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 | 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:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0 | [390, 1] | [423, 15] | exact (div_lt_one h.rp).mpr 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
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 | no goals | 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
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
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0 | [390, 1] | [423, 15] | intro z0 z0s | 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)) | 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
z0 : ℂ
z0s : z0 ∈ sphere c0 r
⊢ HasSum (fun n => w1 ^ n • s • (z0 - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z0, z1))
(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.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))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_n1n0 | [390, 1] | [423, 15] | simp_rw [smul_comm s _] | 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
z0 : ℂ
z0s : z0 ∈ sphere c0 r
⊢ HasSum (fun n => w1 ^ n • s • (z0 - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z0, z1))
(s • (z0 - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - (c1 + w1))⁻¹ • f (z0, z1)) | 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
z0 : ℂ
z0s : z0 ∈ sphere c0 r
⊢ HasSum (fun n => w1 ^ n • (z0 - (c0 + w0))⁻¹ • s • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z0, z1))
((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:
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
z0 : ℂ
z0s : z0 ∈ sphere c0 r
⊢ HasSum (fun n => w1 ^ n • s • (z0 - (c0 + w0))⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z0, z1))
(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 [smul_comm (w1 ^ _) _] | 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
z0 : ℂ
z0s : z0 ∈ sphere c0 r
⊢ HasSum (fun n => w1 ^ n • (z0 - (c0 + w0))⁻¹ • s • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z0, z1))
((z0 - (c0 + w0))⁻¹ • s • ∮ (z1 : ℂ) in C(c1, r), (z1 - (c1 + w1))⁻¹ • f (z0, z1)) | 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
z0 : ℂ
z0s : z0 ∈ sphere c0 r
⊢ HasSum (fun n => (z0 - (c0 + w0))⁻¹ • s • w1 ^ n • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z0, z1))
((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:
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
z0 : ℂ
z0s : z0 ∈ sphere c0 r
⊢ HasSum (fun n => w1 ^ n • (z0 - (c0 + w0))⁻¹ • s • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z0, z1))
((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 | 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
z0 : ℂ
z0s : z0 ∈ sphere c0 r
⊢ HasSum (fun n => (z0 - (c0 + w0))⁻¹ • s • w1 ^ n • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z0, z1))
((z0 - (c0 + w0))⁻¹ • s • ∮ (z1 : ℂ) in C(c1, r), (z1 - (c1 + w1))⁻¹ • f (z0, z1)) | case hf.h.hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
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
z0 : ℂ
z0s : z0 ∈ sphere c0 r
⊢ HasSum (fun i => s • w1 ^ i • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ i • (z1 - c1)⁻¹ • f (z0, z1))
(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:
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
z0 : ℂ
z0s : z0 ∈ sphere c0 r
⊢ HasSum (fun n => (z0 - (c0 + w0))⁻¹ • s • w1 ^ n • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n • (z1 - c1)⁻¹ • f (z0, z1))
((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] | have fcs : ContinuousOn (fun z1 ↦ f (z0, z1)) (sphere c1 r) :=
ContinuousOn.mono (h.fc1 (Metric.sphere_subset_closedBall z0s))
Metric.sphere_subset_closedBall | case hf.h.hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
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
z0 : ℂ
z0s : z0 ∈ sphere c0 r
⊢ HasSum (fun i => s • w1 ^ i • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ i • (z1 - c1)⁻¹ • f (z0, z1))
(s • ∮ (z1 : ℂ) in C(c1, r), (z1 - (c1 + w1))⁻¹ • f (z0, z1)) | case hf.h.hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
