url stringclasses 147 values | commit stringclasses 147 values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | set s := superF d | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ Tendsto (fun n => Complex.abs ((f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) atTop (𝓝 (⋯.potential c ↑z)) | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
⊢ Tendsto (fun n => Complex.abs ((f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) atTop (𝓝 (s.potential c ↑z)) | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ Tendsto (fun n => Complex.abs ((f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) atTop (𝓝 (⋯.potential c ↑z))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | suffices h : Tendsto (fun n ↦ (abs ((f' d c)^[n] z) *
s.potential c ↑((f' d c)^[n] z)) ^ (-((d ^ n : ℕ) : ℝ)⁻¹))
atTop (𝓝 1) by
replace h := h.mul_const (s.potential c z)
simp only [div_mul_cancel₀ _ potential_pos.ne', one_mul, ← f_f'_iter, s.potential_eqn_iter,
Real.mul_rpow (Complex.abs.nonneg _) (pow_nonneg s.potential_nonneg _),
Real.pow_rpow_inv_natCast s.potential_nonneg (pow_ne_zero _ (d_ne_zero d)),
Real.rpow_neg (pow_nonneg s.potential_nonneg _), ← div_eq_mul_inv] at h
exact h | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
⊢ Tendsto (fun n => Complex.abs ((f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) atTop (𝓝 (s.potential c ↑z)) | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z) * s.potential c ↑((f' d c)^[n] z)) ^ (-(↑(d ^ n))⁻¹)) atTop (𝓝 1) | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
⊢ Tendsto (fun n => Complex.abs ((f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) atTop (𝓝 (s.potential c ↑z))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | simp only [← s.abs_bottcher, ← Complex.abs.map_mul, mul_comm _ (s.bottcher _ _)] | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z) * s.potential c ↑((f' d c)^[n] z)) ^ (-(↑(d ^ n))⁻¹)) atTop (𝓝 1) | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
⊢ Tendsto (fun n => Complex.abs (s.bottcher c ↑((f' d c)^[n] z) * (f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) atTop (𝓝 1) | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z) * s.potential c ↑((f' d c)^[n] z)) ^ (-(↑(d ^ n))⁻¹)) atTop (𝓝 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | rw [Metric.tendsto_atTop] | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
⊢ Tendsto (fun n => Complex.abs (s.bottcher c ↑((f' d c)^[n] z) * (f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) atTop (𝓝 1) | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
⊢ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (Complex.abs (s.bottcher c ↑((f' d c)^[n] z) * (f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) 1 < ε | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
⊢ Tendsto (fun n => Complex.abs (s.bottcher c ↑((f' d c)^[n] z) * (f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) atTop (𝓝 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | intro r rp | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
⊢ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (Complex.abs (s.bottcher c ↑((f' d c)^[n] z) * (f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) 1 < ε | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
r : ℝ
rp : r > 0
⊢ ∃ N, ∀ n ≥ N, dist (Complex.abs (s.bottcher c ↑((f' d c)^[n] z) * (f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) 1 < r | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
⊢ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (Complex.abs (s.bottcher c ↑((f' d c)^[n] z) * (f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) 1 < ε
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | rcases Metric.tendsto_atTop.mp ((bottcher_large_approx d c).comp (tendsto_iter_atInf d z3 cz))
(min (1 / 2) (r / 4)) (by bound) with ⟨n, h⟩ | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
r : ℝ
rp : r > 0
⊢ ∃ N, ∀ n ≥ N, dist (Complex.abs (s.bottcher c ↑((f' d c)^[n] z) * (f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) 1 < r | case intro
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
r : ℝ
rp : r > 0
n : ℕ
h : ∀ n_1 ≥ n, dist (((fun z => ⋯.bottcher c ↑z * z) ∘ fun n => (f' d c)^[n] z) n_1) 1 < min (1 / 2) (r / 4)
⊢ ∃ N, ∀ n ≥ N, dist (Complex.abs (s.bottcher c ↑((f' d c)^[n] z) * (f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) 1 < r | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
r : ℝ
rp : r > 0
⊢ ∃ N, ∀ n ≥ N, dist (Complex.abs (s.bottcher c ↑((f' d c)^[n] z) * (f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) 1 < r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | use n | case intro
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
r : ℝ
rp : r > 0
n : ℕ
h : ∀ n_1 ≥ n, dist (((fun z => ⋯.bottcher c ↑z * z) ∘ fun n => (f' d c)^[n] z) n_1) 1 < min (1 / 2) (r / 4)
⊢ ∃ N, ∀ n ≥ N, dist (Complex.abs (s.bottcher c ↑((f' d c)^[n] z) * (f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) 1 < r | case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
r : ℝ
rp : r > 0
n : ℕ
h : ∀ n_1 ≥ n, dist (((fun z => ⋯.bottcher c ↑z * z) ∘ fun n => (f' d c)^[n] z) n_1) 1 < min (1 / 2) (r / 4)
⊢ ∀ n_1 ≥ n, dist (Complex.abs (s.bottcher c ↑((f' d c)^[n_1] z) * (f' d c)^[n_1] z) ^ (-(↑(d ^ n_1))⁻¹)) 1 < r | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
r : ℝ
rp : r > 0
n : ℕ
h : ∀ n_1 ≥ n, dist (((fun z => ⋯.bottcher c ↑z * z) ∘ fun n => (f' d c)^[n] z) n_1) 1 < min (1 / 2) (r / 4)
⊢ ∃ N, ∀ n ≥ N, dist (Complex.abs (s.bottcher c ↑((f' d c)^[n] z) * (f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) 1 < r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | intro k nk | case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
r : ℝ
rp : r > 0
n : ℕ
h : ∀ n_1 ≥ n, dist (((fun z => ⋯.bottcher c ↑z * z) ∘ fun n => (f' d c)^[n] z) n_1) 1 < min (1 / 2) (r / 4)
⊢ ∀ n_1 ≥ n, dist (Complex.abs (s.bottcher c ↑((f' d c)^[n_1] z) * (f' d c)^[n_1] z) ^ (-(↑(d ^ n_1))⁻¹)) 1 < r | case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
r : ℝ
rp : r > 0
n : ℕ
h : ∀ n_1 ≥ n, dist (((fun z => ⋯.bottcher c ↑z * z) ∘ fun n => (f' d c)^[n] z) n_1) 1 < min (1 / 2) (r / 4)
k : ℕ
nk : k ≥ n
⊢ dist (Complex.abs (s.bottcher c ↑((f' d c)^[k] z) * (f' d c)^[k] z) ^ (-(↑(d ^ k))⁻¹)) 1 < r | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
r : ℝ
rp : r > 0
n : ℕ
h : ∀ n_1 ≥ n, dist (((fun z => ⋯.bottcher c ↑z * z) ∘ fun n => (f' d c)^[n] z) n_1) 1 < min (1 / 2) (r / 4)
⊢ ∀ n_1 ≥ n, dist (Complex.abs (s.bottcher c ↑((f' d c)^[n_1] z) * (f' d c)^[n_1] z) ^ (-(↑(d ^ n_1))⁻¹)) 1 < r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | specialize h k nk | case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
r : ℝ
rp : r > 0
n : ℕ
h : ∀ n_1 ≥ n, dist (((fun z => ⋯.bottcher c ↑z * z) ∘ fun n => (f' d c)^[n] z) n_1) 1 < min (1 / 2) (r / 4)
k : ℕ
nk : k ≥ n
⊢ dist (Complex.abs (s.bottcher c ↑((f' d c)^[k] z) * (f' d c)^[k] z) ^ (-(↑(d ^ k))⁻¹)) 1 < r | case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
r : ℝ
rp : r > 0
n k : ℕ
nk : k ≥ n
h : dist (((fun z => ⋯.bottcher c ↑z * z) ∘ fun n => (f' d c)^[n] z) k) 1 < min (1 / 2) (r / 4)
⊢ dist (Complex.abs (s.bottcher c ↑((f' d c)^[k] z) * (f' d c)^[k] z) ^ (-(↑(d ^ k))⁻¹)) 1 < r | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
r : ℝ
rp : r > 0
n : ℕ
h : ∀ n_1 ≥ n, dist (((fun z => ⋯.bottcher c ↑z * z) ∘ fun n => (f' d c)^[n] z) n_1) 1 < min (1 / 2) (r / 4)
k : ℕ
nk : k ≥ n
