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/Hartogs/Hartogs.lean | uneven_bounded | [769, 1] | [798, 63] | ring_nf | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr : s < r1
t : ℝ
ts : s < t
tr : t < r1
tp : t > 0
c : ℝ
cp : c ≥ 0
ch : ∀ (n : ℕ), ∀ z1 ∈ closedBall c1 t, ‖unevenSeries u z1 n‖ ≤ c * t⁻¹ ^ n
z : ℂ × ℂ
zs : dist z.1 c0 < s ∧ dist z.2 c1 < s
z1t : z.2 ∈ closedBall c1 t
z1r : z.2 ∈ closedBall c1 r1
ds' : z.1 - c0 ∈ EMetric.ball 0 (ENNReal.ofReal s)
hs : HasSum (fun n => (z.1 - c0) ^ n • (unevenSeries u z.2).coeff n) (f z)
g : ℕ → ℝ := fun n => c * (s / t) ^ n
gs : HasSum g (c * (1 - s / t)⁻¹)
n : ℕ
ds : Complex.abs (z.1 - c0) < s
⊢ s ^ n * (c * t⁻¹ ^ n) = c * (s ^ n * t⁻¹ ^ n) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr : s < r1
t : ℝ
ts : s < t
tr : t < r1
tp : t > 0
c : ℝ
cp : c ≥ 0
ch : ∀ (n : ℕ), ∀ z1 ∈ closedBall c1 t, ‖unevenSeries u z1 n‖ ≤ c * t⁻¹ ^ n
z : ℂ × ℂ
zs : dist z.1 c0 < s ∧ dist z.2 c1 < s
z1t : z.2 ∈ closedBall c1 t
z1r : z.2 ∈ closedBall c1 r1
ds' : z.1 - c0 ∈ EMetric.ball 0 (ENNReal.ofReal s)
hs : HasSum (fun n => (z.1 - c0) ^ n • (unevenSeries u z.2).coeff n) (f z)
g : ℕ → ℝ := fun n => c * (s / t) ^ n
gs : HasSum g (c * (1 - s / t)⁻¹)
n : ℕ
ds : Complex.abs (z.1 - c0) < s
⊢ s ^ n * (c * t⁻¹ ^ n) = c * (s ^ n * t⁻¹ ^ n)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | uneven_bounded | [769, 1] | [798, 63] | rw [← mul_pow, ← div_eq_mul_inv] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr : s < r1
t : ℝ
ts : s < t
tr : t < r1
tp : t > 0
c : ℝ
cp : c ≥ 0
ch : ∀ (n : ℕ), ∀ z1 ∈ closedBall c1 t, ‖unevenSeries u z1 n‖ ≤ c * t⁻¹ ^ n
z : ℂ × ℂ
zs : dist z.1 c0 < s ∧ dist z.2 c1 < s
z1t : z.2 ∈ closedBall c1 t
z1r : z.2 ∈ closedBall c1 r1
ds' : z.1 - c0 ∈ EMetric.ball 0 (ENNReal.ofReal s)
hs : HasSum (fun n => (z.1 - c0) ^ n • (unevenSeries u z.2).coeff n) (f z)
g : ℕ → ℝ := fun n => c * (s / t) ^ n
gs : HasSum g (c * (1 - s / t)⁻¹)
n : ℕ
ds : Complex.abs (z.1 - c0) < s
⊢ c * (s ^ n * t⁻¹ ^ n) = c * (s / t) ^ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sp : s > 0
sr : s < r1
t : ℝ
ts : s < t
tr : t < r1
tp : t > 0
c : ℝ
cp : c ≥ 0
ch : ∀ (n : ℕ), ∀ z1 ∈ closedBall c1 t, ‖unevenSeries u z1 n‖ ≤ c * t⁻¹ ^ n
z : ℂ × ℂ
zs : dist z.1 c0 < s ∧ dist z.2 c1 < s
z1t : z.2 ∈ closedBall c1 t
z1r : z.2 ∈ closedBall c1 r1
ds' : z.1 - c0 ∈ EMetric.ball 0 (ENNReal.ofReal s)
hs : HasSum (fun n => (z.1 - c0) ^ n • (unevenSeries u z.2).coeff n) (f z)
g : ℕ → ℝ := fun n => c * (s / t) ^ n
gs : HasSum g (c * (1 - s / t)⁻¹)
n : ℕ
ds : Complex.abs (z.1 - c0) < s
⊢ c * (s ^ n * t⁻¹ ^ n) = c * (s / t) ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Pair.hartogs | [803, 1] | [827, 71] | have h : Har f s := ⟨fa0, fa1⟩ | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
⊢ AnalyticOn ℂ f s | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
⊢ AnalyticOn ℂ f s | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
⊢ AnalyticOn ℂ f s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Pair.hartogs | [803, 1] | [827, 71] | intro c cs | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
⊢ AnalyticOn ℂ f s | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
⊢ AnalyticAt ℂ f c | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
⊢ AnalyticOn ℂ f s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Pair.hartogs | [803, 1] | [827, 71] | rcases Metric.isOpen_iff.mp so c cs with ⟨r, rp, rs⟩ | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
⊢ AnalyticAt ℂ f c | case intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
⊢ AnalyticAt ℂ f c | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
⊢ AnalyticAt ℂ f c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Pair.hartogs | [803, 1] | [827, 71] | rcases exists_between rp with ⟨t, tp, tr⟩ | case intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
⊢ AnalyticAt ℂ f c | case intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
⊢ AnalyticAt ℂ f c | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
⊢ AnalyticAt ℂ f c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Pair.hartogs | [803, 1] | [827, 71] | have bs : closedBall (c.1, c.2) t ⊆ s := by
refine _root_.trans ?_ rs; simp only [fst_snd_eq]; exact Metric.closedBall_subset_ball tr | case intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
⊢ AnalyticAt ℂ f c | case intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
bs : closedBall (c.1, c.2) t ⊆ s
⊢ AnalyticAt ℂ f c | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
⊢ AnalyticAt ℂ f c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Pair.hartogs | [803, 1] | [827, 71] | rcases to_uneven (h.mono bs) tp with ⟨c0', r0, r1, us, c0s, u⟩ | case intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
bs : closedBall (c.1, c.2) t ⊆ s
⊢ AnalyticAt ℂ f c | case intro.intro.intro.intro.intro.intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
bs : closedBall (c.1, c.2) t ⊆ s
c0' : ℂ
r0 r1 : ℝ
us : ball c0' r1 ⊆ closedBall c.1 t
c0s : c.1 ∈ ball c0' r1
u : Uneven f c0' c.2 r0 r1
⊢ AnalyticAt ℂ f c | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
bs : closedBall (c.1, c.2) t ⊆ s
⊢ AnalyticAt ℂ f c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Pair.hartogs | [803, 1] | [827, 71] | have cr : abs (c.1 - c0') < r1 := by
simp only [Complex.dist_eq, Metric.mem_ball] at c0s; exact c0s | case intro.intro.intro.intro.intro.intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
bs : closedBall (c.1, c.2) t ⊆ s
c0' : ℂ
r0 r1 : ℝ
us : ball c0' r1 ⊆ closedBall c.1 t
c0s : c.1 ∈ ball c0' r1
u : Uneven f c0' c.2 r0 r1
⊢ AnalyticAt ℂ f c | case intro.intro.intro.intro.intro.intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
bs : closedBall (c.1, c.2) t ⊆ s
c0' : ℂ
r0 r1 : ℝ
us : ball c0' r1 ⊆ closedBall c.1 t
c0s : c.1 ∈ ball c0' r1
u : Uneven f c0' c.2 r0 r1
cr : Complex.abs (c.1 - c0') < r1
⊢ AnalyticAt ℂ f c | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
bs : closedBall (c.1, c.2) t ⊆ s
c0' : ℂ
r0 r1 : ℝ
us : ball c0' r1 ⊆ closedBall c.1 t
c0s : c.1 ∈ ball c0' r1
u : Uneven f c0' c.2 r0 r1
⊢ AnalyticAt ℂ f c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Pair.hartogs | [803, 1] | [827, 71] | rcases exists_between cr with ⟨v, vc, vr⟩ | case intro.intro.intro.intro.intro.intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
bs : closedBall (c.1, c.2) t ⊆ s
c0' : ℂ
r0 r1 : ℝ
us : ball c0' r1 ⊆ closedBall c.1 t
c0s : c.1 ∈ ball c0' r1
u : Uneven f c0' c.2 r0 r1
cr : Complex.abs (c.1 - c0') < r1
⊢ AnalyticAt ℂ f c | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
bs : closedBall (c.1, c.2) t ⊆ s
c0' : ℂ
r0 r1 : ℝ
us : ball c0' r1 ⊆ closedBall c.1 t
c0s : c.1 ∈ ball c0' r1
u : Uneven f c0' c.2 r0 r1
cr : Complex.abs (c.1 - c0') < r1
v : ℝ
vc : Complex.abs (c.1 - c0') < v
vr : v < r1
⊢ AnalyticAt ℂ f c | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
bs : closedBall (c.1, c.2) t ⊆ s
c0' : ℂ
r0 r1 : ℝ
us : ball c0' r1 ⊆ closedBall c.1 t
c0s : c.1 ∈ ball c0' r1
u : Uneven f c0' c.2 r0 r1
cr : Complex.abs (c.1 - c0') < r1
