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/Osgood.lean | Separate.fc1 | [113, 1] | [118, 67] | intro z1 z1m | case refine_2
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
⊢ Set.MapsTo (fun z1 => (w0, z1)) (closedBall c1 r) s | case refine_2
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
z1 : ℂ
z1m : z1 ∈ closedBall c1 r
⊢ (fun z1 => (w0, z1)) z1 ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
⊢ Set.MapsTo (fun z1 => (w0, z1)) (closedBall c1 ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | Separate.fc1 | [113, 1] | [118, 67] | apply h.rs | case refine_2
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
z1 : ℂ
z1m : z1 ∈ closedBall c1 r
⊢ (fun z1 => (w0, z1)) z1 ∈ s | case refine_2.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
z1 : ℂ
z1m : z1 ∈ closedBall c1 r
⊢ (fun z1 => (w0, z1)) z1 ∈ closedBall (c0, c1) r | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
z1 : ℂ
z1m : z1 ∈ closedBall c1 r
⊢ (fun z1 => (w... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | Separate.fc1 | [113, 1] | [118, 67] | rw [← closedBall_prod_same] | case refine_2.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
z1 : ℂ
z1m : z1 ∈ closedBall c1 r
⊢ (fun z1 => (w0, z1)) z1 ∈ closedBall (c0, c1) r | case refine_2.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
z1 : ℂ
z1m : z1 ∈ closedBall c1 r
⊢ (fun z1 => (w0, z1)) z1 ∈ closedBall c0 r ×ˢ closedBall c1 r | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
z1 : ℂ
z1m : z1 ∈ closedBall c1 r
⊢ (fun z1 => ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | Separate.fc1 | [113, 1] | [118, 67] | exact Set.mem_prod.mpr ⟨w0m, z1m⟩ | case refine_2.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
z1 : ℂ
z1m : z1 ∈ closedBall c1 r
⊢ (fun z1 => (w0, z1)) z1 ∈ closedBall c0 r ×ˢ closedBall c1 r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
z1 : ℂ
z1m : z1 ∈ closedBall c1 r
⊢ (fun z1 => ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | Separate.fd0 | [121, 1] | [125, 40] | apply h.rs | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
w1m : w1 ∈ closedBall c1 r
⊢ (w0, w1) ∈ s | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
w1m : w1 ∈ closedBall c1 r
⊢ (w0, w1) ∈ closedBall (c0, c1) r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
w1m : w1 ∈ closedBall c1 r
⊢ (w0, w1) ∈ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | Separate.fd0 | [121, 1] | [125, 40] | rw [←closedBall_prod_same] | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
w1m : w1 ∈ closedBall c1 r
⊢ (w0, w1) ∈ closedBall (c0, c1) r | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
w1m : w1 ∈ closedBall c1 r
⊢ (w0, w1) ∈ closedBall c0 r ×ˢ closedBall c1 r | Please generate a tactic in lean4 to solve the state.
STATE:
case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
w1m : w1 ∈ closedBall c1 r
⊢ (w0, w1) ∈ closedBall (c0, ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | Separate.fd0 | [121, 1] | [125, 40] | exact Set.mem_prod.mpr ⟨w0m, w1m⟩ | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
w1m : w1 ∈ closedBall c1 r
⊢ (w0, w1) ∈ closedBall c0 r ×ˢ closedBall c1 r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
w1m : w1 ∈ closedBall c1 r
⊢ (w0, w1) ∈ closedBall c0 r ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | Separate.fd1 | [128, 1] | [132, 40] | apply h.rs | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
w1m : w1 ∈ closedBall c1 r
⊢ (w0, w1) ∈ s | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
w1m : w1 ∈ closedBall c1 r
⊢ (w0, w1) ∈ closedBall (c0, c1) r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
w1m : w1 ∈ closedBall c1 r
⊢ (w0, w1) ∈ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | Separate.fd1 | [128, 1] | [132, 40] | rw [←closedBall_prod_same] | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
w1m : w1 ∈ closedBall c1 r
⊢ (w0, w1) ∈ closedBall (c0, c1) r | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
w1m : w1 ∈ closedBall c1 r
⊢ (w0, w1) ∈ closedBall c0 r ×ˢ closedBall c1 r | Please generate a tactic in lean4 to solve the state.
