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 | diam_ball_eq | [295, 1] | [326, 50] | linarith | 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
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r
m : ∀ (t : ℝ), |t| ≤ 1 → c + ↑t * (↑r - ↑e / 2) ∈ ball c r
lo : dist (c + ↑1 * (↑r - ↑e / 2)) (c + ↑(-1) * (↑r - ↑e / 2)) ≤ Metric.diam (ball c r)
⊢ 2 * r - e ≥ r - e
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
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r
m : ∀ (t : ℝ), |t| ≤ 1 → c + ↑t * (↑r - ↑e / 2) ∈ ball c r
lo : dist (c + ↑1 * (↑r - ↑e / 2)) (c + ↑(-1) * (↑r - ↑e / 2)) ≤ Metric.diam (ball c r)
⊢ r - e ≥ 0 | 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
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r
m : ∀ (t : ℝ), |t| ≤ 1 → c + ↑t * (↑r - ↑e / 2) ∈ ball c r
lo : dist (c + ↑1 * (↑r - ↑e / 2)) (c + ↑(-1) * (↑r - ↑e / 2)) ≤ Metric.diam (ball c r)
⊢ r - e ≥ 0 | 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
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r
m : ∀ (t : ℝ), |t| ≤ 1 → c + ↑t * (↑r - ↑e / 2) ∈ ball c r
lo : dist (c + ↑1 * (↑r - ↑e / 2)) (c + ↑(-1) * (↑r - ↑e / 2)) ≤ Metric.diam (ball c r)
⊢ 2 * r - e ≥ r - e
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
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r
m : ∀ (t : ℝ), |t| ≤ 1 → c + ↑t * (↑r - ↑e / 2) ∈ ball c r
lo : dist (c + ↑1 * (↑r - ↑e / 2)) (c + ↑(-1) * (↑r - ↑e / 2)) ≤ Metric.diam (ball c r)
⊢ r - e ≥ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | diam_ball_eq | [295, 1] | [326, 50] | simp only [sub_nonneg, er.le] | 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
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r
m : ∀ (t : ℝ), |t| ≤ 1 → c + ↑t * (↑r - ↑e / 2) ∈ ball c r
lo : dist (c + ↑1 * (↑r - ↑e / 2)) (c + ↑(-1) * (↑r - ↑e / 2)) ≤ Metric.diam (ball c r)
⊢ r - e ≥ 0 | 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
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r
m : ∀ (t : ℝ), |t| ≤ 1 → c + ↑t * (↑r - ↑e / 2) ∈ ball c r
lo : dist (c + ↑1 * (↑r - ↑e / 2)) (c + ↑(-1) * (↑r - ↑e / 2)) ≤ Metric.diam (ball c r)
⊢ r - e ≥ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | diam_ball_eq | [295, 1] | [326, 50] | simp only [Complex.ofReal_sub, Complex.ofReal_mul, Complex.ofReal_one] | 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
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r
m : ∀ (t : ℝ), |t| ≤ 1 → c + ↑t * (↑r - ↑e / 2) ∈ ball c r
lo : dist (c + ↑1 * (↑r - ↑e / 2)) (c + ↑(-1) * (↑r - ↑e / 2)) ≤ Metric.diam (ball c r)
re : 2 * r - e ≥ 0
⊢ Complex.abs (2 * ↑r - ↑e) = Complex.abs ↑(2 * r - e) | 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
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r
m : ∀ (t : ℝ), |t| ≤ 1 → c + ↑t * (↑r - ↑e / 2) ∈ ball c r
lo : dist (c + ↑1 * (↑r - ↑e / 2)) (c + ↑(-1) * (↑r - ↑e / 2)) ≤ Metric.diam (ball c r)
re : 2 * r - e ≥ 0
⊢ Complex.abs (2 * ↑r - ↑e) = Complex.abs (↑2 * ↑r - ↑e) | 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
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r
m : ∀ (t : ℝ), |t| ≤ 1 → c + ↑t * (↑r - ↑e / 2) ∈ ball c r
lo : dist (c + ↑1 * (↑r - ↑e / 2)) (c + ↑(-1) * (↑r - ↑e / 2)) ≤ Metric.diam (ball c r)
re : 2 * r - e ≥ 0
⊢ Complex.abs (2 * ↑r - ↑e) = Complex.abs ↑(2 * r - e)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | diam_ball_eq | [295, 1] | [326, 50] | norm_num | 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
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r
m : ∀ (t : ℝ), |t| ≤ 1 → c + ↑t * (↑r - ↑e / 2) ∈ ball c r
lo : dist (c + ↑1 * (↑r - ↑e / 2)) (c + ↑(-1) * (↑r - ↑e / 2)) ≤ Metric.diam (ball c r)
re : 2 * r - e ≥ 0
⊢ Complex.abs (2 * ↑r - ↑e) = Complex.abs (↑2 * ↑r - ↑e) | 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
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r
m : ∀ (t : ℝ), |t| ≤ 1 → c + ↑t * (↑r - ↑e / 2) ∈ ball c r
lo : dist (c + ↑1 * (↑r - ↑e / 2)) (c + ↑(-1) * (↑r - ↑e / 2)) ≤ Metric.diam (ball c r)
re : 2 * r - e ≥ 0
⊢ Complex.abs (2 * ↑r - ↑e) = Complex.abs (↑2 * ↑r - ↑e)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | diam_ball_eq | [295, 1] | [326, 50] | rw [Complex.abs_ofReal, abs_of_nonneg re] | 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
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r
m : ∀ (t : ℝ), |t| ≤ 1 → c + ↑t * (↑r - ↑e / 2) ∈ ball c r
lo : dist (c + ↑1 * (↑r - ↑e / 2)) (c + ↑(-1) * (↑r - ↑e / 2)) ≤ Metric.diam (ball c r)
re : 2 * r - e ≥ 0
⊢ Complex.abs ↑(2 * r - e) = 2 * r - e | 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
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r
m : ∀ (t : ℝ), |t| ≤ 1 → c + ↑t * (↑r - ↑e / 2) ∈ ball c r
lo : dist (c + ↑1 * (↑r - ↑e / 2)) (c + ↑(-1) * (↑r - ↑e / 2)) ≤ Metric.diam (ball c r)
re : 2 * r - e ≥ 0
⊢ Complex.abs ↑(2 * r - e) = 2 * r - e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | diam_closedBall_eq | [329, 1] | [333, 83] | apply le_antisymm (Metric.diam_closedBall rp) | 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
c : ℂ
r : ℝ
rp : r ≥ 0
⊢ Metric.diam (closedBall c r) = 2 * 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)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
⊢ 2 * r ≤ Metric.diam (closedBall 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)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
⊢ Metric.diam (closedBall c r) = 2 * r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | diam_closedBall_eq | [329, 1] | [333, 83] | trans Metric.diam (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)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
⊢ 2 * r ≤ Metric.diam (closedBall 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)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
⊢ 2 * r ≤ Metric.diam (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)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
⊢ Metric.diam (ball c r) ≤ Metric.diam (closedBall 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)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
⊢ 2 * r ≤ Metric.diam (closedBall c r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | diam_closedBall_eq | [329, 1] | [333, 83] | rw [diam_ball_eq rp] | 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
c : ℂ
r : ℝ
rp : r ≥ 0
⊢ 2 * r ≤ Metric.diam (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)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
⊢ Metric.diam (ball c r) ≤ Metric.diam (closedBall 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)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
⊢ Metric.diam (ball c r) ≤ Metric.diam (closedBall 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)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
⊢ 2 * r ≤ Metric.diam (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)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
⊢ Metric.diam (ball c r) ≤ Metric.diam (closedBall c r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | diam_closedBall_eq | [329, 1] | [333, 83] | exact Metric.diam_mono Metric.ball_subset_closedBall Metric.isBounded_closedBall | 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
c : ℂ
r : ℝ
rp : r ≥ 0
⊢ Metric.diam (ball c r) ≤ Metric.diam (closedBall 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)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
⊢ Metric.diam (ball c r) ≤ Metric.diam (closedBall c r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | le_of_ball_subset_closedBall | [336, 1] | [340, 63] | intro 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)
c0 c0' c1 z0✝ z1✝ w0 w1 : ℂ
r r0✝ r1✝ b e : ℝ
z0 z1 : ℂ
r0 r1 : ℝ
r0p : r0 ≥ 0
r1p : r1 ≥ 0
⊢ ball z0 r0 ⊆ closedBall z1 r1 → r0 ≤ 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)
c0 c0' c1 z0✝ z1✝ w0 w1 : ℂ
r r0✝ r1✝ b e : ℝ
z0 z1 : ℂ
r0 r1 : ℝ
r0p : r0 ≥ 0
r1p : r1 ≥ 0
s : ball z0 r0 ⊆ closedBall z1 r1
⊢ r0 ≤ 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)
c0 c0' c1 z0✝ z1✝ w0 w1 : ℂ
r r0✝ r1✝ b e : ℝ
z0 z1 : ℂ
r0 r1 : ℝ
r0p : r0 ≥ 0
r1p : r1 ≥ 0
⊢ ball z0 r0 ⊆ closedBall z1 r1 → r0 ≤ r1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | le_of_ball_subset_closedBall | [336, 1] | [340, 63] | have m := Metric.diam_mono s Metric.isBounded_closedBall | 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
z0 z1 : ℂ
r0 r1 : ℝ
r0p : r0 ≥ 0
r1p : r1 ≥ 0
s : ball z0 r0 ⊆ closedBall z1 r1
⊢ r0 ≤ 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)
c0 c0' c1 z0✝ z1✝ w0 w1 : ℂ
r r0✝ r1✝ b e : ℝ
z0 z1 : ℂ
r0 r1 : ℝ
r0p : r0 ≥ 0
r1p : r1 ≥ 0
s : ball z0 r0 ⊆ closedBall z1 r1
m : Metric.diam (ball z0 r0) ≤ Metric.diam (closedBall z1 r1)
⊢ r0 ≤ 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)
c0 c0' c1 z0✝ z1✝ w0 w1 : ℂ
r r0✝ r1✝ b e : ℝ
z0 z1 : ℂ
r0 r1 : ℝ
r0p : r0 ≥ 0
r1p : r1 ≥ 0
s : ball z0 r0 ⊆ closedBall z1 r1
⊢ r0 ≤ r1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | le_of_ball_subset_closedBall | [336, 1] | [340, 63] | rw [diam_ball_eq r0p, diam_closedBall_eq r1p] at m | 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
z0 z1 : ℂ
r0 r1 : ℝ
r0p : r0 ≥ 0
r1p : r1 ≥ 0
s : ball z0 r0 ⊆ closedBall z1 r1