⊢ AnalyticAt ℂ f c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Pair.hartogs | [803, 1] | [827, 71] | rcases uneven_bounded u (lt_of_le_of_lt (Complex.abs.nonneg _) vc) vr with ⟨b, _, fb⟩ | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
bs : closedBall (c.1, c.2) t ⊆ s
c0' : ℂ
r0 r1 : ℝ
us : ball c0' r1 ⊆ closedBall c.1 t
c0s : c.1 ∈ ball c0' r1
u : Uneven f c0' c.2 r0 r1
cr : Complex.abs (c.1 - c0') < r1
v : ℝ
vc : Complex.abs (c.1 - c0') < v
vr : v < r1
⊢ AnalyticAt ℂ f c | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
bs : closedBall (c.1, c.2) t ⊆ s
c0' : ℂ
r0 r1 : ℝ
us : ball c0' r1 ⊆ closedBall c.1 t
c0s : c.1 ∈ ball c0' r1
u : Uneven f c0' c.2 r0 r1
cr : Complex.abs (c.1 - c0') < r1
v : ℝ
vc : Complex.abs (c.1 - c0') < v
vr : v < r1
b : ℝ
left✝ : b ≥ 0
fb : ∀ z ∈ ball (c0', c.2) v, ‖f z‖ ≤ b
⊢ AnalyticAt ℂ f c | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
bs : closedBall (c.1, c.2) t ⊆ s
c0' : ℂ
r0 r1 : ℝ
us : ball c0' r1 ⊆ closedBall c.1 t
c0s : c.1 ∈ ball c0' r1
u : Uneven f c0' c.2 r0 r1
cr : Complex.abs (c.1 - c0') < r1
v : ℝ
vc : Complex.abs (c.1 - c0') < v
vr : v < r1
⊢ AnalyticAt ℂ f c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Pair.hartogs | [803, 1] | [827, 71] | have fa := of_bounded (h.mono ?_) Metric.isOpen_ball fb | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
bs : closedBall (c.1, c.2) t ⊆ s
c0' : ℂ
r0 r1 : ℝ
us : ball c0' r1 ⊆ closedBall c.1 t
c0s : c.1 ∈ ball c0' r1
u : Uneven f c0' c.2 r0 r1
cr : Complex.abs (c.1 - c0') < r1
v : ℝ
vc : Complex.abs (c.1 - c0') < v
vr : v < r1
b : ℝ
left✝ : b ≥ 0
fb : ∀ z ∈ ball (c0', c.2) v, ‖f z‖ ≤ b
⊢ AnalyticAt ℂ f c | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
bs : closedBall (c.1, c.2) t ⊆ s
c0' : ℂ
r0 r1 : ℝ
us : ball c0' r1 ⊆ closedBall c.1 t
c0s : c.1 ∈ ball c0' r1
u : Uneven f c0' c.2 r0 r1
cr : Complex.abs (c.1 - c0') < r1
v : ℝ
vc : Complex.abs (c.1 - c0') < v
vr : v < r1
b : ℝ
left✝ : b ≥ 0
fb : ∀ z ∈ ball (c0', c.2) v, ‖f z‖ ≤ b
fa : AnalyticOn ℂ f (ball (c0', c.2) v)
⊢ AnalyticAt ℂ f c
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
bs : closedBall (c.1, c.2) t ⊆ s
c0' : ℂ
r0 r1 : ℝ
us : ball c0' r1 ⊆ closedBall c.1 t
c0s : c.1 ∈ ball c0' r1
u : Uneven f c0' c.2 r0 r1
cr : Complex.abs (c.1 - c0') < r1
v : ℝ
vc : Complex.abs (c.1 - c0') < v
vr : v < r1
b : ℝ
left✝ : b ≥ 0
fb : ∀ z ∈ ball (c0', c.2) v, ‖f z‖ ≤ b
⊢ ball (c0', c.2) v ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
bs : closedBall (c.1, c.2) t ⊆ s
c0' : ℂ
r0 r1 : ℝ
us : ball c0' r1 ⊆ closedBall c.1 t
c0s : c.1 ∈ ball c0' r1
u : Uneven f c0' c.2 r0 r1
cr : Complex.abs (c.1 - c0') < r1
v : ℝ
vc : Complex.abs (c.1 - c0') < v
vr : v < r1
b : ℝ
left✝ : b ≥ 0
fb : ∀ z ∈ ball (c0', c.2) v, ‖f z‖ ≤ b
⊢ AnalyticAt ℂ f c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Pair.hartogs | [803, 1] | [827, 71] | refine _root_.trans ?_ rs | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
⊢ closedBall (c.1, c.2) t ⊆ s | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
⊢ closedBall (c.1, c.2) t ⊆ ball c r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
⊢ closedBall (c.1, c.2) t ⊆ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Pair.hartogs | [803, 1] | [827, 71] | simp only [fst_snd_eq] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
⊢ closedBall (c.1, c.2) t ⊆ ball c r | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
⊢ closedBall c t ⊆ ball c r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
⊢ closedBall (c.1, c.2) t ⊆ ball c r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Pair.hartogs | [803, 1] | [827, 71] | exact Metric.closedBall_subset_ball tr | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
⊢ closedBall c t ⊆ ball c r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
⊢ closedBall c t ⊆ ball c r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Pair.hartogs | [803, 1] | [827, 71] | simp only [Complex.dist_eq, Metric.mem_ball] at c0s | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
bs : closedBall (c.1, c.2) t ⊆ s
c0' : ℂ
r0 r1 : ℝ
us : ball c0' r1 ⊆ closedBall c.1 t
c0s : c.1 ∈ ball c0' r1
u : Uneven f c0' c.2 r0 r1
⊢ Complex.abs (c.1 - c0') < r1 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
bs : closedBall (c.1, c.2) t ⊆ s
c0' : ℂ
r0 r1 : ℝ
us : ball c0' r1 ⊆ closedBall c.1 t
u : Uneven f c0' c.2 r0 r1
c0s : Complex.abs (c.1 - c0') < r1
⊢ Complex.abs (c.1 - c0') < r1 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
bs : closedBall (c.1, c.2) t ⊆ s
c0' : ℂ
r0 r1 : ℝ
us : ball c0' r1 ⊆ closedBall c.1 t
c0s : c.1 ∈ ball c0' r1
u : Uneven f c0' c.2 r0 r1
⊢ Complex.abs (c.1 - c0') < r1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Pair.hartogs | [803, 1] | [827, 71] | exact c0s | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
bs : closedBall (c.1, c.2) t ⊆ s
c0' : ℂ
r0 r1 : ℝ
us : ball c0' r1 ⊆ closedBall c.1 t
u : Uneven f c0' c.2 r0 r1
c0s : Complex.abs (c.1 - c0') < r1
⊢ Complex.abs (c.1 - c0') < r1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
bs : closedBall (c.1, c.2) t ⊆ s
c0' : ℂ
r0 r1 : ℝ
us : ball c0' r1 ⊆ closedBall c.1 t
u : Uneven f c0' c.2 r0 r1
c0s : Complex.abs (c.1 - c0') < r1
⊢ Complex.abs (c.1 - c0') < r1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Pair.hartogs | [803, 1] | [827, 71] | apply fa | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
bs : closedBall (c.1, c.2) t ⊆ s
c0' : ℂ
r0 r1 : ℝ
us : ball c0' r1 ⊆ closedBall c.1 t
c0s : c.1 ∈ ball c0' r1
u : Uneven f c0' c.2 r0 r1
cr : Complex.abs (c.1 - c0') < r1
v : ℝ
vc : Complex.abs (c.1 - c0') < v
vr : v < r1
b : ℝ
left✝ : b ≥ 0
fb : ∀ z ∈ ball (c0', c.2) v, ‖f z‖ ≤ b
fa : AnalyticOn ℂ f (ball (c0', c.2) v)
⊢ AnalyticAt ℂ f c | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2.a
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
bs : closedBall (c.1, c.2) t ⊆ s
c0' : ℂ
r0 r1 : ℝ
us : ball c0' r1 ⊆ closedBall c.1 t
c0s : c.1 ∈ ball c0' r1
u : Uneven f c0' c.2 r0 r1
cr : Complex.abs (c.1 - c0') < r1
v : ℝ
vc : Complex.abs (c.1 - c0') < v
vr : v < r1
b : ℝ
left✝ : b ≥ 0
fb : ∀ z ∈ ball (c0', c.2) v, ‖f z‖ ≤ b
fa : AnalyticOn ℂ f (ball (c0', c.2) v)
⊢ c ∈ ball (c0', c.2) v | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
bs : closedBall (c.1, c.2) t ⊆ s
c0' : ℂ
r0 r1 : ℝ
us : ball c0' r1 ⊆ closedBall c.1 t
c0s : c.1 ∈ ball c0' r1
u : Uneven f c0' c.2 r0 r1
cr : Complex.abs (c.1 - c0') < r1
v : ℝ
vc : Complex.abs (c.1 - c0') < v
vr : v < r1
b : ℝ
left✝ : b ≥ 0
fb : ∀ z ∈ ball (c0', c.2) v, ‖f z‖ ≤ b
fa : AnalyticOn ℂ f (ball (c0', c.2) v)
⊢ AnalyticAt ℂ f c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Pair.hartogs | [803, 1] | [827, 71] | simp only [Metric.mem_ball, Prod.dist_eq, Complex.dist_eq, dist_self, ge_iff_le, apply_nonneg,
max_eq_left, vc] | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2.a
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
bs : closedBall (c.1, c.2) t ⊆ s
c0' : ℂ
r0 r1 : ℝ
us : ball c0' r1 ⊆ closedBall c.1 t
c0s : c.1 ∈ ball c0' r1
u : Uneven f c0' c.2 r0 r1
cr : Complex.abs (c.1 - c0') < r1
v : ℝ
