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/PotentialLower.lean | pass_through | [28, 1] | [54, 19] | apply Nat.find_min | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h : p n
k : ℕ
hk : Nat.find ⋯ = k
k4 : p k
k0 : k ≠ 0
k1 : 1 ≤ k
⊢ ¬p (k - 1) | case a
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h : p n
k : ℕ
hk : Nat.find ⋯ = k
k4 : p k
k0 : k ≠ 0
k1 : 1 ≤ k
⊢ k - 1 < Nat.find ?H
case H
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h : p n
k : ℕ
hk : Nat.find ⋯ = k
k4 : p k
k0 : k ≠ 0
k1 : 1 ≤ k
⊢ ∃ n, p n | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h : p n
k : ℕ
hk : Nat.find ⋯ = k
k4 : p k
k0 : k ≠ 0
k1 : 1 ≤ k
⊢ ¬p (k - 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | pass_through | [28, 1] | [54, 19] | rw [hk] | case a
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h : p n
k : ℕ
hk : Nat.find ⋯ = k
k4 : p k
k0 : k ≠ 0
k1 : 1 ≤ k
⊢ k - 1 < Nat.find ?H
case H
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h : p n
k : ℕ
hk : Nat.find ⋯ = k
k4 : p k
k0 : k ≠ 0
k1 : 1 ≤ k
⊢ ∃ n, p n | case a
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h : p n
k : ℕ
hk : Nat.find ⋯ = k
k4 : p k
k0 : k ≠ 0
k1 : 1 ≤ k
⊢ k - 1 < k | Please generate a tactic in lean4 to solve the state.
STATE:
case a
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h : p n
k : ℕ
hk : Nat.find ⋯ = k
k4 : p k
k0 : k ≠ 0
k1 : 1 ≤ k
⊢ k - 1 < Nat.find ?H
case H
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h : p n
k : ℕ
hk : Nat.find ⋯ = k
k4 : p k
k0 : k ≠ 0
k1 : 1 ≤ k
⊢ ∃ n, p n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | pass_through | [28, 1] | [54, 19] | omega | case a
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h : p n
k : ℕ
hk : Nat.find ⋯ = k
k4 : p k
k0 : k ≠ 0
k1 : 1 ≤ k
⊢ k - 1 < k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h : p n
k : ℕ
hk : Nat.find ⋯ = k
k4 : p k
k0 : k ≠ 0
k1 : 1 ≤ k
⊢ k - 1 < k
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | pass_through | [28, 1] | [54, 19] | norm_num | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h : p n
k : ℕ
hk : Nat.find ⋯ = k
k0 : k ≠ 0
k1 : 1 ≤ k
k4 : 4 < Complex.abs ((f' 2 c)^[k] z)
lt : Complex.abs ((f' 2 c)^[k - 1] z) ≤ 4
fs : Complex.abs ((f' 2 c)^[k] z) ≤ Complex.abs ((f' 2 c)^[k - 1] z) ^ 2 + Complex.abs c
⊢ 4 ^ 2 + 4 ≤ 20 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h : p n
k : ℕ
hk : Nat.find ⋯ = k
k0 : k ≠ 0
k1 : 1 ≤ k
k4 : 4 < Complex.abs ((f' 2 c)^[k] z)
lt : Complex.abs ((f' 2 c)^[k - 1] z) ≤ 4
fs : Complex.abs ((f' 2 c)^[k] z) ≤ Complex.abs ((f' 2 c)^[k - 1] z) ^ 2 + Complex.abs c
⊢ 4 ^ 2 + 4 ≤ 20
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | lower_anti | [56, 1] | [84, 26] | have hd : ∀ x, 4 ≤ x → HasDerivAt (fun x : ℝ ↦ 1 / x - k / x^p)
(-(x^2)⁻¹ - k * (-(p * x^(p-1)) / (x^p)^2)) x := by
intro x x2
simp only [one_div, div_eq_mul_inv, one_mul]
refine (hasDerivAt_inv (by positivity)).sub (HasDerivAt.const_mul _ ?_)
exact (Real.hasDerivAt_rpow_const (Or.inl (by positivity))).inv (by positivity) | c z : ℂ
k p : ℝ
kp : autoParam (k * p ≤ 2) _auto✝
hp : autoParam (3 / 2 ≤ p) _auto✝
⊢ AntitoneOn (fun x => 1 / x - k / x ^ p) (Ici 4) | c z : ℂ
k p : ℝ
kp : autoParam (k * p ≤ 2) _auto✝
hp : autoParam (3 / 2 ≤ p) _auto✝
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
⊢ AntitoneOn (fun x => 1 / x - k / x ^ p) (Ici 4) | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
k p : ℝ
kp : autoParam (k * p ≤ 2) _auto✝
hp : autoParam (3 / 2 ≤ p) _auto✝
⊢ AntitoneOn (fun x => 1 / x - k / x ^ p) (Ici 4)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | lower_anti | [56, 1] | [84, 26] | simp only [ge_iff_le] at kp hp | c z : ℂ
k p : ℝ
kp : autoParam (k * p ≤ 2) _auto✝
hp : autoParam (3 / 2 ≤ p) _auto✝
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
⊢ AntitoneOn (fun x => 1 / x - k / x ^ p) (Ici 4) | c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
⊢ AntitoneOn (fun x => 1 / x - k / x ^ p) (Ici 4) | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
k p : ℝ
kp : autoParam (k * p ≤ 2) _auto✝
hp : autoParam (3 / 2 ≤ p) _auto✝
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
⊢ AntitoneOn (fun x => 1 / x - k / x ^ p) (Ici 4)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | lower_anti | [56, 1] | [84, 26] | have d : DifferentiableOn ℝ (fun x ↦ 1 / x - k / x^p) (Ici 4) :=
fun x m ↦ (hd x m).differentiableAt.differentiableWithinAt | c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
⊢ AntitoneOn (fun x => 1 / x - k / x ^ p) (Ici 4) | c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
⊢ AntitoneOn (fun x => 1 / x - k / x ^ p) (Ici 4) | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
⊢ AntitoneOn (fun x => 1 / x - k / x ^ p) (Ici 4)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | lower_anti | [56, 1] | [84, 26] | apply antitoneOn_of_deriv_nonpos (convex_Ici _) | c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
⊢ AntitoneOn (fun x => 1 / x - k / x ^ p) (Ici 4) | case hf
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
⊢ ContinuousOn (fun x => 1 / x - k / x ^ p) (Ici 4)
case hf'
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
⊢ DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (interior (Ici 4))
case hf'_nonpos
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
⊢ ∀ x ∈ interior (Ici 4), deriv (fun x => 1 / x - k / x ^ p) x ≤ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
⊢ AntitoneOn (fun x => 1 / x - k / x ^ p) (Ici 4)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | lower_anti | [56, 1] | [84, 26] | intro x x2 | c z : ℂ
k p : ℝ
kp : autoParam (k * p ≤ 2) _auto✝
hp : autoParam (3 / 2 ≤ p) _auto✝
⊢ ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x | c z : ℂ
k p : ℝ
kp : autoParam (k * p ≤ 2) _auto✝
hp : autoParam (3 / 2 ≤ p) _auto✝
x : ℝ
x2 : 4 ≤ x
⊢ HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
k p : ℝ
kp : autoParam (k * p ≤ 2) _auto✝
hp : autoParam (3 / 2 ≤ p) _auto✝
⊢ ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | lower_anti | [56, 1] | [84, 26] | simp only [one_div, div_eq_mul_inv, one_mul] | c z : ℂ
k p : ℝ
kp : autoParam (k * p ≤ 2) _auto✝
hp : autoParam (3 / 2 ≤ p) _auto✝
x : ℝ
x2 : 4 ≤ x
⊢ HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x | c z : ℂ
k p : ℝ
kp : autoParam (k * p ≤ 2) _auto✝
hp : autoParam (3 / 2 ≤ p) _auto✝
x : ℝ
x2 : 4 ≤ x
⊢ HasDerivAt (fun x => x⁻¹ - k * (x ^ p)⁻¹) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) * ((x ^ p) ^ 2)⁻¹)) x | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
k p : ℝ
kp : autoParam (k * p ≤ 2) _auto✝
hp : autoParam (3 / 2 ≤ p) _auto✝
x : ℝ
x2 : 4 ≤ x
⊢ HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | lower_anti | [56, 1] | [84, 26] | refine (hasDerivAt_inv (by positivity)).sub (HasDerivAt.const_mul _ ?_) | c z : ℂ