m : Metric.diam (ball z0 r0) ≤ Metric.diam (closedBall z1 r1)
⊢ r0 ≤ 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)
c0 c0' c1 z0✝ z1✝ w0 w1 : ℂ
r r0✝ r1✝ b e : ℝ
z0 z1 : ℂ
r0 r1 : ℝ
r0p : r0 ≥ 0
r1p : r1 ≥ 0
s : ball z0 r0 ⊆ closedBall z1 r1
m : 2 * r0 ≤ 2 * r1
⊢ r0 ≤ 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)
c0 c0' c1 z0✝ z1✝ w0 w1 : ℂ
r r0✝ r1✝ b e : ℝ
z0 z1 : ℂ
r0 r1 : ℝ
r0p : r0 ≥ 0
r1p : r1 ≥ 0
s : ball z0 r0 ⊆ closedBall z1 r1
m : Metric.diam (ball z0 r0) ≤ Metric.diam (closedBall z1 r1)
⊢ r0 ≤ r1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | le_of_ball_subset_closedBall | [336, 1] | [340, 63] | linarith | 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
z0 z1 : ℂ
r0 r1 : ℝ
r0p : r0 ≥ 0
r1p : r1 ≥ 0
s : ball z0 r0 ⊆ closedBall z1 r1
m : 2 * r0 ≤ 2 * r1
⊢ r0 ≤ 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)
c0 c0' c1 z0✝ z1✝ w0 w1 : ℂ
r r0✝ r1✝ b e : ℝ
z0 z1 : ℂ
r0 r1 : ℝ
r0p : r0 ≥ 0
r1p : r1 ≥ 0
s : ball z0 r0 ⊆ closedBall z1 r1
m : 2 * r0 ≤ 2 * r1
⊢ r0 ≤ r1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | to_uneven | [344, 1] | [373, 16] | have r4p : r / 4 > 0 := by linarith | 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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
⊢ ∃ c0' r0 r1, ball c0' r1 ⊆ closedBall c0 r ∧ c0 ∈ ball c0' r1 ∧ Uneven f c0' c1 r0 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)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
⊢ ∃ c0' r0 r1, ball c0' r1 ⊆ closedBall c0 r ∧ c0 ∈ ball c0' r1 ∧ Uneven f c0' c1 r0 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)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
⊢ ∃ c0' r0 r1, ball c0' r1 ⊆ closedBall c0 r ∧ c0 ∈ ball c0' r1 ∧ Uneven f c0' c1 r0 r1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | to_uneven | [344, 1] | [373, 16] | rcases on_subdisk h rp r4p with ⟨c0', r0, r0p, m, 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)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
⊢ ∃ c0' r0 r1, ball c0' r1 ⊆ closedBall c0 r ∧ c0 ∈ ball c0' r1 ∧ Uneven f c0' c1 r0 r1 | 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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
m : c0' ∈ closedBall c0 (r / 4)
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
⊢ ∃ c0' r0 r1, ball c0' r1 ⊆ closedBall c0 r ∧ c0 ∈ ball c0' r1 ∧ Uneven f c0' c1 r0 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)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
⊢ ∃ c0' r0 r1, ball c0' r1 ⊆ closedBall c0 r ∧ c0 ∈ ball c0' r1 ∧ Uneven f c0' c1 r0 r1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | to_uneven | [344, 1] | [373, 16] | simp only [Metric.mem_closedBall] at m | 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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
m : c0' ∈ closedBall c0 (r / 4)
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
⊢ ∃ c0' r0 r1, ball c0' r1 ⊆ closedBall c0 r ∧ c0 ∈ ball c0' r1 ∧ Uneven f c0' c1 r0 r1 | 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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
⊢ ∃ c0' r0 r1, ball c0' r1 ⊆ closedBall c0 r ∧ c0 ∈ ball c0' r1 ∧ Uneven f c0' c1 r0 r1 | 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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
m : c0' ∈ closedBall c0 (r / 4)
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
⊢ ∃ c0' r0 r1, ball c0' r1 ⊆ closedBall c0 r ∧ c0 ∈ ball c0' r1 ∧ Uneven f c0' c1 r0 r1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | to_uneven | [344, 1] | [373, 16] | have sub : closedBall c0' (r / 2) ⊆ closedBall c0 r := by
apply Metric.closedBall_subset_closedBall'
calc
r / 2 + dist c0' c0 ≤ r / 2 + r / 4 := by linarith
_ = 3 / 4 * r := by ring
_ ≤ 1 * r := by linarith
_ = r := by ring | 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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
⊢ ∃ c0' r0 r1, ball c0' r1 ⊆ closedBall c0 r ∧ c0 ∈ ball c0' r1 ∧ Uneven f c0' c1 r0 r1 | 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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
⊢ ∃ c0' r0 r1, ball c0' r1 ⊆ closedBall c0 r ∧ c0 ∈ ball c0' r1 ∧ Uneven f c0' c1 r0 r1 | 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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
⊢ ∃ c0' r0 r1, ball c0' r1 ⊆ closedBall c0 r ∧ c0 ∈ ball c0' r1 ∧ Uneven f c0' c1 r0 r1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | to_uneven | [344, 1] | [373, 16] | have r01 : min r0 (r / 2) ≤ r / 2 := by bound | 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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
⊢ ∃ c0' r0 r1, ball c0' r1 ⊆ closedBall c0 r ∧ c0 ∈ ball c0' r1 ∧ Uneven f c0' c1 r0 r1 | 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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
⊢ ∃ c0' r0 r1, ball c0' r1 ⊆ closedBall c0 r ∧ c0 ∈ ball c0' r1 ∧ Uneven f c0' c1 r0 r1 | 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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
⊢ ∃ c0' r0 r1, ball c0' r1 ⊆ closedBall c0 r ∧ c0 ∈ ball c0' r1 ∧ Uneven f c0' c1 r0 r1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | to_uneven | [344, 1] | [373, 16] | have c0m : c0 ∈ ball c0' (r / 2) := by
simp only [Metric.mem_ball]; rw [dist_comm]; apply lt_of_le_of_lt m; bound | 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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
⊢ ∃ c0' r0 r1, ball c0' r1 ⊆ closedBall c0 r ∧ c0 ∈ ball c0' r1 ∧ Uneven f c0' c1 r0 r1 | 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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
⊢ ∃ c0' r0 r1, ball c0' r1 ⊆ closedBall c0 r ∧ c0 ∈ ball c0' r1 ∧ Uneven f c0' c1 r0 r1 | 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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
⊢ ∃ c0' r0 r1, ball c0' r1 ⊆ closedBall c0 r ∧ c0 ∈ ball c0' r1 ∧ Uneven f c0' c1 r0 r1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | to_uneven | [344, 1] | [373, 16] | have h' : Har f (closedBall (c0', c1) (r / 2)) := by
refine Har.mono ?_ h; simp only [← closedBall_prod_same]; apply Set.prod_mono
assumption; apply Metric.closedBall_subset_closedBall; linarith | 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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
⊢ ∃ c0' r0 r1, ball c0' r1 ⊆ closedBall c0 r ∧ c0 ∈ ball c0' r1 ∧ Uneven f c0' c1 r0 r1 | 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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
h' : Har f (closedBall (c0', c1) (r / 2))
⊢ ∃ c0' r0 r1, ball c0' r1 ⊆ closedBall c0 r ∧ c0 ∈ ball c0' r1 ∧ Uneven f c0' c1 r0 r1 | 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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
⊢ ∃ c0' r0 r1, ball c0' r1 ⊆ closedBall c0 r ∧ c0 ∈ ball c0' r1 ∧ Uneven f c0' c1 r0 r1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | to_uneven | [344, 1] | [373, 16] | have a' : AnalyticOn ℂ f (ball c0' (min r0 (r / 2)) ×ˢ ball c1 (r / 2)) := by
apply a.mono; apply Set.prod_mono
apply Metric.ball_subset_ball'
simp only [dist_self, add_zero, min_le_iff, le_refl, true_or_iff]
apply Metric.ball_subset_ball; linarith | 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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
h' : Har f (closedBall (c0', c1) (r / 2))
⊢ ∃ c0' r0 r1, ball c0' r1 ⊆ closedBall c0 r ∧ c0 ∈ ball c0' r1 ∧ Uneven f c0' c1 r0 r1 | 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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
h' : Har f (closedBall (c0', c1) (r / 2))
a' : AnalyticOn ℂ f (ball c0' (min r0 (r / 2)) ×ˢ ball c1 (r / 2))
⊢ ∃ c0' r0 r1, ball c0' r1 ⊆ closedBall c0 r ∧ c0 ∈ ball c0' r1 ∧ Uneven f c0' c1 r0 r1 | 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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
h' : Har f (closedBall (c0', c1) (r / 2))
⊢ ∃ c0' r0 r1, ball c0' r1 ⊆ closedBall c0 r ∧ c0 ∈ ball c0' r1 ∧ Uneven f c0' c1 r0 r1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | to_uneven | [344, 1] | [373, 16] | use c0', min r0 (r / 2), r / 2, _root_.trans Metric.ball_subset_closedBall sub, c0m | 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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
h' : Har f (closedBall (c0', c1) (r / 2))
a' : AnalyticOn ℂ f (ball c0' (min r0 (r / 2)) ×ˢ ball c1 (r / 2))
⊢ ∃ c0' r0 r1, ball c0' r1 ⊆ closedBall c0 r ∧ c0 ∈ ball c0' r1 ∧ Uneven f c0' c1 r0 r1 | 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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
h' : Har f (closedBall (c0', c1) (r / 2))
a' : AnalyticOn ℂ f (ball c0' (min r0 (r / 2)) ×ˢ ball c1 (r / 2))
⊢ Uneven f c0' c1 (min r0 (r / 2)) (r / 2) | 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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
h' : Har f (closedBall (c0', c1) (r / 2))
a' : AnalyticOn ℂ f (ball c0' (min r0 (r / 2)) ×ˢ ball c1 (r / 2))
⊢ ∃ c0' r0 r1, ball c0' r1 ⊆ closedBall c0 r ∧ c0 ∈ ball c0' r1 ∧ Uneven f c0' c1 r0 r1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | to_uneven | [344, 1] | [373, 16] | exact
{ r0p := by bound
r1p := by bound