vc : Complex.abs (c.1 - c0') < v
vr : v < r1
b : ℝ
left✝ : b ≥ 0
fb : ∀ z ∈ ball (c0', c.2) v, ‖f z‖ ≤ b
fa : AnalyticOn ℂ f (ball (c0', c.2) v)
⊢ c ∈ ball (c0', c.2) v | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2.a
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
bs : closedBall (c.1, c.2) t ⊆ s
c0' : ℂ
r0 r1 : ℝ
us : ball c0' r1 ⊆ closedBall c.1 t
c0s : c.1 ∈ ball c0' r1
u : Uneven f c0' c.2 r0 r1
cr : Complex.abs (c.1 - c0') < r1
v : ℝ
vc : Complex.abs (c.1 - c0') < v
vr : v < r1
b : ℝ
left✝ : b ≥ 0
fb : ∀ z ∈ ball (c0', c.2) v, ‖f z‖ ≤ b
fa : AnalyticOn ℂ f (ball (c0', c.2) v)
⊢ c ∈ ball (c0', c.2) v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Pair.hartogs | [803, 1] | [827, 71] | refine _root_.trans ?_ bs | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
bs : closedBall (c.1, c.2) t ⊆ s
c0' : ℂ
r0 r1 : ℝ
us : ball c0' r1 ⊆ closedBall c.1 t
c0s : c.1 ∈ ball c0' r1
u : Uneven f c0' c.2 r0 r1
cr : Complex.abs (c.1 - c0') < r1
v : ℝ
vc : Complex.abs (c.1 - c0') < v
vr : v < r1
b : ℝ
left✝ : b ≥ 0
fb : ∀ z ∈ ball (c0', c.2) v, ‖f z‖ ≤ b
⊢ ball (c0', c.2) v ⊆ s | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
bs : closedBall (c.1, c.2) t ⊆ s
c0' : ℂ
r0 r1 : ℝ
us : ball c0' r1 ⊆ closedBall c.1 t
c0s : c.1 ∈ ball c0' r1
u : Uneven f c0' c.2 r0 r1
cr : Complex.abs (c.1 - c0') < r1
v : ℝ
vc : Complex.abs (c.1 - c0') < v
vr : v < r1
b : ℝ
left✝ : b ≥ 0
fb : ∀ z ∈ ball (c0', c.2) v, ‖f z‖ ≤ b
⊢ ball (c0', c.2) v ⊆ closedBall (c.1, c.2) t | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
bs : closedBall (c.1, c.2) t ⊆ s
c0' : ℂ
r0 r1 : ℝ
us : ball c0' r1 ⊆ closedBall c.1 t
c0s : c.1 ∈ ball c0' r1
u : Uneven f c0' c.2 r0 r1
cr : Complex.abs (c.1 - c0') < r1
v : ℝ
vc : Complex.abs (c.1 - c0') < v
vr : v < r1
b : ℝ
left✝ : b ≥ 0
fb : ∀ z ∈ ball (c0', c.2) v, ‖f z‖ ≤ b
⊢ ball (c0', c.2) v ⊆ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Pair.hartogs | [803, 1] | [827, 71] | simp_rw [← ball_prod_same, ← closedBall_prod_same, Set.prod_subset_prod_iff] | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
bs : closedBall (c.1, c.2) t ⊆ s
c0' : ℂ
r0 r1 : ℝ
us : ball c0' r1 ⊆ closedBall c.1 t
c0s : c.1 ∈ ball c0' r1
u : Uneven f c0' c.2 r0 r1
cr : Complex.abs (c.1 - c0') < r1
v : ℝ
vc : Complex.abs (c.1 - c0') < v
vr : v < r1
b : ℝ
left✝ : b ≥ 0
fb : ∀ z ∈ ball (c0', c.2) v, ‖f z‖ ≤ b
⊢ ball (c0', c.2) v ⊆ closedBall (c.1, c.2) t | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
bs : closedBall (c.1, c.2) t ⊆ s
c0' : ℂ
r0 r1 : ℝ
us : ball c0' r1 ⊆ closedBall c.1 t
c0s : c.1 ∈ ball c0' r1
u : Uneven f c0' c.2 r0 r1
cr : Complex.abs (c.1 - c0') < r1
v : ℝ
vc : Complex.abs (c.1 - c0') < v
vr : v < r1
b : ℝ
left✝ : b ≥ 0
fb : ∀ z ∈ ball (c0', c.2) v, ‖f z‖ ≤ b
⊢ ball c0' v ⊆ closedBall c.1 t ∧ ball c.2 v ⊆ closedBall c.2 t ∨ ball c0' v = ∅ ∨ ball c.2 v = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
bs : closedBall (c.1, c.2) t ⊆ s
c0' : ℂ
r0 r1 : ℝ
us : ball c0' r1 ⊆ closedBall c.1 t
c0s : c.1 ∈ ball c0' r1
u : Uneven f c0' c.2 r0 r1
cr : Complex.abs (c.1 - c0') < r1
v : ℝ
vc : Complex.abs (c.1 - c0') < v
vr : v < r1
b : ℝ
left✝ : b ≥ 0
fb : ∀ z ∈ ball (c0', c.2) v, ‖f z‖ ≤ b
⊢ ball (c0', c.2) v ⊆ closedBall (c.1, c.2) t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Pair.hartogs | [803, 1] | [827, 71] | apply Or.inl | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
bs : closedBall (c.1, c.2) t ⊆ s
c0' : ℂ
r0 r1 : ℝ
us : ball c0' r1 ⊆ closedBall c.1 t
c0s : c.1 ∈ ball c0' r1
u : Uneven f c0' c.2 r0 r1
cr : Complex.abs (c.1 - c0') < r1
v : ℝ
vc : Complex.abs (c.1 - c0') < v
vr : v < r1
b : ℝ
left✝ : b ≥ 0
fb : ∀ z ∈ ball (c0', c.2) v, ‖f z‖ ≤ b
⊢ ball c0' v ⊆ closedBall c.1 t ∧ ball c.2 v ⊆ closedBall c.2 t ∨ ball c0' v = ∅ ∨ ball c.2 v = ∅ | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1.h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
bs : closedBall (c.1, c.2) t ⊆ s
c0' : ℂ
r0 r1 : ℝ
us : ball c0' r1 ⊆ closedBall c.1 t
c0s : c.1 ∈ ball c0' r1
u : Uneven f c0' c.2 r0 r1
cr : Complex.abs (c.1 - c0') < r1
v : ℝ
vc : Complex.abs (c.1 - c0') < v
vr : v < r1
b : ℝ
left✝ : b ≥ 0
fb : ∀ z ∈ ball (c0', c.2) v, ‖f z‖ ≤ b
⊢ ball c0' v ⊆ closedBall c.1 t ∧ ball c.2 v ⊆ closedBall c.2 t | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
bs : closedBall (c.1, c.2) t ⊆ s
c0' : ℂ
r0 r1 : ℝ
us : ball c0' r1 ⊆ closedBall c.1 t
c0s : c.1 ∈ ball c0' r1
u : Uneven f c0' c.2 r0 r1
cr : Complex.abs (c.1 - c0') < r1
v : ℝ
vc : Complex.abs (c.1 - c0') < v
vr : v < r1
b : ℝ
left✝ : b ≥ 0
fb : ∀ z ∈ ball (c0', c.2) v, ‖f z‖ ≤ b
⊢ ball c0' v ⊆ closedBall c.1 t ∧ ball c.2 v ⊆ closedBall c.2 t ∨ ball c0' v = ∅ ∨ ball c.2 v = ∅
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Pair.hartogs | [803, 1] | [827, 71] | use _root_.trans (Metric.ball_subset_ball vr.le) us | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1.h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
bs : closedBall (c.1, c.2) t ⊆ s
c0' : ℂ
r0 r1 : ℝ
us : ball c0' r1 ⊆ closedBall c.1 t
c0s : c.1 ∈ ball c0' r1
u : Uneven f c0' c.2 r0 r1
cr : Complex.abs (c.1 - c0') < r1
v : ℝ
vc : Complex.abs (c.1 - c0') < v
vr : v < r1
b : ℝ
left✝ : b ≥ 0
fb : ∀ z ∈ ball (c0', c.2) v, ‖f z‖ ≤ b
⊢ ball c0' v ⊆ closedBall c.1 t ∧ ball c.2 v ⊆ closedBall c.2 t | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
bs : closedBall (c.1, c.2) t ⊆ s
c0' : ℂ
r0 r1 : ℝ
us : ball c0' r1 ⊆ closedBall c.1 t
c0s : c.1 ∈ ball c0' r1
u : Uneven f c0' c.2 r0 r1
cr : Complex.abs (c.1 - c0') < r1
v : ℝ
vc : Complex.abs (c.1 - c0') < v
vr : v < r1
b : ℝ
left✝ : b ≥ 0
fb : ∀ z ∈ ball (c0', c.2) v, ‖f z‖ ≤ b
⊢ ball c.2 v ⊆ closedBall c.2 t | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1.h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
bs : closedBall (c.1, c.2) t ⊆ s
c0' : ℂ
r0 r1 : ℝ
us : ball c0' r1 ⊆ closedBall c.1 t
c0s : c.1 ∈ ball c0' r1
u : Uneven f c0' c.2 r0 r1
cr : Complex.abs (c.1 - c0') < r1
v : ℝ
vc : Complex.abs (c.1 - c0') < v
vr : v < r1
b : ℝ
left✝ : b ≥ 0
fb : ∀ z ∈ ball (c0', c.2) v, ‖f z‖ ≤ b
⊢ ball c0' v ⊆ closedBall c.1 t ∧ ball c.2 v ⊆ closedBall c.2 t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Pair.hartogs | [803, 1] | [827, 71] | have r1t := le_of_ball_subset_closedBall u.r1p.le tp.le us | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
bs : closedBall (c.1, c.2) t ⊆ s
c0' : ℂ
r0 r1 : ℝ
us : ball c0' r1 ⊆ closedBall c.1 t
c0s : c.1 ∈ ball c0' r1
u : Uneven f c0' c.2 r0 r1
cr : Complex.abs (c.1 - c0') < r1
v : ℝ
vc : Complex.abs (c.1 - c0') < v
vr : v < r1
b : ℝ
left✝ : b ≥ 0
fb : ∀ z ∈ ball (c0', c.2) v, ‖f z‖ ≤ b
⊢ ball c.2 v ⊆ closedBall c.2 t | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
bs : closedBall (c.1, c.2) t ⊆ s
c0' : ℂ
r0 r1 : ℝ
us : ball c0' r1 ⊆ closedBall c.1 t
c0s : c.1 ∈ ball c0' r1
u : Uneven f c0' c.2 r0 r1
cr : Complex.abs (c.1 - c0') < r1
v : ℝ
vc : Complex.abs (c.1 - c0') < v
vr : v < r1
b : ℝ
left✝ : b ≥ 0
fb : ∀ z ∈ ball (c0', c.2) v, ‖f z‖ ≤ b
r1t : r1 ≤ t
⊢ ball c.2 v ⊆ closedBall c.2 t | Please generate a tactic in lean4 to solve the state.