⊢ dist (Complex.abs (s.bottcher c ↑((f' d c)^[k] z) * (f' d c)^[k] z) ^ (-(↑(d ^ k))⁻¹)) 1 < r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | generalize hw : (f' d c)^[k] z = w | case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
r : ℝ
rp : r > 0
n k : ℕ
nk : k ≥ n
h : dist (((fun z => ⋯.bottcher c ↑z * z) ∘ fun n => (f' d c)^[n] z) k) 1 < min (1 / 2) (r / 4)
⊢ dist (Complex.abs (s.bottcher c ↑((f' d c)^[k] z) * (f' d c)^[k] z) ^ (-(↑(d ^ k))⁻¹)) 1 < r | case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
r : ℝ
rp : r > 0
n k : ℕ
nk : k ≥ n
h : dist (((fun z => ⋯.bottcher c ↑z * z) ∘ fun n => (f' d c)^[n] z) k) 1 < min (1 / 2) (r / 4)
w : ℂ
hw : (f' d c)^[k] z = w
⊢ dist (Complex.abs (s.bottcher c ↑w * w) ^ (-(↑(d ^ k))⁻¹)) 1 < r | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
r : ℝ
rp : r > 0
n k : ℕ
nk : k ≥ n
h : dist (((fun z => ⋯.bottcher c ↑z * z) ∘ fun n => (f' d c)^[n] z) k) 1 < min (1 / 2) (r / 4)
⊢ dist (Complex.abs (s.bottcher c ↑((f' d c)^[k] z) * (f' d c)^[k] z) ^ (-(↑(d ^ k))⁻¹)) 1 < r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | generalize hp : s.bottcher c w * w = p | case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
r : ℝ
rp : r > 0
n k : ℕ
nk : k ≥ n
h : dist (((fun z => ⋯.bottcher c ↑z * z) ∘ fun n => (f' d c)^[n] z) k) 1 < min (1 / 2) (r / 4)
w : ℂ
hw : (f' d c)^[k] z = w
⊢ dist (Complex.abs (s.bottcher c ↑w * w) ^ (-(↑(d ^ k))⁻¹)) 1 < r | case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
r : ℝ
rp : r > 0
n k : ℕ
nk : k ≥ n
h : dist (((fun z => ⋯.bottcher c ↑z * z) ∘ fun n => (f' d c)^[n] z) k) 1 < min (1 / 2) (r / 4)
w : ℂ
hw : (f' d c)^[k] z = w
p : ℂ
hp : s.bottcher c ↑w * w = p
⊢ dist (Complex.abs p ^ (-(↑(d ^ k))⁻¹)) 1 < r | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
r : ℝ
rp : r > 0
n k : ℕ
nk : k ≥ n
h : dist (((fun z => ⋯.bottcher c ↑z * z) ∘ fun n => (f' d c)^[n] z) k) 1 < min (1 / 2) (r / 4)
w : ℂ
hw : (f' d c)^[k] z = w
⊢ dist (Complex.abs (s.bottcher c ↑w * w) ^ (-(↑(d ^ k))⁻¹)) 1 < r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | simp only [hw, hp, Function.comp, Complex.dist_eq, Real.dist_eq] at h ⊢ | case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
r : ℝ
rp : r > 0
n k : ℕ
nk : k ≥ n
h : dist (((fun z => ⋯.bottcher c ↑z * z) ∘ fun n => (f' d c)^[n] z) k) 1 < min (1 / 2) (r / 4)
w : ℂ
hw : (f' d c)^[k] z = w
p : ℂ
hp : s.bottcher c ↑w * w = p
⊢ dist (Complex.abs p ^ (-(↑(d ^ k))⁻¹)) 1 < r | case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
r : ℝ
rp : r > 0
n k : ℕ
nk : k ≥ n
w : ℂ
hw : (f' d c)^[k] z = w
p : ℂ
hp : s.bottcher c ↑w * w = p
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
⊢ |Complex.abs p ^ (-(↑(d ^ k))⁻¹) - 1| < r | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
r : ℝ
rp : r > 0
n k : ℕ
nk : k ≥ n
h : dist (((fun z => ⋯.bottcher c ↑z * z) ∘ fun n => (f' d c)^[n] z) k) 1 < min (1 / 2) (r / 4)
w : ℂ
hw : (f' d c)^[k] z = w
p : ℂ
hp : s.bottcher c ↑w * w = p
⊢ dist (Complex.abs p ^ (-(↑(d ^ k))⁻¹)) 1 < r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | clear hp w hw nk n s cz z3 | case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
r : ℝ
rp : r > 0
n k : ℕ
nk : k ≥ n
w : ℂ
hw : (f' d c)^[k] z = w
p : ℂ
hp : s.bottcher c ↑w * w = p
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
⊢ |Complex.abs p ^ (-(↑(d ^ k))⁻¹) - 1| < r | case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
⊢ |Complex.abs p ^ (-(↑(d ^ k))⁻¹) - 1| < r | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
r : ℝ
rp : r > 0
n k : ℕ
nk : k ≥ n
w : ℂ
hw : (f' d c)^[k] z = w
p : ℂ
hp : s.bottcher c ↑w * w = p
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
⊢ |Complex.abs p ^ (-(↑(d ^ k))⁻¹) - 1| < r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | generalize ha : abs p = a | case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
⊢ |Complex.abs p ^ (-(↑(d ^ k))⁻¹) - 1| < r | case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
⊢ |a ^ (-(↑(d ^ k))⁻¹) - 1| < r | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
⊢ |Complex.abs p ^ (-(↑(d ^ k))⁻¹) - 1| < r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | generalize hb : ((d ^ k : ℕ) : ℝ)⁻¹ = b | case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
⊢ |a ^ (-(↑(d ^ k))⁻¹) - 1| < r | case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
⊢ |a ^ (-b) - 1| < r | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
⊢ |a ^ (-(↑(d ^ k))⁻¹) - 1| < r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | have a1 : |a - 1| < min (1 / 2) (r / 4) := by
rw [← ha]; refine lt_of_le_of_lt ?_ h
rw [← Complex.abs.map_one]; apply Complex.abs.abs_abv_sub_le_abv_sub | case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
⊢ |a ^ (-b) - 1| < r | case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
⊢ |a ^ (-b) - 1| < r | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
⊢ |a ^ (-b) - 1| < r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | have am : a ∈ ball (1 : ℝ) (1 / 2) := by
simp only [mem_ball, Real.dist_eq]; exact (lt_min_iff.mp a1).1 | case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
⊢ |a ^ (-b) - 1| < r | case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
⊢ |a ^ (-b) - 1| < r | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
⊢ |a ^ (-b) - 1| < r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | have b0 : 0 ≤ b := by rw [← hb]; bound | case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
⊢ |a ^ (-b) - 1| < r | case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
⊢ |a ^ (-b) - 1| < r | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
⊢ |a ^ (-b) - 1| < r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | have b1 : b ≤ 1 := by rw [← hb]; bound | case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
⊢ |a ^ (-b) - 1| < r | case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
⊢ |a ^ (-b) - 1| < r | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
⊢ |a ^ (-b) - 1| < r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | have hd : ∀ x, x ∈ ball (1 : ℝ) (1 / 2) →
HasDerivAt (fun x ↦ x ^ (-b)) (1 * -b * x ^ (-b - 1) + 0 * x ^ (-b) * log x) x := by
intro x m; apply HasDerivAt.rpow (hasDerivAt_id _) (hasDerivAt_const _ _)
simp only [mem_ball, Real.dist_eq, id] at m ⊢; linarith [abs_lt.mp m] | case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
⊢ |a ^ (-b) - 1| < r | case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (1 * -b * x ^ (-b - 1) + 0 * x ^ (-b) * x.log) x
⊢ |a ^ (-b) - 1| < r | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
⊢ |a ^ (-b) - 1| < r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | simp only [MulZeroClass.zero_mul, add_zero, one_mul] at hd | case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (1 * -b * x ^ (-b - 1) + 0 * x ^ (-b) * x.log) x
⊢ |a ^ (-b) - 1| < r | case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
⊢ |a ^ (-b) - 1| < r | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (1 * -b * x ^ (-b - 1) + 0 * x ^ (-b) * x.log) x
⊢ |a ^ (-b) - 1| < r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | have bound : ∀ x, x ∈ ball (1 : ℝ) (1 / 2) → ‖deriv (fun x ↦ x ^ (-b)) x‖ ≤ 4 := by
intro x m
simp only [(hd x m).deriv, Real.norm_eq_abs, abs_mul, abs_neg, abs_of_nonneg b0]
simp only [mem_ball, Real.dist_eq, abs_lt, lt_sub_iff_add_lt, sub_lt_iff_lt_add] at m
norm_num at m
have x0 : 0 < x := by linarith
calc b * |x ^ (-b - 1)|
_ ≤ 1 * |x| ^ (-b - 1) := by bound
