : ℂ), 4 ≤ Complex.abs z → Complex.abs c ≤ Complex.abs z → iter_error d c z ≤ 0.15
iec : iter_error d c c ≤ 0.15
⊢ 4 ≤ Complex.abs w | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z).log.log + 0.548 ≤ (Complex.abs w).log.log
ie : ∀ (z : ℂ), 4 ≤ Complex.abs z → Complex.abs c ≤ Complex.abs z → iter_error d c z ≤ 0.15
iec : iter_error d c c ≤ 0.15
⊢ 4 ≤ Complex.abs w
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | eqn_near | [71, 1] | [79, 61] | have m : ∀ᶠ y : ℂ × ℂ in 𝓝 (c, x), (y.1, (f y.1)^[n] (r y.1 y.2)) ∈ s.near := by
refine ContinuousAt.eventually_mem ?_ (s.isOpen_near.mem_nhds mem)
exact continuousAt_fst.prod (s.continuousAt_iter continuousAt_fst holo.continuousAt) | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
⊢ ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
m : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), (y.1, (f y.1)^[n] (r y.1 y.2)) ∈ s.near
⊢ ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
⊢ ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | eqn_near | [71, 1] | [79, 61] | apply holo.eventually.mp | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
m : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), (y.1, (f y.1)^[n] (r y.1 y.2)) ∈ s.near
⊢ ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
m : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), (y.1, (f y.1)^[n] (r y.1 y.2)) ∈ s.near
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, x), HolomorphicAt (I.prod I) I (uncurry r) x → Eqn s n r x | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
m : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), (y.1, (f y.1)^[n] (r y.1 y.2)) ∈ s.near
⊢ ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | eqn_near | [71, 1] | [79, 61] | apply loc.mp | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
m : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), (y.1, (f y.1)^[n] (r y.1 y.2)) ∈ s.near
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, x), HolomorphicAt (I.prod I) I (uncurry r) x → Eqn s n r x | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
m : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), (y.1, (f y.1)^[n] (r y.1 y.2)) ∈ s.near
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, x),
s.bottcherNear x.1 ((f x.1)^[n] (r x.1 x.2)) = x.2 ^ d ^ n → HolomorphicAt (I.prod I) I (uncurry r) x → Eqn s n r x | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
m : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), (y.1, (f y.1)^[n] (r y.1 y.2)) ∈ s.near
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, x), HolomorphicAt (I.prod I) I (uncurry r) x → Eqn s n r x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | eqn_near | [71, 1] | [79, 61] | apply m.mp | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
m : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), (y.1, (f y.1)^[n] (r y.1 y.2)) ∈ s.near
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, x),
s.bottcherNear x.1 ((f x.1)^[n] (r x.1 x.2)) = x.2 ^ d ^ n → HolomorphicAt (I.prod I) I (uncurry r) x → Eqn s n r x | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
m : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), (y.1, (f y.1)^[n] (r y.1 y.2)) ∈ s.near
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, x),
(x.1, (f x.1)^[n] (r x.1 x.2)) ∈ s.near →
s.bottcherNear x.1 ((f x.1)^[n] (r x.1 x.2)) = x.2 ^ d ^ n →
HolomorphicAt (I.prod I) I (uncurry r) x → Eqn s n r x | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
m : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), (y.1, (f y.1)^[n] (r y.1 y.2)) ∈ s.near
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, x),
s.bottcherNear x.1 ((f x.1)^[n] (r x.1 x.2)) = x.2 ^ d ^ n → HolomorphicAt (I.prod I) I (uncurry r) x → Eqn s n r x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | eqn_near | [71, 1] | [79, 61] | apply eventually_of_forall | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
m : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), (y.1, (f y.1)^[n] (r y.1 y.2)) ∈ s.near