STATE:
case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
bs : closedBall (c.1, c.2) t ⊆ s
c0' : ℂ
r0 r1 : ℝ
us : ball c0' r1 ⊆ closedBall c.1 t
c0s : c.1 ∈ ball c0' r1
u : Uneven f c0' c.2 r0 r1
cr : Complex.abs (c.1 - c0') < r1
v : ℝ
vc : Complex.abs (c.1 - c0') < v
vr : v < r1
b : ℝ
left✝ : b ≥ 0
fb : ∀ z ∈ ball (c0', c.2) v, ‖f z‖ ≤ b
⊢ ball c.2 v ⊆ closedBall c.2 t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Pair.hartogs | [803, 1] | [827, 71] | exact _root_.trans Metric.ball_subset_closedBall
(Metric.closedBall_subset_closedBall (_root_.trans vr.le r1t)) | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
bs : closedBall (c.1, c.2) t ⊆ s
c0' : ℂ
r0 r1 : ℝ
us : ball c0' r1 ⊆ closedBall c.1 t
c0s : c.1 ∈ ball c0' r1
u : Uneven f c0' c.2 r0 r1
cr : Complex.abs (c.1 - c0') < r1
v : ℝ
vc : Complex.abs (c.1 - c0') < v
vr : v < r1
b : ℝ
left✝ : b ≥ 0
fb : ∀ z ∈ ball (c0', c.2) v, ‖f z‖ ≤ b
r1t : r1 ≤ t
⊢ ball c.2 v ⊆ closedBall c.2 t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
so : IsOpen s
fa0 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, c1)) c0
fa1 : ∀ (c0 c1 : ℂ), (c0, c1) ∈ s → AnalyticAt ℂ (fun z1 => f (c0, z1)) c1
h : Har f s
c : ℂ × ℂ
cs : c ∈ s
r : ℝ
rp : r > 0
rs : ball c r ⊆ s
t : ℝ
tp : 0 < t
tr : t < r
bs : closedBall (c.1, c.2) t ⊆ s
c0' : ℂ
r0 r1 : ℝ
us : ball c0' r1 ⊆ closedBall c.1 t
c0s : c.1 ∈ ball c0' r1
u : Uneven f c0' c.2 r0 r1
cr : Complex.abs (c.1 - c0') < r1
v : ℝ
vc : Complex.abs (c.1 - c0') < v
vr : v < r1
b : ℝ
left✝ : b ≥ 0
fb : ∀ z ∈ ball (c0', c.2) v, ‖f z‖ ≤ b
r1t : r1 ≤ t
⊢ ball c.2 v ⊆ closedBall c.2 t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Pair.hartogs_at | [830, 1] | [835, 82] | rcases eventually_nhds_iff.mp (fa0.and fa1) with ⟨s, fa, o, cs⟩ | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
c : ℂ × ℂ
fa0 : ∀ᶠ (p : ℂ × ℂ) in 𝓝 c, AnalyticAt ℂ (fun z0 => f (z0, p.2)) p.1
fa1 : ∀ᶠ (p : ℂ × ℂ) in 𝓝 c, AnalyticAt ℂ (fun z1 => f (p.1, z1)) p.2
⊢ AnalyticAt ℂ f c | case intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
c : ℂ × ℂ
fa0 : ∀ᶠ (p : ℂ × ℂ) in 𝓝 c, AnalyticAt ℂ (fun z0 => f (z0, p.2)) p.1
fa1 : ∀ᶠ (p : ℂ × ℂ) in 𝓝 c, AnalyticAt ℂ (fun z1 => f (p.1, z1)) p.2
s : Set (ℂ × ℂ)
fa : ∀ x ∈ s, AnalyticAt ℂ (fun z0 => f (z0, x.2)) x.1 ∧ AnalyticAt ℂ (fun z1 => f (x.1, z1)) x.2
o : IsOpen s
cs : c ∈ s
⊢ AnalyticAt ℂ f c | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
c : ℂ × ℂ
fa0 : ∀ᶠ (p : ℂ × ℂ) in 𝓝 c, AnalyticAt ℂ (fun z0 => f (z0, p.2)) p.1
fa1 : ∀ᶠ (p : ℂ × ℂ) in 𝓝 c, AnalyticAt ℂ (fun z1 => f (p.1, z1)) p.2
⊢ AnalyticAt ℂ f c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Pair.hartogs_at | [830, 1] | [835, 82] | exact Pair.hartogs o (fun c0 c1 m ↦ (fa _ m).1) (fun c0 c1 m ↦ (fa _ m).2) c cs | case intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
c : ℂ × ℂ
fa0 : ∀ᶠ (p : ℂ × ℂ) in 𝓝 c, AnalyticAt ℂ (fun z0 => f (z0, p.2)) p.1
fa1 : ∀ᶠ (p : ℂ × ℂ) in 𝓝 c, AnalyticAt ℂ (fun z1 => f (p.1, z1)) p.2
s : Set (ℂ × ℂ)
fa : ∀ x ∈ s, AnalyticAt ℂ (fun z0 => f (z0, x.2)) x.1 ∧ AnalyticAt ℂ (fun z1 => f (x.1, z1)) x.2
o : IsOpen s
cs : c ∈ s
⊢ AnalyticAt ℂ f c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
c : ℂ × ℂ
fa0 : ∀ᶠ (p : ℂ × ℂ) in 𝓝 c, AnalyticAt ℂ (fun z0 => f (z0, p.2)) p.1
fa1 : ∀ᶠ (p : ℂ × ℂ) in 𝓝 c, AnalyticAt ℂ (fun z1 => f (p.1, z1)) p.2
s : Set (ℂ × ℂ)
fa : ∀ x ∈ s, AnalyticAt ℂ (fun z0 => f (z0, x.2)) x.1 ∧ AnalyticAt ℂ (fun z1 => f (x.1, z1)) x.2
o : IsOpen s
cs : c ∈ s
⊢ AnalyticAt ℂ f c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | d_pos | [41, 1] | [41, 71] | linarith [two_le_d d] | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ 0 < d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ 0 < d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | d_gt_one | [43, 1] | [43, 74] | linarith [two_le_d d] | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ 1 < d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ 1 < d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | d_minus_one_pos | [45, 1] | [45, 91] | have h := two_le_d d | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ 0 < d - 1 | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
h : 2 ≤ d
⊢ 0 < d - 1 | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ 0 < d - 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | d_minus_one_pos | [45, 1] | [45, 91] | omega | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
h : 2 ≤ d
⊢ 0 < d - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
h : 2 ≤ d
⊢ 0 < d - 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | one_le_d_minus_one | [46, 1] | [46, 94] | have h := two_le_d d | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ 1 ≤ d - 1 | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
h : 2 ≤ d
⊢ 1 ≤ d - 1 | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ 1 ≤ d - 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | one_le_d_minus_one | [46, 1] | [46, 94] | omega | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
h : 2 ≤ d
⊢ 1 ≤ d - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
h : 2 ≤ d
⊢ 1 ≤ d - 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | two_le_cast_d | [47, 1] | [48, 56] | norm_num | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ 2 ≤ ↑2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ 2 ≤ ↑2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrotExt_coe | [77, 1] | [80, 72] | simp only [multibrotExt, mem_union, mem_singleton_iff, coe_eq_inf_iff, or_false_iff, mem_image,
mem_compl_iff, coe_eq_coe, not_iff_not] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
⊢ ↑c ∈ multibrotExt d ↔ c ∉ multibrot d | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
⊢ (∃ x ∈ multibrot d, x = c) ↔ c ∈ multibrot d | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
⊢ ↑c ∈ multibrotExt d ↔ c ∉ multibrot d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrotExt_coe | [77, 1] | [80, 72] | constructor | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
⊢ (∃ x ∈ multibrot d, x = c) ↔ c ∈ multibrot d | case mp
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
⊢ (∃ x ∈ multibrot d, x = c) → c ∈ multibrot d
case mpr
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
⊢ c ∈ multibrot d → ∃ x ∈ multibrot d, x = c | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
⊢ (∃ x ∈ multibrot d, x = c) ↔ c ∈ multibrot d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrotExt_coe | [77, 1] | [80, 72] | intro ⟨x, m, e⟩ | case mp
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
⊢ (∃ x ∈ multibrot d, x = c) → c ∈ multibrot d
case mpr
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
⊢ c ∈ multibrot d → ∃ x ∈ multibrot d, x = c | case mp
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c x : ℂ
m : x ∈ multibrot d
e : x = c
⊢ c ∈ multibrot d
case mpr
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
⊢ c ∈ multibrot d → ∃ x ∈ multibrot d, x = c | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
⊢ (∃ x ∈ multibrot d, x = c) → c ∈ multibrot d
case mpr
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
⊢ c ∈ multibrot d → ∃ x ∈ multibrot d, x = c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrotExt_coe | [77, 1] | [80, 72] | rw [e] at m | case mp
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c x : ℂ
m : x ∈ multibrot d
e : x = c
⊢ c ∈ multibrot d
case mpr
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
⊢ c ∈ multibrot d → ∃ x ∈ multibrot d, x = c | case mp
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c x : ℂ
m : c ∈ multibrot d
e : x = c
⊢ c ∈ multibrot d
case mpr
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
⊢ c ∈ multibrot d → ∃ x ∈ multibrot d, x = c | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c x : ℂ
m : x ∈ multibrot d
e : x = c
⊢ c ∈ multibrot d
case mpr
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
⊢ c ∈ multibrot d → ∃ x ∈ multibrot d, x = c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrotExt_coe | [77, 1] | [80, 72] | exact m | case mp
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c x : ℂ
m : c ∈ multibrot d
e : x = c
⊢ c ∈ multibrot d
case mpr
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