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
z0 : ℂ
z0s : z0 ∈ sphere c0 r
fcs : ContinuousOn (fun z1 => f (z0, z1)) (sphere c1 r)
⊢ HasSum (fun i => s • w1 ^ i • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ i • (z1 - c1)⁻¹ • f (z0, z1))
(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:
case hf.h.hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
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
z0 : ℂ
z0s : z0 ∈ sphere c0 r
⊢ HasSum (fun i => s • w1 ^ i • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ i • (z1 - c1)⁻¹ • f (z0, z1))
(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] | have hs1 := cauchy1_hasSum h.rp fcs w1m | case hf.h.hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
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
z0 : ℂ
z0s : z0 ∈ sphere c0 r
fcs : ContinuousOn (fun z1 => f (z0, z1)) (sphere c1 r)
⊢ HasSum (fun i => s • w1 ^ i • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ i • (z1 - c1)⁻¹ • f (z0, z1))
(s • ∮ (z1 : ℂ) in C(c1, r), (z1 - (c1 + w1))⁻¹ • f (z0, z1)) | case hf.h.hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
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
z0 : ℂ
z0s : z0 ∈ sphere c0 r
fcs : ContinuousOn (fun z1 => f (z0, z1)) (sphere c1 r)
hs1 :
HasSum (fun n => w1 ^ n • (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z))
((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c1, r), (z - (c1 + w1))⁻¹ • f (z0, z))
⊢ HasSum (fun i => s • w1 ^ i • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ i • (z1 - c1)⁻¹ • f (z0, z1))
(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:
case hf.h.hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
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
z0 : ℂ
z0s : z0 ∈ sphere c0 r
fcs : ContinuousOn (fun z1 => f (z0, z1)) (sphere c1 r)
⊢ HasSum (fun i => s • w1 ^ i • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ i • (z1 - c1)⁻¹ • f (z0, z1))
(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] | simp_rw [hs, smul_comm _ s] at hs1 | case hf.h.hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
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
z0 : ℂ
z0s : z0 ∈ sphere c0 r
fcs : ContinuousOn (fun z1 => f (z0, z1)) (sphere c1 r)
hs1 :
HasSum (fun n => w1 ^ n • (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z))
((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c1, r), (z - (c1 + w1))⁻¹ • f (z0, z))
⊢ HasSum (fun i => s • w1 ^ i • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ i • (z1 - c1)⁻¹ • f (z0, z1))
(s • ∮ (z1 : ℂ) in C(c1, r), (z1 - (c1 + w1))⁻¹ • f (z0, z1)) | case hf.h.hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
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
z0 : ℂ
z0s : z0 ∈ sphere c0 r
fcs : ContinuousOn (fun z1 => f (z0, z1)) (sphere c1 r)
hs1 :
HasSum (fun n => s • w1 ^ n • ∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z))
(s • ∮ (z : ℂ) in C(c1, r), (z - (c1 + w1))⁻¹ • f (z0, z))
⊢ HasSum (fun i => s • w1 ^ i • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ i • (z1 - c1)⁻¹ • f (z0, z1))
(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:
case hf.h.hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
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
z0 : ℂ
z0s : z0 ∈ sphere c0 r
fcs : ContinuousOn (fun z1 => f (z0, z1)) (sphere c1 r)
hs1 :
HasSum (fun n => w1 ^ n • (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z))
((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c1, r), (z - (c1 + w1))⁻¹ • f (z0, z))
⊢ HasSum (fun i => s • w1 ^ i • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ i • (z1 - c1)⁻¹ • f (z0, z1))
(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] | assumption | case hf.h.hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
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
z0 : ℂ
z0s : z0 ∈ sphere c0 r
fcs : ContinuousOn (fun z1 => f (z0, z1)) (sphere c1 r)
hs1 :
HasSum (fun n => s • w1 ^ n • ∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z))