k p : ℝ
kp : autoParam (k * p ≤ 2) _auto✝
hp : autoParam (3 / 2 ≤ p) _auto✝
x : ℝ
x2 : 4 ≤ x
⊢ HasDerivAt (fun x => x⁻¹ - k * (x ^ p)⁻¹) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) * ((x ^ p) ^ 2)⁻¹)) x | c z : ℂ
k p : ℝ
kp : autoParam (k * p ≤ 2) _auto✝
hp : autoParam (3 / 2 ≤ p) _auto✝
x : ℝ
x2 : 4 ≤ x
⊢ HasDerivAt (fun x => (x ^ p)⁻¹) (-(p * x ^ (p - 1)) * ((x ^ p) ^ 2)⁻¹) x | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
k p : ℝ
kp : autoParam (k * p ≤ 2) _auto✝
hp : autoParam (3 / 2 ≤ p) _auto✝
x : ℝ
x2 : 4 ≤ x
⊢ HasDerivAt (fun x => x⁻¹ - k * (x ^ p)⁻¹) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) * ((x ^ p) ^ 2)⁻¹)) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | lower_anti | [56, 1] | [84, 26] | exact (Real.hasDerivAt_rpow_const (Or.inl (by positivity))).inv (by positivity) | c z : ℂ
k p : ℝ
kp : autoParam (k * p ≤ 2) _auto✝
hp : autoParam (3 / 2 ≤ p) _auto✝
x : ℝ
x2 : 4 ≤ x
⊢ HasDerivAt (fun x => (x ^ p)⁻¹) (-(p * x ^ (p - 1)) * ((x ^ p) ^ 2)⁻¹) x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
k p : ℝ
kp : autoParam (k * p ≤ 2) _auto✝
hp : autoParam (3 / 2 ≤ p) _auto✝
x : ℝ
x2 : 4 ≤ x
⊢ HasDerivAt (fun x => (x ^ p)⁻¹) (-(p * x ^ (p - 1)) * ((x ^ p) ^ 2)⁻¹) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | lower_anti | [56, 1] | [84, 26] | positivity | c z : ℂ
k p : ℝ
kp : autoParam (k * p ≤ 2) _auto✝
hp : autoParam (3 / 2 ≤ p) _auto✝
x : ℝ
x2 : 4 ≤ x
⊢ x ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
k p : ℝ
kp : autoParam (k * p ≤ 2) _auto✝
hp : autoParam (3 / 2 ≤ p) _auto✝
x : ℝ
x2 : 4 ≤ x
⊢ x ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | lower_anti | [56, 1] | [84, 26] | positivity | c z : ℂ
k p : ℝ
kp : autoParam (k * p ≤ 2) _auto✝
hp : autoParam (3 / 2 ≤ p) _auto✝
x : ℝ
x2 : 4 ≤ x
⊢ x ^ p ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
k p : ℝ
kp : autoParam (k * p ≤ 2) _auto✝
hp : autoParam (3 / 2 ≤ p) _auto✝
x : ℝ
x2 : 4 ≤ x
⊢ x ^ p ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | lower_anti | [56, 1] | [84, 26] | exact d.continuousOn | case hf
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
⊢ ContinuousOn (fun x => 1 / x - k / x ^ p) (Ici 4) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
⊢ ContinuousOn (fun x => 1 / x - k / x ^ p) (Ici 4)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | lower_anti | [56, 1] | [84, 26] | exact d.mono interior_subset | case hf'
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
⊢ DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (interior (Ici 4)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf'
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
⊢ DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (interior (Ici 4))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | lower_anti | [56, 1] | [84, 26] | intro x x4 | case hf'_nonpos
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
⊢ ∀ x ∈ interior (Ici 4), deriv (fun x => 1 / x - k / x ^ p) x ≤ 0 | case hf'_nonpos
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : x ∈ interior (Ici 4)
⊢ deriv (fun x => 1 / x - k / x ^ p) x ≤ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hf'_nonpos
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
⊢ ∀ x ∈ interior (Ici 4), deriv (fun x => 1 / x - k / x ^ p) x ≤ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | lower_anti | [56, 1] | [84, 26] | simp only [nonempty_Iio, interior_Ici', mem_Ioi] at x4 | case hf'_nonpos
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : x ∈ interior (Ici 4)
⊢ deriv (fun x => 1 / x - k / x ^ p) x ≤ 0 | case hf'_nonpos
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
⊢ deriv (fun x => 1 / x - k / x ^ p) x ≤ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hf'_nonpos
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : x ∈ interior (Ici 4)
⊢ deriv (fun x => 1 / x - k / x ^ p) x ≤ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | lower_anti | [56, 1] | [84, 26] | have x0 : 0 < x := by linarith | case hf'_nonpos
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
⊢ deriv (fun x => 1 / x - k / x ^ p) x ≤ 0 | case hf'_nonpos
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
x0 : 0 < x
⊢ deriv (fun x => 1 / x - k / x ^ p) x ≤ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hf'_nonpos
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
⊢ deriv (fun x => 1 / x - k / x ^ p) x ≤ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | lower_anti | [56, 1] | [84, 26] | simp only [(hd x x4.le).deriv] | case hf'_nonpos
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
x0 : 0 < x
⊢ deriv (fun x => 1 / x - k / x ^ p) x ≤ 0 | case hf'_nonpos
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
x0 : 0 < x
⊢ -(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2) ≤ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hf'_nonpos
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
x0 : 0 < x
⊢ deriv (fun x => 1 / x - k / x ^ p) x ≤ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | lower_anti | [56, 1] | [84, 26] | simp only [←Real.rpow_mul x0.le, Nat.cast_ofNat, neg_div, mul_div_assoc p, ←
Real.rpow_sub x0, mul_neg, ←mul_assoc k p, sub_neg_eq_add, neg_add_le_iff_le_add, add_zero,
←Real.rpow_two, ←Real.rpow_neg x0.le] | case hf'_nonpos
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
x0 : 0 < x
⊢ -(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2) ≤ 0 | case hf'_nonpos
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
x0 : 0 < x
⊢ k * p * x ^ (p - 1 - p * 2) ≤ x ^ (-2) | Please generate a tactic in lean4 to solve the state.
STATE:
case hf'_nonpos
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
x0 : 0 < x
⊢ -(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2) ≤ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | lower_anti | [56, 1] | [84, 26] | ring_nf | case hf'_nonpos
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
x0 : 0 < x
⊢ k * p * x ^ (p - 1 - p * 2) ≤ x ^ (-2) | case hf'_nonpos
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
x0 : 0 < x
⊢ k * p * x ^ (-1 - p) ≤ x ^ (-2) | Please generate a tactic in lean4 to solve the state.
STATE:
case hf'_nonpos
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
x0 : 0 < x
⊢ k * p * x ^ (p - 1 - p * 2) ≤ x ^ (-2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | lower_anti | [56, 1] | [84, 26] | simp only [←neg_add', Real.rpow_neg x0.le (1 + p)] | case hf'_nonpos
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
x0 : 0 < x
⊢ k * p * x ^ (-1 - p) ≤ x ^ (-2) | case hf'_nonpos
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
x0 : 0 < x
⊢ k * p * (x ^ (1 + p))⁻¹ ≤ x ^ (-2) | Please generate a tactic in lean4 to solve the state.