⊢ c ∈ multibrot d → ∃ x ∈ multibrot d, x = c | case mpr
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
⊢ c ∈ multibrot d → ∃ x ∈ multibrot d, x = c | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c x : ℂ
m : c ∈ multibrot d
e : x = c
⊢ c ∈ multibrot d
case mpr
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
⊢ c ∈ multibrot d → ∃ x ∈ multibrot d, x = c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrotExt_coe | [77, 1] | [80, 72] | intro m | case mpr
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
⊢ c ∈ multibrot d → ∃ x ∈ multibrot d, x = c | case mpr
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
m : c ∈ multibrot d
⊢ ∃ x ∈ multibrot d, x = c | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
⊢ c ∈ multibrot d → ∃ x ∈ multibrot d, x = c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrotExt_coe | [77, 1] | [80, 72] | use c, m | case mpr
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
m : c ∈ multibrot d
⊢ ∃ x ∈ multibrot d, x = c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
m : c ∈ multibrot d
⊢ ∃ x ∈ multibrot d, x = c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | coe_preimage_multibrotExt | [81, 1] | [82, 84] | apply Set.ext | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ (fun z => ↑z) ⁻¹' multibrotExt d = (multibrot d)ᶜ | case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ∀ (x : ℂ), x ∈ (fun z => ↑z) ⁻¹' multibrotExt d ↔ x ∈ (multibrot d)ᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ (fun z => ↑z) ⁻¹' multibrotExt d = (multibrot d)ᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | coe_preimage_multibrotExt | [81, 1] | [82, 84] | intro z | case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ∀ (x : ℂ), x ∈ (fun z => ↑z) ⁻¹' multibrotExt d ↔ x ∈ (multibrot d)ᶜ | case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
⊢ z ∈ (fun z => ↑z) ⁻¹' multibrotExt d ↔ z ∈ (multibrot d)ᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ∀ (x : ℂ), x ∈ (fun z => ↑z) ⁻¹' multibrotExt d ↔ x ∈ (multibrot d)ᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | coe_preimage_multibrotExt | [81, 1] | [82, 84] | simp only [mem_compl_iff, mem_preimage, multibrotExt_coe] | case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
⊢ z ∈ (fun z => ↑z) ⁻¹' multibrotExt d ↔ z ∈ (multibrot d)ᶜ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
⊢ z ∈ (fun z => ↑z) ⁻¹' multibrotExt d ↔ z ∈ (multibrot d)ᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | f_0' | [91, 1] | [92, 62] | simp only [lift_coe', f', zero_pow (d_ne_zero _), zero_add] | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ f' d c 0 = c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ f' d c 0 = c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | f_0 | [94, 1] | [95, 77] | simp only [f, ← coe_zero, lift_coe', f', zero_pow (d_ne_zero _), zero_add] | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ f d c 0 = ↑c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ f d c 0 = ↑c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | deriv_f' | [100, 1] | [103, 43] | have h : HasDerivAt (f' d c) (d * z ^ (d - 1) + 0) z :=
(hasDerivAt_pow _ _).add (hasDerivAt_const _ _) | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
⊢ deriv (f' d c) z = ↑d * z ^ (d - 1) | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
h : HasDerivAt (f' d c) (↑d * z ^ (d - 1) + 0) z
⊢ deriv (f' d c) z = ↑d * z ^ (d - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
⊢ deriv (f' d c) z = ↑d * z ^ (d - 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | deriv_f' | [100, 1] | [103, 43] | simp only [add_zero] at h | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
h : HasDerivAt (f' d c) (↑d * z ^ (d - 1) + 0) z
⊢ deriv (f' d c) z = ↑d * z ^ (d - 1) | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
h : HasDerivAt (f' d c) (↑d * z ^ (d - 1)) z
⊢ deriv (f' d c) z = ↑d * z ^ (d - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
h : HasDerivAt (f' d c) (↑d * z ^ (d - 1) + 0) z
⊢ deriv (f' d c) z = ↑d * z ^ (d - 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | deriv_f' | [100, 1] | [103, 43] | exact h.deriv | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
h : HasDerivAt (f' d c) (↑d * z ^ (d - 1)) z
⊢ deriv (f' d c) z = ↑d * z ^ (d - 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
h : HasDerivAt (f' d c) (↑d * z ^ (d - 1)) z
⊢ deriv (f' d c) z = ↑d * z ^ (d - 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | tendsto_f'_atInf | [105, 1] | [125, 31] | simp only [atInf_basis.tendsto_right_iff, Complex.norm_eq_abs, Set.mem_setOf_eq,
forall_true_left, uncurry, Metric.eventually_nhds_prod_iff] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
⊢ Tendsto (uncurry (f' d)) (𝓝 c ×ˢ atInf) atInf | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
⊢ ∀ (i : ℝ),
∃ ε > 0,
∃ pa, (∀ᶠ (i : ℂ) in atInf, pa i) ∧ ∀ {x : ℂ}, dist x c < ε → ∀ {i_1 : ℂ}, pa i_1 → i < Complex.abs (f' d x i_1) | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
⊢ Tendsto (uncurry (f' d)) (𝓝 c ×ˢ atInf) atInf
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | tendsto_f'_atInf | [105, 1] | [125, 31] | intro r | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
⊢ ∀ (i : ℝ),
∃ ε > 0,
∃ pa, (∀ᶠ (i : ℂ) in atInf, pa i) ∧ ∀ {x : ℂ}, dist x c < ε → ∀ {i_1 : ℂ}, pa i_1 → i < Complex.abs (f' d x i_1) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
⊢ ∃ ε > 0, ∃ pa, (∀ᶠ (i : ℂ) in atInf, pa i) ∧ ∀ {x : ℂ}, dist x c < ε → ∀ {i : ℂ}, pa i → r < Complex.abs (f' d x i) | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
⊢ ∀ (i : ℝ),
∃ ε > 0,
∃ pa, (∀ᶠ (i : ℂ) in atInf, pa i) ∧ ∀ {x : ℂ}, dist x c < ε → ∀ {i_1 : ℂ}, pa i_1 → i < Complex.abs (f' d x i_1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | tendsto_f'_atInf | [105, 1] | [125, 31] | use 1, zero_lt_one, fun z ↦ max r 0 + abs c + 1 < abs z | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
⊢ ∃ ε > 0, ∃ pa, (∀ᶠ (i : ℂ) in atInf, pa i) ∧ ∀ {x : ℂ}, dist x c < ε → ∀ {i : ℂ}, pa i → r < Complex.abs (f' d x i) | case h
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
⊢ (∀ᶠ (i : ℂ) in atInf, max r 0 + Complex.abs c + 1 < Complex.abs i) ∧
∀ {x : ℂ}, dist x c < 1 → ∀ {i : ℂ}, max r 0 + Complex.abs c + 1 < Complex.abs i → r < Complex.abs (f' d x i) | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
⊢ ∃ ε > 0, ∃ pa, (∀ᶠ (i : ℂ) in atInf, pa i) ∧ ∀ {x : ℂ}, dist x c < ε → ∀ {i : ℂ}, pa i → r < Complex.abs (f' d x i)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | tendsto_f'_atInf | [105, 1] | [125, 31] | constructor | case h
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
⊢ (∀ᶠ (i : ℂ) in atInf, max r 0 + Complex.abs c + 1 < Complex.abs i) ∧
∀ {x : ℂ}, dist x c < 1 → ∀ {i : ℂ}, max r 0 + Complex.abs c + 1 < Complex.abs i → r < Complex.abs (f' d x i) | case h.left
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
⊢ ∀ᶠ (i : ℂ) in atInf, max r 0 + Complex.abs c + 1 < Complex.abs i
case h.right
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
⊢ ∀ {x : ℂ}, dist x c < 1 → ∀ {i : ℂ}, max r 0 + Complex.abs c + 1 < Complex.abs i → r < Complex.abs (f' d x i) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
⊢ (∀ᶠ (i : ℂ) in atInf, max r 0 + Complex.abs c + 1 < Complex.abs i) ∧
∀ {x : ℂ}, dist x c < 1 → ∀ {i : ℂ}, max r 0 + Complex.abs c + 1 < Complex.abs i → r < Complex.abs (f' d x i)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | tendsto_f'_atInf | [105, 1] | [125, 31] | refine (eventually_atInf (max r 0 + abs c + 1)).mp (eventually_of_forall fun w h ↦ ?_) | case h.left
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
⊢ ∀ᶠ (i : ℂ) in atInf, max r 0 + Complex.abs c + 1 < Complex.abs i | case h.left
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
w : ℂ
h : ‖w‖ > max r 0 + Complex.abs c + 1
⊢ max r 0 + Complex.abs c + 1 < Complex.abs w | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
⊢ ∀ᶠ (i : ℂ) in atInf, max r 0 + Complex.abs c + 1 < Complex.abs i
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | tendsto_f'_atInf | [105, 1] | [125, 31] | simp only [Complex.norm_eq_abs] at h | case h.left
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