STATE:
case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
w1m : w1 ∈ closedBall c1 r
⊢ (w0, w1) ∈ closedBall (c0, ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | Separate.fd1 | [128, 1] | [132, 40] | exact Set.mem_prod.mpr ⟨w0m, w1m⟩ | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
w1m : w1 ∈ closedBall c1 r
⊢ (w0, w1) ∈ closedBall c0 r ×ˢ closedBall c1 r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
w1m : w1 ∈ closedBall c1 r
⊢ (w0, w1) ∈ closedBall c0 r ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1 | [135, 1] | [140, 92] | refine Complex.two_pi_I_inv_smul_circleIntegral_sub_inv_smul_of_differentiable_on_off_countable
Set.countable_empty wm fc ?_ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b r : ℝ
c w : ℂ
f : ℂ → E
wm : w ∈ ball c r
fc : ContinuousOn f (closedBall c r)
fd : ∀ z ∈ ball c r, DifferentiableAt ℂ f z
⊢ ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, r), (z - w)⁻¹ • f z... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b r : ℝ
c w : ℂ
f : ℂ → E
wm : w ∈ ball c r
fc : ContinuousOn f (closedBall c r)
fd : ∀ z ∈ ball c r, DifferentiableAt ℂ f z
⊢ ∀ x ∈ ball c r \ ∅, DifferentiableAt ℂ (fun z => f z) x | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b r : ℝ
c w : ℂ
f : ℂ → E
wm : w ∈ ball c r
fc : ContinuousOn f (closedBall c r)
fd : ∀ z ∈ ball c r, DifferentiableAt ℂ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1 | [135, 1] | [140, 92] | intro z zm | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b r : ℝ
c w : ℂ
f : ℂ → E
wm : w ∈ ball c r
fc : ContinuousOn f (closedBall c r)
fd : ∀ z ∈ ball c r, DifferentiableAt ℂ f z
⊢ ∀ x ∈ ball c r \ ∅, DifferentiableAt ℂ (fun z => f z) x | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b r : ℝ
c w : ℂ
f : ℂ → E
wm : w ∈ ball c r
fc : ContinuousOn f (closedBall c r)
fd : ∀ z ∈ ball c r, DifferentiableAt ℂ f z
z : ℂ
zm : z ∈ ball c r \ ∅
⊢ DifferentiableAt ℂ (fun z =... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b r : ℝ
c w : ℂ
f : ℂ → E
wm : w ∈ ball c r
fc : ContinuousOn f (closedBall c r)
fd : ∀ z ∈ ball c r, DifferentiableAt ℂ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1 | [135, 1] | [140, 92] | apply fd z _ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b r : ℝ
c w : ℂ
f : ℂ → E
wm : w ∈ ball c r
fc : ContinuousOn f (closedBall c r)
fd : ∀ z ∈ ball c r, DifferentiableAt ℂ f z
z : ℂ
zm : z ∈ ball c r \ ∅
⊢ DifferentiableAt ℂ (fun z =... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b r : ℝ
c w : ℂ
f : ℂ → E
wm : w ∈ ball c r
fc : ContinuousOn f (closedBall c r)
fd : ∀ z ∈ ball c r, DifferentiableAt ℂ f z
z : ℂ
zm : z ∈ ball c r \ ∅
⊢ z ∈ ball c r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b r : ℝ
c w : ℂ
f : ℂ → E
wm : w ∈ ball c r
fc : ContinuousOn f (closedBall c r)
fd : ∀ z ∈ ball c r, DifferentiableAt ℂ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1 | [135, 1] | [140, 92] | simp only [Metric.mem_ball, Set.diff_empty] at zm ⊢ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b r : ℝ
c w : ℂ
f : ℂ → E
wm : w ∈ ball c r
fc : ContinuousOn f (closedBall c r)
fd : ∀ z ∈ ball c r, DifferentiableAt ℂ f z
z : ℂ
zm : z ∈ ball c r \ ∅
⊢ z ∈ ball c r | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b r : ℝ
c w : ℂ
f : ℂ → E
wm : w ∈ ball c r
fc : ContinuousOn f (closedBall c r)
fd : ∀ z ∈ ball c r, DifferentiableAt ℂ f z
z : ℂ
zm : dist z c < r
⊢ dist z c < r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b r : ℝ
c w : ℂ
f : ℂ → E
wm : w ∈ ball c r
fc : ContinuousOn f (closedBall c r)
fd : ∀ z ∈ ball c r, DifferentiableAt ℂ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1 | [135, 1] | [140, 92] | assumption | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b r : ℝ
c w : ℂ
f : ℂ → E
wm : w ∈ ball c r
fc : ContinuousOn f (closedBall c r)
fd : ∀ z ∈ ball c r, DifferentiableAt ℂ f z
z : ℂ
zm : dist z c < r
⊢ dist z c < r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b r : ℝ
c w : ℂ
f : ℂ → E
wm : w ∈ ball c r
fc : ContinuousOn f (closedBall c r)
fd : ∀ z ∈ ball c r, DifferentiableAt ℂ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2 | [143, 1] | [160, 55] | have h1 := fun z0 (z0m : z0 ∈ closedBall c0 r) ↦
cauchy1 w1m (h.fc1 z0m) fun z1 z1m ↦ h.fd1 z0m (mem_open_closed z1m) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
⊢ ((2 * ↑π * I)⁻¹ •
∮ (z0 : ℂ) in C(c0, r), (z0 - w0)⁻¹ • (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 ... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c1, r), (z - w1)⁻¹ • f (z0, z)) = f (z0, w1)
⊢ ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
⊢ ((2 * ↑π * I)⁻¹ •
∮ (z0 : ℂ) in C(c0, r)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2 | [143, 1] | [160, 55] | have ic1 : ContinuousOn (fun z0 ↦ (2 * π * I : ℂ)⁻¹ • ∮ z1 in C(c1, r), (z1 - w1)⁻¹ • f (z0, z1))
(closedBall c0 r) :=
(h.fc0 w1m).congr h1 | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c1, r), (z - w1)⁻¹ • f (z0, z)) = f (z0, w1)
⊢ ... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c1, r), (z - w1)⁻¹ • f (z0, z)) = f (z0, w1)
ic... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I)⁻¹ • ∮... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2 | [143, 1] | [160, 55] | have id1 : DifferentiableOn ℂ (fun z0 ↦ (2 * π * I : ℂ)⁻¹ • ∮ z1 in C(c1, r), (z1 - w1)⁻¹
• f (z0, z1)) (ball c0 r) := by