r01
h := h'
a := a' } | 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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
h' : Har f (closedBall (c0', c1) (r / 2))
a' : AnalyticOn ℂ f (ball c0' (min r0 (r / 2)) ×ˢ ball c1 (r / 2))
⊢ Uneven f c0' c1 (min r0 (r / 2)) (r / 2) | 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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
h' : Har f (closedBall (c0', c1) (r / 2))
a' : AnalyticOn ℂ f (ball c0' (min r0 (r / 2)) ×ˢ ball c1 (r / 2))
⊢ Uneven f c0' c1 (min r0 (r / 2)) (r / 2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | to_uneven | [344, 1] | [373, 16] | linarith | 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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
⊢ r / 4 > 0 | 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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
⊢ r / 4 > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | to_uneven | [344, 1] | [373, 16] | apply Metric.closedBall_subset_closedBall' | 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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
⊢ closedBall c0' (r / 2) ⊆ closedBall c0 r | case 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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
⊢ r / 2 + dist c0' c0 ≤ 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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
⊢ closedBall c0' (r / 2) ⊆ closedBall c0 r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | to_uneven | [344, 1] | [373, 16] | calc
r / 2 + dist c0' c0 ≤ r / 2 + r / 4 := by linarith
_ = 3 / 4 * r := by ring
_ ≤ 1 * r := by linarith
_ = r := by ring | case 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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
⊢ r / 2 + dist c0' c0 ≤ r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case 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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
⊢ r / 2 + dist c0' c0 ≤ r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | to_uneven | [344, 1] | [373, 16] | linarith | 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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
⊢ r / 2 + dist c0' c0 ≤ r / 2 + r / 4 | 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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
⊢ r / 2 + dist c0' c0 ≤ r / 2 + r / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | to_uneven | [344, 1] | [373, 16] | ring | 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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
⊢ r / 2 + r / 4 = 3 / 4 * 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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
⊢ r / 2 + r / 4 = 3 / 4 * r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | to_uneven | [344, 1] | [373, 16] | linarith | 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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
⊢ 3 / 4 * r ≤ 1 * 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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
⊢ 3 / 4 * r ≤ 1 * r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | to_uneven | [344, 1] | [373, 16] | ring | 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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
⊢ 1 * r = 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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
⊢ 1 * r = r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | to_uneven | [344, 1] | [373, 16] | bound | 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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
⊢ min r0 (r / 2) ≤ r / 2 | 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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
⊢ min r0 (r / 2) ≤ r / 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | to_uneven | [344, 1] | [373, 16] | simp only [Metric.mem_ball] | 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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
⊢ c0 ∈ ball c0' (r / 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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
⊢ dist c0 c0' < r / 2 | 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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
⊢ c0 ∈ ball c0' (r / 2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | to_uneven | [344, 1] | [373, 16] | rw [dist_comm] | 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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
⊢ dist c0 c0' < r / 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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
⊢ dist c0' c0 < r / 2 | 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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
⊢ dist c0 c0' < r / 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | to_uneven | [344, 1] | [373, 16] | apply lt_of_le_of_lt m | 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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
⊢ dist c0' c0 < r / 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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
⊢ r / 4 < r / 2 | 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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
⊢ dist c0' c0 < r / 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | to_uneven | [344, 1] | [373, 16] | bound | 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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
⊢ r / 4 < r / 2 | 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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
⊢ r / 4 < r / 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | to_uneven | [344, 1] | [373, 16] | refine Har.mono ?_ 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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
⊢ Har f (closedBall (c0', c1) (r / 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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
⊢ closedBall (c0', c1) (r / 2) ⊆ closedBall (c0, c1) 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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
⊢ Har f (closedBall (c0', c1) (r / 2))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | to_uneven | [344, 1] | [373, 16] | simp only [← closedBall_prod_same] | 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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
⊢ closedBall (c0', c1) (r / 2) ⊆ closedBall (c0, c1) 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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
⊢ closedBall c0' (r / 2) ×ˢ closedBall c1 (r / 2) ⊆ closedBall c0 r ×ˢ closedBall c1 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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
⊢ closedBall (c0', c1) (r / 2) ⊆ closedBall (c0, c1) r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | to_uneven | [344, 1] | [373, 16] | apply Set.prod_mono | 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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
⊢ closedBall c0' (r / 2) ×ˢ closedBall c1 (r / 2) ⊆ closedBall c0 r ×ˢ closedBall c1 r | case hs
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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
⊢ closedBall c0' (r / 2) ⊆ closedBall c0 r
case ht
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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
⊢ closedBall c1 (r / 2) ⊆ closedBall c1 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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
⊢ closedBall c0' (r / 2) ×ˢ closedBall c1 (r / 2) ⊆ closedBall c0 r ×ˢ closedBall c1 r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | to_uneven | [344, 1] | [373, 16] | assumption | case hs
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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
⊢ closedBall c0' (r / 2) ⊆ closedBall c0 r
case ht
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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
⊢ closedBall c1 (r / 2) ⊆ closedBall c1 r | case ht
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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
⊢ closedBall c1 (r / 2) ⊆ closedBall c1 r | Please generate a tactic in lean4 to solve the state.
STATE:
case hs
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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
⊢ closedBall c0' (r / 2) ⊆ closedBall c0 r
case ht
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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
⊢ closedBall c1 (r / 2) ⊆ closedBall c1 r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | to_uneven | [344, 1] | [373, 16] | apply Metric.closedBall_subset_closedBall | case ht
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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
⊢ closedBall c1 (r / 2) ⊆ closedBall c1 r | case ht.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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
⊢ r / 2 ≤ r | Please generate a tactic in lean4 to solve the state.
