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 | sum_integral_commute | [274, 1] | [294, 71] | intro t _ | case h_lim.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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summable b
h : ∀ z ∈ sphere c r, HasSum (fun n => f n z) (g z)
⊢ ∀ a ∈ Ι 0 (2 * π),
HasSum (fun n => (circleMap 0 r a * I) • f n (circleMap c r a)) ((circleMap 0 r a * I) • g (circleMap c r a)) | case h_lim.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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summable b
h : ∀ z ∈ sphere c r, HasSum (fun n => f n z) (g z)
t : ℝ
a✝ : t ∈ Ι 0 (2 * π)
⊢ HasSum (fun n => (circleMap 0 r t * I) • f n (circleMap c r t)) ((circleMap 0 r t * I) • g (circleMap c r t)) | Please generate a tactic in lean4 to solve the state.
STATE:
case h_lim.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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summable b
h : ∀ z ∈ sphere c r, HasSum (fun n => f n z) (g z)
⊢ ∀ a ∈ Ι 0 (2 * π),
HasSum (fun n => (circleMap 0 r a * I) • f n (circleMap c r a)) ((circleMap 0 r a * I) • g (circleMap c r a))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | sum_integral_commute | [274, 1] | [294, 71] | apply HasSum.const_smul | case h_lim.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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summable b
h : ∀ z ∈ sphere c r, HasSum (fun n => f n z) (g z)
t : ℝ
a✝ : t ∈ Ι 0 (2 * π)
⊢ HasSum (fun n => (circleMap 0 r t * I) • f n (circleMap c r t)) ((circleMap 0 r t * I) • g (circleMap c r t)) | case h_lim.a.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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summable b
h : ∀ z ∈ sphere c r, HasSum (fun n => f n z) (g z)
t : ℝ
a✝ : t ∈ Ι 0 (2 * π)
⊢ HasSum (fun i => f i (circleMap c r t)) (g (circleMap c r t)) | Please generate a tactic in lean4 to solve the state.
STATE:
case h_lim.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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summable b
h : ∀ z ∈ sphere c r, HasSum (fun n => f n z) (g z)
t : ℝ
a✝ : t ∈ Ι 0 (2 * π)
⊢ HasSum (fun n => (circleMap 0 r t * I) • f n (circleMap c r t)) ((circleMap 0 r t * I) • g (circleMap c r t))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | sum_integral_commute | [274, 1] | [294, 71] | exact h (circleMap c r t) (circleMap_mem_sphere _ (by linarith) _) | case h_lim.a.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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summable b
h : ∀ z ∈ sphere c r, HasSum (fun n => f n z) (g z)
t : ℝ
a✝ : t ∈ Ι 0 (2 * π)
⊢ HasSum (fun i => f i (circleMap c r t)) (g (circleMap c r t)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h_lim.a.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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summable b
h : ∀ z ∈ sphere c r, HasSum (fun n => f n z) (g z)
t : ℝ
a✝ : t ∈ Ι 0 (2 * π)
⊢ HasSum (fun i => f i (circleMap c r t)) (g (circleMap c r t))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | sum_integral_commute | [274, 1] | [294, 71] | 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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summable b
h : ∀ z ∈ sphere c r, HasSum (fun n => f n z) (g z)
t : ℝ
a✝ : t ∈ Ι 0 (2 * π)
⊢ 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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summable b
h : ∀ z ∈ sphere c r, HasSum (fun n => f n z) (g z)
t : ℝ
a✝ : t ∈ Ι 0 (2 * π)
⊢ 0 ≤ r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | bounded_circleIntegral | [297, 1] | [325, 33] | rw [circleIntegral] | E : Type
inst✝² : NormedAddCommGroup E
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 : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
⊢ ‖∮ (z : ℂ) in C(c, r), f z‖ ≤ 2 * π * r * b | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
⊢ ‖∫ (θ : ℝ) in 0 ..2 * π, deriv (circleMap c r) θ • f (circleMap c r θ)‖ ≤ 2 * π * r * 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 → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
⊢ ‖∮ (z : ℂ) in C(c, r), f z‖ ≤ 2 * π * r * b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | bounded_circleIntegral | [297, 1] | [325, 33] | simp only [deriv_circleMap] | E : Type
inst✝² : NormedAddCommGroup E
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 : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
⊢ ‖∫ (θ : ℝ) in 0 ..2 * π, deriv (circleMap c r) θ • f (circleMap c r θ)‖ ≤ 2 * π * r * b | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
⊢ ‖∫ (θ : ℝ) in 0 ..2 * π, (circleMap 0 r θ * I) • f (circleMap c r θ)‖ ≤ 2 * π * r * 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 → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
⊢ ‖∫ (θ : ℝ) in 0 ..2 * π, deriv (circleMap c r) θ • f (circleMap c r θ)‖ ≤ 2 * π * r * b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | bounded_circleIntegral | [297, 1] | [325, 33] | have nonneg_2π := Real.two_pi_pos.le | E : Type
inst✝² : NormedAddCommGroup E
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 : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
⊢ ‖∫ (θ : ℝ) in 0 ..2 * π, (circleMap 0 r θ * I) • f (circleMap c r θ)‖ ≤ 2 * π * r * b | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
⊢ ‖∫ (θ : ℝ) in 0 ..2 * π, (circleMap 0 r θ * I) • f (circleMap c r θ)‖ ≤ 2 * π * r * 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 → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
⊢ ‖∫ (θ : ℝ) in 0 ..2 * π, (circleMap 0 r θ * I) • f (circleMap c r θ)‖ ≤ 2 * π * r * b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | bounded_circleIntegral | [297, 1] | [325, 33] | have ib : ‖(∫ t in (0)..(2*π), (circleMap 0 r t * I) • f (circleMap c r t))‖ ≤
(∫ t in (0)..(2*π), ‖(circleMap 0 r t * I) • f (circleMap c r t)‖) :=
intervalIntegral.norm_integral_le_integral_norm nonneg_2π | E : Type
inst✝² : NormedAddCommGroup E
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 : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
⊢ ‖∫ (θ : ℝ) in 0 ..2 * π, (circleMap 0 r θ * I) • f (circleMap c r θ)‖ ≤ 2 * π * r * b | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
ib :
‖∫ (t : ℝ) in 0 ..2 * π, (circleMap 0 r t * I) • f (circleMap c r t)‖ ≤
∫ (t : ℝ) in 0 ..2 * π, ‖(circleMap 0 r t * I) • f (circleMap c r t)‖
⊢ ‖∫ (θ : ℝ) in 0 ..2 * π, (circleMap 0 r θ * I) • f (circleMap c r θ)‖ ≤ 2 * π * r * 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 → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
⊢ ‖∫ (θ : ℝ) in 0 ..2 * π, (circleMap 0 r θ * I) • f (circleMap c r θ)‖ ≤ 2 * π * r * b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | bounded_circleIntegral | [297, 1] | [325, 33] | refine le_trans ib ?_ | E : Type
inst✝² : NormedAddCommGroup E
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 : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
ib :
‖∫ (t : ℝ) in 0 ..2 * π, (circleMap 0 r t * I) • f (circleMap c r t)‖ ≤
∫ (t : ℝ) in 0 ..2 * π, ‖(circleMap 0 r t * I) • f (circleMap c r t)‖
⊢ ‖∫ (θ : ℝ) in 0 ..2 * π, (circleMap 0 r θ * I) • f (circleMap c r θ)‖ ≤ 2 * π * r * b | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
ib :
‖∫ (t : ℝ) in 0 ..2 * π, (circleMap 0 r t * I) • f (circleMap c r t)‖ ≤
∫ (t : ℝ) in 0 ..2 * π, ‖(circleMap 0 r t * I) • f (circleMap c r t)‖
⊢ ∫ (t : ℝ) in 0 ..2 * π, ‖(circleMap 0 r t * I) • f (circleMap c r t)‖ ≤ 2 * π * r * 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 → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
ib :
‖∫ (t : ℝ) in 0 ..2 * π, (circleMap 0 r t * I) • f (circleMap c r t)‖ ≤
∫ (t : ℝ) in 0 ..2 * π, ‖(circleMap 0 r t * I) • f (circleMap c r t)‖
⊢ ‖∫ (θ : ℝ) in 0 ..2 * π, (circleMap 0 r θ * I) • f (circleMap c r θ)‖ ≤ 2 * π * r * b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | bounded_circleIntegral | [297, 1] | [325, 33] | clear ib | E : Type
inst✝² : NormedAddCommGroup E
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 : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
ib :
‖∫ (t : ℝ) in 0 ..2 * π, (circleMap 0 r t * I) • f (circleMap c r t)‖ ≤
∫ (t : ℝ) in 0 ..2 * π, ‖(circleMap 0 r t * I) • f (circleMap c r t)‖
⊢ ∫ (t : ℝ) in 0 ..2 * π, ‖(circleMap 0 r t * I) • f (circleMap c r t)‖ ≤ 2 * π * r * b | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
⊢ ∫ (t : ℝ) in 0 ..2 * π, ‖(circleMap 0 r t * I) • f (circleMap c r t)‖ ≤ 2 * π * r * 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 → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
ib :
‖∫ (t : ℝ) in 0 ..2 * π, (circleMap 0 r t * I) • f (circleMap c r t)‖ ≤
∫ (t : ℝ) in 0 ..2 * π, ‖(circleMap 0 r t * I) • f (circleMap c r t)‖
⊢ ∫ (t : ℝ) in 0 ..2 * π, ‖(circleMap 0 r t * I) • f (circleMap c r t)‖ ≤ 2 * π * r * b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | bounded_circleIntegral | [297, 1] | [325, 33] | simp_rw [norm_smul, Complex.norm_eq_abs] | E : Type
inst✝² : NormedAddCommGroup E
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 : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
⊢ ∫ (t : ℝ) in 0 ..2 * π, ‖(circleMap 0 r t * I) • f (circleMap c r t)‖ ≤ 2 * π * r * b | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
⊢ ∫ (t : ℝ) in 0 ..2 * π, Complex.abs (circleMap 0 r t * I) * ‖f (circleMap c r t)‖ ≤ 2 * π * r * 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 → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
⊢ ∫ (t : ℝ) in 0 ..2 * π, ‖(circleMap 0 r t * I) • f (circleMap c r t)‖ ≤ 2 * π * r * b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | bounded_circleIntegral | [297, 1] | [325, 33] | simp only [map_mul, abs_circleMap_zero, Complex.abs_I, mul_one, integral_const_mul] | E : Type
inst✝² : NormedAddCommGroup E
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 : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
⊢ ∫ (t : ℝ) in 0 ..2 * π, Complex.abs (circleMap 0 r t * I) * ‖f (circleMap c r t)‖ ≤ 2 * π * r * b | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
⊢ |r| * ∫ (x : ℝ) in 0 ..2 * π, ‖f (circleMap c r x)‖ ≤ 2 * π * r * 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 → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
⊢ ∫ (t : ℝ) in 0 ..2 * π, Complex.abs (circleMap 0 r t * I) * ‖f (circleMap c r t)‖ ≤ 2 * π * r * b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | bounded_circleIntegral | [297, 1] | [325, 33] | have mo : ∀ t, t ∈ Set.Icc 0 (2 * π) → ‖f (circleMap c r t)‖ ≤ b := fun t _ ↦
fb (circleMap c r t) (circleMap_mem_sphere c (by linarith) t) | E : Type
inst✝² : NormedAddCommGroup E
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 : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
⊢ |r| * ∫ (x : ℝ) in 0 ..2 * π, ‖f (circleMap c r x)‖ ≤ 2 * π * r * b | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
⊢ |r| * ∫ (x : ℝ) in 0 ..2 * π, ‖f (circleMap c r x)‖ ≤ 2 * π * r * 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 → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
⊢ |r| * ∫ (x : ℝ) in 0 ..2 * π, ‖f (circleMap c r x)‖ ≤ 2 * π * r * b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | bounded_circleIntegral | [297, 1] | [325, 33] | have i0 : IntervalIntegrable (fun t ↦ ‖f (circleMap c r t)‖) Real.measureSpace.volume
0 (2*π) := by
apply ContinuousOn.intervalIntegrable
have ca : ContinuousOn (norm : E → ℝ) Set.univ := Continuous.continuousOn continuous_norm
refine ContinuousOn.comp ca ?_ (Set.mapsTo_univ _ _)
apply ContinuousOn.comp fc
exact Continuous.continuousOn (continuous_circleMap _ _)
intro t _; exact circleMap_mem_sphere _ (by 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 : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