rw [differentiableOn_congr fun z zs ↦ h1 z (mem_open_closed zs)]
intro z0 z0m; apply DifferentiableAt.differentiableWithinAt
exact h.fd0 (mem_open_closed z0m) (mem_open_closed w1m) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c1, r), (z - w1)⁻¹ • f (z0, z)) = f (z0, w1)
ic... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c1, r), (z - w1)⁻¹ • f (z0, z)) = f (z0, w1)
ic... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I)⁻¹ • ∮... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2 | [143, 1] | [160, 55] | have h01 :=
cauchy1 w0m ic1 fun z0 z0m ↦
DifferentiableOn.differentiableAt id1 (IsOpen.mem_nhds isOpen_ball z0m) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c1, r), (z - w1)⁻¹ • f (z0, z)) = f (z0, w1)
ic... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c1, r), (z - w1)⁻¹ • f (z0, z)) = f (z0, w1)
ic... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I)⁻¹ • ∮... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2 | [143, 1] | [160, 55] | exact _root_.trans h01 (h1 w0 (mem_open_closed w0m)) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c1, r), (z - w1)⁻¹ • f (z0, z)) = f (z0, w1)
ic... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I)⁻¹ • ∮... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2 | [143, 1] | [160, 55] | rw [differentiableOn_congr fun z zs ↦ h1 z (mem_open_closed zs)] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c1, r), (z - w1)⁻¹ • f (z0, z)) = f (z0, w1)
ic... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c1, r), (z - w1)⁻¹ • f (z0, z)) = f (z0, w1)
ic... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I)⁻¹ • ∮... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2 | [143, 1] | [160, 55] | intro z0 z0m | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c1, r), (z - w1)⁻¹ • f (z0, z)) = f (z0, w1)
ic... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c1, r), (z - w1)⁻¹ • f (z0, z)) = f (z0, w1)
ic... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I)⁻¹ • ∮... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2 | [143, 1] | [160, 55] | apply DifferentiableAt.differentiableWithinAt | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c1, r), (z - w1)⁻¹ • f (z0, z)) = f (z0, w1)
ic... | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c1, r), (z - w1)⁻¹ • f (z0, z)) = f (z0,... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I)⁻¹ • ∮... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2 | [143, 1] | [160, 55] | exact h.fd0 (mem_open_closed z0m) (mem_open_closed w1m) | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c1, r), (z - w1)⁻¹ • f (z0, z)) = f (z0,... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_hasSum | [176, 1] | [188, 13] | simp at wm | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : w ∈ ball 0 r
⊢ HasSum (fun n => w ^ n • (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, r), (z - c)⁻¹ ^ n • (z - c)⁻... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
⊢ HasSum (fun n => w ^ n • (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, r), (z - c)⁻¹ ^ n • (z ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : w ∈ ball 0 r
⊢ HasSum (fun n => w ^ n • (2... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_hasSum | [176, 1] | [188, 13] | have ci : CircleIntegrable f c r := ContinuousOn.circleIntegrable (by linarith) fc | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
⊢ HasSum (fun n => w ^ n • (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, r), (z - c)⁻¹ ^ n • (z ... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f c r
⊢ HasSum (fun n => w ^ n • (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
⊢ HasSum (fun n => w ^ n... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_hasSum | [176, 1] | [188, 13] | have h := hasSum_cauchyPowerSeries_integral ci wm | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f c r
⊢ HasSum (fun n => w ^ n • (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in ... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f c r
h :
HasSum (fun n => (cauchyPowerSeries f c r n) fun x ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_hasSum | [176, 1] | [188, 13] | simp_rw [cauchyPowerSeries_apply] at h | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f c r
h :
HasSum (fun n => (cauchyPowerSeries f c r n) fun x ... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f c r
h :
HasSum (fun n => (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c,... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_hasSum | [176, 1] | [188, 13] | generalize hs : (2*π*I : ℂ)⁻¹ = s | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f c r
h :
HasSum (fun n => (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c,... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f c r
h :
HasSum (fun n => (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_hasSum | [176, 1] | [188, 13] | simp_rw [hs] at h | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f c r
h :
HasSum (fun n => (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f c r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
h :
HasSum (fun n => s •... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_hasSum | [176, 1] | [188, 13] | generalize hg : (s • ∮ z : ℂ in C(c, r), (z - (c + w))⁻¹ • f z) = g | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f c r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