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, x),
(x.1, (f x.1)^[n] (r x.1 x.2)) ∈ s.near →
s.bottcherNear x.1 ((f x.1)^[n] (r x.1 x.2)) = x.2 ^ d ^ n →
HolomorphicAt (I.prod I) I (uncurry r) x → Eqn s n r x | case hp
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
m : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), (y.1, (f y.1)^[n] (r y.1 y.2)) ∈ s.near
⊢ ∀ (x : ℂ × ℂ),
(x.1, (f x.1)^[n] (r x.1 x.2)) ∈ s.near →
s.bottcherNear x.1 ((f x.1)^[n] (r x.1 x.2)) = x.2 ^ d ^ n →
HolomorphicAt (I.prod I) I (uncurry r) x → Eqn s n r x | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
m : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), (y.1, (f y.1)^[n] (r y.1 y.2)) ∈ s.near
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, x),
(x.1, (f x.1)^[n] (r x.1 x.2)) ∈ s.near →
s.bottcherNear x.1 ((f x.1)^[n] (r x.1 x.2)) = x.2 ^ d ^ n →
HolomorphicAt (I.prod I) I (uncurry r) x → Eqn s n r x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | eqn_near | [71, 1] | [79, 61] | intro _ m l h | case hp
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
m : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), (y.1, (f y.1)^[n] (r y.1 y.2)) ∈ s.near
⊢ ∀ (x : ℂ × ℂ),
(x.1, (f x.1)^[n] (r x.1 x.2)) ∈ s.near →
s.bottcherNear x.1 ((f x.1)^[n] (r x.1 x.2)) = x.2 ^ d ^ n →
HolomorphicAt (I.prod I) I (uncurry r) x → Eqn s n r x | case hp
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
m✝ : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), (y.1, (f y.1)^[n] (r y.1 y.2)) ∈ s.near
x✝ : ℂ × ℂ
m : (x✝.1, (f x✝.1)^[n] (r x✝.1 x✝.2)) ∈ s.near
l : s.bottcherNear x✝.1 ((f x✝.1)^[n] (r x✝.1 x✝.2)) = x✝.2 ^ d ^ n
h : HolomorphicAt (I.prod I) I (uncurry r) x✝
⊢ Eqn s n r x✝ | Please generate a tactic in lean4 to solve the state.
STATE:
case hp
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
m : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), (y.1, (f y.1)^[n] (r y.1 y.2)) ∈ s.near
⊢ ∀ (x : ℂ × ℂ),
(x.1, (f x.1)^[n] (r x.1 x.2)) ∈ s.near →
s.bottcherNear x.1 ((f x.1)^[n] (r x.1 x.2)) = x.2 ^ d ^ n →
HolomorphicAt (I.prod I) I (uncurry r) x → Eqn s n r x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | eqn_near | [71, 1] | [79, 61] | exact ⟨h, m, l⟩ | case hp
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
m✝ : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), (y.1, (f y.1)^[n] (r y.1 y.2)) ∈ s.near
x✝ : ℂ × ℂ
m : (x✝.1, (f x✝.1)^[n] (r x✝.1 x✝.2)) ∈ s.near
l : s.bottcherNear x✝.1 ((f x✝.1)^[n] (r x✝.1 x✝.2)) = x✝.2 ^ d ^ n
h : HolomorphicAt (I.prod I) I (uncurry r) x✝
⊢ Eqn s n r x✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hp
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
m✝ : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), (y.1, (f y.1)^[n] (r y.1 y.2)) ∈ s.near
x✝ : ℂ × ℂ
m : (x✝.1, (f x✝.1)^[n] (r x✝.1 x✝.2)) ∈ s.near
l : s.bottcherNear x✝.1 ((f x✝.1)^[n] (r x✝.1 x✝.2)) = x✝.2 ^ d ^ n
h : HolomorphicAt (I.prod I) I (uncurry r) x✝
⊢ Eqn s n r x✝
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | eqn_near | [71, 1] | [79, 61] | refine ContinuousAt.eventually_mem ?_ (s.isOpen_near.mem_nhds mem) | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
⊢ ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), (y.1, (f y.1)^[n] (r y.1 y.2)) ∈ s.near | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
⊢ ContinuousAt (fun y => (y.1, (f y.1)^[n] (r y.1 y.2))) (c, x) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
⊢ ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), (y.1, (f y.1)^[n] (r y.1 y.2)) ∈ s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | eqn_near | [71, 1] | [79, 61] | exact continuousAt_fst.prod (s.continuousAt_iter continuousAt_fst holo.continuousAt) | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
⊢ ContinuousAt (fun y => (y.1, (f y.1)^[n] (r y.1 y.2))) (c, x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
⊢ ContinuousAt (fun y => (y.1, (f y.1)^[n] (r y.1 y.2))) (c, x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Eqn.congr | [82, 1] | [88, 41] | have s := loc.self_of_nhds | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