⊢ |r| * ∫ (x : ℝ) in 0 ..2 * π, ‖f (circleMap c r x)‖ ≤ 2 * π * r * b | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
i0 : IntervalIntegrable (fun t => ‖f (circleMap c r t)‖) MeasureTheory.volume 0 (2 * π)
⊢ |r| * ∫ (x : ℝ) in 0 ..2 * π, ‖f (circleMap c r x)‖ ≤ 2 * π * r * 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 → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
⊢ |r| * ∫ (x : ℝ) in 0 ..2 * π, ‖f (circleMap c r x)‖ ≤ 2 * π * r * b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | bounded_circleIntegral | [297, 1] | [325, 33] | have i1 : IntervalIntegrable (fun _ ↦ b) Real.measureSpace.volume 0 (2 * π) :=
intervalIntegrable_const | E : Type
inst✝² : NormedAddCommGroup E
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 : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
i0 : IntervalIntegrable (fun t => ‖f (circleMap c r t)‖) MeasureTheory.volume 0 (2 * π)
⊢ |r| * ∫ (x : ℝ) in 0 ..2 * π, ‖f (circleMap c r x)‖ ≤ 2 * π * r * b | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
i0 : IntervalIntegrable (fun t => ‖f (circleMap c r t)‖) MeasureTheory.volume 0 (2 * π)
i1 : IntervalIntegrable (fun x => b) MeasureTheory.volume 0 (2 * π)
⊢ |r| * ∫ (x : ℝ) in 0 ..2 * π, ‖f (circleMap c r x)‖ ≤ 2 * π * r * 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 → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
i0 : IntervalIntegrable (fun t => ‖f (circleMap c r t)‖) MeasureTheory.volume 0 (2 * π)
⊢ |r| * ∫ (x : ℝ) in 0 ..2 * π, ‖f (circleMap c r x)‖ ≤ 2 * π * r * b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | bounded_circleIntegral | [297, 1] | [325, 33] | have im := intervalIntegral.integral_mono_on nonneg_2π i0 i1 mo | E : Type
inst✝² : NormedAddCommGroup E
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 : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
i0 : IntervalIntegrable (fun t => ‖f (circleMap c r t)‖) MeasureTheory.volume 0 (2 * π)
i1 : IntervalIntegrable (fun x => b) MeasureTheory.volume 0 (2 * π)
⊢ |r| * ∫ (x : ℝ) in 0 ..2 * π, ‖f (circleMap c r x)‖ ≤ 2 * π * r * b | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
i0 : IntervalIntegrable (fun t => ‖f (circleMap c r t)‖) MeasureTheory.volume 0 (2 * π)
i1 : IntervalIntegrable (fun x => b) MeasureTheory.volume 0 (2 * π)
im : ∫ (u : ℝ) in 0 ..2 * π, ‖f (circleMap c r u)‖ ≤ ∫ (u : ℝ) in 0 ..2 * π, b
⊢ |r| * ∫ (x : ℝ) in 0 ..2 * π, ‖f (circleMap c r x)‖ ≤ 2 * π * r * 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 → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
i0 : IntervalIntegrable (fun t => ‖f (circleMap c r t)‖) MeasureTheory.volume 0 (2 * π)
i1 : IntervalIntegrable (fun x => b) MeasureTheory.volume 0 (2 * π)
⊢ |r| * ∫ (x : ℝ) in 0 ..2 * π, ‖f (circleMap c r x)‖ ≤ 2 * π * r * b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | bounded_circleIntegral | [297, 1] | [325, 33] | simp only [integral_const, sub_zero, smul_eq_mul] at im | E : Type
inst✝² : NormedAddCommGroup E
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 : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
i0 : IntervalIntegrable (fun t => ‖f (circleMap c r t)‖) MeasureTheory.volume 0 (2 * π)
i1 : IntervalIntegrable (fun x => b) MeasureTheory.volume 0 (2 * π)
im : ∫ (u : ℝ) in 0 ..2 * π, ‖f (circleMap c r u)‖ ≤ ∫ (u : ℝ) in 0 ..2 * π, b
⊢ |r| * ∫ (x : ℝ) in 0 ..2 * π, ‖f (circleMap c r x)‖ ≤ 2 * π * r * b | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
i0 : IntervalIntegrable (fun t => ‖f (circleMap c r t)‖) MeasureTheory.volume 0 (2 * π)
i1 : IntervalIntegrable (fun x => b) MeasureTheory.volume 0 (2 * π)
im : ∫ (u : ℝ) in 0 ..2 * π, ‖f (circleMap c r u)‖ ≤ 2 * π * b
⊢ |r| * ∫ (x : ℝ) in 0 ..2 * π, ‖f (circleMap c r x)‖ ≤ 2 * π * r * 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 → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
i0 : IntervalIntegrable (fun t => ‖f (circleMap c r t)‖) MeasureTheory.volume 0 (2 * π)
i1 : IntervalIntegrable (fun x => b) MeasureTheory.volume 0 (2 * π)
im : ∫ (u : ℝ) in 0 ..2 * π, ‖f (circleMap c r u)‖ ≤ ∫ (u : ℝ) in 0 ..2 * π, b
⊢ |r| * ∫ (x : ℝ) in 0 ..2 * π, ‖f (circleMap c r x)‖ ≤ 2 * π * r * b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | bounded_circleIntegral | [297, 1] | [325, 33] | calc |r| * ∫ t in (0)..(2*π), ‖f (circleMap c r t)‖
_ ≤ |r| * (2 * π * b) := by bound
_ = r * (2 * π * b) := by rw [abs_of_pos rp]
_ = 2 * π * r * b := by 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
f : ℂ → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
i0 : IntervalIntegrable (fun t => ‖f (circleMap c r t)‖) MeasureTheory.volume 0 (2 * π)
i1 : IntervalIntegrable (fun x => b) MeasureTheory.volume 0 (2 * π)
im : ∫ (u : ℝ) in 0 ..2 * π, ‖f (circleMap c r u)‖ ≤ 2 * π * b
⊢ |r| * ∫ (x : ℝ) in 0 ..2 * π, ‖f (circleMap c r x)‖ ≤ 2 * π * r * b | 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 : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
i0 : IntervalIntegrable (fun t => ‖f (circleMap c r t)‖) MeasureTheory.volume 0 (2 * π)
i1 : IntervalIntegrable (fun x => b) MeasureTheory.volume 0 (2 * π)
im : ∫ (u : ℝ) in 0 ..2 * π, ‖f (circleMap c r u)‖ ≤ 2 * π * b
⊢ |r| * ∫ (x : ℝ) in 0 ..2 * π, ‖f (circleMap c r x)‖ ≤ 2 * π * r * b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | bounded_circleIntegral | [297, 1] | [325, 33] | 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 : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
t : ℝ
x✝ : t ∈ Set.Icc 0 (2 * π)
⊢ 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 : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
t : ℝ
x✝ : t ∈ Set.Icc 0 (2 * π)
⊢ 0 ≤ r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | bounded_circleIntegral | [297, 1] | [325, 33] | apply ContinuousOn.intervalIntegrable | E : Type
inst✝² : NormedAddCommGroup E
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 : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
⊢ IntervalIntegrable (fun t => ‖f (circleMap c r t)‖) MeasureTheory.volume 0 (2 * π) | case hu
E : Type
inst✝² : NormedAddCommGroup E
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 : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
⊢ ContinuousOn (fun t => ‖f (circleMap c r t)‖) (Set.uIcc 0 (2 * π)) | 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 : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
⊢ IntervalIntegrable (fun t => ‖f (circleMap c r t)‖) MeasureTheory.volume 0 (2 * π)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | bounded_circleIntegral | [297, 1] | [325, 33] | have ca : ContinuousOn (norm : E → ℝ) Set.univ := Continuous.continuousOn continuous_norm | case hu
E : Type
inst✝² : NormedAddCommGroup E
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 : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
⊢ ContinuousOn (fun t => ‖f (circleMap c r t)‖) (Set.uIcc 0 (2 * π)) | case hu
E : Type
inst✝² : NormedAddCommGroup E
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 : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
ca : ContinuousOn norm Set.univ
⊢ ContinuousOn (fun t => ‖f (circleMap c r t)‖) (Set.uIcc 0 (2 * π)) | Please generate a tactic in lean4 to solve the state.
STATE:
case hu
E : Type
inst✝² : NormedAddCommGroup E
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 : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
⊢ ContinuousOn (fun t => ‖f (circleMap c r t)‖) (Set.uIcc 0 (2 * π))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | bounded_circleIntegral | [297, 1] | [325, 33] | refine ContinuousOn.comp ca ?_ (Set.mapsTo_univ _ _) | case hu
E : Type
inst✝² : NormedAddCommGroup E
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 : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
ca : ContinuousOn norm Set.univ
⊢ ContinuousOn (fun t => ‖f (circleMap c r t)‖) (Set.uIcc 0 (2 * π)) | case hu
E : Type
inst✝² : NormedAddCommGroup E
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 : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
ca : ContinuousOn norm Set.univ
⊢ ContinuousOn (fun t => f (circleMap c r t)) (Set.uIcc 0 (2 * π)) | Please generate a tactic in lean4 to solve the state.
STATE:
case hu
E : Type
inst✝² : NormedAddCommGroup E
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 : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
ca : ContinuousOn norm Set.univ
⊢ ContinuousOn (fun t => ‖f (circleMap c r t)‖) (Set.uIcc 0 (2 * π))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | bounded_circleIntegral | [297, 1] | [325, 33] | apply ContinuousOn.comp fc | case hu
E : Type
inst✝² : NormedAddCommGroup E
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 : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
ca : ContinuousOn norm Set.univ
⊢ ContinuousOn (fun t => f (circleMap c r t)) (Set.uIcc 0 (2 * π)) | case hu.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 → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
ca : ContinuousOn norm Set.univ
⊢ ContinuousOn (fun t => circleMap c r t) (Set.uIcc 0 (2 * π))
case hu.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 : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
ca : ContinuousOn norm Set.univ
⊢ Set.MapsTo (fun t => circleMap c r t) (Set.uIcc 0 (2 * π)) (sphere c r) | Please generate a tactic in lean4 to solve the state.
STATE:
case hu
E : Type
inst✝² : NormedAddCommGroup E
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 : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
ca : ContinuousOn norm Set.univ
⊢ ContinuousOn (fun t => f (circleMap c r t)) (Set.uIcc 0 (2 * π))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | bounded_circleIntegral | [297, 1] | [325, 33] | exact Continuous.continuousOn (continuous_circleMap _ _) | case hu.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 → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
ca : ContinuousOn norm Set.univ
⊢ ContinuousOn (fun t => circleMap c r t) (Set.uIcc 0 (2 * π))
case hu.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 : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
ca : ContinuousOn norm Set.univ
⊢ Set.MapsTo (fun t => circleMap c r t) (Set.uIcc 0 (2 * π)) (sphere c r) | case hu.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 : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
ca : ContinuousOn norm Set.univ
⊢ Set.MapsTo (fun t => circleMap c r t) (Set.uIcc 0 (2 * π)) (sphere c r) | Please generate a tactic in lean4 to solve the state.
STATE:
case hu.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 → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
ca : ContinuousOn norm Set.univ
⊢ ContinuousOn (fun t => circleMap c r t) (Set.uIcc 0 (2 * π))
case hu.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 : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
ca : ContinuousOn norm Set.univ
⊢ Set.MapsTo (fun t => circleMap c r t) (Set.uIcc 0 (2 * π)) (sphere c r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | bounded_circleIntegral | [297, 1] | [325, 33] | intro t _ | case hu.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 : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
ca : ContinuousOn norm Set.univ
⊢ Set.MapsTo (fun t => circleMap c r t) (Set.uIcc 0 (2 * π)) (sphere c r) | case hu.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 : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
ca : ContinuousOn norm Set.univ
t : ℝ
a✝ : t ∈ Set.uIcc 0 (2 * π)
⊢ (fun t => circleMap c r t) t ∈ sphere c r | Please generate a tactic in lean4 to solve the state.