STATE:
case hf'_nonpos
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
x0 : 0 < x
⊢ k * p * x ^ (-1 - p) ≤ x ^ (-2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | lower_anti | [56, 1] | [84, 26] | rw [mul_inv_le_iff (by positivity), ←Real.rpow_add x0, (by ring_nf : 1 + p + -2 = p - 1)] | case hf'_nonpos
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
x0 : 0 < x
⊢ k * p * (x ^ (1 + p))⁻¹ ≤ x ^ (-2) | case hf'_nonpos
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
x0 : 0 < x
⊢ k * p ≤ x ^ (p - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case hf'_nonpos
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
x0 : 0 < x
⊢ k * p * (x ^ (1 + p))⁻¹ ≤ x ^ (-2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | lower_anti | [56, 1] | [84, 26] | have p1' : 1/2 ≤ p - 1 := by linarith | case hf'_nonpos
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
x0 : 0 < x
⊢ k * p ≤ x ^ (p - 1) | case hf'_nonpos
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
x0 : 0 < x
p1' : 1 / 2 ≤ p - 1
⊢ k * p ≤ x ^ (p - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case hf'_nonpos
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
x0 : 0 < x
⊢ k * p ≤ x ^ (p - 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | lower_anti | [56, 1] | [84, 26] | refine le_trans kp (le_trans ?_ (Real.rpow_le_rpow_of_exponent_le (by linarith) p1')) | case hf'_nonpos
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
x0 : 0 < x
p1' : 1 / 2 ≤ p - 1
⊢ k * p ≤ x ^ (p - 1) | case hf'_nonpos
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
x0 : 0 < x
p1' : 1 / 2 ≤ p - 1
⊢ 2 ≤ x ^ (1 / 2) | Please generate a tactic in lean4 to solve the state.
STATE:
case hf'_nonpos
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
x0 : 0 < x
p1' : 1 / 2 ≤ p - 1
⊢ k * p ≤ x ^ (p - 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | lower_anti | [56, 1] | [84, 26] | rw [one_div, Real.le_rpow_inv_iff_of_pos (by norm_num) x0.le (by norm_num)] | case hf'_nonpos
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
x0 : 0 < x
p1' : 1 / 2 ≤ p - 1
⊢ 2 ≤ x ^ (1 / 2) | case hf'_nonpos
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
x0 : 0 < x
p1' : 1 / 2 ≤ p - 1
⊢ 2 ^ 2 ≤ x | Please generate a tactic in lean4 to solve the state.
STATE:
case hf'_nonpos
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
x0 : 0 < x
p1' : 1 / 2 ≤ p - 1
⊢ 2 ≤ x ^ (1 / 2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | lower_anti | [56, 1] | [84, 26] | norm_num | case hf'_nonpos
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
x0 : 0 < x
p1' : 1 / 2 ≤ p - 1
⊢ 2 ^ 2 ≤ x | case hf'_nonpos
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
x0 : 0 < x
p1' : 1 / 2 ≤ p - 1
⊢ 4 ≤ x | Please generate a tactic in lean4 to solve the state.
STATE:
case hf'_nonpos
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
x0 : 0 < x
p1' : 1 / 2 ≤ p - 1
⊢ 2 ^ 2 ≤ x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | lower_anti | [56, 1] | [84, 26] | exact x4.le | case hf'_nonpos
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
x0 : 0 < x
p1' : 1 / 2 ≤ p - 1
⊢ 4 ≤ x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf'_nonpos
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
x0 : 0 < x
p1' : 1 / 2 ≤ p - 1
⊢ 4 ≤ x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | lower_anti | [56, 1] | [84, 26] | linarith | c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
⊢ 0 < x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
⊢ 0 < x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | lower_anti | [56, 1] | [84, 26] | positivity | c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
x0 : 0 < x
⊢ 0 < x ^ (1 + p) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
x0 : 0 < x
⊢ 0 < x ^ (1 + p)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | lower_anti | [56, 1] | [84, 26] | ring_nf | c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
x0 : 0 < x
⊢ 1 + p + -2 = p - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
x0 : 0 < x
⊢ 1 + p + -2 = p - 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | lower_anti | [56, 1] | [84, 26] | linarith | c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
x0 : 0 < x
⊢ 1 / 2 ≤ p - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
x0 : 0 < x
⊢ 1 / 2 ≤ p - 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | lower_anti | [56, 1] | [84, 26] | linarith | c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
x0 : 0 < x
p1' : 1 / 2 ≤ p - 1
⊢ 1 ≤ x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
x0 : 0 < x
p1' : 1 / 2 ≤ p - 1
⊢ 1 ≤ x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | lower_anti | [56, 1] | [84, 26] | norm_num | c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
x0 : 0 < x
p1' : 1 / 2 ≤ p - 1
⊢ 0 ≤ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
x0 : 0 < x
p1' : 1 / 2 ≤ p - 1
⊢ 0 ≤ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | lower_anti | [56, 1] | [84, 26] | norm_num | c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
x0 : 0 < x
p1' : 1 / 2 ≤ p - 1
⊢ 0 < 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
k p : ℝ
hd : ∀ (x : ℝ), 4 ≤ x → HasDerivAt (fun x => 1 / x - k / x ^ p) (-(x ^ 2)⁻¹ - k * (-(p * x ^ (p - 1)) / (x ^ p) ^ 2)) x
kp : k * p ≤ 2
hp : 3 / 2 ≤ p
d : DifferentiableOn ℝ (fun x => 1 / x - k / x ^ p) (Ici 4)
x : ℝ
x4 : 4 < x
x0 : 0 < x
p1' : 1 / 2 ≤ p - 1
⊢ 0 < 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | le_potential | [86, 1] | [119, 13] | set s := superF 2 | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
⊢ 0.216 ≤ ⋯.potential c ↑z | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
⊢ 0.216 ≤ s.potential c ↑z | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
⊢ 0.216 ≤ ⋯.potential c ↑z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | le_potential | [86, 1] | [119, 13] | by_cases m : (c,↑z) ∈ s.basin | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
⊢ 0.216 ≤ s.potential c ↑z | case pos
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
⊢ 0.216 ≤ s.potential c ↑z
case neg
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∉ s.basin
⊢ 0.216 ≤ s.potential c ↑z | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
⊢ 0.216 ≤ s.potential c ↑z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | le_potential | [86, 1] | [119, 13] | rcases pass_through c4 z4 m with ⟨n,w4,w20⟩ | case pos
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
⊢ 0.216 ≤ s.potential c ↑z | case pos.intro.intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w4 : 4 < Complex.abs ((f' 2 c)^[n + 1] z)
w20 : Complex.abs ((f' 2 c)^[n + 1] z) ≤ 20
⊢ 0.216 ≤ s.potential c ↑z | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
⊢ 0.216 ≤ s.potential c ↑z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | le_potential | [86, 1] | [119, 13] | generalize hw : (f' 2 c)^[n+1] z = w at w4 w20 | case pos.intro.intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w4 : 4 < Complex.abs ((f' 2 c)^[n + 1] z)
w20 : Complex.abs ((f' 2 c)^[n + 1] z) ≤ 20
⊢ 0.216 ≤ s.potential c ↑z | case pos.intro.intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