_ = (x ^ (b + 1))⁻¹ := by rw [← Real.rpow_neg x0.le, neg_add', one_mul, abs_of_pos x0]
_ ≤ ((1 / 2 : ℝ) ^ (b + 1))⁻¹ := by bound
_ = 2 ^ (b + 1) := by rw [one_div, Real.inv_rpow zero_le_two, inv_inv]
_ ≤ 2 ^ (1 + 1 : ℝ) := by bound
_ ≤ 4 := by norm_num | case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
⊢ |a ^ (-b) - 1| < r | case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
bound : ∀ x ∈ ball 1 (1 / 2), ‖deriv (fun x => x ^ (-b)) x‖ ≤ 4
⊢ |a ^ (-b) - 1| < r | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
⊢ |a ^ (-b) - 1| < r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | have le := Convex.norm_image_sub_le_of_norm_deriv_le (fun x m ↦ (hd x m).differentiableAt) bound
(convex_ball _ _) (mem_ball_self (by norm_num)) am | case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
bound : ∀ x ∈ ball 1 (1 / 2), ‖deriv (fun x => x ^ (-b)) x‖ ≤ 4
⊢ |a ^ (-b) - 1| < r | case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
bound : ∀ x ∈ ball 1 (1 / 2), ‖deriv (fun x => x ^ (-b)) x‖ ≤ 4
le : ‖a ^ (-b) - 1 ^ (-b)‖ ≤ 4 * ‖a - 1‖
⊢ |a ^ (-b) - 1| < r | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
bound : ∀ x ∈ ball 1 (1 / 2), ‖deriv (fun x => x ^ (-b)) x‖ ≤ 4
⊢ |a ^ (-b) - 1| < r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | simp only [Real.norm_eq_abs, Real.one_rpow] at le | case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
bound : ∀ x ∈ ball 1 (1 / 2), ‖deriv (fun x => x ^ (-b)) x‖ ≤ 4
le : ‖a ^ (-b) - 1 ^ (-b)‖ ≤ 4 * ‖a - 1‖
⊢ |a ^ (-b) - 1| < r | case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
bound : ∀ x ∈ ball 1 (1 / 2), ‖deriv (fun x => x ^ (-b)) x‖ ≤ 4
le : |a ^ (-b) - 1| ≤ 4 * |a - 1|
⊢ |a ^ (-b) - 1| < r | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
bound : ∀ x ∈ ball 1 (1 / 2), ‖deriv (fun x => x ^ (-b)) x‖ ≤ 4
le : ‖a ^ (-b) - 1 ^ (-b)‖ ≤ 4 * ‖a - 1‖
⊢ |a ^ (-b) - 1| < r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | calc |a ^ (-b) - 1|
_ ≤ 4 * |a - 1| := le
_ < 4 * (r / 4) := by linarith [(lt_min_iff.mp a1).2]
_ = r := by ring | case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
bound : ∀ x ∈ ball 1 (1 / 2), ‖deriv (fun x => x ^ (-b)) x‖ ≤ 4
le : |a ^ (-b) - 1| ≤ 4 * |a - 1|
⊢ |a ^ (-b) - 1| < r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
bound : ∀ x ∈ ball 1 (1 / 2), ‖deriv (fun x => x ^ (-b)) x‖ ≤ 4
le : |a ^ (-b) - 1| ≤ 4 * |a - 1|
⊢ |a ^ (-b) - 1| < r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | replace h := h.mul_const (s.potential c z) | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
h : Tendsto (fun n => (Complex.abs ((f' d c)^[n] z) * s.potential c ↑((f' d c)^[n] z)) ^ (-(↑(d ^ n))⁻¹)) atTop (𝓝 1)
⊢ Tendsto (fun n => Complex.abs ((f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) atTop (𝓝 (s.potential c ↑z)) | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
h :
Tendsto
(fun k => (Complex.abs ((f' d c)^[k] z) * s.potential c ↑((f' d c)^[k] z)) ^ (-(↑(d ^ k))⁻¹) * s.potential c ↑z)
atTop (𝓝 (1 * s.potential c ↑z))
⊢ Tendsto (fun n => Complex.abs ((f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) atTop (𝓝 (s.potential c ↑z)) | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
h : Tendsto (fun n => (Complex.abs ((f' d c)^[n] z) * s.potential c ↑((f' d c)^[n] z)) ^ (-(↑(d ^ n))⁻¹)) atTop (𝓝 1)
⊢ Tendsto (fun n => Complex.abs ((f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) atTop (𝓝 (s.potential c ↑z))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | simp only [div_mul_cancel₀ _ potential_pos.ne', one_mul, ← f_f'_iter, s.potential_eqn_iter,
Real.mul_rpow (Complex.abs.nonneg _) (pow_nonneg s.potential_nonneg _),
Real.pow_rpow_inv_natCast s.potential_nonneg (pow_ne_zero _ (d_ne_zero d)),
Real.rpow_neg (pow_nonneg s.potential_nonneg _), ← div_eq_mul_inv] at h | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
h :
Tendsto
(fun k => (Complex.abs ((f' d c)^[k] z) * s.potential c ↑((f' d c)^[k] z)) ^ (-(↑(d ^ k))⁻¹) * s.potential c ↑z)
atTop (𝓝 (1 * s.potential c ↑z))
⊢ Tendsto (fun n => Complex.abs ((f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) atTop (𝓝 (s.potential c ↑z)) | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
h : Tendsto (fun k => Complex.abs ((f' d c)^[k] z) ^ (-(↑(d ^ k))⁻¹)) atTop (𝓝 (s.potential c ↑z))
⊢ Tendsto (fun n => Complex.abs ((f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) atTop (𝓝 (s.potential c ↑z)) | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
h :
Tendsto
(fun k => (Complex.abs ((f' d c)^[k] z) * s.potential c ↑((f' d c)^[k] z)) ^ (-(↑(d ^ k))⁻¹) * s.potential c ↑z)
atTop (𝓝 (1 * s.potential c ↑z))
⊢ Tendsto (fun n => Complex.abs ((f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) atTop (𝓝 (s.potential c ↑z))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | exact h | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
h : Tendsto (fun k => Complex.abs ((f' d c)^[k] z) ^ (-(↑(d ^ k))⁻¹)) atTop (𝓝 (s.potential c ↑z))
⊢ Tendsto (fun n => Complex.abs ((f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) atTop (𝓝 (s.potential c ↑z)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
h : Tendsto (fun k => Complex.abs ((f' d c)^[k] z) ^ (-(↑(d ^ k))⁻¹)) atTop (𝓝 (s.potential c ↑z))
⊢ Tendsto (fun n => Complex.abs ((f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) atTop (𝓝 (s.potential c ↑z))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | bound | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
r : ℝ
rp : r > 0
⊢ min (1 / 2) (r / 4) > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
r : ℝ
rp : r > 0
⊢ min (1 / 2) (r / 4) > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | rw [← ha] | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
⊢ |a - 1| < min (1 / 2) (r / 4) | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
⊢ |Complex.abs p - 1| < min (1 / 2) (r / 4) | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
⊢ |a - 1| < min (1 / 2) (r / 4)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | refine lt_of_le_of_lt ?_ h | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
⊢ |Complex.abs p - 1| < min (1 / 2) (r / 4) | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
⊢ |Complex.abs p - 1| ≤ Complex.abs (p - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
⊢ |Complex.abs p - 1| < min (1 / 2) (r / 4)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | rw [← Complex.abs.map_one] | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
⊢ |Complex.abs p - 1| ≤ Complex.abs (p - 1) | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
⊢ |Complex.abs p - Complex.abs 1| ≤ Complex.abs (p - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
⊢ |Complex.abs p - 1| ≤ Complex.abs (p - 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | apply Complex.abs.abs_abv_sub_le_abv_sub | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
⊢ |Complex.abs p - Complex.abs 1| ≤ Complex.abs (p - 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
⊢ |Complex.abs p - Complex.abs 1| ≤ Complex.abs (p - 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | simp only [mem_ball, Real.dist_eq] | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
⊢ a ∈ ball 1 (1 / 2) | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
⊢ |a - 1| < 1 / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
⊢ a ∈ ball 1 (1 / 2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | exact (lt_min_iff.mp a1).1 | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