STATE:
case hu.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 : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
ca : ContinuousOn norm Set.univ
⊢ Set.MapsTo (fun t => circleMap c r t) (Set.uIcc 0 (2 * π)) (sphere c r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | bounded_circleIntegral | [297, 1] | [325, 33] | exact circleMap_mem_sphere _ (by linarith) _ | case hu.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 : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
ca : ContinuousOn norm Set.univ
t : ℝ
a✝ : t ∈ Set.uIcc 0 (2 * π)
⊢ (fun t => circleMap c r t) t ∈ sphere c r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hu.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 : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
ca : ContinuousOn norm Set.univ
t : ℝ
a✝ : t ∈ Set.uIcc 0 (2 * π)
⊢ (fun t => circleMap c r t) t ∈ sphere c r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | bounded_circleIntegral | [297, 1] | [325, 33] | 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 : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
ca : ContinuousOn norm Set.univ
t : ℝ
a✝ : t ∈ Set.uIcc 0 (2 * π)
⊢ 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 : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
ca : ContinuousOn norm Set.univ
t : ℝ
a✝ : t ∈ Set.uIcc 0 (2 * π)
⊢ 0 ≤ r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | bounded_circleIntegral | [297, 1] | [325, 33] | 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 → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
i0 : IntervalIntegrable (fun t => ‖f (circleMap c r t)‖) MeasureTheory.volume 0 (2 * π)
i1 : IntervalIntegrable (fun x => b) MeasureTheory.volume 0 (2 * π)
im : ∫ (u : ℝ) in 0 ..2 * π, ‖f (circleMap c r u)‖ ≤ 2 * π * b
⊢ |r| * ∫ (t : ℝ) in 0 ..2 * π, ‖f (circleMap c r t)‖ ≤ |r| * (2 * π * b) | 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 : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
i0 : IntervalIntegrable (fun t => ‖f (circleMap c r t)‖) MeasureTheory.volume 0 (2 * π)
i1 : IntervalIntegrable (fun x => b) MeasureTheory.volume 0 (2 * π)
im : ∫ (u : ℝ) in 0 ..2 * π, ‖f (circleMap c r u)‖ ≤ 2 * π * b
⊢ |r| * ∫ (t : ℝ) in 0 ..2 * π, ‖f (circleMap c r t)‖ ≤ |r| * (2 * π * b)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | bounded_circleIntegral | [297, 1] | [325, 33] | 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 → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
i0 : IntervalIntegrable (fun t => ‖f (circleMap c r t)‖) MeasureTheory.volume 0 (2 * π)
i1 : IntervalIntegrable (fun x => b) MeasureTheory.volume 0 (2 * π)
im : ∫ (u : ℝ) in 0 ..2 * π, ‖f (circleMap c r u)‖ ≤ 2 * π * b
⊢ |r| * (2 * π * b) = r * (2 * π * b) | 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 : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
i0 : IntervalIntegrable (fun t => ‖f (circleMap c r t)‖) MeasureTheory.volume 0 (2 * π)
i1 : IntervalIntegrable (fun x => b) MeasureTheory.volume 0 (2 * π)
im : ∫ (u : ℝ) in 0 ..2 * π, ‖f (circleMap c r u)‖ ≤ 2 * π * b
⊢ |r| * (2 * π * b) = r * (2 * π * b)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | bounded_circleIntegral | [297, 1] | [325, 33] | 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
f : ℂ → E
c : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
i0 : IntervalIntegrable (fun t => ‖f (circleMap c r t)‖) MeasureTheory.volume 0 (2 * π)
i1 : IntervalIntegrable (fun x => b) MeasureTheory.volume 0 (2 * π)
im : ∫ (u : ℝ) in 0 ..2 * π, ‖f (circleMap c r u)‖ ≤ 2 * π * b
⊢ r * (2 * π * b) = 2 * π * r * b | 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 : ℂ
r b : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
fb : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
nonneg_2π : 0 ≤ 2 * π
mo : ∀ t ∈ Set.Icc 0 (2 * π), ‖f (circleMap c r t)‖ ≤ b
i0 : IntervalIntegrable (fun t => ‖f (circleMap c r t)‖) MeasureTheory.volume 0 (2 * π)
i1 : IntervalIntegrable (fun x => b) MeasureTheory.volume 0 (2 * π)
im : ∫ (u : ℝ) in 0 ..2 * π, ‖f (circleMap c r u)‖ ≤ 2 * π * b
⊢ r * (2 * π * b) = 2 * π * r * b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_bound | [328, 1] | [342, 74] | have sb : ∀ z, z ∈ sphere c r → ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b := by
intro z zs; have fb := bh z zs
rw [norm_smul, norm_smul, Complex.norm_eq_abs, Complex.norm_eq_abs]
simp only [inv_pow, map_inv₀, map_pow, ge_iff_le, Metric.mem_sphere, Complex.dist_eq] at zs ⊢
rw [zs]; ring_nf; 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 → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
⊢ ‖∮ (z : ℂ) in C(c, r), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ 2 * π * b * r⁻¹ ^ 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
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
sb : ∀ z ∈ sphere c r, ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b
⊢ ‖∮ (z : ℂ) in C(c, r), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ 2 * π * b * r⁻¹ ^ 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
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
⊢ ‖∮ (z : ℂ) in C(c, r), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ 2 * π * b * r⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_bound | [328, 1] | [342, 74] | have isb := bounded_circleIntegral rp ?_ sb | E : Type
inst✝² : NormedAddCommGroup E
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
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
sb : ∀ z ∈ sphere c r, ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b
⊢ ‖∮ (z : ℂ) in C(c, r), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ 2 * π * b * r⁻¹ ^ n | 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
f : ℂ → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
sb : ∀ z ∈ sphere c r, ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b
isb : ‖∮ (z : ℂ) in C(c, r), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ 2 * π * r * (r⁻¹ ^ n * r⁻¹ * b)
⊢ ‖∮ (z : ℂ) in C(c, r), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ 2 * π * b * r⁻¹ ^ n
case refine_1
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
sb : ∀ z ∈ sphere c r, ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b
⊢ ContinuousOn (fun z => (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z) (sphere 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
f : ℂ → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
sb : ∀ z ∈ sphere c r, ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b
⊢ ‖∮ (z : ℂ) in C(c, r), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ 2 * π * b * r⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_bound | [328, 1] | [342, 74] | intro 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
f : ℂ → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
⊢ ∀ z ∈ sphere c r, ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
z : ℂ
zs : z ∈ sphere c r
⊢ ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * 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 → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
⊢ ∀ z ∈ sphere c r, ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_bound | [328, 1] | [342, 74] | have fb := bh 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
f : ℂ → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
z : ℂ
zs : z ∈ sphere c r
⊢ ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
z : ℂ
zs : z ∈ sphere c r
fb : ‖f z‖ ≤ b
⊢ ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * 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 → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
z : ℂ
zs : z ∈ sphere c r
⊢ ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_bound | [328, 1] | [342, 74] | rw [norm_smul, norm_smul, Complex.norm_eq_abs, Complex.norm_eq_abs] | E : Type
inst✝² : NormedAddCommGroup E
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
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
z : ℂ
zs : z ∈ sphere c r
fb : ‖f z‖ ≤ b
⊢ ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
z : ℂ
zs : z ∈ sphere c r
fb : ‖f z‖ ≤ b
⊢ Complex.abs ((z - c)⁻¹ ^ n) * (Complex.abs (z - c)⁻¹ * ‖f z‖) ≤ r⁻¹ ^ n * r⁻¹ * 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 → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
z : ℂ
zs : z ∈ sphere c r
fb : ‖f z‖ ≤ b
⊢ ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_bound | [328, 1] | [342, 74] | simp only [inv_pow, map_inv₀, map_pow, ge_iff_le, Metric.mem_sphere, Complex.dist_eq] at zs ⊢ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
z : ℂ
zs : z ∈ sphere c r
fb : ‖f z‖ ≤ b
⊢ Complex.abs ((z - c)⁻¹ ^ n) * (Complex.abs (z - c)⁻¹ * ‖f z‖) ≤ r⁻¹ ^ n * r⁻¹ * b | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
z : ℂ
fb : ‖f z‖ ≤ b
zs : Complex.abs (z - c) = r
⊢ (Complex.abs (z - c) ^ n)⁻¹ * ((Complex.abs (z - c))⁻¹ * ‖f z‖) ≤ (r ^ n)⁻¹ * r⁻¹ * 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 → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
z : ℂ
zs : z ∈ sphere c r
fb : ‖f z‖ ≤ b
⊢ Complex.abs ((z - c)⁻¹ ^ n) * (Complex.abs (z - c)⁻¹ * ‖f z‖) ≤ r⁻¹ ^ n * r⁻¹ * b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_bound | [328, 1] | [342, 74] | 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
f : ℂ → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
z : ℂ
fb : ‖f z‖ ≤ b
zs : Complex.abs (z - c) = r
⊢ (Complex.abs (z - c) ^ n)⁻¹ * ((Complex.abs (z - c))⁻¹ * ‖f z‖) ≤ (r ^ n)⁻¹ * r⁻¹ * b | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
z : ℂ
fb : ‖f z‖ ≤ b
zs : Complex.abs (z - c) = r
⊢ (r ^ n)⁻¹ * (r⁻¹ * ‖f z‖) ≤ (r ^ n)⁻¹ * r⁻¹ * 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 → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
z : ℂ
fb : ‖f z‖ ≤ b
zs : Complex.abs (z - c) = r
⊢ (Complex.abs (z - c) ^ n)⁻¹ * ((Complex.abs (z - c))⁻¹ * ‖f z‖) ≤ (r ^ n)⁻¹ * r⁻¹ * b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_bound | [328, 1] | [342, 74] | 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
f : ℂ → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
z : ℂ
fb : ‖f z‖ ≤ b
zs : Complex.abs (z - c) = r
⊢ (r ^ n)⁻¹ * (r⁻¹ * ‖f z‖) ≤ (r ^ n)⁻¹ * r⁻¹ * b | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
z : ℂ
fb : ‖f z‖ ≤ b
zs : Complex.abs (z - c) = r
⊢ r⁻¹ * r⁻¹ ^ n * ‖f z‖ ≤ r⁻¹ * r⁻¹ ^ n * 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 → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
z : ℂ
fb : ‖f z‖ ≤ b
zs : Complex.abs (z - c) = r
⊢ (r ^ n)⁻¹ * (r⁻¹ * ‖f z‖) ≤ (r ^ n)⁻¹ * r⁻¹ * b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_bound | [328, 1] | [342, 74] | 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 → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
z : ℂ
fb : ‖f z‖ ≤ b
zs : Complex.abs (z - c) = r
⊢ r⁻¹ * r⁻¹ ^ n * ‖f z‖ ≤ r⁻¹ * r⁻¹ ^ n * b | 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
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
z : ℂ
fb : ‖f z‖ ≤ b
zs : Complex.abs (z - c) = r
⊢ r⁻¹ * r⁻¹ ^ n * ‖f z‖ ≤ r⁻¹ * r⁻¹ ^ n * b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_bound | [328, 1] | [342, 74] | calc ‖∮ z in C(c, r), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖
_ ≤ 2 * π * r * (r⁻¹ ^ n * r⁻¹ * b) := isb
_ = 2 * π * b * r⁻¹ ^ n * (r * r⁻¹) := by ring
_ = 2 * π * b * r⁻¹ ^ n := by rw [mul_inv_cancel rp.ne']; simp | 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
f : ℂ → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
sb : ∀ z ∈ sphere c r, ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b
isb : ‖∮ (z : ℂ) in C(c, r), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ 2 * π * r * (r⁻¹ ^ n * r⁻¹ * b)
⊢ ‖∮ (z : ℂ) in C(c, r), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ 2 * π * b * r⁻¹ ^ n | no goals | 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
f : ℂ → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
sb : ∀ z ∈ sphere c r, ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b
isb : ‖∮ (z : ℂ) in C(c, r), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ 2 * π * r * (r⁻¹ ^ n * r⁻¹ * b)