⊢ 0.216 ≤ s.potential c ↑z | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w4 : 4 < Complex.abs ((f' 2 c)^[n + 1] z)
w20 : Complex.abs ((f' 2 c)^[n + 1] z) ≤ 20
⊢ 0.216 ≤ s.potential c ↑z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | le_potential | [86, 1] | [119, 13] | have cw : abs c ≤ abs w := by linarith | case pos.intro.intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
⊢ 0.216 ≤ s.potential c ↑z | case pos.intro.intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
⊢ 0.216 ≤ s.potential c ↑z | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
⊢ 0.216 ≤ s.potential c ↑z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | le_potential | [86, 1] | [119, 13] | have pw : 0.0469 ≤ s.potential c w := by
have pw := (abs_le.mp (le_trans (potential_approx 2 w4.le cw)
(potential_error_le_of_z4 2 w4.le cw))).1
rw [le_sub_iff_add_le, neg_add_eq_sub] at pw
have anti := lower_anti 0.8095 1.864 (by norm_num) (by norm_num)
(a := abs w) (b := 20) w4.le (by norm_num) w20
refine le_trans ?_ (le_trans anti pw)
norm_num
have le : (266 : ℝ) ≤ 20 ^ (233 / 125 : ℝ) := by
rw [div_eq_mul_inv, Real.rpow_mul (by positivity), Real.le_rpow_inv_iff_of_pos (by norm_num)
(by positivity) (by positivity)]
norm_num
exact le_trans (by norm_num) (sub_le_sub_left (div_le_div_of_nonneg_left (by norm_num)
(by norm_num) le) _) | case pos.intro.intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
⊢ 0.216 ≤ s.potential c ↑z | case pos.intro.intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 469e-4 ≤ s.potential c ↑w
⊢ 0.216 ≤ s.potential c ↑z | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
⊢ 0.216 ≤ s.potential c ↑z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | le_potential | [86, 1] | [119, 13] | have pwz : s.potential c w = s.potential c z ^ 2^(n+1) := by
simp only [←hw, ←f_f'_iter, s.potential_eqn_iter] | case pos.intro.intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 469e-4 ≤ s.potential c ↑w
⊢ 0.216 ≤ s.potential c ↑z | case pos.intro.intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 469e-4 ≤ s.potential c ↑w
pwz : s.potential c ↑w = s.potential c ↑z ^ 2 ^ (n + 1)
⊢ 0.216 ≤ s.potential c ↑z | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 469e-4 ≤ s.potential c ↑w
⊢ 0.216 ≤ s.potential c ↑z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | le_potential | [86, 1] | [119, 13] | rw [←Real.rpow_natCast] at pwz | case pos.intro.intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 469e-4 ≤ s.potential c ↑w
pwz : s.potential c ↑w = s.potential c ↑z ^ 2 ^ (n + 1)
⊢ 0.216 ≤ s.potential c ↑z | case pos.intro.intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 469e-4 ≤ s.potential c ↑w
pwz : s.potential c ↑w = s.potential c ↑z ^ ↑(2 ^ (n + 1))
⊢ 0.216 ≤ s.potential c ↑z | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 469e-4 ≤ s.potential c ↑w
pwz : s.potential c ↑w = s.potential c ↑z ^ 2 ^ (n + 1)
⊢ 0.216 ≤ s.potential c ↑z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | le_potential | [86, 1] | [119, 13] | rw [←Real.rpow_inv_eq s.potential_nonneg s.potential_nonneg
(NeZero.natCast_ne (2^(n+1)) ℝ)] at pwz | case pos.intro.intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 469e-4 ≤ s.potential c ↑w
pwz : s.potential c ↑w = s.potential c ↑z ^ ↑(2 ^ (n + 1))
⊢ 0.216 ≤ s.potential c ↑z | case pos.intro.intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 469e-4 ≤ s.potential c ↑w
pwz : s.potential c ↑w ^ (↑(2 ^ (n + 1)))⁻¹ = s.potential c ↑z
⊢ 0.216 ≤ s.potential c ↑z | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 469e-4 ≤ s.potential c ↑w
pwz : s.potential c ↑w = s.potential c ↑z ^ ↑(2 ^ (n + 1))
⊢ 0.216 ≤ s.potential c ↑z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | le_potential | [86, 1] | [119, 13] | rw [←pwz] | case pos.intro.intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 469e-4 ≤ s.potential c ↑w
pwz : s.potential c ↑w ^ (↑(2 ^ (n + 1)))⁻¹ = s.potential c ↑z
⊢ 0.216 ≤ s.potential c ↑z | case pos.intro.intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 469e-4 ≤ s.potential c ↑w
pwz : s.potential c ↑w ^ (↑(2 ^ (n + 1)))⁻¹ = s.potential c ↑z
⊢ 0.216 ≤ s.potential c ↑w ^ (↑(2 ^ (n + 1)))⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 469e-4 ≤ s.potential c ↑w
pwz : s.potential c ↑w ^ (↑(2 ^ (n + 1)))⁻¹ = s.potential c ↑z
⊢ 0.216 ≤ s.potential c ↑z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | le_potential | [86, 1] | [119, 13] | refine le_trans ?_ (Real.rpow_le_rpow (by norm_num) pw (by positivity)) | case pos.intro.intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 469e-4 ≤ s.potential c ↑w
pwz : s.potential c ↑w ^ (↑(2 ^ (n + 1)))⁻¹ = s.potential c ↑z
⊢ 0.216 ≤ s.potential c ↑w ^ (↑(2 ^ (n + 1)))⁻¹ | case pos.intro.intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 469e-4 ≤ s.potential c ↑w
pwz : s.potential c ↑w ^ (↑(2 ^ (n + 1)))⁻¹ = s.potential c ↑z
⊢ 0.216 ≤ 469e-4 ^ (↑(2 ^ (n + 1)))⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 469e-4 ≤ s.potential c ↑w
pwz : s.potential c ↑w ^ (↑(2 ^ (n + 1)))⁻¹ = s.potential c ↑z
⊢ 0.216 ≤ s.potential c ↑w ^ (↑(2 ^ (n + 1)))⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | le_potential | [86, 1] | [119, 13] | rw [Real.le_rpow_inv_iff_of_pos (by norm_num) (by norm_num) (by positivity), Real.rpow_natCast] | case pos.intro.intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 469e-4 ≤ s.potential c ↑w
pwz : s.potential c ↑w ^ (↑(2 ^ (n + 1)))⁻¹ = s.potential c ↑z
⊢ 0.216 ≤ 469e-4 ^ (↑(2 ^ (n + 1)))⁻¹ | case pos.intro.intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 469e-4 ≤ s.potential c ↑w
pwz : s.potential c ↑w ^ (↑(2 ^ (n + 1)))⁻¹ = s.potential c ↑z
⊢ 0.216 ^ 2 ^ (n + 1) ≤ 469e-4 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 469e-4 ≤ s.potential c ↑w
pwz : s.potential c ↑w ^ (↑(2 ^ (n + 1)))⁻¹ = s.potential c ↑z
⊢ 0.216 ≤ 469e-4 ^ (↑(2 ^ (n + 1)))⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | le_potential | [86, 1] | [119, 13] | refine le_trans (pow_le_pow_of_le_one (by norm_num) (by norm_num)
(pow_le_pow_right (by norm_num) (Nat.le_add_left 1 n))) ?_ | case pos.intro.intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 469e-4 ≤ s.potential c ↑w
pwz : s.potential c ↑w ^ (↑(2 ^ (n + 1)))⁻¹ = s.potential c ↑z
⊢ 0.216 ^ 2 ^ (n + 1) ≤ 469e-4 | case pos.intro.intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 469e-4 ≤ s.potential c ↑w
pwz : s.potential c ↑w ^ (↑(2 ^ (n + 1)))⁻¹ = s.potential c ↑z
⊢ 0.216 ^ 2 ^ 1 ≤ 469e-4 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 469e-4 ≤ s.potential c ↑w
pwz : s.potential c ↑w ^ (↑(2 ^ (n + 1)))⁻¹ = s.potential c ↑z
⊢ 0.216 ^ 2 ^ (n + 1) ≤ 469e-4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | le_potential | [86, 1] | [119, 13] | norm_num | case pos.intro.intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 469e-4 ≤ s.potential c ↑w
pwz : s.potential c ↑w ^ (↑(2 ^ (n + 1)))⁻¹ = s.potential c ↑z
⊢ 0.216 ^ 2 ^ 1 ≤ 469e-4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 469e-4 ≤ s.potential c ↑w