STATE:
case ht
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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
⊢ closedBall c1 (r / 2) ⊆ closedBall c1 r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | to_uneven | [344, 1] | [373, 16] | linarith | case ht.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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
⊢ r / 2 ≤ r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case ht.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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
⊢ r / 2 ≤ r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | to_uneven | [344, 1] | [373, 16] | apply a.mono | 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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
h' : Har f (closedBall (c0', c1) (r / 2))
⊢ AnalyticOn ℂ f (ball c0' (min r0 (r / 2)) ×ˢ ball c1 (r / 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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
h' : Har f (closedBall (c0', c1) (r / 2))
⊢ ball c0' (min r0 (r / 2)) ×ˢ ball c1 (r / 2) ⊆ ball c0' r0 ×ˢ ball c1 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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
h' : Har f (closedBall (c0', c1) (r / 2))
⊢ AnalyticOn ℂ f (ball c0' (min r0 (r / 2)) ×ˢ ball c1 (r / 2))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | to_uneven | [344, 1] | [373, 16] | apply Set.prod_mono | 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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
h' : Har f (closedBall (c0', c1) (r / 2))
⊢ ball c0' (min r0 (r / 2)) ×ˢ ball c1 (r / 2) ⊆ ball c0' r0 ×ˢ ball c1 r | case hs
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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
h' : Har f (closedBall (c0', c1) (r / 2))
⊢ ball c0' (min r0 (r / 2)) ⊆ ball c0' r0
case ht
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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
h' : Har f (closedBall (c0', c1) (r / 2))
⊢ ball c1 (r / 2) ⊆ ball c1 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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
h' : Har f (closedBall (c0', c1) (r / 2))
⊢ ball c0' (min r0 (r / 2)) ×ˢ ball c1 (r / 2) ⊆ ball c0' r0 ×ˢ ball c1 r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | to_uneven | [344, 1] | [373, 16] | apply Metric.ball_subset_ball' | case hs
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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
h' : Har f (closedBall (c0', c1) (r / 2))
⊢ ball c0' (min r0 (r / 2)) ⊆ ball c0' r0
case ht
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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
h' : Har f (closedBall (c0', c1) (r / 2))
⊢ ball c1 (r / 2) ⊆ ball c1 r | case hs.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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
h' : Har f (closedBall (c0', c1) (r / 2))
⊢ min r0 (r / 2) + dist c0' c0' ≤ r0
case ht
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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
h' : Har f (closedBall (c0', c1) (r / 2))
⊢ ball c1 (r / 2) ⊆ ball c1 r | Please generate a tactic in lean4 to solve the state.
STATE:
case hs
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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
h' : Har f (closedBall (c0', c1) (r / 2))
⊢ ball c0' (min r0 (r / 2)) ⊆ ball c0' r0
case ht
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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
h' : Har f (closedBall (c0', c1) (r / 2))
⊢ ball c1 (r / 2) ⊆ ball c1 r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | to_uneven | [344, 1] | [373, 16] | simp only [dist_self, add_zero, min_le_iff, le_refl, true_or_iff] | case hs.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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
h' : Har f (closedBall (c0', c1) (r / 2))
⊢ min r0 (r / 2) + dist c0' c0' ≤ r0
case ht
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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
h' : Har f (closedBall (c0', c1) (r / 2))
⊢ ball c1 (r / 2) ⊆ ball c1 r | case ht
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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
h' : Har f (closedBall (c0', c1) (r / 2))
⊢ ball c1 (r / 2) ⊆ ball c1 r | Please generate a tactic in lean4 to solve the state.
STATE:
case hs.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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
h' : Har f (closedBall (c0', c1) (r / 2))
⊢ min r0 (r / 2) + dist c0' c0' ≤ r0
case ht
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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
h' : Har f (closedBall (c0', c1) (r / 2))
⊢ ball c1 (r / 2) ⊆ ball c1 r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | to_uneven | [344, 1] | [373, 16] | apply Metric.ball_subset_ball | case ht
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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
h' : Har f (closedBall (c0', c1) (r / 2))
⊢ ball c1 (r / 2) ⊆ ball c1 r | case ht.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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
h' : Har f (closedBall (c0', c1) (r / 2))
⊢ r / 2 ≤ r | Please generate a tactic in lean4 to solve the state.
STATE:
case ht
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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
h' : Har f (closedBall (c0', c1) (r / 2))
⊢ ball c1 (r / 2) ⊆ ball c1 r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | to_uneven | [344, 1] | [373, 16] | linarith | case ht.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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
h' : Har f (closedBall (c0', c1) (r / 2))
⊢ r / 2 ≤ r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case ht.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)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0✝ r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
h' : Har f (closedBall (c0', c1) (r / 2))
⊢ r / 2 ≤ r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | to_uneven | [344, 1] | [373, 16] | bound | 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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
h' : Har f (closedBall (c0', c1) (r / 2))
a' : AnalyticOn ℂ f (ball c0' (min r0 (r / 2)) ×ˢ ball c1 (r / 2))
⊢ min r0 (r / 2) > 0 | 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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
h' : Har f (closedBall (c0', c1) (r / 2))
a' : AnalyticOn ℂ f (ball c0' (min r0 (r / 2)) ×ˢ ball c1 (r / 2))
⊢ min r0 (r / 2) > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | to_uneven | [344, 1] | [373, 16] | bound | 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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
h' : Har f (closedBall (c0', c1) (r / 2))
a' : AnalyticOn ℂ f (ball c0' (min r0 (r / 2)) ×ˢ ball c1 (r / 2))
⊢ r / 2 > 0 | 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
h : Har f (closedBall (c0, c1) r)
rp : r > 0
r4p : r / 4 > 0
c0' : ℂ
r0 : ℝ
r0p : r0 > 0
a : AnalyticOn ℂ f (ball c0' r0 ×ˢ ball c1 r)
m : dist c0' c0 ≤ r / 4
sub : closedBall c0' (r / 2) ⊆ closedBall c0 r
r01 : min r0 (r / 2) ≤ r / 2
c0m : c0 ∈ ball c0' (r / 2)
h' : Har f (closedBall (c0', c1) (r / 2))
a' : AnalyticOn ℂ f (ball c0' (min r0 (r / 2)) ×ˢ ball c1 (r / 2))
⊢ r / 2 > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_apply | [392, 1] | [395, 14] | simp only [unevenSeries', ContinuousMultilinearMap.mkPiRing_apply, Finset.prod_const_one,
one_smul] | 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
r : ℝ
z1 : ℂ
n : ℕ
⊢ ((unevenSeries' u r z1 n) fun x => 1) = unevenTerm' u r z1 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
r : ℝ
z1 : ℂ
n : ℕ
⊢ ((unevenSeries' u r z1 n) fun x => 1) = unevenTerm' u r z1 n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | uneven_is_cauchy | [397, 1] | [399, 53] | funext | 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
r : ℝ
⊢ unevenSeries' u r z1 = cauchyPowerSeries (fun z0 => f (z0, z1)) c0 r | case 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)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
r : ℝ
x✝ : ℕ
⊢ unevenSeries' u r z1 x✝ = cauchyPowerSeries (fun z0 => f (z0, z1)) c0 r x✝ | 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
r : ℝ
⊢ unevenSeries' u r z1 = cauchyPowerSeries (fun z0 => f (z0, z1)) c0 r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | uneven_is_cauchy | [397, 1] | [399, 53] | rw [unevenSeries', cauchyPowerSeries] | case 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)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
r : ℝ
x✝ : ℕ
⊢ unevenSeries' u r z1 x✝ = cauchyPowerSeries (fun z0 => f (z0, z1)) c0 r x✝ | case 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)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
r : ℝ
x✝ : ℕ
⊢ ContinuousMultilinearMap.mkPiRing ℂ (Fin x✝) (unevenTerm' u r z1 x✝) =
ContinuousMultilinearMap.mkPiRing ℂ (Fin x✝)
((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c0, r), (z - c0)⁻¹ ^ x✝ • (z - c0)⁻¹ • f (z, z1)) | Please generate a tactic in lean4 to solve the state.