⊢ ‖∮ (z : ℂ) in C(c, r), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ 2 * π * b * r⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_bound | [328, 1] | [342, 74] | 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
f : ℂ → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
sb : ∀ z ∈ sphere c r, ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b
isb : ‖∮ (z : ℂ) in C(c, r), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ 2 * π * r * (r⁻¹ ^ n * r⁻¹ * b)
⊢ 2 * π * r * (r⁻¹ ^ n * r⁻¹ * b) = 2 * π * b * r⁻¹ ^ n * (r * r⁻¹) | 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
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
sb : ∀ z ∈ sphere c r, ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b
isb : ‖∮ (z : ℂ) in C(c, r), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ 2 * π * r * (r⁻¹ ^ n * r⁻¹ * b)
⊢ 2 * π * r * (r⁻¹ ^ n * r⁻¹ * b) = 2 * π * b * r⁻¹ ^ n * (r * r⁻¹)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_bound | [328, 1] | [342, 74] | rw [mul_inv_cancel rp.ne'] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
sb : ∀ z ∈ sphere c r, ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b
isb : ‖∮ (z : ℂ) in C(c, r), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ 2 * π * r * (r⁻¹ ^ n * r⁻¹ * b)
⊢ 2 * π * b * r⁻¹ ^ n * (r * r⁻¹) = 2 * π * b * r⁻¹ ^ 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
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
sb : ∀ z ∈ sphere c r, ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b
isb : ‖∮ (z : ℂ) in C(c, r), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ 2 * π * r * (r⁻¹ ^ n * r⁻¹ * b)
⊢ 2 * π * b * r⁻¹ ^ n * 1 = 2 * π * b * r⁻¹ ^ 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
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
sb : ∀ z ∈ sphere c r, ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b
isb : ‖∮ (z : ℂ) in C(c, r), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ 2 * π * r * (r⁻¹ ^ n * r⁻¹ * b)
⊢ 2 * π * b * r⁻¹ ^ n * (r * r⁻¹) = 2 * π * b * r⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_bound | [328, 1] | [342, 74] | 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 → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
sb : ∀ z ∈ sphere c r, ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b
isb : ‖∮ (z : ℂ) in C(c, r), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ 2 * π * r * (r⁻¹ ^ n * r⁻¹ * b)
⊢ 2 * π * b * r⁻¹ ^ n * 1 = 2 * π * b * r⁻¹ ^ n | 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
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
sb : ∀ z ∈ sphere c r, ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b
isb : ‖∮ (z : ℂ) in C(c, r), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ 2 * π * r * (r⁻¹ ^ n * r⁻¹ * b)
⊢ 2 * π * b * r⁻¹ ^ n * 1 = 2 * π * b * r⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_bound | [328, 1] | [342, 74] | apply ContinuousOn.smul | case refine_1
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
sb : ∀ z ∈ sphere c r, ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b
⊢ ContinuousOn (fun z => (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z) (sphere c r) | case refine_1.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 → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
sb : ∀ z ∈ sphere c r, ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b
⊢ ContinuousOn (fun x => (x - c)⁻¹ ^ n) (sphere c r)
case refine_1.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 → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
sb : ∀ z ∈ sphere c r, ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b
⊢ ContinuousOn (fun x => (x - c)⁻¹ • f x) (sphere c r) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
sb : ∀ z ∈ sphere c r, ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b
⊢ ContinuousOn (fun z => (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z) (sphere c r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_bound | [328, 1] | [342, 74] | apply ContinuousOn.pow | case refine_1.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 → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
sb : ∀ z ∈ sphere c r, ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b
⊢ ContinuousOn (fun x => (x - c)⁻¹ ^ n) (sphere c r)
case refine_1.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 → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
sb : ∀ z ∈ sphere c r, ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b
⊢ ContinuousOn (fun x => (x - c)⁻¹ • f x) (sphere c r) | case refine_1.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 → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
sb : ∀ z ∈ sphere c r, ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b
⊢ ContinuousOn (fun x => (x - c)⁻¹) (sphere c r)
case refine_1.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 → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
sb : ∀ z ∈ sphere c r, ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b
⊢ ContinuousOn (fun x => (x - c)⁻¹ • f x) (sphere c r) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.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 → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
sb : ∀ z ∈ sphere c r, ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b
⊢ ContinuousOn (fun x => (x - c)⁻¹ ^ n) (sphere c r)
case refine_1.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 → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
sb : ∀ z ∈ sphere c r, ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b
⊢ ContinuousOn (fun x => (x - c)⁻¹ • f x) (sphere c r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_bound | [328, 1] | [342, 74] | exact ContinuousOn.inv_sphere rp | case refine_1.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 → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
sb : ∀ z ∈ sphere c r, ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b
⊢ ContinuousOn (fun x => (x - c)⁻¹) (sphere c r)
case refine_1.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 → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
sb : ∀ z ∈ sphere c r, ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b
⊢ ContinuousOn (fun x => (x - c)⁻¹ • f x) (sphere c r) | case refine_1.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 → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
sb : ∀ z ∈ sphere c r, ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b
⊢ ContinuousOn (fun x => (x - c)⁻¹ • f x) (sphere c r) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.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 → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
sb : ∀ z ∈ sphere c r, ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b
⊢ ContinuousOn (fun x => (x - c)⁻¹) (sphere c r)
case refine_1.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 → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
sb : ∀ z ∈ sphere c r, ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b
⊢ ContinuousOn (fun x => (x - c)⁻¹ • f x) (sphere c r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_bound | [328, 1] | [342, 74] | apply ContinuousOn.smul | case refine_1.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 → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
sb : ∀ z ∈ sphere c r, ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b
⊢ ContinuousOn (fun x => (x - c)⁻¹ • f x) (sphere c r) | case refine_1.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 → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
sb : ∀ z ∈ sphere c r, ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b
⊢ ContinuousOn (fun x => (x - c)⁻¹) (sphere c r)
case refine_1.hg.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 → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
sb : ∀ z ∈ sphere c r, ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b
⊢ ContinuousOn (fun x => f x) (sphere c r) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.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 → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
sb : ∀ z ∈ sphere c r, ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b
⊢ ContinuousOn (fun x => (x - c)⁻¹ • f x) (sphere c r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_bound | [328, 1] | [342, 74] | exact ContinuousOn.inv_sphere rp | case refine_1.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 → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
sb : ∀ z ∈ sphere c r, ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b
⊢ ContinuousOn (fun x => (x - c)⁻¹) (sphere c r)
case refine_1.hg.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 → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
sb : ∀ z ∈ sphere c r, ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b
⊢ ContinuousOn (fun x => f x) (sphere c r) | case refine_1.hg.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 → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
sb : ∀ z ∈ sphere c r, ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b
⊢ ContinuousOn (fun x => f x) (sphere c r) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.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 → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
sb : ∀ z ∈ sphere c r, ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b
⊢ ContinuousOn (fun x => (x - c)⁻¹) (sphere c r)
case refine_1.hg.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 → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
sb : ∀ z ∈ sphere c r, ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b
⊢ ContinuousOn (fun x => f x) (sphere c r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_bound | [328, 1] | [342, 74] | assumption | case refine_1.hg.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 → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
sb : ∀ z ∈ sphere c r, ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b
⊢ ContinuousOn (fun x => f x) (sphere c r) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hg.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 → E
b r : ℝ
c : ℂ
rp : r > 0
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
sb : ∀ z ∈ sphere c r, ‖(z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ r⁻¹ ^ n * r⁻¹ * b
⊢ ContinuousOn (fun x => f x) (sphere c r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_bound' | [345, 1] | [357, 55] | have a : abs (2*π*I : ℂ)⁻¹ = (2*π)⁻¹ := by
simp only [mul_inv_rev, Complex.inv_I, neg_mul, map_neg_eq_map, map_mul, Complex.abs_I,
map_inv₀, Complex.abs_ofReal, Complex.abs_two, one_mul, mul_eq_mul_right_iff, inv_inj,
abs_eq_self, inv_eq_zero, OfNat.ofNat_ne_zero, or_false]
exact Real.pi_pos.le | E : Type
inst✝² : NormedAddCommGroup E
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
r : ℝ
c : ℂ
rp : r > 0
b : ℝ
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
⊢ ‖(2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, r), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ b * r⁻¹ ^ 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
r : ℝ
c : ℂ
rp : r > 0
b : ℝ
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
a : Complex.abs (2 * ↑π * I)⁻¹ = (2 * π)⁻¹
⊢ ‖(2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, r), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ b * r⁻¹ ^ 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
r : ℝ
c : ℂ
rp : r > 0
b : ℝ
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
⊢ ‖(2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, r), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ b * r⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_bound' | [345, 1] | [357, 55] | rw [norm_smul, Complex.norm_eq_abs, 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 → E
r : ℝ
c : ℂ
rp : r > 0
b : ℝ
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
a : Complex.abs (2 * ↑π * I)⁻¹ = (2 * π)⁻¹
⊢ ‖(2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, r), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ b * r⁻¹ ^ 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