w : ℂ
h : ‖w‖ > max r 0 + Complex.abs c + 1
⊢ max r 0 + Complex.abs c + 1 < Complex.abs w | case h.left
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
w : ℂ
h : Complex.abs w > max r 0 + Complex.abs c + 1
⊢ max r 0 + Complex.abs c + 1 < Complex.abs w | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
w : ℂ
h : ‖w‖ > max r 0 + Complex.abs c + 1
⊢ max r 0 + Complex.abs c + 1 < Complex.abs w
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | tendsto_f'_atInf | [105, 1] | [125, 31] | exact h | case h.left
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
w : ℂ
h : Complex.abs w > max r 0 + Complex.abs c + 1
⊢ max r 0 + Complex.abs c + 1 < Complex.abs w | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
w : ℂ
h : Complex.abs w > max r 0 + Complex.abs c + 1
⊢ max r 0 + Complex.abs c + 1 < Complex.abs w
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | tendsto_f'_atInf | [105, 1] | [125, 31] | intro e ec z h | case h.right
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
⊢ ∀ {x : ℂ}, dist x c < 1 → ∀ {i : ℂ}, max r 0 + Complex.abs c + 1 < Complex.abs i → r < Complex.abs (f' d x i) | case h.right
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
e : ℂ
ec : dist e c < 1
z : ℂ
h : max r 0 + Complex.abs c + 1 < Complex.abs z
⊢ r < Complex.abs (f' d e z) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
⊢ ∀ {x : ℂ}, dist x c < 1 → ∀ {i : ℂ}, max r 0 + Complex.abs c + 1 < Complex.abs i → r < Complex.abs (f' d x i)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | tendsto_f'_atInf | [105, 1] | [125, 31] | simp only [Complex.dist_eq] at ec | case h.right
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
e : ℂ
ec : dist e c < 1
z : ℂ
h : max r 0 + Complex.abs c + 1 < Complex.abs z
⊢ r < Complex.abs (f' d e z) | case h.right
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
e : ℂ
ec : Complex.abs (e - c) < 1
z : ℂ
h : max r 0 + Complex.abs c + 1 < Complex.abs z
⊢ r < Complex.abs (f' d e z) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
e : ℂ
ec : dist e c < 1
z : ℂ
h : max r 0 + Complex.abs c + 1 < Complex.abs z
⊢ r < Complex.abs (f' d e z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | tendsto_f'_atInf | [105, 1] | [125, 31] | have zz : abs z ≤ abs (z ^ d) := by
rw [Complex.abs.map_pow]
refine le_self_pow ?_ (d_ne_zero _)
exact le_trans (le_add_of_nonneg_left (add_nonneg (le_max_right _ _) (Complex.abs.nonneg _)))
h.le | case h.right
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
e : ℂ
ec : Complex.abs (e - c) < 1
z : ℂ
h : max r 0 + Complex.abs c + 1 < Complex.abs z
⊢ r < Complex.abs (f' d e z) | case h.right
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
e : ℂ
ec : Complex.abs (e - c) < 1
z : ℂ
h : max r 0 + Complex.abs c + 1 < Complex.abs z
zz : Complex.abs z ≤ Complex.abs (z ^ d)
⊢ r < Complex.abs (f' d e z) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
e : ℂ
ec : Complex.abs (e - c) < 1
z : ℂ
h : max r 0 + Complex.abs c + 1 < Complex.abs z
⊢ r < Complex.abs (f' d e z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | tendsto_f'_atInf | [105, 1] | [125, 31] | calc abs (f' d e z)
_ = abs (z ^ d + e) := rfl
_ = abs (z ^ d + (c + (e - c))) := by ring_nf
_ ≥ abs (z ^ d) - abs (c + (e - c)) := by bound
_ ≥ abs (z ^ d) - (abs c + abs (e - c)) := by bound
_ ≥ abs z - (abs c + 1) := by bound
_ > max r 0 + abs c + 1 - (abs c + 1) := by bound
_ = max r 0 := by ring_nf
_ ≥ r := le_max_left _ _ | case h.right
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
e : ℂ
ec : Complex.abs (e - c) < 1
z : ℂ
h : max r 0 + Complex.abs c + 1 < Complex.abs z
zz : Complex.abs z ≤ Complex.abs (z ^ d)
⊢ r < Complex.abs (f' d e z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
e : ℂ
ec : Complex.abs (e - c) < 1
z : ℂ
h : max r 0 + Complex.abs c + 1 < Complex.abs z
zz : Complex.abs z ≤ Complex.abs (z ^ d)
⊢ r < Complex.abs (f' d e z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | tendsto_f'_atInf | [105, 1] | [125, 31] | rw [Complex.abs.map_pow] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
e : ℂ
ec : Complex.abs (e - c) < 1
z : ℂ
h : max r 0 + Complex.abs c + 1 < Complex.abs z
⊢ Complex.abs z ≤ Complex.abs (z ^ d) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
e : ℂ
ec : Complex.abs (e - c) < 1
z : ℂ
h : max r 0 + Complex.abs c + 1 < Complex.abs z
⊢ Complex.abs z ≤ Complex.abs z ^ d | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
e : ℂ
ec : Complex.abs (e - c) < 1
z : ℂ
h : max r 0 + Complex.abs c + 1 < Complex.abs z
⊢ Complex.abs z ≤ Complex.abs (z ^ d)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | tendsto_f'_atInf | [105, 1] | [125, 31] | refine le_self_pow ?_ (d_ne_zero _) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
e : ℂ
ec : Complex.abs (e - c) < 1
z : ℂ
h : max r 0 + Complex.abs c + 1 < Complex.abs z
⊢ Complex.abs z ≤ Complex.abs z ^ d | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
e : ℂ
ec : Complex.abs (e - c) < 1
z : ℂ
h : max r 0 + Complex.abs c + 1 < Complex.abs z
⊢ 1 ≤ Complex.abs z | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
e : ℂ
ec : Complex.abs (e - c) < 1
z : ℂ
h : max r 0 + Complex.abs c + 1 < Complex.abs z
⊢ Complex.abs z ≤ Complex.abs z ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | tendsto_f'_atInf | [105, 1] | [125, 31] | exact le_trans (le_add_of_nonneg_left (add_nonneg (le_max_right _ _) (Complex.abs.nonneg _)))
h.le | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
e : ℂ
ec : Complex.abs (e - c) < 1
z : ℂ
h : max r 0 + Complex.abs c + 1 < Complex.abs z
⊢ 1 ≤ Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
e : ℂ
ec : Complex.abs (e - c) < 1
z : ℂ
h : max r 0 + Complex.abs c + 1 < Complex.abs z
⊢ 1 ≤ Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | tendsto_f'_atInf | [105, 1] | [125, 31] | ring_nf | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
e : ℂ
ec : Complex.abs (e - c) < 1
z : ℂ
h : max r 0 + Complex.abs c + 1 < Complex.abs z
zz : Complex.abs z ≤ Complex.abs (z ^ d)
⊢ Complex.abs (z ^ d + e) = Complex.abs (z ^ d + (c + (e - c))) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
e : ℂ
ec : Complex.abs (e - c) < 1
z : ℂ
h : max r 0 + Complex.abs c + 1 < Complex.abs z
zz : Complex.abs z ≤ Complex.abs (z ^ d)
⊢ Complex.abs (z ^ d + e) = Complex.abs (z ^ d + (c + (e - c)))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | tendsto_f'_atInf | [105, 1] | [125, 31] | bound | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
e : ℂ
ec : Complex.abs (e - c) < 1
z : ℂ
h : max r 0 + Complex.abs c + 1 < Complex.abs z
zz : Complex.abs z ≤ Complex.abs (z ^ d)
⊢ Complex.abs (z ^ d + (c + (e - c))) ≥ Complex.abs (z ^ d) - Complex.abs (c + (e - c)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
e : ℂ
ec : Complex.abs (e - c) < 1
z : ℂ
h : max r 0 + Complex.abs c + 1 < Complex.abs z
zz : Complex.abs z ≤ Complex.abs (z ^ d)
⊢ Complex.abs (z ^ d + (c + (e - c))) ≥ Complex.abs (z ^ d) - Complex.abs (c + (e - c))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | tendsto_f'_atInf | [105, 1] | [125, 31] | bound | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
e : ℂ
ec : Complex.abs (e - c) < 1
z : ℂ
h : max r 0 + Complex.abs c + 1 < Complex.abs z
zz : Complex.abs z ≤ Complex.abs (z ^ d)
⊢ Complex.abs (z ^ d) - Complex.abs (c + (e - c)) ≥ Complex.abs (z ^ d) - (Complex.abs c + Complex.abs (e - c)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
e : ℂ
ec : Complex.abs (e - c) < 1
z : ℂ
h : max r 0 + Complex.abs c + 1 < Complex.abs z
zz : Complex.abs z ≤ Complex.abs (z ^ d)
⊢ Complex.abs (z ^ d) - Complex.abs (c + (e - c)) ≥ Complex.abs (z ^ d) - (Complex.abs c + Complex.abs (e - c))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | tendsto_f'_atInf | [105, 1] | [125, 31] | bound | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
e : ℂ
ec : Complex.abs (e - c) < 1
z : ℂ
h : max r 0 + Complex.abs c + 1 < Complex.abs z
zz : Complex.abs z ≤ Complex.abs (z ^ d)
⊢ Complex.abs (z ^ d) - (Complex.abs c + Complex.abs (e - c)) ≥ Complex.abs z - (Complex.abs c + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
e : ℂ
ec : Complex.abs (e - c) < 1
z : ℂ
h : max r 0 + Complex.abs c + 1 < Complex.abs z