h :
HasSum (fun n => s •... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f c r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
h :
HasSum (fun n => s •... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_hasSum | [176, 1] | [188, 13] | rw [hg] at h | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f c r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
h :
HasSum (fun n => s •... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f c r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
g : E
h : HasSum (fun n =>... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_hasSum | [176, 1] | [188, 13] | simp_rw [div_eq_mul_inv, mul_pow, ← smul_smul, circleIntegral.integral_smul, smul_comm s _] at h | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f c r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
g : E
h : HasSum (fun n =>... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f c r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
g : E
hg : (s • ∮ (z : ℂ) ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_hasSum | [176, 1] | [188, 13] | assumption | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f c r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
g : E
hg : (s • ∮ (z : ℂ) ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_hasSum | [176, 1] | [188, 13] | linarith | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
⊢ 0 ≤ r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
⊢ 0 ≤ r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | rcases (IsCompact.prod cs (isCompact_sphere _ _)).bddAbove_image fc.norm with ⟨b, bh⟩ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
⊢ ContinuousOn (fun z0 => ∮ (z1 : ℂ) in C(c1, r), f z0 z1) s | case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : b ∈ upperBounds ((fun x => ‖uncurry f x‖) '' s ×ˢ sp... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
⊢ ContinuousOn... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | simp only [mem_upperBounds, Set.forall_mem_image] at bh | case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : b ∈ upperBounds ((fun x => ‖uncurry f x‖) '' s ×ˢ sp... | case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | intro z1 z1s | case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ... | case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | have fb : ∀ᶠ x : ℂ in 𝓝[s] z1, ∀ᵐ t : ℝ, t ∈ Set.uIoc 0 (2 * π) →
‖deriv (circleMap c1 r) t • (fun z1 : ℂ ↦ f x z1) (circleMap c1 r t)‖ ≤ r * b := by
apply eventually_nhdsWithin_of_forall; intro x xs
apply MeasureTheory.ae_of_all _; intro t _; simp only [deriv_circleMap]
rw [norm_smul, Complex.norm_eq_abs]
... | case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ... | case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | refine intervalIntegral.continuousWithinAt_of_dominated_interval ?_ fb (by simp) ?_ | case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ... | case intro.refine_1
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncu... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | apply eventually_nhdsWithin_of_forall | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ≤ b
z1 : ℂ
... | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ≤ b
... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | intro x xs | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ≤ b
... | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ≤ b
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | apply MeasureTheory.ae_of_all _ | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ≤ b
... | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ≤ b
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | intro t _ | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ≤ b
... | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ≤ b
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | simp only [deriv_circleMap] | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ≤ b
... | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ≤ b
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | rw [norm_smul, Complex.norm_eq_abs] | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ≤ b
... | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ≤ b
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | simp only [map_mul, abs_circleMap_zero, Complex.abs_I, mul_one] | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ≤ b
... | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ≤ b
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | have bx := @bh (x, circleMap c1 r t) (Set.mk_mem_prod xs (circleMap_mem_sphere c1
(by linarith) t)) | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ≤ b
... | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ≤ b
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | simp only [uncurry] at bx | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ≤ b
... | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ≤ b
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | calc |r| * ‖f x (circleMap c1 r t)‖ ≤ |r| * b := by bound
_ = r * b := by rw [abs_of_pos rp] | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ≤ b
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | linarith | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ≤ b
z1 : ℂ