⊢ |a - 1| < 1 / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
⊢ |a - 1| < 1 / 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | rw [← hb] | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
⊢ 0 ≤ b | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
⊢ 0 ≤ (↑(d ^ k))⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
⊢ 0 ≤ b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | bound | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
⊢ 0 ≤ (↑(d ^ k))⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
⊢ 0 ≤ (↑(d ^ k))⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | rw [← hb] | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
⊢ b ≤ 1 | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
⊢ (↑(d ^ k))⁻¹ ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
⊢ b ≤ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | bound | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
⊢ (↑(d ^ k))⁻¹ ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
⊢ (↑(d ^ k))⁻¹ ≤ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | intro x m | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
⊢ ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (1 * -b * x ^ (-b - 1) + 0 * x ^ (-b) * x.log) x | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
x : ℝ
m : x ∈ ball 1 (1 / 2)
⊢ HasDerivAt (fun x => x ^ (-b)) (1 * -b * x ^ (-b - 1) + 0 * x ^ (-b) * x.log) x | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
⊢ ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (1 * -b * x ^ (-b - 1) + 0 * x ^ (-b) * x.log) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | apply HasDerivAt.rpow (hasDerivAt_id _) (hasDerivAt_const _ _) | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
x : ℝ
m : x ∈ ball 1 (1 / 2)
⊢ HasDerivAt (fun x => x ^ (-b)) (1 * -b * x ^ (-b - 1) + 0 * x ^ (-b) * x.log) x | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
x : ℝ
m : x ∈ ball 1 (1 / 2)
⊢ 0 < id x | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
x : ℝ
m : x ∈ ball 1 (1 / 2)
⊢ HasDerivAt (fun x => x ^ (-b)) (1 * -b * x ^ (-b - 1) + 0 * x ^ (-b) * x.log) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | simp only [mem_ball, Real.dist_eq, id] at m ⊢ | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
x : ℝ
m : x ∈ ball 1 (1 / 2)
⊢ 0 < id x | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
x : ℝ
m : |x - 1| < 1 / 2
⊢ 0 < x | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
x : ℝ
m : x ∈ ball 1 (1 / 2)
⊢ 0 < id x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | linarith [abs_lt.mp m] | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
x : ℝ
m : |x - 1| < 1 / 2
⊢ 0 < x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
x : ℝ
m : |x - 1| < 1 / 2
⊢ 0 < x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | intro x m | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
⊢ ∀ x ∈ ball 1 (1 / 2), ‖deriv (fun x => x ^ (-b)) x‖ ≤ 4 | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : x ∈ ball 1 (1 / 2)
⊢ ‖deriv (fun x => x ^ (-b)) x‖ ≤ 4 | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
⊢ ∀ x ∈ ball 1 (1 / 2), ‖deriv (fun x => x ^ (-b)) x‖ ≤ 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | simp only [(hd x m).deriv, Real.norm_eq_abs, abs_mul, abs_neg, abs_of_nonneg b0] | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : x ∈ ball 1 (1 / 2)
⊢ ‖deriv (fun x => x ^ (-b)) x‖ ≤ 4 | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : x ∈ ball 1 (1 / 2)
⊢ b * |x ^ (-b - 1)| ≤ 4 | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : x ∈ ball 1 (1 / 2)
⊢ ‖deriv (fun x => x ^ (-b)) x‖ ≤ 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | simp only [mem_ball, Real.dist_eq, abs_lt, lt_sub_iff_add_lt, sub_lt_iff_lt_add] at m | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : x ∈ ball 1 (1 / 2)
⊢ b * |x ^ (-b - 1)| ≤ 4 | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : -(1 / 2) + 1 < x ∧ x < 1 / 2 + 1
⊢ b * |x ^ (-b - 1)| ≤ 4 | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : x ∈ ball 1 (1 / 2)
⊢ b * |x ^ (-b - 1)| ≤ 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | norm_num at m | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : -(1 / 2) + 1 < x ∧ x < 1 / 2 + 1
⊢ b * |x ^ (-b - 1)| ≤ 4 | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : 1 / 2 < x ∧ x < 3 / 2
⊢ b * |x ^ (-b - 1)| ≤ 4 | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : -(1 / 2) + 1 < x ∧ x < 1 / 2 + 1
⊢ b * |x ^ (-b - 1)| ≤ 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | have x0 : 0 < x := by linarith | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : 1 / 2 < x ∧ x < 3 / 2
⊢ b * |x ^ (-b - 1)| ≤ 4 | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : 1 / 2 < x ∧ x < 3 / 2
x0 : 0 < x
⊢ b * |x ^ (-b - 1)| ≤ 4 | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : 1 / 2 < x ∧ x < 3 / 2
⊢ b * |x ^ (-b - 1)| ≤ 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | calc b * |x ^ (-b - 1)|
_ ≤ 1 * |x| ^ (-b - 1) := by bound
_ = (x ^ (b + 1))⁻¹ := by rw [← Real.rpow_neg x0.le, neg_add', one_mul, abs_of_pos x0]
_ ≤ ((1 / 2 : ℝ) ^ (b + 1))⁻¹ := by bound
_ = 2 ^ (b + 1) := by rw [one_div, Real.inv_rpow zero_le_two, inv_inv]
_ ≤ 2 ^ (1 + 1 : ℝ) := by bound
_ ≤ 4 := by norm_num | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : 1 / 2 < x ∧ x < 3 / 2
x0 : 0 < x
⊢ b * |x ^ (-b - 1)| ≤ 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : 1 / 2 < x ∧ x < 3 / 2
x0 : 0 < x
⊢ b * |x ^ (-b - 1)| ≤ 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | linarith | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : 1 / 2 < x ∧ x < 3 / 2
⊢ 0 < x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : 1 / 2 < x ∧ x < 3 / 2
⊢ 0 < x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | bound | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : 1 / 2 < x ∧ x < 3 / 2
x0 : 0 < x
⊢ b * |x ^ (-b - 1)| ≤ 1 * |x| ^ (-b - 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : 1 / 2 < x ∧ x < 3 / 2
x0 : 0 < x
⊢ b * |x ^ (-b - 1)| ≤ 1 * |x| ^ (-b - 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | rw [← Real.rpow_neg x0.le, neg_add', one_mul, abs_of_pos x0] | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : 1 / 2 < x ∧ x < 3 / 2
x0 : 0 < x
⊢ 1 * |x| ^ (-b - 1) = (x ^ (b + 1))⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : 1 / 2 < x ∧ x < 3 / 2
x0 : 0 < x
⊢ 1 * |x| ^ (-b - 1) = (x ^ (b + 1))⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | bound | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : 1 / 2 < x ∧ x < 3 / 2
x0 : 0 < x
⊢ (x ^ (b + 1))⁻¹ ≤ ((1 / 2) ^ (b + 1))⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : 1 / 2 < x ∧ x < 3 / 2
x0 : 0 < x
⊢ (x ^ (b + 1))⁻¹ ≤ ((1 / 2) ^ (b + 1))⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | rw [one_div, Real.inv_rpow zero_le_two, inv_inv] | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : 1 / 2 < x ∧ x < 3 / 2
x0 : 0 < x
⊢ ((1 / 2) ^ (b + 1))⁻¹ = 2 ^ (b + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : 1 / 2 < x ∧ x < 3 / 2
x0 : 0 < x
⊢ ((1 / 2) ^ (b + 1))⁻¹ = 2 ^ (b + 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | bound | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : 1 / 2 < x ∧ x < 3 / 2
x0 : 0 < x
⊢ 2 ^ (b + 1) ≤ 2 ^ (1 + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : 1 / 2 < x ∧ x < 3 / 2
x0 : 0 < x
⊢ 2 ^ (b + 1) ≤ 2 ^ (1 + 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | norm_num | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : 1 / 2 < x ∧ x < 3 / 2