STATE:
case 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)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
r : ℝ
x✝ : ℕ
⊢ unevenSeries' u r z1 x✝ = cauchyPowerSeries (fun z0 => f (z0, z1)) c0 r x✝
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | uneven_is_cauchy | [397, 1] | [399, 53] | rfl | case 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)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
r : ℝ
x✝ : ℕ
⊢ ContinuousMultilinearMap.mkPiRing ℂ (Fin x✝) (unevenTerm' u r z1 x✝) =
ContinuousMultilinearMap.mkPiRing ℂ (Fin x✝)
((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c0, r), (z - c0)⁻¹ ^ x✝ • (z - c0)⁻¹ • f (z, z1)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case 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)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
r : ℝ
x✝ : ℕ
⊢ ContinuousMultilinearMap.mkPiRing ℂ (Fin x✝) (unevenTerm' u r z1 x✝) =
ContinuousMultilinearMap.mkPiRing ℂ (Fin x✝)
((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c0, r), (z - c0)⁻¹ ^ x✝ • (z - c0)⁻¹ • f (z, z1))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Uneven.has_series | [401, 1] | [412, 50] | set sn := s.toNNReal | 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
sr1 : s ≤ r1
z1s : z1 ∈ closedBall c1 r1
⊢ HasFPowerSeriesOnBall (fun z0 => f (z0, z1)) (unevenSeries' u s z1) c0 (ENNReal.ofReal 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)
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
sr1 : s ≤ r1
z1s : z1 ∈ closedBall c1 r1
sn : ℝ≥0 := s.toNNReal
⊢ HasFPowerSeriesOnBall (fun z0 => f (z0, z1)) (unevenSeries' u s z1) c0 (ENNReal.ofReal 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)
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
sr1 : s ≤ r1
z1s : z1 ∈ closedBall c1 r1
⊢ HasFPowerSeriesOnBall (fun z0 => f (z0, z1)) (unevenSeries' u s z1) c0 (ENNReal.ofReal s)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Uneven.has_series | [401, 1] | [412, 50] | have sns : s = sn := by simp only [Real.coe_toNNReal', sp.le, max_eq_left, sn] | 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
sr1 : s ≤ r1
z1s : z1 ∈ closedBall c1 r1
sn : ℝ≥0 := s.toNNReal
⊢ HasFPowerSeriesOnBall (fun z0 => f (z0, z1)) (unevenSeries' u s z1) c0 (ENNReal.ofReal 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)
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
sr1 : s ≤ r1
z1s : z1 ∈ closedBall c1 r1
sn : ℝ≥0 := s.toNNReal
sns : s = ↑sn
⊢ HasFPowerSeriesOnBall (fun z0 => f (z0, z1)) (unevenSeries' u s z1) c0 (ENNReal.ofReal 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)
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
sr1 : s ≤ r1
z1s : z1 ∈ closedBall c1 r1
sn : ℝ≥0 := s.toNNReal
⊢ HasFPowerSeriesOnBall (fun z0 => f (z0, z1)) (unevenSeries' u s z1) c0 (ENNReal.ofReal s)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Uneven.has_series | [401, 1] | [412, 50] | have snp : sn > 0 := Real.toNNReal_pos.mpr sp | 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
sr1 : s ≤ r1
z1s : z1 ∈ closedBall c1 r1
sn : ℝ≥0 := s.toNNReal
sns : s = ↑sn
⊢ HasFPowerSeriesOnBall (fun z0 => f (z0, z1)) (unevenSeries' u s z1) c0 (ENNReal.ofReal 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)
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
sr1 : s ≤ r1
z1s : z1 ∈ closedBall c1 r1
sn : ℝ≥0 := s.toNNReal
sns : s = ↑sn
snp : sn > 0
⊢ HasFPowerSeriesOnBall (fun z0 => f (z0, z1)) (unevenSeries' u s z1) c0 (ENNReal.ofReal 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)
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
sr1 : s ≤ r1
z1s : z1 ∈ closedBall c1 r1
sn : ℝ≥0 := s.toNNReal
sns : s = ↑sn
⊢ HasFPowerSeriesOnBall (fun z0 => f (z0, z1)) (unevenSeries' u s z1) c0 (ENNReal.ofReal s)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Uneven.has_series | [401, 1] | [412, 50] | rw [uneven_is_cauchy] | 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
sr1 : s ≤ r1
z1s : z1 ∈ closedBall c1 r1
sn : ℝ≥0 := s.toNNReal
sns : s = ↑sn
snp : sn > 0
⊢ HasFPowerSeriesOnBall (fun z0 => f (z0, z1)) (unevenSeries' u s z1) c0 (ENNReal.ofReal 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)
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
sr1 : s ≤ r1
z1s : z1 ∈ closedBall c1 r1
sn : ℝ≥0 := s.toNNReal
sns : s = ↑sn
snp : sn > 0
⊢ HasFPowerSeriesOnBall (fun z0 => f (z0, z1)) (cauchyPowerSeries (fun z0 => f (z0, z1)) c0 s) c0 (ENNReal.ofReal 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)
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
sr1 : s ≤ r1
z1s : z1 ∈ closedBall c1 r1
sn : ℝ≥0 := s.toNNReal
sns : s = ↑sn
snp : sn > 0
⊢ HasFPowerSeriesOnBall (fun z0 => f (z0, z1)) (unevenSeries' u s z1) c0 (ENNReal.ofReal s)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Uneven.has_series | [401, 1] | [412, 50] | rw [sns, ← ENNReal.coe_nnreal_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)
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
sr1 : s ≤ r1
z1s : z1 ∈ closedBall c1 r1
sn : ℝ≥0 := s.toNNReal
sns : s = ↑sn
snp : sn > 0
⊢ HasFPowerSeriesOnBall (fun z0 => f (z0, z1)) (cauchyPowerSeries (fun z0 => f (z0, z1)) c0 s) c0 (ENNReal.ofReal 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)
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
sr1 : s ≤ r1
z1s : z1 ∈ closedBall c1 r1
sn : ℝ≥0 := s.toNNReal
sns : s = ↑sn
snp : sn > 0
⊢ HasFPowerSeriesOnBall (fun z0 => f (z0, z1)) (cauchyPowerSeries (fun z0 => f (z0, z1)) c0 ↑sn) c0 ↑sn | 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
sr1 : s ≤ r1
z1s : z1 ∈ closedBall c1 r1
sn : ℝ≥0 := s.toNNReal
sns : s = ↑sn
snp : sn > 0
⊢ HasFPowerSeriesOnBall (fun z0 => f (z0, z1)) (cauchyPowerSeries (fun z0 => f (z0, z1)) c0 s) c0 (ENNReal.ofReal s)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Uneven.has_series | [401, 1] | [412, 50] | refine DifferentiableOn.hasFPowerSeriesOnBall ?_ snp | 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
sr1 : s ≤ r1
z1s : z1 ∈ closedBall c1 r1
sn : ℝ≥0 := s.toNNReal
sns : s = ↑sn
snp : sn > 0
⊢ HasFPowerSeriesOnBall (fun z0 => f (z0, z1)) (cauchyPowerSeries (fun z0 => f (z0, z1)) c0 ↑sn) c0 ↑sn | 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
sr1 : s ≤ r1
z1s : z1 ∈ closedBall c1 r1
sn : ℝ≥0 := s.toNNReal
sns : s = ↑sn
snp : sn > 0
⊢ DifferentiableOn ℂ (fun z0 => f (z0, z1)) (closedBall c0 ↑sn) | 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
sr1 : s ≤ r1
z1s : z1 ∈ closedBall c1 r1
sn : ℝ≥0 := s.toNNReal
sns : s = ↑sn
snp : sn > 0
⊢ HasFPowerSeriesOnBall (fun z0 => f (z0, z1)) (cauchyPowerSeries (fun z0 => f (z0, z1)) c0 ↑sn) c0 ↑sn
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Uneven.has_series | [401, 1] | [412, 50] | rw [← sns] | 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
sr1 : s ≤ r1
z1s : z1 ∈ closedBall c1 r1
sn : ℝ≥0 := s.toNNReal
sns : s = ↑sn
snp : sn > 0
⊢ DifferentiableOn ℂ (fun z0 => f (z0, z1)) (closedBall c0 ↑sn) | 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
sr1 : s ≤ r1
z1s : z1 ∈ closedBall c1 r1
sn : ℝ≥0 := s.toNNReal
sns : s = ↑sn
snp : sn > 0
⊢ DifferentiableOn ℂ (fun z0 => f (z0, z1)) (closedBall c0 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)
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
sr1 : s ≤ r1
z1s : z1 ∈ closedBall c1 r1
sn : ℝ≥0 := s.toNNReal
sns : s = ↑sn
snp : sn > 0
⊢ DifferentiableOn ℂ (fun z0 => f (z0, z1)) (closedBall c0 ↑sn)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Uneven.has_series | [401, 1] | [412, 50] | refine DifferentiableOn.mono ?_ (Metric.closedBall_subset_closedBall sr1) | 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
sr1 : s ≤ r1
z1s : z1 ∈ closedBall c1 r1
sn : ℝ≥0 := s.toNNReal
sns : s = ↑sn
snp : sn > 0
⊢ DifferentiableOn ℂ (fun z0 => f (z0, z1)) (closedBall c0 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)
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
sr1 : s ≤ r1
z1s : z1 ∈ closedBall c1 r1
sn : ℝ≥0 := s.toNNReal
sns : s = ↑sn
snp : sn > 0
⊢ DifferentiableOn ℂ (fun z0 => f (z0, z1)) (closedBall 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)
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
sr1 : s ≤ r1
z1s : z1 ∈ closedBall c1 r1
sn : ℝ≥0 := s.toNNReal
sns : s = ↑sn
snp : sn > 0
⊢ DifferentiableOn ℂ (fun z0 => f (z0, z1)) (closedBall c0 s)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Uneven.has_series | [401, 1] | [412, 50] | exact AnalyticOn.differentiableOn (u.h.on0 z1s) | 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
sr1 : s ≤ r1
z1s : z1 ∈ closedBall c1 r1
sn : ℝ≥0 := s.toNNReal
sns : s = ↑sn
snp : sn > 0
⊢ DifferentiableOn ℂ (fun z0 => f (z0, z1)) (closedBall 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)
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
sr1 : s ≤ r1
z1s : z1 ∈ closedBall c1 r1
sn : ℝ≥0 := s.toNNReal
sns : s = ↑sn
snp : sn > 0
⊢ DifferentiableOn ℂ (fun z0 => f (z0, z1)) (closedBall c0 r1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Uneven.has_series | [401, 1] | [412, 50] | simp only [Real.coe_toNNReal', sp.le, max_eq_left, sn] | 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
sr1 : s ≤ r1
z1s : z1 ∈ closedBall c1 r1
sn : ℝ≥0 := s.toNNReal
⊢ s = ↑sn | 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
sr1 : s ≤ r1
z1s : z1 ∈ closedBall c1 r1
sn : ℝ≥0 := s.toNNReal
⊢ s = ↑sn
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenTerm_eq | [414, 1] | [420, 62] | intro z1s | 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
r : ℝ
rp : r > 0
rr1 : r ≤ r1
z1 : ℂ
⊢ z1 ∈ closedBall c1 r1 → unevenTerm' u r z1 = unevenTerm u z1 | 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
r : ℝ
rp : r > 0
rr1 : r ≤ r1
z1 : ℂ
z1s : z1 ∈ closedBall c1 r1
⊢ unevenTerm' u r z1 = unevenTerm u z1 | 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
r : ℝ
rp : r > 0
rr1 : r ≤ r1
z1 : ℂ
⊢ z1 ∈ closedBall c1 r1 → unevenTerm' u r z1 = unevenTerm u z1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenTerm_eq | [414, 1] | [420, 62] | funext x | 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
r : ℝ
rp : r > 0
rr1 : r ≤ r1
z1 : ℂ
z1s : z1 ∈ closedBall c1 r1
⊢ unevenTerm' u r z1 = unevenTerm u z1 | case 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
r : ℝ
rp : r > 0
rr1 : r ≤ r1
z1 : ℂ
z1s : z1 ∈ closedBall c1 r1
x : ℕ
⊢ unevenTerm' u r z1 x = unevenTerm u z1 x | 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
r : ℝ
rp : r > 0
rr1 : r ≤ r1
z1 : ℂ
z1s : z1 ∈ closedBall c1 r1
⊢ unevenTerm' u r z1 = unevenTerm u z1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenTerm_eq | [414, 1] | [420, 62] | have p0 := u.has_series rp rr1 z1s | case 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
r : ℝ
rp : r > 0
rr1 : r ≤ r1
z1 : ℂ
z1s : z1 ∈ closedBall c1 r1
x : ℕ
⊢ unevenTerm' u r z1 x = unevenTerm u z1 x | case 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
r : ℝ
rp : r > 0
rr1 : r ≤ r1
z1 : ℂ
z1s : z1 ∈ closedBall c1 r1
x : ℕ
p0 : HasFPowerSeriesOnBall (fun z0 => f (z0, z1)) (unevenSeries' u r z1) c0 (ENNReal.ofReal r)
⊢ unevenTerm' u r z1 x = unevenTerm u z1 x | Please generate a tactic in lean4 to solve the state.