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | bound | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ≤ b
z1 : ℂ
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | rw [abs_of_pos rp] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ≤ b
z1 : ℂ
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | simp | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ≤ b
z1 : ℂ
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | apply eventually_nhdsWithin_of_forall | case intro.refine_1
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncu... | case intro.refine_1.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖un... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_1
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | intro x xs | case intro.refine_1.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖un... | case intro.refine_1.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖un... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_1.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphe... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | apply ContinuousOn.aestronglyMeasurable | case intro.refine_1.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖un... | case intro.refine_1.h.hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_1.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphe... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | apply ContinuousOn.smul | case intro.refine_1.h.hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ... | case intro.refine_1.h.hf.hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_1.h.hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ s... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | simp | case intro.refine_1.h.hf.hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r... | case intro.refine_1.h.hf.hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_1.h.hf.hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | exact ContinuousOn.mul (Continuous.continuousOn (continuous_circleMap _ _)) continuousOn_const | case intro.refine_1.h.hf.hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r... | case intro.refine_1.h.hf.hg
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_1.h.hf.hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | have comp : (fun t ↦ f x (circleMap c1 r t)) = uncurry f ∘ fun t ↦ (x, circleMap c1 r t) := by
apply funext; intro t; simp | case intro.refine_1.h.hf.hg
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r... | case intro.refine_1.h.hf.hg
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_1.h.hf.hg
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | simp | case intro.refine_1.h.hf.hg
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r... | case intro.refine_1.h.hf.hg
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_1.h.hf.hg
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | rw [comp] | case intro.refine_1.h.hf.hg
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r... | case intro.refine_1.h.hf.hg
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_1.h.hf.hg
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | apply ContinuousOn.comp fc | case intro.refine_1.h.hf.hg
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r... | case intro.refine_1.h.hf.hg.hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_1.h.hf.hg
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | exact ContinuousOn.prod continuousOn_const (Continuous.continuousOn (continuous_circleMap _ _)) | case intro.refine_1.h.hf.hg.hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c... | case intro.refine_1.h.hf.hg.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_1.h.hf.hg.hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | intro t _ | case intro.refine_1.h.hf.hg.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1... | case intro.refine_1.h.hf.hg.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_1.h.hf.hg.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | simp | case intro.refine_1.h.hf.hg.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1... | case intro.refine_1.h.hf.hg.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_1.h.hf.hg.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | exact ⟨xs, by linarith⟩ | case intro.refine_1.h.hf.hg.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1... | case intro.refine_1.h.hs
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_1.h.hf.hg.h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | exact measurableSet_uIoc | case intro.refine_1.h.hs
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_1.h.hs
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ s... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | apply funext | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ≤ b
z1 : ℂ
... | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ≤ b
... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | intro t | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ≤ b
... | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ≤ b
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | simp | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ≤ b
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | linarith | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ≤ b
z1 : ℂ
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | apply MeasureTheory.ae_of_all _ | case intro.refine_2