x0 : 0 < x
⊢ 2 ^ (1 + 1) ≤ 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : 1 / 2 < x ∧ x < 3 / 2
x0 : 0 < x
⊢ 2 ^ (1 + 1) ≤ 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | norm_num | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
bound : ∀ x ∈ ball 1 (1 / 2), ‖deriv (fun x => x ^ (-b)) x‖ ≤ 4
⊢ 0 < 1 / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
bound : ∀ x ∈ ball 1 (1 / 2), ‖deriv (fun x => x ^ (-b)) x‖ ≤ 4
⊢ 0 < 1 / 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | linarith [(lt_min_iff.mp a1).2] | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
bound : ∀ x ∈ ball 1 (1 / 2), ‖deriv (fun x => x ^ (-b)) x‖ ≤ 4
le : |a ^ (-b) - 1| ≤ 4 * |a - 1|
⊢ 4 * |a - 1| < 4 * (r / 4) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
bound : ∀ x ∈ ball 1 (1 / 2), ‖deriv (fun x => x ^ (-b)) x‖ ≤ 4
le : |a ^ (-b) - 1| ≤ 4 * |a - 1|
⊢ 4 * |a - 1| < 4 * (r / 4)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_potential | [31, 1] | [86, 21] | ring | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
bound : ∀ x ∈ ball 1 (1 / 2), ‖deriv (fun x => x ^ (-b)) x‖ ≤ 4
le : |a ^ (-b) - 1| ≤ 4 * |a - 1|
⊢ 4 * (r / 4) = r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
bound : ∀ x ∈ ball 1 (1 / 2), ‖deriv (fun x => x ^ (-b)) x‖ ≤ 4
le : |a ^ (-b) - 1| ≤ 4 * |a - 1|
⊢ 4 * (r / 4) = r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_log_neg_log_potential | [88, 1] | [106, 82] | set s := superF d | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(⋯.potential c ↑z).log).log) | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log) | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(⋯.potential c ↑z).log).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_log_neg_log_potential | [88, 1] | [106, 82] | have zn1 : ∀ {n}, 1 < abs ((f' d c)^[n] z) := by
intro n; exact lt_of_lt_of_le (by norm_num) (le_trans z3 (le_self_iter d z3 cz _)) | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log) | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log) | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_log_neg_log_potential | [88, 1] | [106, 82] | have zn0 : ∀ {n}, 0 < abs ((f' d c)^[n] z) := fun {_} ↦ lt_trans zero_lt_one zn1 | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log) | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log) | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_log_neg_log_potential | [88, 1] | [106, 82] | have ln0 : ∀ {n}, 0 < log (abs ((f' d c)^[n] z)) := fun {_} ↦ Real.log_pos zn1 | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log) | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log) | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_log_neg_log_potential | [88, 1] | [106, 82] | have dn0 : ∀ {n}, (d:ℝ)^n ≠ 0 := fun {_} ↦ pow_ne_zero _ (Nat.cast_ne_zero.mpr (d_ne_zero d)) | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log) | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
dn0 : ∀ {n : ℕ}, ↑d ^ n ≠ 0
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log) | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_log_neg_log_potential | [88, 1] | [106, 82] | have p0 : 0 < s.potential c z := potential_pos | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
dn0 : ∀ {n : ℕ}, ↑d ^ n ≠ 0
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log) | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
dn0 : ∀ {n : ℕ}, ↑d ^ n ≠ 0
p0 : 0 < s.potential c ↑z
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log) | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
dn0 : ∀ {n : ℕ}, ↑d ^ n ≠ 0
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_log_neg_log_potential | [88, 1] | [106, 82] | have p1 : s.potential c z < 1 := potential_lt_one_of_two_lt (by linarith) cz | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
dn0 : ∀ {n : ℕ}, ↑d ^ n ≠ 0
p0 : 0 < s.potential c ↑z
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log) | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
dn0 : ∀ {n : ℕ}, ↑d ^ n ≠ 0
p0 : 0 < s.potential c ↑z
p1 : s.potential c ↑z < 1
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log) | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
dn0 : ∀ {n : ℕ}, ↑d ^ n ≠ 0
p0 : 0 < s.potential c ↑z
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_log_neg_log_potential | [88, 1] | [106, 82] | set f := fun x ↦ log (log x⁻¹) | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
dn0 : ∀ {n : ℕ}, ↑d ^ n ≠ 0
p0 : 0 < s.potential c ↑z
p1 : s.potential c ↑z < 1
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log) | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (_root_.f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
dn0 : ∀ {n : ℕ}, ↑d ^ n ≠ 0
p0 : 0 < s.potential c ↑z
p1 : s.potential c ↑z < 1
f : ℝ → ℝ := fun x => x⁻¹.log.log
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log) | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
dn0 : ∀ {n : ℕ}, ↑d ^ n ≠ 0
p0 : 0 < s.potential c ↑z
p1 : s.potential c ↑z < 1
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_log_neg_log_potential | [88, 1] | [106, 82] | have fc : ContinuousAt f ((superF d).potential c z) := by
refine ((NormedField.continuousAt_inv.mpr p0.ne').log (inv_ne_zero p0.ne')).log ?_
exact Real.log_ne_zero_of_pos_of_ne_one (inv_pos.mpr p0) (inv_ne_one.mpr p1.ne) | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (_root_.f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
dn0 : ∀ {n : ℕ}, ↑d ^ n ≠ 0
p0 : 0 < s.potential c ↑z
p1 : s.potential c ↑z < 1
f : ℝ → ℝ := fun x => x⁻¹.log.log
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log) | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (_root_.f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
dn0 : ∀ {n : ℕ}, ↑d ^ n ≠ 0
p0 : 0 < s.potential c ↑z
p1 : s.potential c ↑z < 1
f : ℝ → ℝ := fun x => x⁻¹.log.log
fc : ContinuousAt f (⋯.potential c ↑z)
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log) | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (_root_.f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
dn0 : ∀ {n : ℕ}, ↑d ^ n ≠ 0
p0 : 0 < s.potential c ↑z
p1 : s.potential c ↑z < 1
f : ℝ → ℝ := fun x => x⁻¹.log.log
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_log_neg_log_potential | [88, 1] | [106, 82] | have t := Tendsto.comp fc (tendsto_potential d z3 cz) | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (_root_.f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
dn0 : ∀ {n : ℕ}, ↑d ^ n ≠ 0
p0 : 0 < s.potential c ↑z
p1 : s.potential c ↑z < 1
f : ℝ → ℝ := fun x => x⁻¹.log.log
fc : ContinuousAt f (⋯.potential c ↑z)
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log) | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (_root_.f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
dn0 : ∀ {n : ℕ}, ↑d ^ n ≠ 0
p0 : 0 < s.potential c ↑z
p1 : s.potential c ↑z < 1
f : ℝ → ℝ := fun x => x⁻¹.log.log
fc : ContinuousAt f (⋯.potential c ↑z)
t : Tendsto (f ∘ fun n => Complex.abs ((f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) atTop (𝓝 (f (⋯.potential c ↑z)))
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log) | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (_root_.f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
dn0 : ∀ {n : ℕ}, ↑d ^ n ≠ 0
p0 : 0 < s.potential c ↑z
p1 : s.potential c ↑z < 1
f : ℝ → ℝ := fun x => x⁻¹.log.log
fc : ContinuousAt f (⋯.potential c ↑z)