pwz : s.potential c ↑w ^ (↑(2 ^ (n + 1)))⁻¹ = s.potential c ↑z
⊢ 0.216 ^ 2 ^ 1 ≤ 469e-4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | le_potential | [86, 1] | [119, 13] | linarith | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
⊢ Complex.abs c ≤ Complex.abs w | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
⊢ Complex.abs c ≤ Complex.abs w
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | le_potential | [86, 1] | [119, 13] | have pw := (abs_le.mp (le_trans (potential_approx 2 w4.le cw)
(potential_error_le_of_z4 2 w4.le cw))).1 | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
⊢ 469e-4 ≤ s.potential c ↑w | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : -(0.8095 / Complex.abs w ^ 1.864) ≤ ⋯.potential c ↑w - 1 / Complex.abs w
⊢ 469e-4 ≤ s.potential c ↑w | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
⊢ 469e-4 ≤ s.potential c ↑w
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | le_potential | [86, 1] | [119, 13] | rw [le_sub_iff_add_le, neg_add_eq_sub] at pw | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : -(0.8095 / Complex.abs w ^ 1.864) ≤ ⋯.potential c ↑w - 1 / Complex.abs w
⊢ 469e-4 ≤ s.potential c ↑w | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 1 / Complex.abs w - 0.8095 / Complex.abs w ^ 1.864 ≤ ⋯.potential c ↑w
⊢ 469e-4 ≤ s.potential c ↑w | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : -(0.8095 / Complex.abs w ^ 1.864) ≤ ⋯.potential c ↑w - 1 / Complex.abs w
⊢ 469e-4 ≤ s.potential c ↑w
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | le_potential | [86, 1] | [119, 13] | have anti := lower_anti 0.8095 1.864 (by norm_num) (by norm_num)
(a := abs w) (b := 20) w4.le (by norm_num) w20 | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 1 / Complex.abs w - 0.8095 / Complex.abs w ^ 1.864 ≤ ⋯.potential c ↑w
⊢ 469e-4 ≤ s.potential c ↑w | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 1 / Complex.abs w - 0.8095 / Complex.abs w ^ 1.864 ≤ ⋯.potential c ↑w
anti : (fun x => 1 / x - 0.8095 / x ^ 1.864) 20 ≤ (fun x => 1 / x - 0.8095 / x ^ 1.864) (Complex.abs w)
⊢ 469e-4 ≤ s.potential c ↑w | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 1 / Complex.abs w - 0.8095 / Complex.abs w ^ 1.864 ≤ ⋯.potential c ↑w
⊢ 469e-4 ≤ s.potential c ↑w
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | le_potential | [86, 1] | [119, 13] | refine le_trans ?_ (le_trans anti pw) | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 1 / Complex.abs w - 0.8095 / Complex.abs w ^ 1.864 ≤ ⋯.potential c ↑w
anti : (fun x => 1 / x - 0.8095 / x ^ 1.864) 20 ≤ (fun x => 1 / x - 0.8095 / x ^ 1.864) (Complex.abs w)
⊢ 469e-4 ≤ s.potential c ↑w | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 1 / Complex.abs w - 0.8095 / Complex.abs w ^ 1.864 ≤ ⋯.potential c ↑w
anti : (fun x => 1 / x - 0.8095 / x ^ 1.864) 20 ≤ (fun x => 1 / x - 0.8095 / x ^ 1.864) (Complex.abs w)
⊢ 469e-4 ≤ (fun x => 1 / x - 0.8095 / x ^ 1.864) 20 | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 1 / Complex.abs w - 0.8095 / Complex.abs w ^ 1.864 ≤ ⋯.potential c ↑w
anti : (fun x => 1 / x - 0.8095 / x ^ 1.864) 20 ≤ (fun x => 1 / x - 0.8095 / x ^ 1.864) (Complex.abs w)
⊢ 469e-4 ≤ s.potential c ↑w
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | le_potential | [86, 1] | [119, 13] | norm_num | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 1 / Complex.abs w - 0.8095 / Complex.abs w ^ 1.864 ≤ ⋯.potential c ↑w
anti : (fun x => 1 / x - 0.8095 / x ^ 1.864) 20 ≤ (fun x => 1 / x - 0.8095 / x ^ 1.864) (Complex.abs w)
⊢ 469e-4 ≤ (fun x => 1 / x - 0.8095 / x ^ 1.864) 20 | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 1 / Complex.abs w - 0.8095 / Complex.abs w ^ 1.864 ≤ ⋯.potential c ↑w
anti : (fun x => 1 / x - 0.8095 / x ^ 1.864) 20 ≤ (fun x => 1 / x - 0.8095 / x ^ 1.864) (Complex.abs w)
⊢ 469 / 10000 ≤ 1 / 20 - 1619 / 2000 / 20 ^ (233 / 125) | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 1 / Complex.abs w - 0.8095 / Complex.abs w ^ 1.864 ≤ ⋯.potential c ↑w
anti : (fun x => 1 / x - 0.8095 / x ^ 1.864) 20 ≤ (fun x => 1 / x - 0.8095 / x ^ 1.864) (Complex.abs w)
⊢ 469e-4 ≤ (fun x => 1 / x - 0.8095 / x ^ 1.864) 20
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | le_potential | [86, 1] | [119, 13] | have le : (266 : ℝ) ≤ 20 ^ (233 / 125 : ℝ) := by
rw [div_eq_mul_inv, Real.rpow_mul (by positivity), Real.le_rpow_inv_iff_of_pos (by norm_num)
(by positivity) (by positivity)]
norm_num | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 1 / Complex.abs w - 0.8095 / Complex.abs w ^ 1.864 ≤ ⋯.potential c ↑w
anti : (fun x => 1 / x - 0.8095 / x ^ 1.864) 20 ≤ (fun x => 1 / x - 0.8095 / x ^ 1.864) (Complex.abs w)
⊢ 469 / 10000 ≤ 1 / 20 - 1619 / 2000 / 20 ^ (233 / 125) | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 1 / Complex.abs w - 0.8095 / Complex.abs w ^ 1.864 ≤ ⋯.potential c ↑w
anti : (fun x => 1 / x - 0.8095 / x ^ 1.864) 20 ≤ (fun x => 1 / x - 0.8095 / x ^ 1.864) (Complex.abs w)
le : 266 ≤ 20 ^ (233 / 125)
⊢ 469 / 10000 ≤ 1 / 20 - 1619 / 2000 / 20 ^ (233 / 125) | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 1 / Complex.abs w - 0.8095 / Complex.abs w ^ 1.864 ≤ ⋯.potential c ↑w
anti : (fun x => 1 / x - 0.8095 / x ^ 1.864) 20 ≤ (fun x => 1 / x - 0.8095 / x ^ 1.864) (Complex.abs w)
⊢ 469 / 10000 ≤ 1 / 20 - 1619 / 2000 / 20 ^ (233 / 125)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | le_potential | [86, 1] | [119, 13] | exact le_trans (by norm_num) (sub_le_sub_left (div_le_div_of_nonneg_left (by norm_num)
(by norm_num) le) _) | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 1 / Complex.abs w - 0.8095 / Complex.abs w ^ 1.864 ≤ ⋯.potential c ↑w
anti : (fun x => 1 / x - 0.8095 / x ^ 1.864) 20 ≤ (fun x => 1 / x - 0.8095 / x ^ 1.864) (Complex.abs w)
le : 266 ≤ 20 ^ (233 / 125)
⊢ 469 / 10000 ≤ 1 / 20 - 1619 / 2000 / 20 ^ (233 / 125) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 1 / Complex.abs w - 0.8095 / Complex.abs w ^ 1.864 ≤ ⋯.potential c ↑w
anti : (fun x => 1 / x - 0.8095 / x ^ 1.864) 20 ≤ (fun x => 1 / x - 0.8095 / x ^ 1.864) (Complex.abs w)
le : 266 ≤ 20 ^ (233 / 125)
⊢ 469 / 10000 ≤ 1 / 20 - 1619 / 2000 / 20 ^ (233 / 125)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | le_potential | [86, 1] | [119, 13] | norm_num | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 1 / Complex.abs w - 0.8095 / Complex.abs w ^ 1.864 ≤ ⋯.potential c ↑w
⊢ 0.8095 * 1.864 ≤ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 1 / Complex.abs w - 0.8095 / Complex.abs w ^ 1.864 ≤ ⋯.potential c ↑w
⊢ 0.8095 * 1.864 ≤ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | le_potential | [86, 1] | [119, 13] | norm_num | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 1 / Complex.abs w - 0.8095 / Complex.abs w ^ 1.864 ≤ ⋯.potential c ↑w
⊢ 3 / 2 ≤ 1.864 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 1 / Complex.abs w - 0.8095 / Complex.abs w ^ 1.864 ≤ ⋯.potential c ↑w
⊢ 3 / 2 ≤ 1.864
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | le_potential | [86, 1] | [119, 13] | norm_num | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 1 / Complex.abs w - 0.8095 / Complex.abs w ^ 1.864 ≤ ⋯.potential c ↑w