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncu... | case intro.refine_2
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncu... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_2
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | intro t _ | case intro.refine_2
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncu... | case intro.refine_2
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncu... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_2
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | simp | case intro.refine_2
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncu... | case intro.refine_2
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncu... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_2
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | apply ContinuousOn.smul continuousOn_const | case intro.refine_2
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncu... | case intro.refine_2.hg
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖u... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_2
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | have comp : (fun x ↦ f x (circleMap c1 r t)) = uncurry f ∘ fun x ↦ (x, circleMap c1 r t) := by
apply funext; intro t; simp | case intro.refine_2.hg
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖u... | case intro.refine_2.hg
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖u... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_2.hg
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sph... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | rw [comp] | case intro.refine_2.hg
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖u... | case intro.refine_2.hg
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖u... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_2.hg
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sph... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | apply ContinuousOn.comp fc (ContinuousOn.prod continuousOn_id continuousOn_const) | case intro.refine_2.hg
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖u... | case intro.refine_2.hg
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖u... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_2.hg
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sph... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | intro x xs | case intro.refine_2.hg
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖u... | case intro.refine_2.hg
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖u... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_2.hg
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sph... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | simp | case intro.refine_2.hg
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖u... | case intro.refine_2.hg
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖u... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_2.hg
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sph... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | exact ⟨xs, by linarith⟩ | case intro.refine_2.hg
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖u... | case intro.refine_2.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖un... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_2.hg
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sph... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | exact z1s | case intro.refine_2.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖un... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_2.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphe... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | apply funext | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ≤ b
z1 : ℂ
... | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ≤ b
... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | intro t | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ≤ b
... | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ≤ b
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | simp | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ≤ b
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | linarith | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ≤ b
z1 : ℂ
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.inv_sphere_ball | [234, 1] | [246, 21] | refine ContinuousOn.inv₀ (ContinuousOn.sub continuousOn_id continuousOn_const) fun z zs ↦ ?_ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
c w : ℂ
r : ℝ
wr : w ∈ ball 0 r
⊢ ContinuousOn (fun z => (z - (c + w))⁻¹) (sphere c r) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
c w : ℂ
r : ℝ
wr : w ∈ ball 0 r
z : ℂ
zs : z ∈ sphere c r
⊢ z - (c + w) ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
c w : ℂ
r : ℝ
wr : w ∈ ball 0 r
⊢ ContinuousOn (fun z => (z - (c + w))⁻¹) (sphere c r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.inv_sphere_ball | [234, 1] | [246, 21] | rw [←Complex.abs.ne_zero_iff] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
c w : ℂ
r : ℝ
wr : w ∈ ball 0 r
z : ℂ
zs : z ∈ sphere c r
⊢ z - (c + w) ≠ 0 | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
c w : ℂ
r : ℝ
wr : w ∈ ball 0 r
z : ℂ
zs : z ∈ sphere c r
⊢ Complex.abs (z - (c + w)) ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