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_log_neg_log_potential | [88, 1] | [106, 82] | simpa only [Real.log_inv, Real.log_neg_eq_log, Nat.cast_pow, Function.comp_def, Real.log_rpow zn0,
neg_mul, ← div_eq_inv_mul, Real.log_div ln0.ne' dn0, Real.log_pow, f] using t | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (_root_.f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
dn0 : ∀ {n : ℕ}, ↑d ^ n ≠ 0
p0 : 0 < s.potential c ↑z
p1 : s.potential c ↑z < 1
f : ℝ → ℝ := fun x => x⁻¹.log.log
fc : ContinuousAt f (⋯.potential c ↑z)
t : Tendsto (f ∘ fun n => Complex.abs ((f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) atTop (𝓝 (f (⋯.potential c ↑z)))
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (_root_.f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
dn0 : ∀ {n : ℕ}, ↑d ^ n ≠ 0
p0 : 0 < s.potential c ↑z
p1 : s.potential c ↑z < 1
f : ℝ → ℝ := fun x => x⁻¹.log.log
fc : ContinuousAt f (⋯.potential c ↑z)
t : Tendsto (f ∘ fun n => Complex.abs ((f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) atTop (𝓝 (f (⋯.potential c ↑z)))
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_log_neg_log_potential | [88, 1] | [106, 82] | intro n | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
⊢ ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z) | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
n : ℕ
⊢ 1 < Complex.abs ((f' d c)^[n] z) | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
⊢ ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_log_neg_log_potential | [88, 1] | [106, 82] | exact lt_of_lt_of_le (by norm_num) (le_trans z3 (le_self_iter d z3 cz _)) | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
n : ℕ
⊢ 1 < Complex.abs ((f' d c)^[n] z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
n : ℕ
⊢ 1 < Complex.abs ((f' d c)^[n] z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_log_neg_log_potential | [88, 1] | [106, 82] | norm_num | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
n : ℕ
⊢ 1 < 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
n : ℕ
⊢ 1 < 3
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_log_neg_log_potential | [88, 1] | [106, 82] | linarith | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
dn0 : ∀ {n : ℕ}, ↑d ^ n ≠ 0
p0 : 0 < s.potential c ↑z
⊢ 2 < Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
dn0 : ∀ {n : ℕ}, ↑d ^ n ≠ 0
p0 : 0 < s.potential c ↑z
⊢ 2 < Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_log_neg_log_potential | [88, 1] | [106, 82] | refine ((NormedField.continuousAt_inv.mpr p0.ne').log (inv_ne_zero p0.ne')).log ?_ | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (_root_.f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
dn0 : ∀ {n : ℕ}, ↑d ^ n ≠ 0
p0 : 0 < s.potential c ↑z
p1 : s.potential c ↑z < 1
f : ℝ → ℝ := fun x => x⁻¹.log.log
⊢ ContinuousAt f (⋯.potential c ↑z) | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (_root_.f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
dn0 : ∀ {n : ℕ}, ↑d ^ n ≠ 0
p0 : 0 < s.potential c ↑z
p1 : s.potential c ↑z < 1
f : ℝ → ℝ := fun x => x⁻¹.log.log
⊢ (s.potential c ↑z)⁻¹.log ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (_root_.f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
dn0 : ∀ {n : ℕ}, ↑d ^ n ≠ 0
p0 : 0 < s.potential c ↑z
p1 : s.potential c ↑z < 1
f : ℝ → ℝ := fun x => x⁻¹.log.log
⊢ ContinuousAt f (⋯.potential c ↑z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | tendsto_log_neg_log_potential | [88, 1] | [106, 82] | exact Real.log_ne_zero_of_pos_of_ne_one (inv_pos.mpr p0) (inv_ne_one.mpr p1.ne) | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (_root_.f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
dn0 : ∀ {n : ℕ}, ↑d ^ n ≠ 0
p0 : 0 < s.potential c ↑z
p1 : s.potential c ↑z < 1
f : ℝ → ℝ := fun x => x⁻¹.log.log
⊢ (s.potential c ↑z)⁻¹.log ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (_root_.f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
dn0 : ∀ {n : ℕ}, ↑d ^ n ≠ 0
p0 : 0 < s.potential c ↑z
p1 : s.potential c ↑z < 1
f : ℝ → ℝ := fun x => x⁻¹.log.log
⊢ (s.potential c ↑z)⁻¹.log ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | log_neg_log_potential_approx | [108, 1] | [122, 47] | apply le_of_forall_pos_lt_add | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ |(-(⋯.potential c ↑z).log).log - (Complex.abs z).log.log| ≤ iter_error d c z | case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ ∀ (ε : ℝ), 0 < ε → |(-(⋯.potential c ↑z).log).log - (Complex.abs z).log.log| < iter_error d c z + ε | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ |(-(⋯.potential c ↑z).log).log - (Complex.abs z).log.log| ≤ iter_error d c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | log_neg_log_potential_approx | [108, 1] | [122, 47] | intro e ep | case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ ∀ (ε : ℝ), 0 < ε → |(-(⋯.potential c ↑z).log).log - (Complex.abs z).log.log| < iter_error d c z + ε | case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
e : ℝ
ep : 0 < e
⊢ |(-(⋯.potential c ↑z).log).log - (Complex.abs z).log.log| < iter_error d c z + e | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ ∀ (ε : ℝ), 0 < ε → |(-(⋯.potential c ↑z).log).log - (Complex.abs z).log.log| < iter_error d c z + ε
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | log_neg_log_potential_approx | [108, 1] | [122, 47] | rcases (Metric.tendsto_nhds.mp (tendsto_log_neg_log_potential d z3 cz) e ep).exists with ⟨n,t⟩ | case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
e : ℝ
ep : 0 < e
⊢ |(-(⋯.potential c ↑z).log).log - (Complex.abs z).log.log| < iter_error d c z + e | case h.intro
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
e : ℝ
ep : 0 < e
n : ℕ
t : dist ((Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) (-(⋯.potential c ↑z).log).log < e
⊢ |(-(⋯.potential c ↑z).log).log - (Complex.abs z).log.log| < iter_error d c z + e | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
e : ℝ
ep : 0 < e
⊢ |(-(⋯.potential c ↑z).log).log - (Complex.abs z).log.log| < iter_error d c z + e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | log_neg_log_potential_approx | [108, 1] | [122, 47] | have ie := iter_approx d z3 cz n | case h.intro
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
e : ℝ
ep : 0 < e
n : ℕ
t : dist ((Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) (-(⋯.potential c ↑z).log).log < e
⊢ |(-(⋯.potential c ↑z).log).log - (Complex.abs z).log.log| < iter_error d c z + e | case h.intro
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
e : ℝ
ep : 0 < e
n : ℕ
t : dist ((Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) (-(⋯.potential c ↑z).log).log < e
ie : |(Complex.abs ((f' d c)^[n] z)).log.log - (Complex.abs z).log.log - ↑n * (↑d).log| ≤ iter_error d c z
⊢ |(-(⋯.potential c ↑z).log).log - (Complex.abs z).log.log| < iter_error d c z + e | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
e : ℝ
ep : 0 < e
n : ℕ
t : dist ((Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) (-(⋯.potential c ↑z).log).log < e
⊢ |(-(⋯.potential c ↑z).log).log - (Complex.abs z).log.log| < iter_error d c z + e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | log_neg_log_potential_approx | [108, 1] | [122, 47] | generalize log (-log ((superF d).potential c z)) = p at ie t | case h.intro
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
e : ℝ
ep : 0 < e
n : ℕ
t : dist ((Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) (-(⋯.potential c ↑z).log).log < e
ie : |(Complex.abs ((f' d c)^[n] z)).log.log - (Complex.abs z).log.log - ↑n * (↑d).log| ≤ iter_error d c z