r : ℝ
c : ℂ
rp : r > 0
b : ℝ
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
a : Complex.abs (2 * ↑π * I)⁻¹ = (2 * π)⁻¹
⊢ (2 * π)⁻¹ * ‖∮ (z : ℂ) in C(c, r), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ b * r⁻¹ ^ 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
r : ℝ
c : ℂ
rp : r > 0
b : ℝ
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
a : Complex.abs (2 * ↑π * I)⁻¹ = (2 * π)⁻¹
⊢ ‖(2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, r), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ b * r⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_bound' | [345, 1] | [357, 55] | calc (2*π)⁻¹ * ‖∮ z in C(c, r), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖
_ ≤ (2*π)⁻¹ * (2*π * b * r⁻¹ ^ n) := by bound [cauchy1_bound rp fc bh n]
_ = (2*π)⁻¹ * (2*π) * b * r⁻¹ ^ n := by ring
_ = b * r⁻¹ ^ n := by field_simp [Real.pi_pos.ne'] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
r : ℝ
c : ℂ
rp : r > 0
b : ℝ
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
a : Complex.abs (2 * ↑π * I)⁻¹ = (2 * π)⁻¹
⊢ (2 * π)⁻¹ * ‖∮ (z : ℂ) in C(c, r), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ b * r⁻¹ ^ n | 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
r : ℝ
c : ℂ
rp : r > 0
b : ℝ
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
a : Complex.abs (2 * ↑π * I)⁻¹ = (2 * π)⁻¹
⊢ (2 * π)⁻¹ * ‖∮ (z : ℂ) in C(c, r), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ b * r⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_bound' | [345, 1] | [357, 55] | simp only [mul_inv_rev, Complex.inv_I, neg_mul, map_neg_eq_map, map_mul, Complex.abs_I,
map_inv₀, Complex.abs_ofReal, Complex.abs_two, one_mul, mul_eq_mul_right_iff, inv_inj,
abs_eq_self, inv_eq_zero, OfNat.ofNat_ne_zero, or_false] | E : Type
inst✝² : NormedAddCommGroup E
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
r : ℝ
c : ℂ
rp : r > 0
b : ℝ
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
⊢ Complex.abs (2 * ↑π * I)⁻¹ = (2 * π)⁻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 → E
r : ℝ
c : ℂ
rp : r > 0
b : ℝ
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
⊢ 0 ≤ π | 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
r : ℝ
c : ℂ
rp : r > 0
b : ℝ
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
⊢ Complex.abs (2 * ↑π * I)⁻¹ = (2 * π)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_bound' | [345, 1] | [357, 55] | exact Real.pi_pos.le | E : Type
inst✝² : NormedAddCommGroup E
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
r : ℝ
c : ℂ
rp : r > 0
b : ℝ
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
⊢ 0 ≤ π | 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
r : ℝ
c : ℂ
rp : r > 0
b : ℝ
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
⊢ 0 ≤ π
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_bound' | [345, 1] | [357, 55] | bound [cauchy1_bound rp fc bh 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
r : ℝ
c : ℂ
rp : r > 0
b : ℝ
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
a : Complex.abs (2 * ↑π * I)⁻¹ = (2 * π)⁻¹
⊢ (2 * π)⁻¹ * ‖∮ (z : ℂ) in C(c, r), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ (2 * π)⁻¹ * (2 * π * b * r⁻¹ ^ n) | 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
r : ℝ
c : ℂ
rp : r > 0
b : ℝ
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
a : Complex.abs (2 * ↑π * I)⁻¹ = (2 * π)⁻¹
⊢ (2 * π)⁻¹ * ‖∮ (z : ℂ) in C(c, r), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ≤ (2 * π)⁻¹ * (2 * π * b * r⁻¹ ^ n)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_bound' | [345, 1] | [357, 55] | 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
f : ℂ → E
r : ℝ
c : ℂ
rp : r > 0
b : ℝ
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
a : Complex.abs (2 * ↑π * I)⁻¹ = (2 * π)⁻¹
⊢ (2 * π)⁻¹ * (2 * π * b * r⁻¹ ^ n) = (2 * π)⁻¹ * (2 * π) * b * r⁻¹ ^ n | 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
r : ℝ
c : ℂ
rp : r > 0
b : ℝ
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
a : Complex.abs (2 * ↑π * I)⁻¹ = (2 * π)⁻¹
⊢ (2 * π)⁻¹ * (2 * π * b * r⁻¹ ^ n) = (2 * π)⁻¹ * (2 * π) * b * r⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_bound' | [345, 1] | [357, 55] | field_simp [Real.pi_pos.ne'] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b✝ : ℝ
f : ℂ → E
r : ℝ
c : ℂ
rp : r > 0
b : ℝ
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
a : Complex.abs (2 * ↑π * I)⁻¹ = (2 * π)⁻¹
⊢ (2 * π)⁻¹ * (2 * π) * b * r⁻¹ ^ n = b * r⁻¹ ^ n | 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
r : ℝ
c : ℂ
rp : r > 0
b : ℝ
fc : ContinuousOn f (sphere c r)
bh : ∀ z ∈ sphere c r, ‖f z‖ ≤ b
n : ℕ
a : Complex.abs (2 * ↑π * I)⁻¹ = (2 * π)⁻¹
⊢ (2 * π)⁻¹ * (2 * π) * b * r⁻¹ ^ n = b * r⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0_bound | [360, 1] | [387, 88] | have isb := cauchy1_bound h.rp
(ContinuousOn.mono (h.fc1 (mem_sphere_closed z0s)) Metric.sphere_subset_closedBall)
(fun z1 z1s ↦ h.fb z0s z1s) 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
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
n : ℕ
z0 : ℂ
z0s : z0 ∈ sphere c0 r
⊢ ‖w1 ^ n • (2 * ↑π * I)⁻¹ • (z0 - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z0, z1)‖ ≤
(r - Complex.abs w0)⁻¹ * b * (Complex.abs w1 / r) ^ 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
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
n : ℕ
z0 : ℂ
z0s : z0 ∈ sphere c0 r
isb : ‖∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z)‖ ≤ 2 * π * b * r⁻¹ ^ n
⊢ ‖w1 ^ n • (2 * ↑π * I)⁻¹ • (z0 - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z0, z1)‖ ≤
(r - Complex.abs w0)⁻¹ * b * (Complex.abs w1 / r) ^ 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
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
n : ℕ
z0 : ℂ
z0s : z0 ∈ sphere c0 r
⊢ ‖w1 ^ n • (2 * ↑π * I)⁻¹ • (z0 - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z0, z1)‖ ≤
(r - Complex.abs w0)⁻¹ * b * (Complex.abs w1 / r) ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0_bound | [360, 1] | [387, 88] | simp only [mem_sphere_iff_norm, Complex.norm_eq_abs, Metric.mem_ball, dist_zero_right] at z0s 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 0 r
n : ℕ
z0 : ℂ
z0s : z0 ∈ sphere c0 r
isb : ‖∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z)‖ ≤ 2 * π * b * r⁻¹ ^ n
⊢ ‖w1 ^ n • (2 * ↑π * I)⁻¹ • (z0 - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z0, z1)‖ ≤
(r - Complex.abs w0)⁻¹ * b * (Complex.abs w1 / r) ^ 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
h : Separate f c0 c1 r b s
n : ℕ
z0 : ℂ
isb : ‖∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z)‖ ≤ 2 * π * b * r⁻¹ ^ n
z0s : Complex.abs (z0 - c0) = r
w0m : Complex.abs w0 < r
⊢ ‖w1 ^ n • (2 * ↑π * I)⁻¹ • (z0 - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z0, z1)‖ ≤
(r - Complex.abs w0)⁻¹ * b * (Complex.abs w1 / r) ^ 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
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
n : ℕ
z0 : ℂ
z0s : z0 ∈ sphere c0 r
isb : ‖∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z)‖ ≤ 2 * π * b * r⁻¹ ^ n
⊢ ‖w1 ^ n • (2 * ↑π * I)⁻¹ • (z0 - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z0, z1)‖ ≤
(r - Complex.abs w0)⁻¹ * b * (Complex.abs w1 / r) ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0_bound | [360, 1] | [387, 88] | have zcw : abs (z0 - (c0 + w0)) ≥ r - abs w0 := by
calc abs (z0 - (c0 + w0))
_ = abs (z0 - c0 + -w0) := by ring_nf
_ ≥ abs (z0 - c0) - abs (-w0) := by bound
_ = r - abs w0 := by rw [z0s]; simp only [map_neg_eq_map] | E : Type
inst✝² : NormedAddCommGroup E
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
n : ℕ
z0 : ℂ
isb : ‖∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z)‖ ≤ 2 * π * b * r⁻¹ ^ n
z0s : Complex.abs (z0 - c0) = r
w0m : Complex.abs w0 < r
⊢ ‖w1 ^ n • (2 * ↑π * I)⁻¹ • (z0 - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z0, z1)‖ ≤
(r - Complex.abs w0)⁻¹ * b * (Complex.abs w1 / r) ^ 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
h : Separate f c0 c1 r b s
n : ℕ
z0 : ℂ
isb : ‖∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z)‖ ≤ 2 * π * b * r⁻¹ ^ n
z0s : Complex.abs (z0 - c0) = r
w0m : Complex.abs w0 < r
zcw : Complex.abs (z0 - (c0 + w0)) ≥ r - Complex.abs w0
⊢ ‖w1 ^ n • (2 * ↑π * I)⁻¹ • (z0 - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z0, z1)‖ ≤
(r - Complex.abs w0)⁻¹ * b * (Complex.abs w1 / r) ^ 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
h : Separate f c0 c1 r b s
n : ℕ
z0 : ℂ
isb : ‖∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z)‖ ≤ 2 * π * b * r⁻¹ ^ n
z0s : Complex.abs (z0 - c0) = r
w0m : Complex.abs w0 < r
⊢ ‖w1 ^ n • (2 * ↑π * I)⁻¹ • (z0 - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z0, z1)‖ ≤
(r - Complex.abs w0)⁻¹ * b * (Complex.abs w1 / r) ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0_bound | [360, 1] | [387, 88] | have zcw' : (abs (z0 - (c0 + w0)))⁻¹ ≤ (r - abs w0)⁻¹ := by 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
h : Separate f c0 c1 r b s
n : ℕ
z0 : ℂ
isb : ‖∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z)‖ ≤ 2 * π * b * r⁻¹ ^ n
z0s : Complex.abs (z0 - c0) = r
w0m : Complex.abs w0 < r
zcw : Complex.abs (z0 - (c0 + w0)) ≥ r - Complex.abs w0
⊢ ‖w1 ^ n • (2 * ↑π * I)⁻¹ • (z0 - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z0, z1)‖ ≤
(r - Complex.abs w0)⁻¹ * b * (Complex.abs w1 / r) ^ 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
h : Separate f c0 c1 r b s
n : ℕ
z0 : ℂ
isb : ‖∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z)‖ ≤ 2 * π * b * r⁻¹ ^ n
z0s : Complex.abs (z0 - c0) = r
w0m : Complex.abs w0 < r
zcw : Complex.abs (z0 - (c0 + w0)) ≥ r - Complex.abs w0
zcw' : (Complex.abs (z0 - (c0 + w0)))⁻¹ ≤ (r - Complex.abs w0)⁻¹
⊢ ‖w1 ^ n • (2 * ↑π * I)⁻¹ • (z0 - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z0, z1)‖ ≤
(r - Complex.abs w0)⁻¹ * b * (Complex.abs w1 / r) ^ 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
h : Separate f c0 c1 r b s
n : ℕ
z0 : ℂ
isb : ‖∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z)‖ ≤ 2 * π * b * r⁻¹ ^ n
z0s : Complex.abs (z0 - c0) = r
w0m : Complex.abs w0 < r
zcw : Complex.abs (z0 - (c0 + w0)) ≥ r - Complex.abs w0
⊢ ‖w1 ^ n • (2 * ↑π * I)⁻¹ • (z0 - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z0, z1)‖ ≤
(r - Complex.abs w0)⁻¹ * b * (Complex.abs w1 / r) ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0_bound | [360, 1] | [387, 88] | have a : (abs (2 * π * I : ℂ))⁻¹ = (2 * π)⁻¹ := by
simp only [map_mul, Complex.abs_two, Complex.abs_ofReal, Complex.abs_I, mul_one, mul_inv_rev,
mul_eq_mul_right_iff, inv_inj, abs_eq_self, inv_eq_zero, OfNat.ofNat_ne_zero, or_false]
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
h : Separate f c0 c1 r b s
n : ℕ
z0 : ℂ
isb : ‖∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z)‖ ≤ 2 * π * b * r⁻¹ ^ n
z0s : Complex.abs (z0 - c0) = r
w0m : Complex.abs w0 < r
zcw : Complex.abs (z0 - (c0 + w0)) ≥ r - Complex.abs w0
zcw' : (Complex.abs (z0 - (c0 + w0)))⁻¹ ≤ (r - Complex.abs w0)⁻¹
⊢ ‖w1 ^ n • (2 * ↑π * I)⁻¹ • (z0 - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z0, z1)‖ ≤
(r - Complex.abs w0)⁻¹ * b * (Complex.abs w1 / r) ^ 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
h : Separate f c0 c1 r b s
n : ℕ
z0 : ℂ
isb : ‖∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z)‖ ≤ 2 * π * b * r⁻¹ ^ n
z0s : Complex.abs (z0 - c0) = r
w0m : Complex.abs w0 < r
zcw : Complex.abs (z0 - (c0 + w0)) ≥ r - Complex.abs w0
zcw' : (Complex.abs (z0 - (c0 + w0)))⁻¹ ≤ (r - Complex.abs w0)⁻¹
a : (Complex.abs (2 * ↑π * I))⁻¹ = (2 * π)⁻¹
⊢ ‖w1 ^ n • (2 * ↑π * I)⁻¹ • (z0 - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z0, z1)‖ ≤
(r - Complex.abs w0)⁻¹ * b * (Complex.abs w1 / r) ^ 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
h : Separate f c0 c1 r b s
n : ℕ
z0 : ℂ
isb : ‖∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z)‖ ≤ 2 * π * b * r⁻¹ ^ n
z0s : Complex.abs (z0 - c0) = r
w0m : Complex.abs w0 < r