⊢ 20 ∈ Ici 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 1 / Complex.abs w - 0.8095 / Complex.abs w ^ 1.864 ≤ ⋯.potential c ↑w
⊢ 20 ∈ Ici 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | le_potential | [86, 1] | [119, 13] | rw [div_eq_mul_inv, Real.rpow_mul (by positivity), Real.le_rpow_inv_iff_of_pos (by norm_num)
(by positivity) (by positivity)] | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 1 / Complex.abs w - 0.8095 / Complex.abs w ^ 1.864 ≤ ⋯.potential c ↑w
anti : (fun x => 1 / x - 0.8095 / x ^ 1.864) 20 ≤ (fun x => 1 / x - 0.8095 / x ^ 1.864) (Complex.abs w)
⊢ 266 ≤ 20 ^ (233 / 125) | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 1 / Complex.abs w - 0.8095 / Complex.abs w ^ 1.864 ≤ ⋯.potential c ↑w
anti : (fun x => 1 / x - 0.8095 / x ^ 1.864) 20 ≤ (fun x => 1 / x - 0.8095 / x ^ 1.864) (Complex.abs w)
⊢ 266 ^ 125 ≤ 20 ^ 233 | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 1 / Complex.abs w - 0.8095 / Complex.abs w ^ 1.864 ≤ ⋯.potential c ↑w
anti : (fun x => 1 / x - 0.8095 / x ^ 1.864) 20 ≤ (fun x => 1 / x - 0.8095 / x ^ 1.864) (Complex.abs w)
⊢ 266 ≤ 20 ^ (233 / 125)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | le_potential | [86, 1] | [119, 13] | norm_num | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 1 / Complex.abs w - 0.8095 / Complex.abs w ^ 1.864 ≤ ⋯.potential c ↑w
anti : (fun x => 1 / x - 0.8095 / x ^ 1.864) 20 ≤ (fun x => 1 / x - 0.8095 / x ^ 1.864) (Complex.abs w)
⊢ 266 ^ 125 ≤ 20 ^ 233 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 1 / Complex.abs w - 0.8095 / Complex.abs w ^ 1.864 ≤ ⋯.potential c ↑w
anti : (fun x => 1 / x - 0.8095 / x ^ 1.864) 20 ≤ (fun x => 1 / x - 0.8095 / x ^ 1.864) (Complex.abs w)
⊢ 266 ^ 125 ≤ 20 ^ 233
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | le_potential | [86, 1] | [119, 13] | positivity | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 1 / Complex.abs w - 0.8095 / Complex.abs w ^ 1.864 ≤ ⋯.potential c ↑w
anti : (fun x => 1 / x - 0.8095 / x ^ 1.864) 20 ≤ (fun x => 1 / x - 0.8095 / x ^ 1.864) (Complex.abs w)
⊢ 0 ≤ 20 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 1 / Complex.abs w - 0.8095 / Complex.abs w ^ 1.864 ≤ ⋯.potential c ↑w
anti : (fun x => 1 / x - 0.8095 / x ^ 1.864) 20 ≤ (fun x => 1 / x - 0.8095 / x ^ 1.864) (Complex.abs w)
⊢ 0 ≤ 20
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | le_potential | [86, 1] | [119, 13] | norm_num | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 1 / Complex.abs w - 0.8095 / Complex.abs w ^ 1.864 ≤ ⋯.potential c ↑w
anti : (fun x => 1 / x - 0.8095 / x ^ 1.864) 20 ≤ (fun x => 1 / x - 0.8095 / x ^ 1.864) (Complex.abs w)
⊢ 0 ≤ 266 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 1 / Complex.abs w - 0.8095 / Complex.abs w ^ 1.864 ≤ ⋯.potential c ↑w
anti : (fun x => 1 / x - 0.8095 / x ^ 1.864) 20 ≤ (fun x => 1 / x - 0.8095 / x ^ 1.864) (Complex.abs w)
⊢ 0 ≤ 266
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | le_potential | [86, 1] | [119, 13] | positivity | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 1 / Complex.abs w - 0.8095 / Complex.abs w ^ 1.864 ≤ ⋯.potential c ↑w
anti : (fun x => 1 / x - 0.8095 / x ^ 1.864) 20 ≤ (fun x => 1 / x - 0.8095 / x ^ 1.864) (Complex.abs w)
⊢ 0 ≤ 20 ^ 233 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 1 / Complex.abs w - 0.8095 / Complex.abs w ^ 1.864 ≤ ⋯.potential c ↑w
anti : (fun x => 1 / x - 0.8095 / x ^ 1.864) 20 ≤ (fun x => 1 / x - 0.8095 / x ^ 1.864) (Complex.abs w)
⊢ 0 ≤ 20 ^ 233
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | le_potential | [86, 1] | [119, 13] | positivity | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 1 / Complex.abs w - 0.8095 / Complex.abs w ^ 1.864 ≤ ⋯.potential c ↑w
anti : (fun x => 1 / x - 0.8095 / x ^ 1.864) 20 ≤ (fun x => 1 / x - 0.8095 / x ^ 1.864) (Complex.abs w)
⊢ 0 < 125 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 1 / Complex.abs w - 0.8095 / Complex.abs w ^ 1.864 ≤ ⋯.potential c ↑w
anti : (fun x => 1 / x - 0.8095 / x ^ 1.864) 20 ≤ (fun x => 1 / x - 0.8095 / x ^ 1.864) (Complex.abs w)
⊢ 0 < 125
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | le_potential | [86, 1] | [119, 13] | norm_num | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 1 / Complex.abs w - 0.8095 / Complex.abs w ^ 1.864 ≤ ⋯.potential c ↑w
anti : (fun x => 1 / x - 0.8095 / x ^ 1.864) 20 ≤ (fun x => 1 / x - 0.8095 / x ^ 1.864) (Complex.abs w)
le : 266 ≤ 20 ^ (233 / 125)
⊢ 469 / 10000 ≤ 1 / 20 - 1619 / 2000 / 266 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 1 / Complex.abs w - 0.8095 / Complex.abs w ^ 1.864 ≤ ⋯.potential c ↑w
anti : (fun x => 1 / x - 0.8095 / x ^ 1.864) 20 ≤ (fun x => 1 / x - 0.8095 / x ^ 1.864) (Complex.abs w)
le : 266 ≤ 20 ^ (233 / 125)
⊢ 469 / 10000 ≤ 1 / 20 - 1619 / 2000 / 266
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | le_potential | [86, 1] | [119, 13] | norm_num | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 1 / Complex.abs w - 0.8095 / Complex.abs w ^ 1.864 ≤ ⋯.potential c ↑w
anti : (fun x => 1 / x - 0.8095 / x ^ 1.864) 20 ≤ (fun x => 1 / x - 0.8095 / x ^ 1.864) (Complex.abs w)
le : 266 ≤ 20 ^ (233 / 125)
⊢ 0 ≤ 1619 / 2000 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 1 / Complex.abs w - 0.8095 / Complex.abs w ^ 1.864 ≤ ⋯.potential c ↑w
anti : (fun x => 1 / x - 0.8095 / x ^ 1.864) 20 ≤ (fun x => 1 / x - 0.8095 / x ^ 1.864) (Complex.abs w)
le : 266 ≤ 20 ^ (233 / 125)
⊢ 0 ≤ 1619 / 2000
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | le_potential | [86, 1] | [119, 13] | norm_num | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 1 / Complex.abs w - 0.8095 / Complex.abs w ^ 1.864 ≤ ⋯.potential c ↑w
anti : (fun x => 1 / x - 0.8095 / x ^ 1.864) 20 ≤ (fun x => 1 / x - 0.8095 / x ^ 1.864) (Complex.abs w)
le : 266 ≤ 20 ^ (233 / 125)
⊢ 0 < 266 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 1 / Complex.abs w - 0.8095 / Complex.abs w ^ 1.864 ≤ ⋯.potential c ↑w
anti : (fun x => 1 / x - 0.8095 / x ^ 1.864) 20 ≤ (fun x => 1 / x - 0.8095 / x ^ 1.864) (Complex.abs w)
le : 266 ≤ 20 ^ (233 / 125)
⊢ 0 < 266
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | le_potential | [86, 1] | [119, 13] | simp only [←hw, ←f_f'_iter, s.potential_eqn_iter] | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 469e-4 ≤ s.potential c ↑w
⊢ s.potential c ↑w = s.potential c ↑z ^ 2 ^ (n + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 469e-4 ≤ s.potential c ↑w
⊢ s.potential c ↑w = s.potential c ↑z ^ 2 ^ (n + 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | le_potential | [86, 1] | [119, 13] | norm_num | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 469e-4 ≤ s.potential c ↑w
pwz : s.potential c ↑w ^ (↑(2 ^ (n + 1)))⁻¹ = s.potential c ↑z
⊢ 0 ≤ 469e-4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 469e-4 ≤ s.potential c ↑w
pwz : s.potential c ↑w ^ (↑(2 ^ (n + 1)))⁻¹ = s.potential c ↑z
⊢ 0 ≤ 469e-4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | le_potential | [86, 1] | [119, 13] | positivity | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 469e-4 ≤ s.potential c ↑w