⊢ |(-(⋯.potential c ↑z).log).log - (Complex.abs z).log.log| < iter_error d c z + e | case h.intro
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
e : ℝ
ep : 0 < e
n : ℕ
ie : |(Complex.abs ((f' d c)^[n] z)).log.log - (Complex.abs z).log.log - ↑n * (↑d).log| ≤ iter_error d c z
p : ℝ
t : dist ((Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) p < e
⊢ |p - (Complex.abs z).log.log| < iter_error d c z + e | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
e : ℝ
ep : 0 < e
n : ℕ
t : dist ((Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) (-(⋯.potential c ↑z).log).log < e
ie : |(Complex.abs ((f' d c)^[n] z)).log.log - (Complex.abs z).log.log - ↑n * (↑d).log| ≤ iter_error d c z
⊢ |(-(⋯.potential c ↑z).log).log - (Complex.abs z).log.log| < iter_error d c z + e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | log_neg_log_potential_approx | [108, 1] | [122, 47] | generalize log (log (Complex.abs ((f' d c)^[n] z))) = x at ie t | case h.intro
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
e : ℝ
ep : 0 < e
n : ℕ
ie : |(Complex.abs ((f' d c)^[n] z)).log.log - (Complex.abs z).log.log - ↑n * (↑d).log| ≤ iter_error d c z
p : ℝ
t : dist ((Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) p < e
⊢ |p - (Complex.abs z).log.log| < iter_error d c z + e | case h.intro
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
e : ℝ
ep : 0 < e
n : ℕ
p x : ℝ
ie : |x - (Complex.abs z).log.log - ↑n * (↑d).log| ≤ iter_error d c z
t : dist (x - ↑n * (↑d).log) p < e
⊢ |p - (Complex.abs z).log.log| < iter_error d c z + e | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
e : ℝ
ep : 0 < e
n : ℕ
ie : |(Complex.abs ((f' d c)^[n] z)).log.log - (Complex.abs z).log.log - ↑n * (↑d).log| ≤ iter_error d c z
p : ℝ
t : dist ((Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) p < e
⊢ |p - (Complex.abs z).log.log| < iter_error d c z + e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | log_neg_log_potential_approx | [108, 1] | [122, 47] | generalize log (log (Complex.abs z)) = y at ie t | case h.intro
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
e : ℝ
ep : 0 < e
n : ℕ
p x : ℝ
ie : |x - (Complex.abs z).log.log - ↑n * (↑d).log| ≤ iter_error d c z
t : dist (x - ↑n * (↑d).log) p < e
⊢ |p - (Complex.abs z).log.log| < iter_error d c z + e | case h.intro
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
e : ℝ
ep : 0 < e
n : ℕ
p x : ℝ
t : dist (x - ↑n * (↑d).log) p < e
y : ℝ
ie : |x - y - ↑n * (↑d).log| ≤ iter_error d c z
⊢ |p - y| < iter_error d c z + e | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
e : ℝ
ep : 0 < e
n : ℕ
p x : ℝ
ie : |x - (Complex.abs z).log.log - ↑n * (↑d).log| ≤ iter_error d c z
t : dist (x - ↑n * (↑d).log) p < e
⊢ |p - (Complex.abs z).log.log| < iter_error d c z + e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | log_neg_log_potential_approx | [108, 1] | [122, 47] | rw [Real.dist_eq, abs_sub_comm] at t | case h.intro
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
e : ℝ
ep : 0 < e
n : ℕ
p x : ℝ
t : dist (x - ↑n * (↑d).log) p < e
y : ℝ
ie : |x - y - ↑n * (↑d).log| ≤ iter_error d c z
⊢ |p - y| < iter_error d c z + e | case h.intro
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
e : ℝ
ep : 0 < e
n : ℕ
p x : ℝ
t : |p - (x - ↑n * (↑d).log)| < e
y : ℝ
ie : |x - y - ↑n * (↑d).log| ≤ iter_error d c z
⊢ |p - y| < iter_error d c z + e | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
e : ℝ
ep : 0 < e
n : ℕ
p x : ℝ
t : dist (x - ↑n * (↑d).log) p < e
y : ℝ
ie : |x - y - ↑n * (↑d).log| ≤ iter_error d c z
⊢ |p - y| < iter_error d c z + e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | log_neg_log_potential_approx | [108, 1] | [122, 47] | rw [add_comm] | case h.intro
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
e : ℝ
ep : 0 < e
n : ℕ
p x : ℝ
t : |p - (x - ↑n * (↑d).log)| < e
y : ℝ
ie : |x - y - ↑n * (↑d).log| ≤ iter_error d c z
⊢ |p - y| < iter_error d c z + e | case h.intro
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
e : ℝ
ep : 0 < e
n : ℕ
p x : ℝ
t : |p - (x - ↑n * (↑d).log)| < e
y : ℝ
ie : |x - y - ↑n * (↑d).log| ≤ iter_error d c z
⊢ |p - y| < e + iter_error d c z | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
e : ℝ
ep : 0 < e
n : ℕ
p x : ℝ
t : |p - (x - ↑n * (↑d).log)| < e
y : ℝ
ie : |x - y - ↑n * (↑d).log| ≤ iter_error d c z
⊢ |p - y| < iter_error d c z + e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | log_neg_log_potential_approx | [108, 1] | [122, 47] | calc |p - y|
_ = |(p - (x - n * log d)) + (x - y - n * log d)| := by ring_nf
_ ≤ |p - (x - n * log d)| + |x - y - n * log d| := abs_add _ _
_ < e + _ := add_lt_add_of_lt_of_le t ie | case h.intro
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
e : ℝ
ep : 0 < e
n : ℕ
p x : ℝ
t : |p - (x - ↑n * (↑d).log)| < e
y : ℝ
ie : |x - y - ↑n * (↑d).log| ≤ iter_error d c z
⊢ |p - y| < e + iter_error d c z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
e : ℝ
ep : 0 < e
n : ℕ
p x : ℝ
t : |p - (x - ↑n * (↑d).log)| < e
y : ℝ
ie : |x - y - ↑n * (↑d).log| ≤ iter_error d c z
⊢ |p - y| < e + iter_error d c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | log_neg_log_potential_approx | [108, 1] | [122, 47] | ring_nf | c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
e : ℝ
ep : 0 < e
n : ℕ
p x : ℝ
t : |p - (x - ↑n * (↑d).log)| < e
y : ℝ
ie : |x - y - ↑n * (↑d).log| ≤ iter_error d c z
⊢ |p - y| = |p - (x - ↑n * (↑d).log) + (x - y - ↑n * (↑d).log)| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
e : ℝ
ep : 0 < e
n : ℕ
p x : ℝ
t : |p - (x - ↑n * (↑d).log)| < e
y : ℝ
ie : |x - y - ↑n * (↑d).log| ≤ iter_error d c z
⊢ |p - y| = |p - (x - ↑n * (↑d).log) + (x - y - ↑n * (↑d).log)|
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | hasDerivAt_ene | [136, 1] | [140, 68] | have h : HasDerivAt (fun x ↦ exp (-exp x)) (exp (-exp x) * -exp x) x :=
HasDerivAt.exp (Real.hasDerivAt_exp x).neg | c z : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
⊢ HasDerivAt ene (-dene x) x | c z : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
h : HasDerivAt (fun x => (-x.exp).exp) ((-x.exp).exp * -x.exp) x
⊢ HasDerivAt ene (-dene x) x | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
⊢ HasDerivAt ene (-dene x) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | hasDerivAt_ene | [136, 1] | [140, 68] | simp only [mul_neg, ← Real.exp_add, neg_add_eq_sub] at h | c z : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
h : HasDerivAt (fun x => (-x.exp).exp) ((-x.exp).exp * -x.exp) x
⊢ HasDerivAt ene (-dene x) x | c z : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
h : HasDerivAt (fun x => (-x.exp).exp) (-(x - x.exp).exp) x
⊢ HasDerivAt ene (-dene x) x | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
h : HasDerivAt (fun x => (-x.exp).exp) ((-x.exp).exp * -x.exp) x
⊢ HasDerivAt ene (-dene x) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | hasDerivAt_ene | [136, 1] | [140, 68] | exact h | c z : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
h : HasDerivAt (fun x => (-x.exp).exp) (-(x - x.exp).exp) x
⊢ HasDerivAt ene (-dene x) x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
h : HasDerivAt (fun x => (-x.exp).exp) (-(x - x.exp).exp) x
⊢ HasDerivAt ene (-dene x) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | dene_anti | [148, 1] | [160, 33] | refine Real.exp_le_exp.mpr ?_ | c z : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x y : ℝ
x0 : 0 ≤ x
xy : x ≤ y