r0 r1 : ℂ → ℂ → S
e : Eqn s n r0 x
loc : (𝓝 x).EventuallyEq (uncurry r0) (uncurry r1)
⊢ Eqn s n r1 x | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s✝ : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
r0 r1 : ℂ → ℂ → S
e : Eqn s✝ n r0 x
loc : (𝓝 x).EventuallyEq (uncurry r0) (uncurry r1)
s : uncurry r0 x = uncurry r1 x
⊢ Eqn s✝ n r1 x | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
r0 r1 : ℂ → ℂ → S
e : Eqn s n r0 x
loc : (𝓝 x).EventuallyEq (uncurry r0) (uncurry r1)
⊢ Eqn s n r1 x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Eqn.congr | [82, 1] | [88, 41] | simp only [uncurry] at s | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s✝ : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
r0 r1 : ℂ → ℂ → S
e : Eqn s✝ n r0 x
loc : (𝓝 x).EventuallyEq (uncurry r0) (uncurry r1)
s : uncurry r0 x = uncurry r1 x
⊢ Eqn s✝ n r1 x | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s✝ : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
r0 r1 : ℂ → ℂ → S
e : Eqn s✝ n r0 x
loc : (𝓝 x).EventuallyEq (uncurry r0) (uncurry r1)
s : r0 x.1 x.2 = r1 x.1 x.2
⊢ Eqn s✝ n r1 x | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s✝ : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
r0 r1 : ℂ → ℂ → S
e : Eqn s✝ n r0 x
loc : (𝓝 x).EventuallyEq (uncurry r0) (uncurry r1)
s : uncurry r0 x = uncurry r1 x
⊢ Eqn s✝ n r1 x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Eqn.congr | [82, 1] | [88, 41] | exact
{ holo := e.holo.congr loc
near := by simp only [← s, e.near]
eqn := by simp only [← s, e.eqn] } | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s✝ : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
r0 r1 : ℂ → ℂ → S
e : Eqn s✝ n r0 x
loc : (𝓝 x).EventuallyEq (uncurry r0) (uncurry r1)
s : r0 x.1 x.2 = r1 x.1 x.2
⊢ Eqn s✝ n r1 x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s✝ : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
r0 r1 : ℂ → ℂ → S
e : Eqn s✝ n r0 x
loc : (𝓝 x).EventuallyEq (uncurry r0) (uncurry r1)
s : r0 x.1 x.2 = r1 x.1 x.2
⊢ Eqn s✝ n r1 x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Eqn.congr | [82, 1] | [88, 41] | simp only [← s, e.near] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s✝ : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
r0 r1 : ℂ → ℂ → S
e : Eqn s✝ n r0 x
loc : (𝓝 x).EventuallyEq (uncurry r0) (uncurry r1)
s : r0 x.1 x.2 = r1 x.1 x.2
⊢ (x.1, (f x.1)^[n] (r1 x.1 x.2)) ∈ s✝.near | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s✝ : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
r0 r1 : ℂ → ℂ → S
e : Eqn s✝ n r0 x
loc : (𝓝 x).EventuallyEq (uncurry r0) (uncurry r1)
s : r0 x.1 x.2 = r1 x.1 x.2
⊢ (x.1, (f x.1)^[n] (r1 x.1 x.2)) ∈ s✝.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Eqn.congr | [82, 1] | [88, 41] | simp only [← s, e.eqn] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s✝ : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
r0 r1 : ℂ → ℂ → S
e : Eqn s✝ n r0 x
loc : (𝓝 x).EventuallyEq (uncurry r0) (uncurry r1)
s : r0 x.1 x.2 = r1 x.1 x.2
⊢ s✝.bottcherNear x.1 ((f x.1)^[n] (r1 x.1 x.2)) = x.2 ^ d ^ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s✝ : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
r0 r1 : ℂ → ℂ → S
e : Eqn s✝ n r0 x
loc : (𝓝 x).EventuallyEq (uncurry r0) (uncurry r1)
s : r0 x.1 x.2 = r1 x.1 x.2
⊢ s✝.bottcherNear x.1 ((f x.1)^[n] (r1 x.1 x.2)) = x.2 ^ d ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Eqn.mono | [91, 1] | [97, 80] | refine Nat.le_induction e.eqn ?_ m nm | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
e : Eqn s n r x
m : ℕ
nm : n ≤ m
⊢ s.bottcherNear x.1 ((f x.1)^[m] (r x.1 x.2)) = x.2 ^ d ^ m | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
e : Eqn s n r x
m : ℕ
nm : n ≤ m
⊢ ∀ (n_1 : ℕ),
n ≤ n_1 →
s.bottcherNear x.1 ((f x.1)^[n_1] (r x.1 x.2)) = x.2 ^ d ^ n_1 →
s.bottcherNear x.1 ((f x.1)^[n_1 + 1] (r x.1 x.2)) = x.2 ^ d ^ (n_1 + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