STATE:
case 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
r : ℝ
rp : r > 0
rr1 : r ≤ r1
z1 : ℂ
z1s : z1 ∈ closedBall c1 r1
x : ℕ
⊢ unevenTerm' u r z1 x = unevenTerm u z1 x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenTerm_eq | [414, 1] | [420, 62] | have p1 := u.has_series u.r1p (by rfl) z1s | case 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
r : ℝ
rp : r > 0
rr1 : r ≤ r1
z1 : ℂ
z1s : z1 ∈ closedBall c1 r1
x : ℕ
p0 : HasFPowerSeriesOnBall (fun z0 => f (z0, z1)) (unevenSeries' u r z1) c0 (ENNReal.ofReal r)
⊢ unevenTerm' u r z1 x = unevenTerm u z1 x | case 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
r : ℝ
rp : r > 0
rr1 : r ≤ r1
z1 : ℂ
z1s : z1 ∈ closedBall c1 r1
x : ℕ
p0 : HasFPowerSeriesOnBall (fun z0 => f (z0, z1)) (unevenSeries' u r z1) c0 (ENNReal.ofReal r)
p1 : HasFPowerSeriesOnBall (fun z0 => f (z0, z1)) (unevenSeries' u r1 z1) c0 (ENNReal.ofReal r1)
⊢ unevenTerm' u r z1 x = unevenTerm u z1 x | Please generate a tactic in lean4 to solve the state.
STATE:
case 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
r : ℝ
rp : r > 0
rr1 : r ≤ r1
z1 : ℂ
z1s : z1 ∈ closedBall c1 r1
x : ℕ
p0 : HasFPowerSeriesOnBall (fun z0 => f (z0, z1)) (unevenSeries' u r z1) c0 (ENNReal.ofReal r)
⊢ unevenTerm' u r z1 x = unevenTerm u z1 x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenTerm_eq | [414, 1] | [420, 62] | have h := HasFPowerSeriesAt.eq_formalMultilinearSeries p0.hasFPowerSeriesAt p1.hasFPowerSeriesAt | case 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
r : ℝ
rp : r > 0
rr1 : r ≤ r1
z1 : ℂ
z1s : z1 ∈ closedBall c1 r1
x : ℕ
p0 : HasFPowerSeriesOnBall (fun z0 => f (z0, z1)) (unevenSeries' u r z1) c0 (ENNReal.ofReal r)
p1 : HasFPowerSeriesOnBall (fun z0 => f (z0, z1)) (unevenSeries' u r1 z1) c0 (ENNReal.ofReal r1)
⊢ unevenTerm' u r z1 x = unevenTerm u z1 x | case 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
r : ℝ
rp : r > 0
rr1 : r ≤ r1
z1 : ℂ
z1s : z1 ∈ closedBall c1 r1
x : ℕ
p0 : HasFPowerSeriesOnBall (fun z0 => f (z0, z1)) (unevenSeries' u r z1) c0 (ENNReal.ofReal r)
p1 : HasFPowerSeriesOnBall (fun z0 => f (z0, z1)) (unevenSeries' u r1 z1) c0 (ENNReal.ofReal r1)
h : unevenSeries' u r z1 = unevenSeries' u r1 z1
⊢ unevenTerm' u r z1 x = unevenTerm u z1 x | Please generate a tactic in lean4 to solve the state.
STATE:
case 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
r : ℝ
rp : r > 0
rr1 : r ≤ r1
z1 : ℂ
z1s : z1 ∈ closedBall c1 r1
x : ℕ
p0 : HasFPowerSeriesOnBall (fun z0 => f (z0, z1)) (unevenSeries' u r z1) c0 (ENNReal.ofReal r)
p1 : HasFPowerSeriesOnBall (fun z0 => f (z0, z1)) (unevenSeries' u r1 z1) c0 (ENNReal.ofReal r1)
⊢ unevenTerm' u r z1 x = unevenTerm u z1 x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenTerm_eq | [414, 1] | [420, 62] | clear p0 p1 | case 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
r : ℝ
rp : r > 0
rr1 : r ≤ r1
z1 : ℂ
z1s : z1 ∈ closedBall c1 r1
x : ℕ
p0 : HasFPowerSeriesOnBall (fun z0 => f (z0, z1)) (unevenSeries' u r z1) c0 (ENNReal.ofReal r)
p1 : HasFPowerSeriesOnBall (fun z0 => f (z0, z1)) (unevenSeries' u r1 z1) c0 (ENNReal.ofReal r1)
h : unevenSeries' u r z1 = unevenSeries' u r1 z1
⊢ unevenTerm' u r z1 x = unevenTerm u z1 x | case 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
r : ℝ
rp : r > 0
rr1 : r ≤ r1
z1 : ℂ
z1s : z1 ∈ closedBall c1 r1
x : ℕ
h : unevenSeries' u r z1 = unevenSeries' u r1 z1
⊢ unevenTerm' u r z1 x = unevenTerm u z1 x | Please generate a tactic in lean4 to solve the state.