zz : Complex.abs z ≤ Complex.abs (z ^ d)
⊢ Complex.abs (z ^ d) - (Complex.abs c + Complex.abs (e - c)) ≥ Complex.abs z - (Complex.abs c + 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | tendsto_f'_atInf | [105, 1] | [125, 31] | bound | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
e : ℂ
ec : Complex.abs (e - c) < 1
z : ℂ
h : max r 0 + Complex.abs c + 1 < Complex.abs z
zz : Complex.abs z ≤ Complex.abs (z ^ d)
⊢ Complex.abs z - (Complex.abs c + 1) > max r 0 + Complex.abs c + 1 - (Complex.abs c + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
e : ℂ
ec : Complex.abs (e - c) < 1
z : ℂ
h : max r 0 + Complex.abs c + 1 < Complex.abs z
zz : Complex.abs z ≤ Complex.abs (z ^ d)
⊢ Complex.abs z - (Complex.abs c + 1) > max r 0 + Complex.abs c + 1 - (Complex.abs c + 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | tendsto_f'_atInf | [105, 1] | [125, 31] | ring_nf | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
e : ℂ
ec : Complex.abs (e - c) < 1
z : ℂ
h : max r 0 + Complex.abs c + 1 < Complex.abs z
zz : Complex.abs z ≤ Complex.abs (z ^ d)
⊢ max r 0 + Complex.abs c + 1 - (Complex.abs c + 1) = max r 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
r : ℝ
e : ℂ
ec : Complex.abs (e - c) < 1
z : ℂ
h : max r 0 + Complex.abs c + 1 < Complex.abs z
zz : Complex.abs z ≤ Complex.abs (z ^ d)
⊢ max r 0 + Complex.abs c + 1 - (Complex.abs c + 1) = max r 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | writtenInExtChartAt_coe_f | [130, 1] | [132, 94] | simp only [writtenInExtChartAt, f, Function.comp, lift_coe', RiemannSphere.extChartAt_coe,
PartialEquiv.symm_symm, coePartialEquiv_apply, coePartialEquiv_symm_apply, toComplex_coe] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
⊢ writtenInExtChartAt I I (↑z) (f d c) = f' d c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
⊢ writtenInExtChartAt I I (↑z) (f d c) = f' d c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | fl_f | [134, 1] | [155, 10] | funext c z | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ fl (f d) ∞ = fun c z => z ^ d / (1 + c * z ^ d) | case h.h
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
⊢ fl (f d) ∞ c z = z ^ d / (1 + c * z ^ d) | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ fl (f d) ∞ = fun c z => z ^ d / (1 + c * z ^ d)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | fl_f | [134, 1] | [155, 10] | simp only [fl, RiemannSphere.extChartAt_inf, Function.comp, invEquiv_apply,
PartialEquiv.trans_apply, Equiv.toPartialEquiv_apply, PartialEquiv.coe_trans_symm,
coePartialEquiv_symm_apply, PartialEquiv.symm_symm, coePartialEquiv_apply,
Equiv.toPartialEquiv_symm_apply, invEquiv_symm, RiemannSphere.inv_inf, toComplex_zero,
add_zero, sub_zero] | case h.h
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
⊢ fl (f d) ∞ c z = z ^ d / (1 + c * z ^ d) | case h.h
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
⊢ (f d c (↑z)⁻¹)⁻¹.toComplex = z ^ d / (1 + c * z ^ d) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
⊢ fl (f d) ∞ c z = z ^ d / (1 + c * z ^ d)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | fl_f | [134, 1] | [155, 10] | by_cases z0 : z = 0 | case h.h
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
⊢ (f d c (↑z)⁻¹)⁻¹.toComplex = z ^ d / (1 + c * z ^ d) | case pos
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z0 : z = 0
⊢ (f d c (↑z)⁻¹)⁻¹.toComplex = z ^ d / (1 + c * z ^ d)
case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z0 : ¬z = 0
⊢ (f d c (↑z)⁻¹)⁻¹.toComplex = z ^ d / (1 + c * z ^ d) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
⊢ (f d c (↑z)⁻¹)⁻¹.toComplex = z ^ d / (1 + c * z ^ d)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | fl_f | [134, 1] | [155, 10] | simp only [f, f', inv_coe z0, lift_coe', inv_pow] | case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z0 : ¬z = 0
⊢ (f d c (↑z)⁻¹)⁻¹.toComplex = z ^ d / (1 + c * z ^ d) | case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z0 : ¬z = 0
⊢ (↑((z ^ d)⁻¹ + c))⁻¹.toComplex = z ^ d / (1 + c * z ^ d) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z0 : ¬z = 0
⊢ (f d c (↑z)⁻¹)⁻¹.toComplex = z ^ d / (1 + c * z ^ d)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | fl_f | [134, 1] | [155, 10] | have zd := pow_ne_zero d z0 | case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z0 : ¬z = 0
⊢ (↑((z ^ d)⁻¹ + c))⁻¹.toComplex = z ^ d / (1 + c * z ^ d) | case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z0 : ¬z = 0
zd : z ^ d ≠ 0
⊢ (↑((z ^ d)⁻¹ + c))⁻¹.toComplex = z ^ d / (1 + c * z ^ d) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z0 : ¬z = 0
⊢ (↑((z ^ d)⁻¹ + c))⁻¹.toComplex = z ^ d / (1 + c * z ^ d)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | fl_f | [134, 1] | [155, 10] | by_cases h : (z ^ d)⁻¹ + c = 0 | case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z0 : ¬z = 0
zd : z ^ d ≠ 0
⊢ (↑((z ^ d)⁻¹ + c))⁻¹.toComplex = z ^ d / (1 + c * z ^ d) | case pos
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z0 : ¬z = 0
zd : z ^ d ≠ 0
h : (z ^ d)⁻¹ + c = 0
⊢ (↑((z ^ d)⁻¹ + c))⁻¹.toComplex = z ^ d / (1 + c * z ^ d)
case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z0 : ¬z = 0
zd : z ^ d ≠ 0
h : ¬(z ^ d)⁻¹ + c = 0
⊢ (↑((z ^ d)⁻¹ + c))⁻¹.toComplex = z ^ d / (1 + c * z ^ d) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z0 : ¬z = 0
zd : z ^ d ≠ 0
⊢ (↑((z ^ d)⁻¹ + c))⁻¹.toComplex = z ^ d / (1 + c * z ^ d)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | fl_f | [134, 1] | [155, 10] | rw [inv_coe h, toComplex_coe, eq_div_iff, inv_mul_eq_iff_eq_mul₀ h, right_distrib,
inv_mul_cancel zd] | case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z0 : ¬z = 0
zd : z ^ d ≠ 0
h : ¬(z ^ d)⁻¹ + c = 0
⊢ (↑((z ^ d)⁻¹ + c))⁻¹.toComplex = z ^ d / (1 + c * z ^ d) | case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z0 : ¬z = 0
zd : z ^ d ≠ 0
h : ¬(z ^ d)⁻¹ + c = 0
⊢ 1 + c * z ^ d ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z0 : ¬z = 0
zd : z ^ d ≠ 0
h : ¬(z ^ d)⁻¹ + c = 0
⊢ (↑((z ^ d)⁻¹ + c))⁻¹.toComplex = z ^ d / (1 + c * z ^ d)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | fl_f | [134, 1] | [155, 10] | contrapose h | case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z0 : ¬z = 0
zd : z ^ d ≠ 0
h : ¬(z ^ d)⁻¹ + c = 0
⊢ 1 + c * z ^ d ≠ 0 | case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z0 : ¬z = 0
zd : z ^ d ≠ 0
h : ¬1 + c * z ^ d ≠ 0
⊢ ¬¬(z ^ d)⁻¹ + c = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z0 : ¬z = 0
zd : z ^ d ≠ 0
h : ¬(z ^ d)⁻¹ + c = 0
⊢ 1 + c * z ^ d ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | fl_f | [134, 1] | [155, 10] | rw [not_not] | case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z0 : ¬z = 0
zd : z ^ d ≠ 0
h : ¬1 + c * z ^ d ≠ 0
⊢ ¬¬(z ^ d)⁻¹ + c = 0 | case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z0 : ¬z = 0
zd : z ^ d ≠ 0
h : ¬1 + c * z ^ d ≠ 0
⊢ (z ^ d)⁻¹ + c = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z0 : ¬z = 0
zd : z ^ d ≠ 0
h : ¬1 + c * z ^ d ≠ 0
⊢ ¬¬(z ^ d)⁻¹ + c = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | fl_f | [134, 1] | [155, 10] | rw [not_not, add_comm, add_eq_zero_iff_eq_neg, ← eq_div_iff zd, neg_div, ←
inv_eq_one_div, ← add_eq_zero_iff_eq_neg, add_comm] at h | case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z0 : ¬z = 0
zd : z ^ d ≠ 0
h : ¬1 + c * z ^ d ≠ 0
⊢ (z ^ d)⁻¹ + c = 0 | case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z0 : ¬z = 0
zd : z ^ d ≠ 0
h : (z ^ d)⁻¹ + c = 0
⊢ (z ^ d)⁻¹ + c = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z0 : ¬z = 0
zd : z ^ d ≠ 0
h : ¬1 + c * z ^ d ≠ 0
⊢ (z ^ d)⁻¹ + c = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | fl_f | [134, 1] | [155, 10] | exact h | case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z0 : ¬z = 0
zd : z ^ d ≠ 0
h : (z ^ d)⁻¹ + c = 0
⊢ (z ^ d)⁻¹ + c = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z0 : ¬z = 0
zd : z ^ d ≠ 0
h : (z ^ d)⁻¹ + c = 0
⊢ (z ^ d)⁻¹ + c = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | fl_f | [134, 1] | [155, 10] | simp only [z0, coe_zero, inv_zero', f, lift_inf', RiemannSphere.inv_inf, toComplex_zero,
zero_pow (d_ne_zero _), zero_div] | case pos