e : Eqn s n r x
m : ℕ
nm : n ≤ m
⊢ s.bottcherNear x.1 ((f x.1)^[m] (r x.1 x.2)) = x.2 ^ d ^ m
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Eqn.mono | [91, 1] | [97, 80] | intro k nk h | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
e : Eqn s n r x
m : ℕ
nm : n ≤ m
⊢ ∀ (n_1 : ℕ),
n ≤ n_1 →
s.bottcherNear x.1 ((f x.1)^[n_1] (r x.1 x.2)) = x.2 ^ d ^ n_1 →
s.bottcherNear x.1 ((f x.1)^[n_1 + 1] (r x.1 x.2)) = x.2 ^ d ^ (n_1 + 1) | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
e : Eqn s n r x
m : ℕ
nm : n ≤ m
k : ℕ
nk : n ≤ k
h : s.bottcherNear x.1 ((f x.1)^[k] (r x.1 x.2)) = x.2 ^ d ^ k
⊢ s.bottcherNear x.1 ((f x.1)^[k + 1] (r x.1 x.2)) = x.2 ^ d ^ (k + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
e : Eqn s n r x
m : ℕ
nm : n ≤ m
⊢ ∀ (n_1 : ℕ),
n ≤ n_1 →
s.bottcherNear x.1 ((f x.1)^[n_1] (r x.1 x.2)) = x.2 ^ d ^ n_1 →
s.bottcherNear x.1 ((f x.1)^[n_1 + 1] (r x.1 x.2)) = x.2 ^ d ^ (n_1 + 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Eqn.mono | [91, 1] | [97, 80] | simp only [h, Function.iterate_succ_apply',
s.bottcherNear_eqn (s.iter_stays_near' e.near nk), pow_succ, pow_mul] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
e : Eqn s n r x
m : ℕ
nm : n ≤ m
k : ℕ
nk : n ≤ k
h : s.bottcherNear x.1 ((f x.1)^[k] (r x.1 x.2)) = x.2 ^ d ^ k
⊢ s.bottcherNear x.1 ((f x.1)^[k + 1] (r x.1 x.2)) = x.2 ^ d ^ (k + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
e : Eqn s n r x
m : ℕ
nm : n ≤ m
k : ℕ
nk : n ≤ k
h : s.bottcherNear x.1 ((f x.1)^[k] (r x.1 x.2)) = x.2 ^ d ^ k
⊢ s.bottcherNear x.1 ((f x.1)^[k + 1] (r x.1 x.2)) = x.2 ^ d ^ (k + 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | mem_domain_self | [112, 1] | [115, 37] | simp only [mem_prod_eq, mem_singleton_iff, eq_self_iff_true, mem_closedBall, Complex.dist_eq,
sub_zero, true_and_iff, le_refl] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
c x : ℂ
⊢ (c, x) ∈ {c} ×ˢ closedBall 0 (Complex.abs x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
c x : ℂ
⊢ (c, x) ∈ {c} ×ˢ closedBall 0 (Complex.abs x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | set u := Complex.abs '' (closedBall 0 (p + 1) \ t) | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | by_cases ne : u = ∅ | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t | case pos
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u = ∅
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t
case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : ¬u = ∅
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | replace ne := nonempty_iff_ne_empty.mpr ne | case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : ¬u = ∅
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t | case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : ¬u = ∅
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | have uc : IsClosed u :=
(((isCompact_closedBall _ _).diff ot).image Complex.continuous_abs).isClosed | case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t | case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | have up : ∀ x : ℝ, x ∈ u → p < x := by
intro x m; rcases m with ⟨z, ⟨_, mt⟩, e⟩; rw [← e]; contrapose mt
simp only [not_not, not_lt] at mt ⊢
apply sub; simp only [mem_closedBall, Complex.dist_eq, sub_zero, mt] | case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t | case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | have ub : BddBelow u := ⟨p, fun _ m ↦ (up _ m).le⟩ | case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t | case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | have iu : sInf u ∈ u := IsClosed.csInf_mem uc ne ub | case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t | case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | rcases exists_between (up _ iu) with ⟨q, pq, qi⟩ | case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t | case neg.intro.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | use min q (p + 1), lt_min pq (by linarith) | case neg.intro.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t | case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