STATE:
case 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
r : ℝ
rp : r > 0
rr1 : r ≤ r1
z1 : ℂ
z1s : z1 ∈ closedBall c1 r1
x : ℕ
p0 : HasFPowerSeriesOnBall (fun z0 => f (z0, z1)) (unevenSeries' u r z1) c0 (ENNReal.ofReal r)
p1 : HasFPowerSeriesOnBall (fun z0 => f (z0, z1)) (unevenSeries' u r1 z1) c0 (ENNReal.ofReal r1)
h : unevenSeries' u r z1 = unevenSeries' u r1 z1
⊢ unevenTerm' u r z1 x = unevenTerm u z1 x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenTerm_eq | [414, 1] | [420, 62] | simp only [unevenTerm, ←unevenSeries_apply, h] | case 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
r : ℝ
rp : r > 0
rr1 : r ≤ r1
z1 : ℂ
z1s : z1 ∈ closedBall c1 r1
x : ℕ
h : unevenSeries' u r z1 = unevenSeries' u r1 z1
⊢ unevenTerm' u r z1 x = unevenTerm u z1 x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
r : ℝ
rp : r > 0
rr1 : r ≤ r1
z1 : ℂ
z1s : z1 ∈ closedBall c1 r1
x : ℕ
h : unevenSeries' u r z1 = unevenSeries' u r1 z1
⊢ unevenTerm' u r z1 x = unevenTerm u z1 x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenTerm_eq | [414, 1] | [420, 62] | rfl | 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
r : ℝ
rp : r > 0
rr1 : r ≤ r1
z1 : ℂ
z1s : z1 ∈ closedBall c1 r1
x : ℕ
p0 : HasFPowerSeriesOnBall (fun z0 => f (z0, z1)) (unevenSeries' u r z1) c0 (ENNReal.ofReal r)
⊢ r1 ≤ 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
r : ℝ
rp : r > 0
rr1 : r ≤ r1
z1 : ℂ
z1s : z1 ∈ closedBall c1 r1
x : ℕ
p0 : HasFPowerSeriesOnBall (fun z0 => f (z0, z1)) (unevenSeries' u r z1) c0 (ENNReal.ofReal r)
⊢ r1 ≤ r1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_eq | [422, 1] | [426, 43] | intro z1s | 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
r : ℝ
rp : r > 0
rr1 : r ≤ r1
z1 : ℂ
⊢ z1 ∈ closedBall c1 r1 → unevenSeries' u r z1 = unevenSeries u z1 | 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
r : ℝ
rp : r > 0
rr1 : r ≤ r1
z1 : ℂ
z1s : z1 ∈ closedBall c1 r1
⊢ unevenSeries' u r z1 = unevenSeries u z1 | 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
r : ℝ
rp : r > 0
rr1 : r ≤ r1
z1 : ℂ
⊢ z1 ∈ closedBall c1 r1 → unevenSeries' u r z1 = unevenSeries u z1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_eq | [422, 1] | [426, 43] | funext | 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
r : ℝ
rp : r > 0
rr1 : r ≤ r1
z1 : ℂ
z1s : z1 ∈ closedBall c1 r1
⊢ unevenSeries' u r z1 = unevenSeries u z1 | case 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
r : ℝ
rp : r > 0
rr1 : r ≤ r1
z1 : ℂ
z1s : z1 ∈ closedBall c1 r1
x✝ : ℕ
⊢ unevenSeries' u r z1 x✝ = unevenSeries u z1 x✝ | 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
r : ℝ
rp : r > 0
rr1 : r ≤ r1
z1 : ℂ
z1s : z1 ∈ closedBall c1 r1
⊢ unevenSeries' u r z1 = unevenSeries u z1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_eq | [422, 1] | [426, 43] | simp_rw [unevenSeries, unevenSeries', unevenTerm_eq u rp rr1 z1s,
unevenTerm_eq u u.r1p (le_refl _) z1s] | case 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
r : ℝ
rp : r > 0
rr1 : r ≤ r1
z1 : ℂ
z1s : z1 ∈ closedBall c1 r1
x✝ : ℕ
⊢ unevenSeries' u r z1 x✝ = unevenSeries u z1 x✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
r : ℝ
rp : r > 0
rr1 : r ≤ r1
z1 : ℂ
z1s : z1 ∈ closedBall c1 r1
x✝ : ℕ
⊢ unevenSeries' u r z1 x✝ = unevenSeries u z1 x✝
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_norm | [428, 1] | [430, 87] | rw [unevenSeries, unevenSeries', unevenTerm, ContinuousMultilinearMap.norm_mkPiRing] | 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
n : ℕ
⊢ ‖unevenSeries u z1 n‖ = ‖unevenTerm u z1 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
n : ℕ
⊢ ‖unevenSeries u z1 n‖ = ‖unevenTerm u z1 n‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | rcases (((isCompact_sphere _ _).prod (isCompact_closedBall _ _)).bddAbove_image
fc.norm).exists_ge 0 with ⟨b, bp, fb⟩ | 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
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
⊢ ∃ c a, c > 0 ∧ a > 0 ∧ ∀ (n : ℕ), ∀ z1 ∈ closedBall c1 s, ‖unevenSeries u z1 n‖ ≤ c * a ^ n | 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)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ y ∈ (fun x => ‖f x‖) '' sphere c0 (r0 / 2) ×ˢ closedBall c1 s, y ≤ b
⊢ ∃ c a, c > 0 ∧ a > 0 ∧ ∀ (n : ℕ), ∀ z1 ∈ closedBall c1 s, ‖unevenSeries u z1 n‖ ≤ c * a ^ n | 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
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
⊢ ∃ c a, c > 0 ∧ a > 0 ∧ ∀ (n : ℕ), ∀ z1 ∈ closedBall c1 s, ‖unevenSeries u z1 n‖ ≤ c * a ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | simp only [Set.forall_mem_image] at fb | 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)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ y ∈ (fun x => ‖f x‖) '' sphere c0 (r0 / 2) ×ˢ closedBall c1 s, y ≤ b
⊢ ∃ c a, c > 0 ∧ a > 0 ∧ ∀ (n : ℕ), ∀ z1 ∈ closedBall c1 s, ‖unevenSeries u z1 n‖ ≤ c * a ^ n | 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)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
⊢ ∃ c a, c > 0 ∧ a > 0 ∧ ∀ (n : ℕ), ∀ z1 ∈ closedBall c1 s, ‖unevenSeries u z1 n‖ ≤ c * a ^ n | 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)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ y ∈ (fun x => ‖f x‖) '' sphere c0 (r0 / 2) ×ˢ closedBall c1 s, y ≤ b
⊢ ∃ c a, c > 0 ∧ a > 0 ∧ ∀ (n : ℕ), ∀ z1 ∈ closedBall c1 s, ‖unevenSeries u z1 n‖ ≤ c * a ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | use b + 1, (r0 / 2)⁻¹, lt_of_le_of_lt bp (lt_add_one _), inv_pos.mpr (half_pos u.r0p) | 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)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
⊢ ∃ c a, c > 0 ∧ a > 0 ∧ ∀ (n : ℕ), ∀ z1 ∈ closedBall c1 s, ‖unevenSeries u z1 n‖ ≤ c * a ^ n | 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)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
⊢ ∀ (n : ℕ), ∀ z1 ∈ closedBall c1 s, ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | 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)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
⊢ ∃ c a, c > 0 ∧ a > 0 ∧ ∀ (n : ℕ), ∀ z1 ∈ closedBall c1 s, ‖unevenSeries u z1 n‖ ≤ c * a ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | intro n z1 z1s | 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)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
⊢ ∀ (n : ℕ), ∀ z1 ∈ closedBall c1 s, ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | 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)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
⊢ ∀ (n : ℕ), ∀ z1 ∈ closedBall c1 s, ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | have r0hp : r0 / 2 > 0 := by linarith [u.r0p] | 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | have r0hr1 : r0 / 2 ≤ r1 := _root_.trans (by linarith [u.r0p]) u.r01 | 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | set g := fun z0 ↦ f (z0, z1) | 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
g : ℂ → E := fun z0 => f (z0, z1)
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | have gc : ContinuousOn g (sphere c0 (r0 / 2)) :=
ContinuousOn.comp fc (ContinuousOn.prod continuousOn_id continuousOn_const) fun z0 z0s ↦
Set.mk_mem_prod z0s z1s | 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
g : ℂ → E := fun z0 => f (z0, z1)
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
g : ℂ → E := fun z0 => f (z0, z1)
gc : ContinuousOn g (sphere c0 (r0 / 2))
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
g : ℂ → E := fun z0 => f (z0, z1)
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | have gb : ∀ z0, z0 ∈ sphere c0 (r0 / 2) → ‖g z0‖ ≤ b := fun z0 z0s ↦ fb (Set.mk_mem_prod z0s z1s) | 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
g : ℂ → E := fun z0 => f (z0, z1)
gc : ContinuousOn g (sphere c0 (r0 / 2))
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
g : ℂ → E := fun z0 => f (z0, z1)
gc : ContinuousOn g (sphere c0 (r0 / 2))
gb : ∀ z0 ∈ sphere c0 (r0 / 2), ‖g z0‖ ≤ b
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
g : ℂ → E := fun z0 => f (z0, z1)
gc : ContinuousOn g (sphere c0 (r0 / 2))
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | have cb := cauchy1_bound' r0hp b gc gb n | 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
g : ℂ → E := fun z0 => f (z0, z1)
gc : ContinuousOn g (sphere c0 (r0 / 2))
gb : ∀ z0 ∈ sphere c0 (r0 / 2), ‖g z0‖ ≤ b
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
g : ℂ → E := fun z0 => f (z0, z1)
gc : ContinuousOn g (sphere c0 (r0 / 2))
gb : ∀ z0 ∈ sphere c0 (r0 / 2), ‖g z0‖ ≤ b
cb : ‖(2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c0, r0 / 2), (z - c0)⁻¹ ^ n • (z - c0)⁻¹ • g z‖ ≤ b * (r0 / 2)⁻¹ ^ n
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
g : ℂ → E := fun z0 => f (z0, z1)
gc : ContinuousOn g (sphere c0 (r0 / 2))
gb : ∀ z0 ∈ sphere c0 (r0 / 2), ‖g z0‖ ≤ b
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | clear bp gc gb | 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
g : ℂ → E := fun z0 => f (z0, z1)
gc : ContinuousOn g (sphere c0 (r0 / 2))
gb : ∀ z0 ∈ sphere c0 (r0 / 2), ‖g z0‖ ≤ b
cb : ‖(2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c0, r0 / 2), (z - c0)⁻¹ ^ n • (z - c0)⁻¹ • g z‖ ≤ b * (r0 / 2)⁻¹ ^ n
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