pwz : s.potential c ↑w ^ (↑(2 ^ (n + 1)))⁻¹ = s.potential c ↑z
⊢ 0 ≤ (↑(2 ^ (n + 1)))⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 469e-4 ≤ s.potential c ↑w
pwz : s.potential c ↑w ^ (↑(2 ^ (n + 1)))⁻¹ = s.potential c ↑z
⊢ 0 ≤ (↑(2 ^ (n + 1)))⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | le_potential | [86, 1] | [119, 13] | norm_num | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 469e-4 ≤ s.potential c ↑w
pwz : s.potential c ↑w ^ (↑(2 ^ (n + 1)))⁻¹ = s.potential c ↑z
⊢ 0 ≤ 0.216 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 469e-4 ≤ s.potential c ↑w
pwz : s.potential c ↑w ^ (↑(2 ^ (n + 1)))⁻¹ = s.potential c ↑z
⊢ 0 ≤ 0.216
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | le_potential | [86, 1] | [119, 13] | positivity | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 469e-4 ≤ s.potential c ↑w
pwz : s.potential c ↑w ^ (↑(2 ^ (n + 1)))⁻¹ = s.potential c ↑z
⊢ 0 < ↑(2 ^ (n + 1)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 469e-4 ≤ s.potential c ↑w
pwz : s.potential c ↑w ^ (↑(2 ^ (n + 1)))⁻¹ = s.potential c ↑z
⊢ 0 < ↑(2 ^ (n + 1))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | le_potential | [86, 1] | [119, 13] | norm_num | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 469e-4 ≤ s.potential c ↑w
pwz : s.potential c ↑w ^ (↑(2 ^ (n + 1)))⁻¹ = s.potential c ↑z
⊢ 0.216 ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 469e-4 ≤ s.potential c ↑w
pwz : s.potential c ↑w ^ (↑(2 ^ (n + 1)))⁻¹ = s.potential c ↑z
⊢ 0.216 ≤ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | le_potential | [86, 1] | [119, 13] | norm_num | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 469e-4 ≤ s.potential c ↑w
pwz : s.potential c ↑w ^ (↑(2 ^ (n + 1)))⁻¹ = s.potential c ↑z
⊢ 1 ≤ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
n : ℕ
w : ℂ
hw : (f' 2 c)^[n + 1] z = w
w4 : 4 < Complex.abs w
w20 : Complex.abs w ≤ 20
cw : Complex.abs c ≤ Complex.abs w
pw : 469e-4 ≤ s.potential c ↑w
pwz : s.potential c ↑w ^ (↑(2 ^ (n + 1)))⁻¹ = s.potential c ↑z
⊢ 1 ≤ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | le_potential | [86, 1] | [119, 13] | rw [s.potential_eq_one (not_exists.mp m)] | case neg
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∉ s.basin
⊢ 0.216 ≤ s.potential c ↑z | case neg
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∉ s.basin
⊢ 0.216 ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∉ s.basin
⊢ 0.216 ≤ s.potential c ↑z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | le_potential | [86, 1] | [119, 13] | norm_num | case neg
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∉ s.basin
⊢ 0.216 ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∉ s.basin
⊢ 0.216 ≤ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | spheres_subset_closedBall | [83, 1] | [88, 44] | rw [←closedBall_prod_same, Set.subset_def] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c1✝ w0 w1 : ℂ
r✝ b : ℝ
c0 c1 : ℂ
r : ℝ
⊢ sphere c0 r ×ˢ sphere c1 r ⊆ closedBall (c0, c1) r | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c1✝ w0 w1 : ℂ
r✝ b : ℝ
c0 c1 : ℂ
r : ℝ
⊢ ∀ x ∈ sphere c0 r ×ˢ sphere c1 r, x ∈ closedBall c0 r ×ˢ closedBall c1 r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c1✝ w0 w1 : ℂ
r✝ b : ℝ
c0 c1 : ℂ
r : ℝ
⊢ sphere c0 r ×ˢ sphere c1 r ⊆ closedBall (c0, c1) r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | spheres_subset_closedBall | [83, 1] | [88, 44] | intro z | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c1✝ w0 w1 : ℂ
r✝ b : ℝ
c0 c1 : ℂ
r : ℝ
⊢ ∀ x ∈ sphere c0 r ×ˢ sphere c1 r, x ∈ closedBall c0 r ×ˢ closedBall c1 r | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c1✝ w0 w1 : ℂ
r✝ b : ℝ
c0 c1 : ℂ
r : ℝ
z : ℂ × ℂ
⊢ z ∈ sphere c0 r ×ˢ sphere c1 r → z ∈ closedBall c0 r ×ˢ closedBall c1 r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c1✝ w0 w1 : ℂ
r✝ b : ℝ
c0 c1 : ℂ
r : ℝ
⊢ ∀ x ∈ sphere c0 r ×ˢ sphere c1 r, x ∈ closedBall c0 r ×ˢ closedBall c1 r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | spheres_subset_closedBall | [83, 1] | [88, 44] | simp only [Set.mem_prod, mem_sphere_iff_norm, Complex.norm_eq_abs, Metric.mem_closedBall, and_imp] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c1✝ w0 w1 : ℂ
r✝ b : ℝ
c0 c1 : ℂ
r : ℝ
z : ℂ × ℂ
⊢ z ∈ sphere c0 r ×ˢ sphere c1 r → z ∈ closedBall c0 r ×ˢ closedBall c1 r | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c1✝ w0 w1 : ℂ
r✝ b : ℝ
c0 c1 : ℂ
r : ℝ
z : ℂ × ℂ
⊢ Complex.abs (z.1 - c0) = r → Complex.abs (z.2 - c1) = r → dist z.1 c0 ≤ r ∧ dist z.2 c1 ≤ r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c1✝ w0 w1 : ℂ
r✝ b : ℝ
c0 c1 : ℂ
r : ℝ
z : ℂ × ℂ
⊢ z ∈ sphere c0 r ×ˢ sphere c1 r → z ∈ closedBall c0 r ×ˢ closedBall c1 r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | spheres_subset_closedBall | [83, 1] | [88, 44] | rw [Complex.dist_eq, Complex.dist_eq] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c1✝ w0 w1 : ℂ
r✝ b : ℝ
c0 c1 : ℂ
r : ℝ
z : ℂ × ℂ
⊢ Complex.abs (z.1 - c0) = r → Complex.abs (z.2 - c1) = r → dist z.1 c0 ≤ r ∧ dist z.2 c1 ≤ r | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c1✝ w0 w1 : ℂ
r✝ b : ℝ
c0 c1 : ℂ
r : ℝ
z : ℂ × ℂ
⊢ Complex.abs (z.1 - c0) = r → Complex.abs (z.2 - c1) = r → Complex.abs (z.1 - c0) ≤ r ∧ Complex.abs (z.2 - c1) ≤ r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c1✝ w0 w1 : ℂ
r✝ b : ℝ
c0 c1 : ℂ
r : ℝ
z : ℂ × ℂ
⊢ Complex.abs (z.1 - c0) = r → Complex.abs (z.2 - c1) = r → dist z.1 c0 ≤ r ∧ dist z.2 c1 ≤ r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | spheres_subset_closedBall | [83, 1] | [88, 44] | intro a b | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c1✝ w0 w1 : ℂ
r✝ b : ℝ
c0 c1 : ℂ
r : ℝ
z : ℂ × ℂ
⊢ Complex.abs (z.1 - c0) = r → Complex.abs (z.2 - c1) = r → Complex.abs (z.1 - c0) ≤ r ∧ Complex.abs (z.2 - c1) ≤ r | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c1✝ w0 w1 : ℂ
r✝ b✝ : ℝ
c0 c1 : ℂ
r : ℝ
z : ℂ × ℂ
a : Complex.abs (z.1 - c0) = r
b : Complex.abs (z.2 - c1) = r
⊢ Complex.abs (z.1 - c0) ≤ r ∧ Complex.abs (z.2 - c1) ≤ r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c1✝ w0 w1 : ℂ
r✝ b : ℝ
c0 c1 : ℂ
r : ℝ
z : ℂ × ℂ
⊢ Complex.abs (z.1 - c0) = r → Complex.abs (z.2 - c1) = r → Complex.abs (z.1 - c0) ≤ r ∧ Complex.abs (z.2 - c1) ≤ r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | spheres_subset_closedBall | [83, 1] | [88, 44] | exact ⟨le_of_eq a, le_of_eq b⟩ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c1✝ w0 w1 : ℂ
r✝ b✝ : ℝ
c0 c1 : ℂ
r : ℝ
z : ℂ × ℂ
a : Complex.abs (z.1 - c0) = r
b : Complex.abs (z.2 - c1) = r
⊢ Complex.abs (z.1 - c0) ≤ r ∧ Complex.abs (z.2 - c1) ≤ r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c1✝ w0 w1 : ℂ
r✝ b✝ : ℝ
c0 c1 : ℂ
r : ℝ
z : ℂ × ℂ
a : Complex.abs (z.1 - c0) = r
b : Complex.abs (z.2 - c1) = r
⊢ Complex.abs (z.1 - c0) ≤ r ∧ Complex.abs (z.2 - c1) ≤ r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | mem_open_closed | [93, 1] | [94, 69] | simp only [Metric.mem_ball, Metric.mem_closedBall] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
z c : ℂ
r : ℝ