⊢ closedBall 0 (min q (p + 1)) ⊆ t | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | intro z m | case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
⊢ closedBall 0 (min q (p + 1)) ⊆ t | case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
m : z ∈ closedBall 0 (min q (p + 1))
⊢ z ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
⊢ closedBall 0 (min q (p + 1)) ⊆ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | simp only [mem_closedBall, Complex.dist_eq, sub_zero, le_min_iff] at m | case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
m : z ∈ closedBall 0 (min q (p + 1))
⊢ z ∈ t | case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
m : Complex.abs z ≤ q ∧ Complex.abs z ≤ p + 1
⊢ z ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
m : z ∈ closedBall 0 (min q (p + 1))
⊢ z ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | rcases m with ⟨zq, zp⟩ | case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
m : Complex.abs z ≤ q ∧ Complex.abs z ≤ p + 1
⊢ z ∈ t | case right.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
zq : Complex.abs z ≤ q
zp : Complex.abs z ≤ p + 1
⊢ z ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
m : Complex.abs z ≤ q ∧ Complex.abs z ≤ p + 1
⊢ z ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | have zi := lt_of_le_of_lt zq qi | case right.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
zq : Complex.abs z ≤ q
zp : Complex.abs z ≤ p + 1
⊢ z ∈ t | case right.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
zq : Complex.abs z ≤ q
zp : Complex.abs z ≤ p + 1
zi : Complex.abs z < sInf u
⊢ z ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
zq : Complex.abs z ≤ q
zp : Complex.abs z ≤ p + 1
⊢ z ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | contrapose zi | case right.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
zq : Complex.abs z ≤ q
zp : Complex.abs z ≤ p + 1
zi : Complex.abs z < sInf u
⊢ z ∈ t | case right.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
zq : Complex.abs z ≤ q
zp : Complex.abs z ≤ p + 1
zi : ¬z ∈ t
⊢ ¬Complex.abs z < sInf u | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
zq : Complex.abs z ≤ q
zp : Complex.abs z ≤ p + 1
zi : Complex.abs z < sInf u
⊢ z ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | simp only [not_lt] | case right.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
zq : Complex.abs z ≤ q
zp : Complex.abs z ≤ p + 1
zi : ¬z ∈ t
⊢ ¬Complex.abs z < sInf u | case right.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
zq : Complex.abs z ≤ q
zp : Complex.abs z ≤ p + 1
zi : ¬z ∈ t
⊢ sInf u ≤ Complex.abs z | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
zq : Complex.abs z ≤ q
zp : Complex.abs z ≤ p + 1
zi : ¬z ∈ t
⊢ ¬Complex.abs z < sInf u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | refine csInf_le ub (mem_image_of_mem _ ?_) | case right.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
zq : Complex.abs z ≤ q
zp : Complex.abs z ≤ p + 1
zi : ¬z ∈ t
⊢ sInf u ≤ Complex.abs z | case right.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
zq : Complex.abs z ≤ q
zp : Complex.abs z ≤ p + 1
zi : ¬z ∈ t
⊢ z ∈ closedBall 0 (p + 1) \ t | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
zq : Complex.abs z ≤ q
zp : Complex.abs z ≤ p + 1
zi : ¬z ∈ t
⊢ sInf u ≤ Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | simp only [mem_diff, mem_closedBall, Complex.dist_eq, sub_zero] | case right.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
zq : Complex.abs z ≤ q
zp : Complex.abs z ≤ p + 1
zi : ¬z ∈ t
⊢ z ∈ closedBall 0 (p + 1) \ t | case right.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
zq : Complex.abs z ≤ q
zp : Complex.abs z ≤ p + 1
zi : ¬z ∈ t
⊢ Complex.abs z ≤ p + 1 ∧ z ∉ t | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
zq : Complex.abs z ≤ q
zp : Complex.abs z ≤ p + 1
zi : ¬z ∈ t
⊢ z ∈ closedBall 0 (p + 1) \ t
TACTIC:
|
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