⊢ dene x ≥ dene y | c z : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x y : ℝ
x0 : 0 ≤ x
xy : x ≤ y
⊢ y - y.exp ≤ x - x.exp | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x y : ℝ
x0 : 0 ≤ x
xy : x ≤ y
⊢ dene x ≥ dene y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | dene_anti | [148, 1] | [160, 33] | exact a x0 (le_trans x0 xy) xy | c z : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x y : ℝ
x0 : 0 ≤ x
xy : x ≤ y
a : AntitoneOn (fun x => x - x.exp) (Ici 0)
⊢ y - y.exp ≤ x - x.exp | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x y : ℝ
x0 : 0 ≤ x
xy : x ≤ y
a : AntitoneOn (fun x => x - x.exp) (Ici 0)
⊢ y - y.exp ≤ x - x.exp
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | dene_anti | [148, 1] | [160, 33] | have hd : ∀ x, HasDerivAt (fun x ↦ x - exp x) (1 - exp x) x :=
fun x ↦ (hasDerivAt_id x).sub (Real.hasDerivAt_exp x) | c z : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x y : ℝ
x0 : 0 ≤ x
xy : x ≤ y
⊢ AntitoneOn (fun x => x - x.exp) (Ici 0) | c z : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x y : ℝ
x0 : 0 ≤ x
xy : x ≤ y
hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x
⊢ AntitoneOn (fun x => x - x.exp) (Ici 0) | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x y : ℝ
x0 : 0 ≤ x
xy : x ≤ y
⊢ AntitoneOn (fun x => x - x.exp) (Ici 0)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | dene_anti | [148, 1] | [160, 33] | have d : Differentiable ℝ (fun x ↦ x - exp x) := fun x ↦ (hd x).differentiableAt | c z : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x y : ℝ
x0 : 0 ≤ x
xy : x ≤ y
hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x
⊢ AntitoneOn (fun x => x - x.exp) (Ici 0) | c z : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
x y : ℝ
x0 : 0 ≤ x
xy : x ≤ y
hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x
d : Differentiable ℝ fun x => x - x.exp
⊢ AntitoneOn (fun x => x - x.exp) (Ici 0) | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x y : ℝ
x0 : 0 ≤ x
xy : x ≤ y
hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x
⊢ AntitoneOn (fun x => x - x.exp) (Ici 0)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | dene_anti | [148, 1] | [160, 33] | apply antitoneOn_of_deriv_nonpos (convex_Ici _) | c z : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
x y : ℝ
x0 : 0 ≤ x
xy : x ≤ y
hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x
d : Differentiable ℝ fun x => x - x.exp
⊢ AntitoneOn (fun x => x - x.exp) (Ici 0) | case hf
c z : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
x y : ℝ
x0 : 0 ≤ x
xy : x ≤ y
hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x
d : Differentiable ℝ fun x => x - x.exp
⊢ ContinuousOn (fun x => x - x.exp) (Ici 0)
case hf'
c z : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
x y : ℝ
x0 : 0 ≤ x
xy : x ≤ y
hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x
d : Differentiable ℝ fun x => x - x.exp
⊢ DifferentiableOn ℝ (fun x => x - x.exp) (interior (Ici 0))
case hf'_nonpos
c z : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
x y : ℝ
x0 : 0 ≤ x
xy : x ≤ y
hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x
d : Differentiable ℝ fun x => x - x.exp
⊢ ∀ x ∈ interior (Ici 0), deriv (fun x => x - x.exp) x ≤ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
x y : ℝ
x0 : 0 ≤ x
xy : x ≤ y
hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x
d : Differentiable ℝ fun x => x - x.exp
⊢ AntitoneOn (fun x => x - x.exp) (Ici 0)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | dene_anti | [148, 1] | [160, 33] | exact d.continuous.continuousOn | case hf
c z : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
x y : ℝ
x0 : 0 ≤ x
xy : x ≤ y
hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x
d : Differentiable ℝ fun x => x - x.exp
⊢ ContinuousOn (fun x => x - x.exp) (Ici 0) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
c z : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
x y : ℝ
x0 : 0 ≤ x
xy : x ≤ y
hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x
d : Differentiable ℝ fun x => x - x.exp
⊢ ContinuousOn (fun x => x - x.exp) (Ici 0)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | dene_anti | [148, 1] | [160, 33] | exact d.differentiableOn | case hf'
c z : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
x y : ℝ
x0 : 0 ≤ x
xy : x ≤ y
hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x
d : Differentiable ℝ fun x => x - x.exp
⊢ DifferentiableOn ℝ (fun x => x - x.exp) (interior (Ici 0)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf'
c z : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
x y : ℝ
x0 : 0 ≤ x
xy : x ≤ y
hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x
d : Differentiable ℝ fun x => x - x.exp
⊢ DifferentiableOn ℝ (fun x => x - x.exp) (interior (Ici 0))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | dene_anti | [148, 1] | [160, 33] | intro x m | case hf'_nonpos
c z : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
x y : ℝ
x0 : 0 ≤ x
xy : x ≤ y
hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x
d : Differentiable ℝ fun x => x - x.exp
⊢ ∀ x ∈ interior (Ici 0), deriv (fun x => x - x.exp) x ≤ 0 | case hf'_nonpos
c z : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
x✝ y : ℝ
x0 : 0 ≤ x✝
xy : x✝ ≤ y
hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x
d : Differentiable ℝ fun x => x - x.exp
x : ℝ
m : x ∈ interior (Ici 0)
⊢ deriv (fun x => x - x.exp) x ≤ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hf'_nonpos
c z : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
x y : ℝ
x0 : 0 ≤ x
xy : x ≤ y
hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x
d : Differentiable ℝ fun x => x - x.exp
⊢ ∀ x ∈ interior (Ici 0), deriv (fun x => x - x.exp) x ≤ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | dene_anti | [148, 1] | [160, 33] | simp only [nonempty_Iio, interior_Ici', mem_Ioi] at m | case hf'_nonpos
c z : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
x✝ y : ℝ
x0 : 0 ≤ x✝
xy : x✝ ≤ y
hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x
d : Differentiable ℝ fun x => x - x.exp
x : ℝ
m : x ∈ interior (Ici 0)
⊢ deriv (fun x => x - x.exp) x ≤ 0 | case hf'_nonpos
c z : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
x✝ y : ℝ
x0 : 0 ≤ x✝
xy : x✝ ≤ y
hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x
d : Differentiable ℝ fun x => x - x.exp
x : ℝ
m : 0 < x
⊢ deriv (fun x => x - x.exp) x ≤ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hf'_nonpos
c z : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
x✝ y : ℝ
x0 : 0 ≤ x✝
xy : x✝ ≤ y
hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x
d : Differentiable ℝ fun x => x - x.exp
x : ℝ
m : x ∈ interior (Ici 0)
⊢ deriv (fun x => x - x.exp) x ≤ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Potential.lean | dene_anti | [148, 1] | [160, 33] | simp only [(hd x).deriv, sub_nonpos, Real.one_le_exp m.le] | case hf'_nonpos
c z : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
x✝ y : ℝ
x0 : 0 ≤ x✝
xy : x✝ ≤ y
hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x
d : Differentiable ℝ fun x => x - x.exp
x : ℝ
m : 0 < x
⊢ deriv (fun x => x - x.exp) x ≤ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf'_nonpos
c z : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
x✝ y : ℝ
x0 : 0 ≤ x✝
xy : x✝ ≤ y
hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x
d : Differentiable ℝ fun x => x - x.exp
x : ℝ
m : 0 < x
⊢ deriv (fun x => x - x.exp) x ≤ 0
TACTIC:
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.