⊢ z ∈ ball c r → z ∈ closedBall c r | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
z c : ℂ
r : ℝ
⊢ dist z c < r → dist z c ≤ r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
z c : ℂ
r : ℝ
⊢ z ∈ ball c r → z ∈ closedBall c r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | mem_open_closed | [93, 1] | [94, 69] | exact le_of_lt | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
z c : ℂ
r : ℝ
⊢ dist z c < r → dist z c ≤ r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
z c : ℂ
r : ℝ
⊢ dist z c < r → dist z c ≤ r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | mem_sphere_closed | [96, 1] | [97, 94] | simp only [mem_sphere_iff_norm, Complex.norm_eq_abs, Metric.mem_closedBall] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
z c : ℂ
r : ℝ
⊢ z ∈ sphere c r → z ∈ closedBall c r | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
z c : ℂ
r : ℝ
⊢ Complex.abs (z - c) = r → dist z c ≤ r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
z c : ℂ
r : ℝ
⊢ z ∈ sphere c r → z ∈ closedBall c r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | mem_sphere_closed | [96, 1] | [97, 94] | exact le_of_eq | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
z c : ℂ
r : ℝ
⊢ Complex.abs (z - c) = r → dist z c ≤ r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
z c : ℂ
r : ℝ
⊢ Complex.abs (z - c) = r → dist z c ≤ r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | center_not_in_sphere | [100, 1] | [102, 50] | simp only [mem_sphere_iff_norm, Complex.norm_eq_abs] at zs | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
c z : ℂ
r : ℝ
rp : r > 0
zs : z ∈ sphere c r
⊢ z - c ≠ 0 | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
c z : ℂ
r : ℝ
rp : r > 0
zs : Complex.abs (z - c) = r
⊢ z - c ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
c z : ℂ
r : ℝ
rp : r > 0
zs : z ∈ sphere c r
⊢ z - c ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | center_not_in_sphere | [100, 1] | [102, 50] | rw [←Complex.abs.ne_zero_iff, zs] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
c z : ℂ
r : ℝ
rp : r > 0
zs : Complex.abs (z - c) = r
⊢ z - c ≠ 0 | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
c z : ℂ
r : ℝ
rp : r > 0
zs : Complex.abs (z - c) = r
⊢ r ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
c z : ℂ
r : ℝ
rp : r > 0
zs : Complex.abs (z - c) = r
⊢ z - c ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | center_not_in_sphere | [100, 1] | [102, 50] | exact rp.ne' | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
c z : ℂ
r : ℝ
rp : r > 0
zs : Complex.abs (z - c) = r
⊢ r ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
c z : ℂ
r : ℝ
rp : r > 0
zs : Complex.abs (z - c) = r
⊢ r ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | Separate.fc0 | [105, 1] | [110, 83] | refine ContinuousOn.comp h.fc ?_ ?_ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w1m : w1 ∈ ball c1 r
⊢ ContinuousOn (fun z0 => f (z0, w1)) (closedBall c0 r) | case refine_1
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w1m : w1 ∈ ball c1 r
⊢ ContinuousOn (fun z0 => (z0, w1)) (closedBall c0 r)
case refine_2
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w1m : w1 ∈ ball c1 r
⊢ Set.MapsTo (fun z0 => (z0, w1)) (closedBall c0 r) s | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w1m : w1 ∈ ball c1 r
⊢ ContinuousOn (fun z0 => f (z0, w1)) (closedBall c0 r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | Separate.fc0 | [105, 1] | [110, 83] | exact ContinuousOn.prod continuousOn_id continuousOn_const | case refine_1
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w1m : w1 ∈ ball c1 r
⊢ ContinuousOn (fun z0 => (z0, w1)) (closedBall c0 r) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w1m : w1 ∈ ball c1 r
⊢ ContinuousOn (fun z0 => (z0, w1)) (closedBall c0 r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | Separate.fc0 | [105, 1] | [110, 83] | intro z0 z0m | case refine_2
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w1m : w1 ∈ ball c1 r
⊢ Set.MapsTo (fun z0 => (z0, w1)) (closedBall c0 r) s | case refine_2
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w1m : w1 ∈ ball c1 r
z0 : ℂ
z0m : z0 ∈ closedBall c0 r
⊢ (fun z0 => (z0, w1)) z0 ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w1m : w1 ∈ ball c1 r
⊢ Set.MapsTo (fun z0 => (z0, w1)) (closedBall c0 r) s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | Separate.fc0 | [105, 1] | [110, 83] | apply h.rs | case refine_2
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w1m : w1 ∈ ball c1 r
z0 : ℂ
z0m : z0 ∈ closedBall c0 r
⊢ (fun z0 => (z0, w1)) z0 ∈ s | case refine_2.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w1m : w1 ∈ ball c1 r
z0 : ℂ
z0m : z0 ∈ closedBall c0 r
⊢ (fun z0 => (z0, w1)) z0 ∈ closedBall (c0, c1) r | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w1m : w1 ∈ ball c1 r
z0 : ℂ
z0m : z0 ∈ closedBall c0 r
⊢ (fun z0 => (z0, w1)) z0 ∈ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | Separate.fc0 | [105, 1] | [110, 83] | rw [← closedBall_prod_same] | case refine_2.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w1m : w1 ∈ ball c1 r
z0 : ℂ
z0m : z0 ∈ closedBall c0 r
⊢ (fun z0 => (z0, w1)) z0 ∈ closedBall (c0, c1) r | case refine_2.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w1m : w1 ∈ ball c1 r
z0 : ℂ
z0m : z0 ∈ closedBall c0 r
⊢ (fun z0 => (z0, w1)) z0 ∈ closedBall c0 r ×ˢ closedBall c1 r | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w1m : w1 ∈ ball c1 r
z0 : ℂ
z0m : z0 ∈ closedBall c0 r
⊢ (fun z0 => (z0, w1)) z0 ∈ closedBall (c0, c1) r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | Separate.fc0 | [105, 1] | [110, 83] | exact Set.mem_prod.mpr ⟨z0m, mem_open_closed w1m⟩ | case refine_2.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w1m : w1 ∈ ball c1 r
z0 : ℂ
z0m : z0 ∈ closedBall c0 r
⊢ (fun z0 => (z0, w1)) z0 ∈ closedBall c0 r ×ˢ closedBall c1 r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w1m : w1 ∈ ball c1 r
z0 : ℂ
z0m : z0 ∈ closedBall c0 r
⊢ (fun z0 => (z0, w1)) z0 ∈ closedBall c0 r ×ˢ closedBall c1 r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | Separate.fc1 | [113, 1] | [118, 67] | refine ContinuousOn.comp h.fc ?_ ?_ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
⊢ ContinuousOn (fun z1 => f (w0, z1)) (closedBall c1 r) | case refine_1
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
⊢ ContinuousOn (fun z1 => (w0, z1)) (closedBall c1 r)
case refine_2
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
⊢ Set.MapsTo (fun z1 => (w0, z1)) (closedBall c1 r) s | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
⊢ ContinuousOn (fun z1 => f (w0, z1)) (closedBall c1 r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | Separate.fc1 | [113, 1] | [118, 67] | exact ContinuousOn.prod continuousOn_const continuousOn_id | case refine_1
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
⊢ ContinuousOn (fun z1 => (w0, z1)) (closedBall c1 r) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
⊢ ContinuousOn (fun z1 => (w0, z1)) (closedBall c1 r)
TACTIC:
|
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