zcw : Complex.abs (z0 - (c0 + w0)) ≥ r - Complex.abs w0
zcw' : (Complex.abs (z0 - (c0 + w0)))⁻¹ ≤ (r - Complex.abs w0)⁻¹
⊢ ‖w1 ^ n • (2 * ↑π * I)⁻¹ • (z0 - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z0, z1)‖ ≤
(r - Complex.abs w0)⁻¹ * b * (Complex.abs w1 / r) ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0_bound | [360, 1] | [387, 88] | rw [norm_smul, norm_smul, norm_smul, Complex.norm_eq_abs, Complex.norm_eq_abs,
Complex.norm_eq_abs, Complex.abs.map_pow, map_inv₀, map_inv₀, 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
n : ℕ
z0 : ℂ
isb : ‖∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z)‖ ≤ 2 * π * b * r⁻¹ ^ n
z0s : Complex.abs (z0 - c0) = r
w0m : Complex.abs w0 < r
zcw : Complex.abs (z0 - (c0 + w0)) ≥ r - Complex.abs w0
zcw' : (Complex.abs (z0 - (c0 + w0)))⁻¹ ≤ (r - Complex.abs w0)⁻¹
a : (Complex.abs (2 * ↑π * I))⁻¹ = (2 * π)⁻¹
⊢ ‖w1 ^ n • (2 * ↑π * I)⁻¹ • (z0 - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z0, z1)‖ ≤
(r - Complex.abs w0)⁻¹ * b * (Complex.abs w1 / r) ^ 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
h : Separate f c0 c1 r b s
n : ℕ
z0 : ℂ
isb : ‖∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z)‖ ≤ 2 * π * b * r⁻¹ ^ n
z0s : Complex.abs (z0 - c0) = r
w0m : Complex.abs w0 < r
zcw : Complex.abs (z0 - (c0 + w0)) ≥ r - Complex.abs w0
zcw' : (Complex.abs (z0 - (c0 + w0)))⁻¹ ≤ (r - Complex.abs w0)⁻¹
a : (Complex.abs (2 * ↑π * I))⁻¹ = (2 * π)⁻¹
⊢ Complex.abs w1 ^ n *
((2 * π)⁻¹ *
((Complex.abs (z0 - (c0 + w0)))⁻¹ * ‖∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z0, z1)‖)) ≤
(r - Complex.abs w0)⁻¹ * b * (Complex.abs w1 / r) ^ 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
h : Separate f c0 c1 r b s
n : ℕ
z0 : ℂ
isb : ‖∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z)‖ ≤ 2 * π * b * r⁻¹ ^ n
z0s : Complex.abs (z0 - c0) = r
w0m : Complex.abs w0 < r
zcw : Complex.abs (z0 - (c0 + w0)) ≥ r - Complex.abs w0
zcw' : (Complex.abs (z0 - (c0 + w0)))⁻¹ ≤ (r - Complex.abs w0)⁻¹
a : (Complex.abs (2 * ↑π * I))⁻¹ = (2 * π)⁻¹
⊢ ‖w1 ^ n • (2 * ↑π * I)⁻¹ • (z0 - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z0, z1)‖ ≤
(r - Complex.abs w0)⁻¹ * b * (Complex.abs w1 / r) ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0_bound | [360, 1] | [387, 88] | calc abs w1 ^ n * ((2*π)⁻¹ * ((abs (z0 - (c0 + w0)))⁻¹ *
‖∮ z1 in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z0, z1)‖))
_ ≤ abs w1 ^ n * ((2 * π)⁻¹ * ((abs (z0 - (c0 + w0)))⁻¹ * (2 * π * b * r⁻¹ ^ n))) := by bound
_ ≤ abs w1 ^ n * ((2 * π)⁻¹ * ((r - abs w0)⁻¹ * (2 * π * b * r⁻¹ ^ n))) := by bound
_ = 2 * π * (2 * π)⁻¹ * (r - abs w0)⁻¹ * b * (abs w1 ^ n * r⁻¹ ^ n) := by ring
_ = (r - abs w0)⁻¹ * b * (abs w1 / r) ^ n := by
rw [mul_inv_cancel Real.two_pi_pos.ne', ← mul_pow, ← div_eq_mul_inv _ r, one_mul] | E : Type
inst✝² : NormedAddCommGroup E
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
n : ℕ
z0 : ℂ
isb : ‖∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z)‖ ≤ 2 * π * b * r⁻¹ ^ n
z0s : Complex.abs (z0 - c0) = r
w0m : Complex.abs w0 < r
zcw : Complex.abs (z0 - (c0 + w0)) ≥ r - Complex.abs w0
zcw' : (Complex.abs (z0 - (c0 + w0)))⁻¹ ≤ (r - Complex.abs w0)⁻¹
a : (Complex.abs (2 * ↑π * I))⁻¹ = (2 * π)⁻¹
⊢ Complex.abs w1 ^ n *
((2 * π)⁻¹ *
((Complex.abs (z0 - (c0 + w0)))⁻¹ * ‖∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z0, z1)‖)) ≤
(r - Complex.abs w0)⁻¹ * b * (Complex.abs w1 / r) ^ n | 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
n : ℕ
z0 : ℂ
isb : ‖∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z)‖ ≤ 2 * π * b * r⁻¹ ^ n
z0s : Complex.abs (z0 - c0) = r
w0m : Complex.abs w0 < r
zcw : Complex.abs (z0 - (c0 + w0)) ≥ r - Complex.abs w0
zcw' : (Complex.abs (z0 - (c0 + w0)))⁻¹ ≤ (r - Complex.abs w0)⁻¹
a : (Complex.abs (2 * ↑π * I))⁻¹ = (2 * π)⁻¹
⊢ Complex.abs w1 ^ n *
((2 * π)⁻¹ *
((Complex.abs (z0 - (c0 + w0)))⁻¹ * ‖∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z0, z1)‖)) ≤
(r - Complex.abs w0)⁻¹ * b * (Complex.abs w1 / r) ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0_bound | [360, 1] | [387, 88] | calc abs (z0 - (c0 + w0))
_ = abs (z0 - c0 + -w0) := by ring_nf
_ ≥ abs (z0 - c0) - abs (-w0) := by bound
_ = r - abs w0 := by rw [z0s]; simp only [map_neg_eq_map] | E : Type
inst✝² : NormedAddCommGroup E
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
n : ℕ
z0 : ℂ
isb : ‖∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z)‖ ≤ 2 * π * b * r⁻¹ ^ n
z0s : Complex.abs (z0 - c0) = r
w0m : Complex.abs w0 < r
⊢ Complex.abs (z0 - (c0 + w0)) ≥ r - Complex.abs w0 | 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
n : ℕ
z0 : ℂ
isb : ‖∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z)‖ ≤ 2 * π * b * r⁻¹ ^ n
z0s : Complex.abs (z0 - c0) = r
w0m : Complex.abs w0 < r
⊢ Complex.abs (z0 - (c0 + w0)) ≥ r - Complex.abs w0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0_bound | [360, 1] | [387, 88] | 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
h : Separate f c0 c1 r b s
n : ℕ
z0 : ℂ
isb : ‖∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z)‖ ≤ 2 * π * b * r⁻¹ ^ n
z0s : Complex.abs (z0 - c0) = r
w0m : Complex.abs w0 < r
⊢ Complex.abs (z0 - (c0 + w0)) = Complex.abs (z0 - c0 + -w0) | 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
n : ℕ
z0 : ℂ
isb : ‖∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z)‖ ≤ 2 * π * b * r⁻¹ ^ n
z0s : Complex.abs (z0 - c0) = r
w0m : Complex.abs w0 < r
⊢ Complex.abs (z0 - (c0 + w0)) = Complex.abs (z0 - c0 + -w0)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0_bound | [360, 1] | [387, 88] | 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
h : Separate f c0 c1 r b s
n : ℕ
z0 : ℂ
isb : ‖∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z)‖ ≤ 2 * π * b * r⁻¹ ^ n
z0s : Complex.abs (z0 - c0) = r
w0m : Complex.abs w0 < r
⊢ Complex.abs (z0 - c0 + -w0) ≥ Complex.abs (z0 - c0) - Complex.abs (-w0) | 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
n : ℕ
z0 : ℂ
isb : ‖∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z)‖ ≤ 2 * π * b * r⁻¹ ^ n
z0s : Complex.abs (z0 - c0) = r
w0m : Complex.abs w0 < r
⊢ Complex.abs (z0 - c0 + -w0) ≥ Complex.abs (z0 - c0) - Complex.abs (-w0)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0_bound | [360, 1] | [387, 88] | rw [z0s] | E : Type
inst✝² : NormedAddCommGroup E
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
n : ℕ
z0 : ℂ
isb : ‖∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z)‖ ≤ 2 * π * b * r⁻¹ ^ n
z0s : Complex.abs (z0 - c0) = r
w0m : Complex.abs w0 < r
⊢ Complex.abs (z0 - c0) - Complex.abs (-w0) = r - Complex.abs w0 | E : Type
inst✝² : NormedAddCommGroup E
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
n : ℕ
z0 : ℂ
isb : ‖∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z)‖ ≤ 2 * π * b * r⁻¹ ^ n
z0s : Complex.abs (z0 - c0) = r
w0m : Complex.abs w0 < r
⊢ r - Complex.abs (-w0) = r - Complex.abs w0 | 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
n : ℕ
z0 : ℂ
isb : ‖∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z)‖ ≤ 2 * π * b * r⁻¹ ^ n
z0s : Complex.abs (z0 - c0) = r
w0m : Complex.abs w0 < r
⊢ Complex.abs (z0 - c0) - Complex.abs (-w0) = r - Complex.abs w0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0_bound | [360, 1] | [387, 88] | simp only [map_neg_eq_map] | E : Type
inst✝² : NormedAddCommGroup E
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
n : ℕ
z0 : ℂ
isb : ‖∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z)‖ ≤ 2 * π * b * r⁻¹ ^ n
z0s : Complex.abs (z0 - c0) = r
w0m : Complex.abs w0 < r
⊢ r - Complex.abs (-w0) = r - Complex.abs w0 | 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
n : ℕ
z0 : ℂ
isb : ‖∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z)‖ ≤ 2 * π * b * r⁻¹ ^ n
z0s : Complex.abs (z0 - c0) = r
w0m : Complex.abs w0 < r
⊢ r - Complex.abs (-w0) = r - Complex.abs w0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0_bound | [360, 1] | [387, 88] | 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
h : Separate f c0 c1 r b s
n : ℕ
z0 : ℂ
isb : ‖∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z)‖ ≤ 2 * π * b * r⁻¹ ^ n
z0s : Complex.abs (z0 - c0) = r
w0m : Complex.abs w0 < r
zcw : Complex.abs (z0 - (c0 + w0)) ≥ r - Complex.abs w0
⊢ (Complex.abs (z0 - (c0 + w0)))⁻¹ ≤ (r - Complex.abs w0)⁻¹ | 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
n : ℕ
z0 : ℂ
isb : ‖∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z)‖ ≤ 2 * π * b * r⁻¹ ^ n
z0s : Complex.abs (z0 - c0) = r
w0m : Complex.abs w0 < r
zcw : Complex.abs (z0 - (c0 + w0)) ≥ r - Complex.abs w0
⊢ (Complex.abs (z0 - (c0 + w0)))⁻¹ ≤ (r - Complex.abs w0)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0_bound | [360, 1] | [387, 88] | simp only [map_mul, Complex.abs_two, Complex.abs_ofReal, Complex.abs_I, mul_one, mul_inv_rev,
mul_eq_mul_right_iff, inv_inj, abs_eq_self, inv_eq_zero, OfNat.ofNat_ne_zero, or_false] | E : Type
inst✝² : NormedAddCommGroup E
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
n : ℕ
z0 : ℂ
isb : ‖∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z)‖ ≤ 2 * π * b * r⁻¹ ^ n
z0s : Complex.abs (z0 - c0) = r
w0m : Complex.abs w0 < r
zcw : Complex.abs (z0 - (c0 + w0)) ≥ r - Complex.abs w0
zcw' : (Complex.abs (z0 - (c0 + w0)))⁻¹ ≤ (r - Complex.abs w0)⁻¹
⊢ (Complex.abs (2 * ↑π * I))⁻¹ = (2 * π)⁻¹ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
n : ℕ
z0 : ℂ
isb : ‖∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z)‖ ≤ 2 * π * b * r⁻¹ ^ n
z0s : Complex.abs (z0 - c0) = r
w0m : Complex.abs w0 < r
zcw : Complex.abs (z0 - (c0 + w0)) ≥ r - Complex.abs w0
zcw' : (Complex.abs (z0 - (c0 + w0)))⁻¹ ≤ (r - Complex.abs w0)⁻¹
⊢ 0 ≤ π | 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
n : ℕ
z0 : ℂ
isb : ‖∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z)‖ ≤ 2 * π * b * r⁻¹ ^ n
z0s : Complex.abs (z0 - c0) = r
w0m : Complex.abs w0 < r
zcw : Complex.abs (z0 - (c0 + w0)) ≥ r - Complex.abs w0
zcw' : (Complex.abs (z0 - (c0 + w0)))⁻¹ ≤ (r - Complex.abs w0)⁻¹
⊢ (Complex.abs (2 * ↑π * I))⁻¹ = (2 * π)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0_bound | [360, 1] | [387, 88] | 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
h : Separate f c0 c1 r b s
n : ℕ
z0 : ℂ
isb : ‖∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z)‖ ≤ 2 * π * b * r⁻¹ ^ n
z0s : Complex.abs (z0 - c0) = r
w0m : Complex.abs w0 < r
zcw : Complex.abs (z0 - (c0 + w0)) ≥ r - Complex.abs w0
zcw' : (Complex.abs (z0 - (c0 + w0)))⁻¹ ≤ (r - Complex.abs w0)⁻¹
⊢ 0 ≤ π | 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
n : ℕ
z0 : ℂ
isb : ‖∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z)‖ ≤ 2 * π * b * r⁻¹ ^ n
z0s : Complex.abs (z0 - c0) = r
w0m : Complex.abs w0 < r
zcw : Complex.abs (z0 - (c0 + w0)) ≥ r - Complex.abs w0
zcw' : (Complex.abs (z0 - (c0 + w0)))⁻¹ ≤ (r - Complex.abs w0)⁻¹
⊢ 0 ≤ π
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0_bound | [360, 1] | [387, 88] | 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
h : Separate f c0 c1 r b s
n : ℕ
z0 : ℂ
isb : ‖∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z)‖ ≤ 2 * π * b * r⁻¹ ^ n
z0s : Complex.abs (z0 - c0) = r
w0m : Complex.abs w0 < r
zcw : Complex.abs (z0 - (c0 + w0)) ≥ r - Complex.abs w0
zcw' : (Complex.abs (z0 - (c0 + w0)))⁻¹ ≤ (r - Complex.abs w0)⁻¹
a : (Complex.abs (2 * ↑π * I))⁻¹ = (2 * π)⁻¹
⊢ Complex.abs w1 ^ n *
((2 * π)⁻¹ *
((Complex.abs (z0 - (c0 + w0)))⁻¹ * ‖∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z0, z1)‖)) ≤
Complex.abs w1 ^ n * ((2 * π)⁻¹ * ((Complex.abs (z0 - (c0 + w0)))⁻¹ * (2 * π * b * r⁻¹ ^ n))) | 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
n : ℕ
z0 : ℂ
isb : ‖∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z)‖ ≤ 2 * π * b * r⁻¹ ^ n
z0s : Complex.abs (z0 - c0) = r
w0m : Complex.abs w0 < r
zcw : Complex.abs (z0 - (c0 + w0)) ≥ r - Complex.abs w0
zcw' : (Complex.abs (z0 - (c0 + w0)))⁻¹ ≤ (r - Complex.abs w0)⁻¹
a : (Complex.abs (2 * ↑π * I))⁻¹ = (2 * π)⁻¹
⊢ Complex.abs w1 ^ n *
((2 * π)⁻¹ *