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z0 : z = 0
⊢ (f d c (↑z)⁻¹)⁻¹.toComplex = z ^ d / (1 + c * z ^ d) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z0 : z = 0
⊢ (f d c (↑z)⁻¹)⁻¹.toComplex = z ^ d / (1 + c * z ^ d)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | fl_f | [134, 1] | [155, 10] | simp only [h, coe_zero, inv_zero', toComplex_inf] | case pos
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z0 : ¬z = 0
zd : z ^ d ≠ 0
h : (z ^ d)⁻¹ + c = 0
⊢ (↑((z ^ d)⁻¹ + c))⁻¹.toComplex = z ^ d / (1 + c * z ^ d) | case pos
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z0 : ¬z = 0
zd : z ^ d ≠ 0
h : (z ^ d)⁻¹ + c = 0
⊢ 0 = z ^ d / (1 + c * z ^ d) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z0 : ¬z = 0
zd : z ^ d ≠ 0
h : (z ^ d)⁻¹ + c = 0
⊢ (↑((z ^ d)⁻¹ + c))⁻¹.toComplex = z ^ d / (1 + c * z ^ d)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | fl_f | [134, 1] | [155, 10] | simp only [← add_eq_zero_iff_neg_eq.mp h, neg_mul, inv_mul_cancel zd, ← sub_eq_add_neg,
sub_self, div_zero] | case pos
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z0 : ¬z = 0
zd : z ^ d ≠ 0
h : (z ^ d)⁻¹ + c = 0
⊢ 0 = z ^ d / (1 + c * z ^ d) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z0 : ¬z = 0
zd : z ^ d ≠ 0
h : (z ^ d)⁻¹ + c = 0
⊢ 0 = z ^ d / (1 + c * z ^ d)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | gl_f | [161, 1] | [166, 13] | simp only [fl_f, gl, g] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
⊢ g (fl (f d) ∞ c) d z = gl d c z | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
⊢ (if z = 0 then 1 else z ^ d / (1 + c * z ^ d) / z ^ d) = (1 + c * z ^ d)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
⊢ g (fl (f d) ∞ c) d z = gl d c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | gl_f | [161, 1] | [166, 13] | by_cases z0 : z = 0 | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
⊢ (if z = 0 then 1 else z ^ d / (1 + c * z ^ d) / z ^ d) = (1 + c * z ^ d)⁻¹ | case pos
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z0 : z = 0
⊢ (if z = 0 then 1 else z ^ d / (1 + c * z ^ d) / z ^ d) = (1 + c * z ^ d)⁻¹
case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z0 : ¬z = 0
⊢ (if z = 0 then 1 else z ^ d / (1 + c * z ^ d) / z ^ d) = (1 + c * z ^ d)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
⊢ (if z = 0 then 1 else z ^ d / (1 + c * z ^ d) / z ^ d) = (1 + c * z ^ d)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | gl_f | [161, 1] | [166, 13] | simp only [if_pos, z0, zero_pow (d_ne_zero _), MulZeroClass.mul_zero, add_zero, inv_one] | case pos
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z0 : z = 0
⊢ (if z = 0 then 1 else z ^ d / (1 + c * z ^ d) / z ^ d) = (1 + c * z ^ d)⁻¹
case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z0 : ¬z = 0
⊢ (if z = 0 then 1 else z ^ d / (1 + c * z ^ d) / z ^ d) = (1 + c * z ^ d)⁻¹ | case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z0 : ¬z = 0
⊢ (if z = 0 then 1 else z ^ d / (1 + c * z ^ d) / z ^ d) = (1 + c * z ^ d)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z0 : z = 0
⊢ (if z = 0 then 1 else z ^ d / (1 + c * z ^ d) / z ^ d) = (1 + c * z ^ d)⁻¹
case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z0 : ¬z = 0
⊢ (if z = 0 then 1 else z ^ d / (1 + c * z ^ d) / z ^ d) = (1 + c * z ^ d)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | gl_f | [161, 1] | [166, 13] | rw [if_neg z0, div_eq_mul_inv _ (_ + _), mul_comm, mul_div_assoc, div_self (pow_ne_zero _ z0),
mul_one] | case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z0 : ¬z = 0
⊢ (if z = 0 then 1 else z ^ d / (1 + c * z ^ d) / z ^ d) = (1 + c * z ^ d)⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z0 : ¬z = 0
⊢ (if z = 0 then 1 else z ^ d / (1 + c * z ^ d) / z ^ d) = (1 + c * z ^ d)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | analyticAt_gl | [168, 1] | [171, 36] | apply (analyticAt_const.add (analyticAt_const.mul ((analyticAt_id _ _).pow _))).inv | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ AnalyticAt ℂ (gl d c) 0 | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ((fun x => 1) + fun x => c * id x ^ d) 0 ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ AnalyticAt ℂ (gl d c) 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | analyticAt_gl | [168, 1] | [171, 36] | simp only [Pi.pow_apply, id_eq, Pi.add_apply, ne_eq, zero_pow (d_ne_zero _), mul_zero, add_zero,
one_ne_zero, not_false_eq_true] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ((fun x => 1) + fun x => c * id x ^ d) 0 ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ((fun x => 1) + fun x => c * id x ^ d) 0 ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | fl_f' | [173, 1] | [174, 85] | funext c z | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ fl (f d) ∞ = fun c z => (z - 0) ^ d • gl d c z | case h.h
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
⊢ fl (f d) ∞ c z = (z - 0) ^ d • gl d c z | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ fl (f d) ∞ = fun c z => (z - 0) ^ d • gl d c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | fl_f' | [173, 1] | [174, 85] | simp only [fl_f, gl, sub_zero, Algebra.id.smul_eq_mul, div_eq_mul_inv] | case h.h
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
⊢ fl (f d) ∞ c z = (z - 0) ^ d • gl d c z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
⊢ fl (f d) ∞ c z = (z - 0) ^ d • gl d c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | gl_zero | [176, 1] | [177, 74] | simp only [gl, zero_pow (d_ne_zero _), MulZeroClass.mul_zero] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ gl d c 0 = 1 | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ (1 + 0)⁻¹ = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ gl d c 0 = 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | gl_zero | [176, 1] | [177, 74] | norm_num | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ (1 + 0)⁻¹ = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ (1 + 0)⁻¹ = 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | gl_frequently_ne_zero | [179, 1] | [181, 20] | refine (analyticAt_gl.continuousAt.eventually_ne ?_).frequently | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ∃ᶠ (z : ℂ) in 𝓝 0, gl d c z ≠ 0 | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ gl d c 0 ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ∃ᶠ (z : ℂ) in 𝓝 0, gl d c z ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | gl_frequently_ne_zero | [179, 1] | [181, 20] | simp only [gl_zero] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ gl d c 0 ≠ 0 | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ 1 ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ gl d c 0 ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | gl_frequently_ne_zero | [179, 1] | [181, 20] | exact one_ne_zero | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ 1 ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ 1 ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | fc_f | [183, 1] | [185, 49] | rw [fl_f', analyticAt_gl.monomial_mul_leadingCoeff gl_frequently_ne_zero, leadingCoeff_of_ne_zero] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ leadingCoeff (fl (f d) ∞ c) 0 = 1 | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ gl d c 0 = 1
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ gl d c 0 ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ leadingCoeff (fl (f d) ∞ c) 0 = 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | fc_f | [183, 1] | [185, 49] | exact gl_zero | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ gl d c 0 = 1
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ gl d c 0 ≠ 0 | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ gl d c 0 ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ gl d c 0 = 1
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ gl d c 0 ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | fc_f | [183, 1] | [185, 49] | rw [gl_zero] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ gl d c 0 ≠ 0 | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ 1 ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ gl d c 0 ≠ 0
TACTIC:
|
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