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] | rw [←hk] | 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
⊢ p k | 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
⊢ p (Nat.find ⋯) | 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 :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | pass_through | [28, 1] | [54, 19] | exact Nat.find_spec (p := p) ⟨_,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
⊢ p (Nat.find ⋯) | 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 :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | pass_through | [28, 1] | [54, 19] | contrapose k4 | 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
⊢ k ≠ 0 | 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 : ¬k ≠ 0
⊢ ¬p k | 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 :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | pass_through | [28, 1] | [54, 19] | simp only [not_not] at k4 | 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 : ¬k ≠ 0
⊢ ¬p k | 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 : k = 0
⊢ ¬p k | 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 :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | pass_through | [28, 1] | [54, 19] | simp only [k4, ←hp, not_lt, Function.iterate_zero_apply, z4] | 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 : k = 0
⊢ ¬p k | 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 :... |
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... | 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 ... | 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 :... |
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 ... | 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 ... | 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)) ... |
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 ... | 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)) ... |
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 <... | 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 :... |
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 ... | 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)
TACTI... |
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 : ∀ (... | 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 ... |
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 => ... |
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 (fu... |
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 ... |
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 ∈ in... |
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
... |
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) ^... | 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
... |
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) ^... | 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
... |
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
... |
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
... |
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
... |
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
... |
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
... |
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
... |
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
... |
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
... |
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 <... |
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 +... |
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 /... |
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' :... |
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' :... |
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' :... |
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 ↑... |
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
TA... |
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
re... | 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 ... | 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.... |
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 ... | 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 =... | 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.... |
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 =... | 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 =... | 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.... |
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 =... | 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 ^... | 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.... |
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 ^... | 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 ^... | 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.... |
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 ^... | 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 ^... | 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.... |
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 ^... | 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 ^... | 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.... |
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 ^... | 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 ^... | 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.... |
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 ^... | 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.... |
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 ... | 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.po... |
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 ... | 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
⊢... | 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 /... |
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
⊢... | 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
a... | 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 / Comple... |
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
a... | 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
a... | 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 / Comple... |
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
a... | 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
a... | 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 / Comple... |
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
a... | 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
a... | 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 / Comple... |
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
a... | 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 / Comple... |
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
⊢... | 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 / Comple... |
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
⊢... | 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 / Comple... |
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
⊢... | 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 / Comple... |
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
a... | 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
a... | 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 / Comple... |
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
a... | 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 / Comple... |
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
a... | 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 / Comple... |
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
a... | 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 / Comple... |
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
a... | 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 / Comple... |
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
a... | 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 / Comple... |
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
a... | 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 / Comple... |
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
a... | 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 / Comple... |
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
a... | 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 / Comple... |
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 ^ (... | 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... |
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 =... | 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... |
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 =... | 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... |
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 =... | 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... |
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 =... | 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... |
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 =... | 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... |
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 =... | 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... |
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/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | have e : ((univ : Set X) ×ˢ t)ᶜ = univ ×ˢ tᶜ := by
apply Set.ext; intro ⟨a, x⟩; rw [mem_compl_iff]
simp only [prod_mk_mem_set_prod_eq, mem_univ, mem_compl_iff, true_and_iff] | X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
⊢ Nonseparating (univ ×ˢ t) | X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
⊢ Nonseparating (univ ×ˢ t) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
⊢ Nonseparating (univ ×ˢ t)
TA... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | constructor | X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
⊢ Nonseparating (univ ×ˢ t) | case dense
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
⊢ Dense (univ ×ˢ t)ᶜ
case loc
X : Type
inst✝⁵ : T... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | apply Set.ext | X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
⊢ (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ | case h
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
⊢ ∀ (x : X × Y), x ∈ (univ ×ˢ t)ᶜ ↔ x ∈ univ ×ˢ tᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
⊢ (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
TA... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | intro ⟨a, x⟩ | case h
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
⊢ ∀ (x : X × Y), x ∈ (univ ×ˢ t)ᶜ ↔ x ∈ univ ×ˢ tᶜ | case h
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
a : X
x : Y
⊢ (a, x) ∈ (univ ×ˢ t)ᶜ ↔ (a, x) ∈ univ ×ˢ tᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
⊢ ∀ (x : X × Y), x ∈ (u... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | rw [mem_compl_iff] | case h
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
a : X
x : Y
⊢ (a, x) ∈ (univ ×ˢ t)ᶜ ↔ (a, x) ∈ univ ×ˢ tᶜ | case h
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
a : X
x : Y
⊢ (a, x) ∉ univ ×ˢ t ↔ (a, x) ∈ univ ×ˢ tᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
a : X
x : Y
⊢ (a, x) ∈ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | simp only [prod_mk_mem_set_prod_eq, mem_univ, mem_compl_iff, true_and_iff] | case h
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
a : X
x : Y
⊢ (a, x) ∉ univ ×ˢ t ↔ (a, x) ∈ univ ×ˢ tᶜ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
a : X
x : Y
⊢ (a, x) ∉ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | rw [e] | case dense
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
⊢ Dense (univ ×ˢ t)ᶜ | case dense
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
⊢ Dense (univ ×ˢ tᶜ) | Please generate a tactic in lean4 to solve the state.
STATE:
case dense
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | exact dense_univ.prod n.dense | case dense
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
⊢ Dense (univ ×ˢ tᶜ) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case dense
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | intro ⟨a, x⟩ u m un | case loc
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
⊢ ∀ (x : X × Y) (u : Set (X × Y)),
x ∈ univ ×ˢ t... | case loc
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
m : (a, x) ∈ univ ×ˢ t
u... | Please generate a tactic in lean4 to solve the state.
STATE:
case loc
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = un... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | simp only [mem_prod_eq, mem_univ, true_and_iff] at m | case loc
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
m : (a, x) ∈ univ ×ˢ t
u... | case loc
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
un : u ∈ 𝓝 (a, x)
m : x... | Please generate a tactic in lean4 to solve the state.
STATE:
case loc
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = un... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | rcases mem_nhds_prod_iff.mp un with ⟨u0, n0, u1, n1, uu⟩ | case loc
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
un : u ∈ 𝓝 (a, x)
m : x... | case loc.intro.intro.intro.intro
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
... | Please generate a tactic in lean4 to solve the state.
STATE:
case loc
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = un... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | rcases n.loc x u1 m n1 with ⟨c1, cs1, cn1, cp1⟩ | case loc.intro.intro.intro.intro
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
... | case loc.intro.intro.intro.intro.intro.intro.intro
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case loc.intro.intro.intro.intro
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | rcases locallyConnectedSpace_iff_open_connected_subsets.mp (by infer_instance) a u0 n0 with
⟨c0, cs0, co0, cm0, cc0⟩ | case loc.intro.intro.intro.intro.intro.intro.intro
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : ... | case loc.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ... | Please generate a tactic in lean4 to solve the state.
STATE:
case loc.intro.intro.intro.intro.intro.intro.intro
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | use c0 ×ˢ c1 | case loc.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ... | case h
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
un : u ∈ 𝓝 (a, x)
m : x ∈... | Please generate a tactic in lean4 to solve the state.
STATE:
case loc.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyCo... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | refine ⟨?_, ?_, ?_⟩ | case h
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
un : u ∈ 𝓝 (a, x)
m : x ∈... | case h.refine_1
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
un : u ∈ 𝓝 (a, x... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | infer_instance | X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
un : u ∈ 𝓝 (a, x)
m : x ∈ t
u0 :... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
... |
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