((Complex.abs (z0 - (c0 + w0)))⁻¹ * ‖∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z0, z1)‖)) ≤
Complex.abs w1 ^ n * ((2 * π)⁻¹ * ((Complex.abs (z0 - (c0 + w0)))⁻¹ * (2 * π * b * r⁻¹ ^ n)))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0_bound | [360, 1] | [387, 88] | 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
h : Separate f c0 c1 r b s
n : ℕ
z0 : ℂ
isb : ‖∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z)‖ ≤ 2 * π * b * r⁻¹ ^ n
z0s : Complex.abs (z0 - c0) = r
w0m : Complex.abs w0 < r
zcw : Complex.abs (z0 - (c0 + w0)) ≥ r - Complex.abs w0
zcw' : (Complex.abs (z0 - (c0 + w0)))⁻¹ ≤ (r - Complex.abs w0)⁻¹
a : (Complex.abs (2 * ↑π * I))⁻¹ = (2 * π)⁻¹
⊢ Complex.abs w1 ^ n * ((2 * π)⁻¹ * ((Complex.abs (z0 - (c0 + w0)))⁻¹ * (2 * π * b * r⁻¹ ^ n))) ≤
Complex.abs w1 ^ n * ((2 * π)⁻¹ * ((r - Complex.abs w0)⁻¹ * (2 * π * b * r⁻¹ ^ n))) | 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
n : ℕ
z0 : ℂ
isb : ‖∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z)‖ ≤ 2 * π * b * r⁻¹ ^ n
z0s : Complex.abs (z0 - c0) = r
w0m : Complex.abs w0 < r
zcw : Complex.abs (z0 - (c0 + w0)) ≥ r - Complex.abs w0
zcw' : (Complex.abs (z0 - (c0 + w0)))⁻¹ ≤ (r - Complex.abs w0)⁻¹
a : (Complex.abs (2 * ↑π * I))⁻¹ = (2 * π)⁻¹
⊢ Complex.abs w1 ^ n * ((2 * π)⁻¹ * ((Complex.abs (z0 - (c0 + w0)))⁻¹ * (2 * π * b * r⁻¹ ^ n))) ≤
Complex.abs w1 ^ n * ((2 * π)⁻¹ * ((r - Complex.abs w0)⁻¹ * (2 * π * b * r⁻¹ ^ n)))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0_bound | [360, 1] | [387, 88] | 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
h : Separate f c0 c1 r b s
n : ℕ
z0 : ℂ
isb : ‖∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z)‖ ≤ 2 * π * b * r⁻¹ ^ n
z0s : Complex.abs (z0 - c0) = r
w0m : Complex.abs w0 < r
zcw : Complex.abs (z0 - (c0 + w0)) ≥ r - Complex.abs w0
zcw' : (Complex.abs (z0 - (c0 + w0)))⁻¹ ≤ (r - Complex.abs w0)⁻¹
a : (Complex.abs (2 * ↑π * I))⁻¹ = (2 * π)⁻¹
⊢ Complex.abs w1 ^ n * ((2 * π)⁻¹ * ((r - Complex.abs w0)⁻¹ * (2 * π * b * r⁻¹ ^ n))) =
2 * π * (2 * π)⁻¹ * (r - Complex.abs w0)⁻¹ * b * (Complex.abs w1 ^ n * r⁻¹ ^ n) | 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
n : ℕ
z0 : ℂ
isb : ‖∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z)‖ ≤ 2 * π * b * r⁻¹ ^ n
z0s : Complex.abs (z0 - c0) = r
w0m : Complex.abs w0 < r
zcw : Complex.abs (z0 - (c0 + w0)) ≥ r - Complex.abs w0
zcw' : (Complex.abs (z0 - (c0 + w0)))⁻¹ ≤ (r - Complex.abs w0)⁻¹
a : (Complex.abs (2 * ↑π * I))⁻¹ = (2 * π)⁻¹
⊢ Complex.abs w1 ^ n * ((2 * π)⁻¹ * ((r - Complex.abs w0)⁻¹ * (2 * π * b * r⁻¹ ^ n))) =
2 * π * (2 * π)⁻¹ * (r - Complex.abs w0)⁻¹ * b * (Complex.abs w1 ^ n * r⁻¹ ^ n)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0_bound | [360, 1] | [387, 88] | rw [mul_inv_cancel Real.two_pi_pos.ne', ← mul_pow, ← div_eq_mul_inv _ r, one_mul] | E : Type
inst✝² : NormedAddCommGroup E
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
n : ℕ
z0 : ℂ
isb : ‖∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z)‖ ≤ 2 * π * b * r⁻¹ ^ n
z0s : Complex.abs (z0 - c0) = r
w0m : Complex.abs w0 < r
zcw : Complex.abs (z0 - (c0 + w0)) ≥ r - Complex.abs w0
zcw' : (Complex.abs (z0 - (c0 + w0)))⁻¹ ≤ (r - Complex.abs w0)⁻¹
a : (Complex.abs (2 * ↑π * I))⁻¹ = (2 * π)⁻¹
⊢ 2 * π * (2 * π)⁻¹ * (r - Complex.abs w0)⁻¹ * b * (Complex.abs w1 ^ n * r⁻¹ ^ n) =
(r - Complex.abs w0)⁻¹ * b * (Complex.abs w1 / r) ^ n | 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
n : ℕ
z0 : ℂ
isb : ‖∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z)‖ ≤ 2 * π * b * r⁻¹ ^ n
z0s : Complex.abs (z0 - c0) = r
w0m : Complex.abs w0 < r
zcw : Complex.abs (z0 - (c0 + w0)) ≥ r - Complex.abs w0
zcw' : (Complex.abs (z0 - (c0 + w0)))⁻¹ ≤ (r - Complex.abs w0)⁻¹
a : (Complex.abs (2 * ↑π * I))⁻¹ = (2 * π)⁻¹
⊢ 2 * π * (2 * π)⁻¹ * (r - Complex.abs w0)⁻¹ * b * (Complex.abs w1 ^ n * r⁻¹ ^ n) =
(r - Complex.abs w0)⁻¹ * b * (Complex.abs w1 / r) ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0 | [390, 1] | [423, 15] | have cw0m : c0 + w0 ∈ ball c0 r := by
simpa only [Metric.mem_ball, dist_self_add_left, Complex.norm_eq_abs, Complex.dist_eq,
sub_zero] using 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 0 r
w1m : w1 ∈ ball 0 r
⊢ HasSum (fun n1 => w1 ^ n1 • h.series2CoeffN0Sum n1 w0) (f (c0 + w0, c1 + 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 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
⊢ HasSum (fun n1 => w1 ^ n1 • h.series2CoeffN0Sum n1 w0) (f (c0 + w0, c1 + 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 0 r
w1m : w1 ∈ ball 0 r
⊢ HasSum (fun n1 => w1 ^ n1 • h.series2CoeffN0Sum n1 w0) (f (c0 + w0, c1 + w1))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0 | [390, 1] | [423, 15] | have cw1m : c1 + w1 ∈ ball c1 r := by
simpa only [Metric.mem_ball, dist_self_add_left, Complex.norm_eq_abs, dist_zero_right] using 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 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
⊢ HasSum (fun n1 => w1 ^ n1 • h.series2CoeffN0Sum n1 w0) (f (c0 + w0, c1 + 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 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
⊢ HasSum (fun n1 => w1 ^ n1 • h.series2CoeffN0Sum n1 w0) (f (c0 + w0, c1 + 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 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
⊢ HasSum (fun n1 => w1 ^ n1 • h.series2CoeffN0Sum n1 w0) (f (c0 + w0, c1 + w1))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0 | [390, 1] | [423, 15] | simp_rw [Separate.series2CoeffN0Sum] | E : Type
inst✝² : NormedAddCommGroup E
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 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
⊢ HasSum (fun n1 => w1 ^ n1 • h.series2CoeffN0Sum n1 w0) (f (c0 + w0, c1 + 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 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
⊢ HasSum
(fun n1 =>
w1 ^ n1 •
(2 * ↑π * I)⁻¹ •
∮ (z0 : ℂ) in C(c0, r),
(z0 - (c0 + w0))⁻¹ • (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1))
(f (c0 + w0, c1 + 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 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
⊢ HasSum (fun n1 => w1 ^ n1 • h.series2CoeffN0Sum n1 w0) (f (c0 + w0, c1 + w1))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0 | [390, 1] | [423, 15] | rw [← cauchy2 h cw0m cw1m] | E : Type
inst✝² : NormedAddCommGroup E
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 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
⊢ HasSum
(fun n1 =>
w1 ^ n1 •
(2 * ↑π * I)⁻¹ •
∮ (z0 : ℂ) in C(c0, r),
(z0 - (c0 + w0))⁻¹ • (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1))
(f (c0 + w0, c1 + 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 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
⊢ HasSum
(fun n1 =>
w1 ^ n1 •
(2 * ↑π * I)⁻¹ •
∮ (z0 : ℂ) in C(c0, r),
(z0 - (c0 + w0))⁻¹ • (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1))
((2 * ↑π * I)⁻¹ •
∮ (z0 : ℂ) in C(c0, r),
(z0 - (c0 + w0))⁻¹ • (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - (c1 + w1))⁻¹ • f (z0, z1)) | 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 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
⊢ HasSum
(fun n1 =>
w1 ^ n1 •
(2 * ↑π * I)⁻¹ •
∮ (z0 : ℂ) in C(c0, r),
(z0 - (c0 + w0))⁻¹ • (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1))
(f (c0 + w0, c1 + w1))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0 | [390, 1] | [423, 15] | generalize hs : (2 * ↑π * I)⁻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
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
⊢ HasSum
(fun n1 =>
w1 ^ n1 •
(2 * ↑π * I)⁻¹ •
∮ (z0 : ℂ) in C(c0, r),
(z0 - (c0 + w0))⁻¹ • (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1))
((2 * ↑π * I)⁻¹ •
∮ (z0 : ℂ) in C(c0, r),
(z0 - (c0 + w0))⁻¹ • (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - (c1 + w1))⁻¹ • f (z0, 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 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
⊢ HasSum
(fun n1 =>
w1 ^ n1 •
s •
∮ (z0 : ℂ) in C(c0, r),
(z0 - (c0 + w0))⁻¹ • s • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1))
(s • ∮ (z0 : ℂ) in C(c0, r), (z0 - (c0 + w0))⁻¹ • s • ∮ (z1 : ℂ) in C(c1, r), (z1 - (c1 + w1))⁻¹ • f (z0, z1)) | 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 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
⊢ HasSum
(fun n1 =>
w1 ^ n1 •
(2 * ↑π * I)⁻¹ •
∮ (z0 : ℂ) in C(c0, r),
(z0 - (c0 + w0))⁻¹ • (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1))
((2 * ↑π * I)⁻¹ •
∮ (z0 : ℂ) in C(c0, r),
(z0 - (c0 + w0))⁻¹ • (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - (c1 + w1))⁻¹ • f (z0, z1))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0 | [390, 1] | [423, 15] | simp_rw [smul_comm _ 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
h : Separate f c0 c1 r b s✝
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
⊢ HasSum
(fun n1 =>
w1 ^ n1 •
s •
∮ (z0 : ℂ) in C(c0, r),
(z0 - (c0 + w0))⁻¹ • s • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1))
(s • ∮ (z0 : ℂ) in C(c0, r), (z0 - (c0 + w0))⁻¹ • s • ∮ (z1 : ℂ) in C(c1, r), (z1 - (c1 + w1))⁻¹ • f (z0, 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 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
⊢ HasSum
(fun n1 =>
s •
w1 ^ n1 •
∮ (z0 : ℂ) in C(c0, r),
s • (z0 - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1))
(s • ∮ (z0 : ℂ) in C(c0, r), s • (z0 - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - (c1 + w1))⁻¹ • f (z0, z1)) | 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 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
⊢ HasSum
(fun n1 =>
w1 ^ n1 •
s •
∮ (z0 : ℂ) in C(c0, r),
(z0 - (c0 + w0))⁻¹ • s • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1))
(s • ∮ (z0 : ℂ) in C(c0, r), (z0 - (c0 + w0))⁻¹ • s • ∮ (z1 : ℂ) in C(c1, r), (z1 - (c1 + w1))⁻¹ • f (z0, z1))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0 | [390, 1] | [423, 15] | apply HasSum.const_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
h : Separate f c0 c1 r b s✝
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
⊢ HasSum
(fun n1 =>
s •
w1 ^ n1 •
∮ (z0 : ℂ) in C(c0, r),
s • (z0 - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1))
(s • ∮ (z0 : ℂ) in C(c0, r), s • (z0 - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - (c1 + w1))⁻¹ • f (z0, z1)) | 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
h : Separate f c0 c1 r b s✝
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
⊢ HasSum
(fun i =>
w1 ^ i •
∮ (z0 : ℂ) in C(c0, r),
s • (z0 - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ i • (z1 - c1)⁻¹ • f (z0, z1))
(∮ (z0 : ℂ) in C(c0, r), s • (z0 - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - (c1 + w1))⁻¹ • f (z0, z1)) | 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 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
⊢ HasSum
(fun n1 =>
s •
w1 ^ n1 •
∮ (z0 : ℂ) in C(c0, r),
s • (z0 - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1))
(s • ∮ (z0 : ℂ) in C(c0, r), s • (z0 - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - (c1 + w1))⁻¹ • f (z0, z1))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0 | [390, 1] | [423, 15] | simp_rw [← circleIntegral.integral_smul (w1 ^ _) _ _ _] | 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
h : Separate f c0 c1 r b s✝
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
⊢ HasSum
(fun i =>
w1 ^ i •
∮ (z0 : ℂ) in C(c0, r),
s • (z0 - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ i • (z1 - c1)⁻¹ • f (z0, z1))
(∮ (z0 : ℂ) in C(c0, r), s • (z0 - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - (c1 + w1))⁻¹ • f (z0, z1)) | 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
h : Separate f c0 c1 r b s✝
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
⊢ HasSum
(fun i =>
∮ (z : ℂ) in C(c0, r),
w1 ^ i • s • (z - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ i • (z1 - c1)⁻¹ • f (z, z1))
(∮ (z0 : ℂ) in C(c0, r), s • (z0 - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - (c1 + w1))⁻¹ • f (z0, z1)) | Please generate a tactic in lean4 to solve the state.