c w : ℂ
r : ℝ
wr : w ∈ ball 0 r
z : ℂ
zs : z ∈ sphere c r
⊢ z - (c + w) ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.inv_sphere_ball | [234, 1] | [246, 21] | simp only [mem_ball_zero_iff, Complex.norm_eq_abs, mem_sphere_iff_norm] at zs wr | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
c w : ℂ
r : ℝ
wr : w ∈ ball 0 r
z : ℂ
zs : z ∈ sphere c r
⊢ Complex.abs (z - (c + w)) ≠ 0 | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
c w : ℂ
r : ℝ
z : ℂ
zs : Complex.abs (z - c) = r
wr : Complex.abs w < r
⊢ Complex.abs (z - (c + w)) ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
c w : ℂ
r : ℝ
wr : w ∈ ball 0 r
z : ℂ
zs : z ∈ sphere c r
⊢ Complex.abs (z - (c + w)) ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.inv_sphere_ball | [234, 1] | [246, 21] | apply ne_of_gt | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
c w : ℂ
r : ℝ
z : ℂ
zs : Complex.abs (z - c) = r
wr : Complex.abs w < r
⊢ Complex.abs (z - (c + w)) ≠ 0 | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
c w : ℂ
r : ℝ
z : ℂ
zs : Complex.abs (z - c) = r
wr : Complex.abs w < r
⊢ 0 < Complex.abs (z - (c + w)) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
c w : ℂ
r : ℝ
z : ℂ
zs : Complex.abs (z - c) = r
wr : Complex.abs w < r
⊢ Complex.abs (z - (c + w)) ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.inv_sphere_ball | [234, 1] | [246, 21] | calc abs (z - (c + w))
_ = abs (z - c + -w) := by ring_nf
_ ≥ abs (z - c) - abs (-w) := by bound
_ = r - abs (-w) := by rw [zs]
_ = r - abs w := by rw [Complex.abs.map_neg]
_ > r - r := (sub_lt_sub_left wr _)
_ = 0 := by ring | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
c w : ℂ
r : ℝ
z : ℂ
zs : Complex.abs (z - c) = r
wr : Complex.abs w < r
⊢ 0 < Complex.abs (z - (c + w)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
c w : ℂ
r : ℝ
z : ℂ
zs : Complex.abs (z - c) = r
wr : Complex.abs w < r
⊢ 0 < Complex.abs (z - (c + w))
TACTI... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.inv_sphere_ball | [234, 1] | [246, 21] | ring_nf | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
c w : ℂ
r : ℝ
z : ℂ
zs : Complex.abs (z - c) = r
wr : Complex.abs w < r
⊢ Complex.abs (z - (c + w)) = Complex.abs (z - c + -w) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
c w : ℂ
r : ℝ
z : ℂ
zs : Complex.abs (z - c) = r
wr : Complex.abs w < r
⊢ Complex.abs (z - (c + w)) = Complex.abs (z... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.inv_sphere_ball | [234, 1] | [246, 21] | bound | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
c w : ℂ
r : ℝ
z : ℂ
zs : Complex.abs (z - c) = r
wr : Complex.abs w < r
⊢ Complex.abs (z - c + -w) ≥ Complex.abs (z - c) - Complex.abs (-w) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
c w : ℂ
r : ℝ
z : ℂ
zs : Complex.abs (z - c) = r
wr : Complex.abs w < r
⊢ Complex.abs (z - c + -w) ≥ Complex.abs (z ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.inv_sphere_ball | [234, 1] | [246, 21] | rw [zs] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
c w : ℂ
r : ℝ
z : ℂ
zs : Complex.abs (z - c) = r
wr : Complex.abs w < r
⊢ Complex.abs (z - c) - Complex.abs (-w) = r - Complex.abs (-w) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
c w : ℂ
r : ℝ
z : ℂ
zs : Complex.abs (z - c) = r
wr : Complex.abs w < r
⊢ Complex.abs (z - c) - Complex.abs (-w) = r... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.inv_sphere_ball | [234, 1] | [246, 21] | rw [Complex.abs.map_neg] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
c w : ℂ
r : ℝ
z : ℂ
zs : Complex.abs (z - c) = r
wr : Complex.abs w < r
⊢ r - Complex.abs (-w) = r - Complex.abs w | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
c w : ℂ
r : ℝ
z : ℂ
zs : Complex.abs (z - c) = r
wr : Complex.abs w < r
⊢ r - Complex.abs (-w) = r - Complex.abs w
T... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.inv_sphere_ball | [234, 1] | [246, 21] | ring | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
c w : ℂ
r : ℝ
z : ℂ
zs : Complex.abs (z - c) = r
wr : Complex.abs w < r
⊢ r - r = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
c w : ℂ
r : ℝ
z : ℂ
zs : Complex.abs (z - c) = r
wr : Complex.abs w < r
⊢ r - r = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.cauchy1 | [249, 1] | [262, 17] | apply ContinuousOn.circleIntegral rp (isCompact_sphere _ _) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ ContinuousOn (fun z0 => ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1)) (sphere c0 r) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ ContinuousOn (uncurry fun z0 z1 => (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1)) (sphere c0 r ×ˢ sphere c1 r) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ ContinuousOn (fun z0 => ∮ (z1 : ℂ) in C(c1, r), ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.cauchy1 | [249, 1] | [262, 17] | apply ContinuousOn.smul | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ ContinuousOn (uncurry fun z0 z1 => (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1)) (sphere c0 r ×ˢ sphere c1 r) | case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ ContinuousOn (fun x => (x.2 - c1)⁻¹ ^ n1) (sphere c0 r ×ˢ sphere c1 r)
case hg
E : Type
inst✝² : Nor... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ ContinuousOn (uncurry fun z0 z1 => (z1 - c1)⁻¹ ^... |
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