g : ℂ → E := fun z0 => f (z0, z1)
cb : ‖(2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c0, r0 / 2), (z - c0)⁻¹ ^ n • (z - c0)⁻¹ • g z‖ ≤ b * (r0 / 2)⁻¹ ^ n
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
bp : 0 ≤ b
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
g : ℂ → E := fun z0 => f (z0, z1)
gc : ContinuousOn g (sphere c0 (r0 / 2))
gb : ∀ z0 ∈ sphere c0 (r0 / 2), ‖g z0‖ ≤ b
cb : ‖(2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c0, r0 / 2), (z - c0)⁻¹ ^ n • (z - c0)⁻¹ • g z‖ ≤ b * (r0 / 2)⁻¹ ^ n
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | have e : (2 * π * I : ℂ)⁻¹ • (∮ z0 in C(c0, r0 / 2), (z0 - c0)⁻¹ ^ n • (z0 - c0)⁻¹ • g z0) =
unevenTerm' u (r0 / 2) z1 n := rfl | 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
g : ℂ → E := fun z0 => f (z0, z1)
cb : ‖(2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c0, r0 / 2), (z - c0)⁻¹ ^ n • (z - c0)⁻¹ • g z‖ ≤ b * (r0 / 2)⁻¹ ^ n
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e✝ : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
g : ℂ → E := fun z0 => f (z0, z1)
cb : ‖(2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c0, r0 / 2), (z - c0)⁻¹ ^ n • (z - c0)⁻¹ • g z‖ ≤ b * (r0 / 2)⁻¹ ^ n
e : ((2 * ↑π * I)⁻¹ • ∮ (z0 : ℂ) in C(c0, r0 / 2), (z0 - c0)⁻¹ ^ n • (z0 - c0)⁻¹ • g z0) = unevenTerm' u (r0 / 2) z1 n
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
g : ℂ → E := fun z0 => f (z0, z1)
cb : ‖(2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c0, r0 / 2), (z - c0)⁻¹ ^ n • (z - c0)⁻¹ • g z‖ ≤ b * (r0 / 2)⁻¹ ^ n
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | rw [e] at cb | 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e✝ : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
g : ℂ → E := fun z0 => f (z0, z1)
cb : ‖(2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c0, r0 / 2), (z - c0)⁻¹ ^ n • (z - c0)⁻¹ • g z‖ ≤ b * (r0 / 2)⁻¹ ^ n
e : ((2 * ↑π * I)⁻¹ • ∮ (z0 : ℂ) in C(c0, r0 / 2), (z0 - c0)⁻¹ ^ n • (z0 - c0)⁻¹ • g z0) = unevenTerm' u (r0 / 2) z1 n
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e✝ : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
g : ℂ → E := fun z0 => f (z0, z1)
cb : ‖unevenTerm' u (r0 / 2) z1 n‖ ≤ b * (r0 / 2)⁻¹ ^ n
e : ((2 * ↑π * I)⁻¹ • ∮ (z0 : ℂ) in C(c0, r0 / 2), (z0 - c0)⁻¹ ^ n • (z0 - c0)⁻¹ • g z0) = unevenTerm' u (r0 / 2) z1 n
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e✝ : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
g : ℂ → E := fun z0 => f (z0, z1)
cb : ‖(2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c0, r0 / 2), (z - c0)⁻¹ ^ n • (z - c0)⁻¹ • g z‖ ≤ b * (r0 / 2)⁻¹ ^ n
e : ((2 * ↑π * I)⁻¹ • ∮ (z0 : ℂ) in C(c0, r0 / 2), (z0 - c0)⁻¹ ^ n • (z0 - c0)⁻¹ • g z0) = unevenTerm' u (r0 / 2) z1 n
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | clear e g | 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e✝ : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
g : ℂ → E := fun z0 => f (z0, z1)
cb : ‖unevenTerm' u (r0 / 2) z1 n‖ ≤ b * (r0 / 2)⁻¹ ^ n
e : ((2 * ↑π * I)⁻¹ • ∮ (z0 : ℂ) in C(c0, r0 / 2), (z0 - c0)⁻¹ ^ n • (z0 - c0)⁻¹ • g z0) = unevenTerm' u (r0 / 2) z1 n
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
cb : ‖unevenTerm' u (r0 / 2) z1 n‖ ≤ b * (r0 / 2)⁻¹ ^ n
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e✝ : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
g : ℂ → E := fun z0 => f (z0, z1)
cb : ‖unevenTerm' u (r0 / 2) z1 n‖ ≤ b * (r0 / 2)⁻¹ ^ n
e : ((2 * ↑π * I)⁻¹ • ∮ (z0 : ℂ) in C(c0, r0 / 2), (z0 - c0)⁻¹ ^ n • (z0 - c0)⁻¹ • g z0) = unevenTerm' u (r0 / 2) z1 n
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | rw [unevenTerm_eq u r0hp r0hr1 (Metric.closedBall_subset_closedBall sr.le z1s)] at cb | 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
cb : ‖unevenTerm' u (r0 / 2) z1 n‖ ≤ b * (r0 / 2)⁻¹ ^ n
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
cb : ‖unevenTerm u z1 n‖ ≤ b * (r0 / 2)⁻¹ ^ n
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
cb : ‖unevenTerm' u (r0 / 2) z1 n‖ ≤ b * (r0 / 2)⁻¹ ^ n
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | rw [unevenSeries_norm u] | 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
cb : ‖unevenTerm u z1 n‖ ≤ b * (r0 / 2)⁻¹ ^ n
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
cb : ‖unevenTerm u z1 n‖ ≤ b * (r0 / 2)⁻¹ ^ n
⊢ ‖unevenTerm u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
cb : ‖unevenTerm u z1 n‖ ≤ b * (r0 / 2)⁻¹ ^ n
⊢ ‖unevenSeries u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | apply _root_.trans cb | 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
cb : ‖unevenTerm u z1 n‖ ≤ b * (r0 / 2)⁻¹ ^ n
⊢ ‖unevenTerm u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
cb : ‖unevenTerm u z1 n‖ ≤ b * (r0 / 2)⁻¹ ^ n
⊢ b * (r0 / 2)⁻¹ ^ n ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
cb : ‖unevenTerm u z1 n‖ ≤ b * (r0 / 2)⁻¹ ^ n
⊢ ‖unevenTerm u z1 n‖ ≤ (b + 1) * (r0 / 2)⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | bound | 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
cb : ‖unevenTerm u z1 n‖ ≤ b * (r0 / 2)⁻¹ ^ n
⊢ b * (r0 / 2)⁻¹ ^ n ≤ (b + 1) * (r0 / 2)⁻¹ ^ n | 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)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
fc : ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
b : ℝ
fb : ∀ ⦃x : ℂ × ℂ⦄, x ∈ sphere c0 (r0 / 2) ×ˢ closedBall c1 s → ‖f x‖ ≤ b
n : ℕ
z1 : ℂ
z1s : z1 ∈ closedBall c1 s
r0hp : r0 / 2 > 0
r0hr1 : r0 / 2 ≤ r1
cb : ‖unevenTerm u z1 n‖ ≤ b * (r0 / 2)⁻¹ ^ n
⊢ b * (r0 / 2)⁻¹ ^ n ≤ (b + 1) * (r0 / 2)⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | suffices fa' : AnalyticOn ℂ f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s) by exact fa'.continuousOn | 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
sr : s < r1
⊢ ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 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)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
⊢ AnalyticOn ℂ f (sphere c0 (r0 / 2) ×ˢ closedBall c1 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)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
⊢ ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | refine u.a.mono (Set.prod_mono ?_ ?_) | 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
sr : s < r1
⊢ AnalyticOn ℂ f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s) | case 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)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
⊢ sphere c0 (r0 / 2) ⊆ ball c0 r0
case 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)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
⊢ closedBall c1 s ⊆ ball c1 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)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
⊢ AnalyticOn ℂ f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | exact fa'.continuousOn | 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
sr : s < r1
fa' : AnalyticOn ℂ f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
⊢ ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s) | 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
sr : s < r1
fa' : AnalyticOn ℂ f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
⊢ ContinuousOn f (sphere c0 (r0 / 2) ×ˢ closedBall c1 s)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | have rh : r0 / 2 < r0 := by linarith [u.r0p] | case 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)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
⊢ sphere c0 (r0 / 2) ⊆ ball c0 r0 | case 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)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
rh : r0 / 2 < r0
⊢ sphere c0 (r0 / 2) ⊆ ball c0 r0 | Please generate a tactic in lean4 to solve the state.
STATE:
case 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)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
⊢ sphere c0 (r0 / 2) ⊆ ball c0 r0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | exact _root_.trans Metric.sphere_subset_closedBall (Metric.closedBall_subset_ball rh) | case 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)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
rh : r0 / 2 < r0
⊢ sphere c0 (r0 / 2) ⊆ ball c0 r0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case 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)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
rh : r0 / 2 < r0
⊢ sphere c0 (r0 / 2) ⊆ ball c0 r0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | linarith [u.r0p] | 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
sr : s < r1
⊢ r0 / 2 < r0 | 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
sr : s < r1
⊢ r0 / 2 < r0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | unevenSeries_uniform_bound | [433, 1] | [459, 57] | exact Metric.closedBall_subset_ball (by linarith [u.r1p]) | case 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)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
⊢ closedBall c1 s ⊆ ball c1 r1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case 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)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
u : Uneven f c0 c1 r0 r1
s : ℝ
sr : s < r1
⊢ closedBall c1 s ⊆ ball c1 r1
TACTIC:
|
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