STATE:
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
h : Separate f c0 c1 r b s✝
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
⊢ HasSum
(fun i =>
w1 ^ i •
∮ (z0 : ℂ) in C(c0, r),
s • (z0 - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ i • (z1 - c1)⁻¹ • f (z0, z1))
(∮ (z0 : ℂ) in C(c0, r), s • (z0 - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - (c1 + w1))⁻¹ • f (z0, z1))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0 | [390, 1] | [423, 15] | apply sum_integral_commute (fun n ↦ (r - abs w0)⁻¹ * b * (abs w1 / r) ^ n) h.rp | 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
h : Separate f c0 c1 r b s✝
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
⊢ HasSum
(fun i =>
∮ (z : ℂ) in C(c0, r),
w1 ^ i • s • (z - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ i • (z1 - c1)⁻¹ • f (z, z1))
(∮ (z0 : ℂ) in C(c0, r), s • (z0 - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - (c1 + w1))⁻¹ • f (z0, z1)) | case hf.fc
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s✝
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
⊢ ∀ (n : ℕ),
ContinuousOn
(fun z => w1 ^ n • s • (z - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z, z1))
(sphere c0 r)
case hf.fb
E : Type
inst✝² : NormedAddCommGroup E
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 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
⊢ ∀ (n : ℕ),
∀ z ∈ sphere c0 r,
‖w1 ^ n • s • (z - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z, z1)‖ ≤
(r - Complex.abs w0)⁻¹ * b * (Complex.abs w1 / r) ^ n
case hf.bs
E : Type
inst✝² : NormedAddCommGroup E
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 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
⊢ Summable fun n => (r - Complex.abs w0)⁻¹ * b * (Complex.abs w1 / r) ^ n
case hf.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 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
⊢ ∀ z ∈ sphere c0 r,
HasSum (fun n => w1 ^ n • s • (z - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z, z1))
(s • (z - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - (c1 + w1))⁻¹ • f (z, z1)) | Please generate a tactic in lean4 to solve the state.
STATE:
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
h : Separate f c0 c1 r b s✝
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
⊢ HasSum
(fun i =>
∮ (z : ℂ) in C(c0, r),
w1 ^ i • s • (z - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ i • (z1 - c1)⁻¹ • f (z, z1))
(∮ (z0 : ℂ) in C(c0, r), s • (z0 - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - (c1 + w1))⁻¹ • f (z0, z1))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0 | [390, 1] | [423, 15] | simpa only [Metric.mem_ball, dist_self_add_left, Complex.norm_eq_abs, Complex.dist_eq,
sub_zero] using 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 0 r
w1m : w1 ∈ ball 0 r
⊢ c0 + w0 ∈ ball c0 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
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
⊢ c0 + w0 ∈ ball c0 r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0 | [390, 1] | [423, 15] | simpa only [Metric.mem_ball, dist_self_add_left, Complex.norm_eq_abs, dist_zero_right] using 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 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
⊢ c1 + w1 ∈ ball c1 r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
⊢ c1 + w1 ∈ ball c1 r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0 | [390, 1] | [423, 15] | intro n | case hf.fc
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s✝
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
⊢ ∀ (n : ℕ),
ContinuousOn
(fun z => w1 ^ n • s • (z - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z, z1))
(sphere c0 r) | case hf.fc
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s✝
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
n : ℕ
⊢ ContinuousOn
(fun z => w1 ^ n • s • (z - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z, z1))
(sphere c0 r) | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.fc
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s✝
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
⊢ ∀ (n : ℕ),
ContinuousOn
(fun z => w1 ^ n • s • (z - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z, z1))
(sphere c0 r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0 | [390, 1] | [423, 15] | apply ContinuousOn.smul continuousOn_const | case hf.fc
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s✝
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
n : ℕ
⊢ ContinuousOn
(fun z => w1 ^ n • s • (z - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z, z1))
(sphere c0 r) | case hf.fc
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s✝
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
n : ℕ
⊢ ContinuousOn (fun x => s • (x - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (x, z1))
(sphere c0 r) | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.fc
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s✝
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
n : ℕ
⊢ ContinuousOn
(fun z => w1 ^ n • s • (z - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z, z1))
(sphere c0 r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0 | [390, 1] | [423, 15] | apply ContinuousOn.smul continuousOn_const | case hf.fc
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s✝
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
n : ℕ
⊢ ContinuousOn (fun x => s • (x - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (x, z1))
(sphere c0 r) | case hf.fc
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s✝
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
n : ℕ
⊢ ContinuousOn (fun x => (x - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (x, z1))
(sphere c0 r) | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.fc
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s✝
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
n : ℕ
⊢ ContinuousOn (fun x => s • (x - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (x, z1))
(sphere c0 r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0 | [390, 1] | [423, 15] | apply ContinuousOn.smul | case hf.fc
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s✝
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
n : ℕ
⊢ ContinuousOn (fun x => (x - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (x, z1))
(sphere c0 r) | case hf.fc.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
h : Separate f c0 c1 r b s✝
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
n : ℕ
⊢ ContinuousOn (fun x => (x - (c0 + w0))⁻¹) (sphere c0 r)
case hf.fc.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
h : Separate f c0 c1 r b s✝
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
n : ℕ
⊢ ContinuousOn (fun x => ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (x, z1)) (sphere c0 r) | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.fc
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s✝
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
n : ℕ
⊢ ContinuousOn (fun x => (x - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (x, z1))
(sphere c0 r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0 | [390, 1] | [423, 15] | exact ContinuousOn.inv_sphere_ball w0m | case hf.fc.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
h : Separate f c0 c1 r b s✝
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
n : ℕ
⊢ ContinuousOn (fun x => (x - (c0 + w0))⁻¹) (sphere c0 r)
case hf.fc.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
h : Separate f c0 c1 r b s✝
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
n : ℕ
⊢ ContinuousOn (fun x => ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (x, z1)) (sphere c0 r) | case hf.fc.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
h : Separate f c0 c1 r b s✝
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
n : ℕ
⊢ ContinuousOn (fun x => ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (x, z1)) (sphere c0 r) | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.fc.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
h : Separate f c0 c1 r b s✝
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
n : ℕ
⊢ ContinuousOn (fun x => (x - (c0 + w0))⁻¹) (sphere c0 r)
case hf.fc.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
h : Separate f c0 c1 r b s✝
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
n : ℕ
⊢ ContinuousOn (fun x => ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (x, z1)) (sphere c0 r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0 | [390, 1] | [423, 15] | apply ContinuousOn.cauchy1 h.rp | case hf.fc.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
h : Separate f c0 c1 r b s✝
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
n : ℕ
⊢ ContinuousOn (fun x => ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (x, z1)) (sphere c0 r) | case hf.fc.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
h : Separate f c0 c1 r b s✝
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
n : ℕ
⊢ ContinuousOn f (sphere c0 r ×ˢ sphere c1 r) | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.fc.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
h : Separate f c0 c1 r b s✝
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
n : ℕ
⊢ ContinuousOn (fun x => ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (x, z1)) (sphere c0 r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0 | [390, 1] | [423, 15] | apply ContinuousOn.mono h.fc h.rs' | case hf.fc.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
h : Separate f c0 c1 r b s✝
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
n : ℕ
⊢ ContinuousOn f (sphere c0 r ×ˢ sphere c1 r) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.fc.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
h : Separate f c0 c1 r b s✝
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
n : ℕ
⊢ ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0 | [390, 1] | [423, 15] | rw [← hs] | case hf.fb
E : Type
inst✝² : NormedAddCommGroup E
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 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
⊢ ∀ (n : ℕ),
∀ z ∈ sphere c0 r,
‖w1 ^ n • s • (z - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z, z1)‖ ≤
(r - Complex.abs w0)⁻¹ * b * (Complex.abs w1 / r) ^ n | case hf.fb
E : Type
inst✝² : NormedAddCommGroup E
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 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
⊢ ∀ (n : ℕ),
∀ z ∈ sphere c0 r,
‖w1 ^ n •
(2 * ↑π * I)⁻¹ • (z - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z, z1)‖ ≤
(r - Complex.abs w0)⁻¹ * b * (Complex.abs w1 / r) ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.fb
E : Type
inst✝² : NormedAddCommGroup E
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 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
⊢ ∀ (n : ℕ),
∀ z ∈ sphere c0 r,
‖w1 ^ n • s • (z - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z, z1)‖ ≤
(r - Complex.abs w0)⁻¹ * b * (Complex.abs w1 / r) ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0 | [390, 1] | [423, 15] | exact fun n z0 z0s ↦ cauchy2_hasSum_n1n0_bound h w0m n z0s | case hf.fb
E : Type
inst✝² : NormedAddCommGroup E
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 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
⊢ ∀ (n : ℕ),
∀ z ∈ sphere c0 r,
‖w1 ^ n •
(2 * ↑π * I)⁻¹ • (z - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z, z1)‖ ≤
(r - Complex.abs w0)⁻¹ * b * (Complex.abs w1 / r) ^ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.fb
E : Type
inst✝² : NormedAddCommGroup E
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 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
⊢ ∀ (n : ℕ),
∀ z ∈ sphere c0 r,
‖w1 ^ n •
(2 * ↑π * I)⁻¹ • (z - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z, z1)‖ ≤
(r - Complex.abs w0)⁻¹ * b * (Complex.abs w1 / r) ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0 | [390, 1] | [423, 15] | apply Summable.mul_left | case hf.bs
E : Type
inst✝² : NormedAddCommGroup E
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 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
⊢ Summable fun n => (r - Complex.abs w0)⁻¹ * b * (Complex.abs w1 / r) ^ n | case hf.bs.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
h : Separate f c0 c1 r b s✝
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
⊢ Summable fun i => (Complex.abs w1 / r) ^ i | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.bs
E : Type
inst✝² : NormedAddCommGroup E
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 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
⊢ Summable fun n => (r - Complex.abs w0)⁻¹ * b * (Complex.abs w1 / r) ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0 | [390, 1] | [423, 15] | apply summable_geometric_of_abs_lt_one | case hf.bs.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
h : Separate f c0 c1 r b s✝
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
⊢ Summable fun i => (Complex.abs w1 / r) ^ i | case hf.bs.hf.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 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
⊢ |Complex.abs w1 / r| < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.bs.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
h : Separate f c0 c1 r b s✝
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
⊢ Summable fun i => (Complex.abs w1 / r) ^ i
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0 | [390, 1] | [423, 15] | rw [abs_div, abs_of_pos h.rp] | case hf.bs.hf.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 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
⊢ |Complex.abs w1 / r| < 1 | case hf.bs.hf.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 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
⊢ |Complex.abs w1| / r < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.bs.hf.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 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
⊢ |Complex.abs w1 / r| < 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0 | [390, 1] | [423, 15] | simp at w1m ⊢ | case hf.bs.hf.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 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
⊢ |Complex.abs w1| / r < 1 | case hf.bs.hf.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 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
w1m : Complex.abs w1 < r
⊢ Complex.abs w1 / r < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.bs.hf.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 0 r
w1m : w1 ∈ ball 0 r
cw0m : c0 + w0 ∈ ball c0 r
cw1m : c1 + w1 ∈ ball c1 r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
⊢ |Complex.abs w1| / r < 1
TACTIC:
|
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