(s • ∮ (z : ℂ) in C(c1, r), (z - (c1 + w1))⁻¹ • f (z0, z))
⊢ HasSum (fun i => s • w1 ^ i • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ i • (z1 - c1)⁻¹ • f (z0, z1))
(s • ∮ (z1 : ℂ) in C(c1, r), (z1 - (c1 + w1))⁻¹ • f (z0, z1)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.h.hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
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
z0 : ℂ
z0s : z0 ∈ sphere c0 r
fcs : ContinuousOn (fun z1 => f (z0, z1)) (sphere c1 r)
hs1 :
HasSum (fun n => s • w1 ^ n • ∮ (z : ℂ) in C(c1, r), (z - c1)⁻¹ ^ n • (z - c1)⁻¹ • f (z0, z))
(s • ∮ (z : ℂ) in C(c1, r), (z - (c1 + w1))⁻¹ • f (z0, z))
⊢ HasSum (fun i => s • w1 ^ i • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ i • (z1 - c1)⁻¹ • f (z0, z1))
(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 | series2Coeff_bound | [426, 1] | [441, 59] | have inner_c :
ContinuousOn
(fun z0 ↦ (2 * π * I : ℂ)⁻¹ • ∮ z1 in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1))
(sphere c0 r) :=
ContinuousOn.smul continuousOn_const (ContinuousOn.cauchy1 h.rp (ContinuousOn.mono h.fc h.rs')) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
n0 n1 : ℕ
⊢ ‖h.series2Coeff n0 n1‖ ≤ b * r⁻¹ ^ (n0 + n1) | E : Type
inst✝² : NormedAddCommGroup E
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
n0 n1 : ℕ
inner_c :
ContinuousOn (fun z0 => (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1))
(sphere c0 r)
⊢ ‖h.series2Coeff n0 n1‖ ≤ b * r⁻¹ ^ (n0 + n1) | 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
n0 n1 : ℕ
⊢ ‖h.series2Coeff n0 n1‖ ≤ b * r⁻¹ ^ (n0 + n1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | series2Coeff_bound | [426, 1] | [441, 59] | have inner_b : ∀ z0 _, ‖(2*π*I : ℂ)⁻¹ • ∮ z1 in C(c1, r),
(z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0,z1)‖ ≤ b * r⁻¹ ^ n1 :=
fun z0 z0s ↦ cauchy1_bound' h.rp b
(ContinuousOn.mono (h.fc1 (mem_sphere_closed z0s)) Metric.sphere_subset_closedBall)
(fun z1 ↦ h.fb z0s) n1 | E : Type
inst✝² : NormedAddCommGroup E
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
n0 n1 : ℕ
inner_c :
ContinuousOn (fun z0 => (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1))
(sphere c0 r)
⊢ ‖h.series2Coeff n0 n1‖ ≤ b * r⁻¹ ^ (n0 + n1) | E : Type
inst✝² : NormedAddCommGroup E
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
n0 n1 : ℕ
inner_c :
ContinuousOn (fun z0 => (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1))
(sphere c0 r)
inner_b :
∀ z0 ∈ sphere c0 r,
‖(2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1)‖ ≤ b * r⁻¹ ^ n1
⊢ ‖h.series2Coeff n0 n1‖ ≤ b * r⁻¹ ^ (n0 + n1) | 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
n0 n1 : ℕ
inner_c :
ContinuousOn (fun z0 => (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1))
(sphere c0 r)
⊢ ‖h.series2Coeff n0 n1‖ ≤ b * r⁻¹ ^ (n0 + n1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | series2Coeff_bound | [426, 1] | [441, 59] | have outer := cauchy1_bound' h.rp _ inner_c inner_b n0 | E : Type
inst✝² : NormedAddCommGroup E
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
n0 n1 : ℕ
inner_c :
ContinuousOn (fun z0 => (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1))
(sphere c0 r)
inner_b :
∀ z0 ∈ sphere c0 r,
‖(2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1)‖ ≤ b * r⁻¹ ^ n1
⊢ ‖h.series2Coeff n0 n1‖ ≤ b * r⁻¹ ^ (n0 + n1) | E : Type
inst✝² : NormedAddCommGroup E
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
n0 n1 : ℕ
inner_c :
ContinuousOn (fun z0 => (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1))
(sphere c0 r)
inner_b :
∀ z0 ∈ sphere c0 r,
‖(2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1)‖ ≤ b * r⁻¹ ^ n1
outer :
‖(2 * ↑π * I)⁻¹ •
∮ (z : ℂ) in C(c0, r),
(z - c0)⁻¹ ^ n0 •
(z - c0)⁻¹ • (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z, z1)‖ ≤
b * r⁻¹ ^ n1 * r⁻¹ ^ n0
⊢ ‖h.series2Coeff n0 n1‖ ≤ b * r⁻¹ ^ (n0 + n1) | 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
n0 n1 : ℕ
inner_c :
ContinuousOn (fun z0 => (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1))
(sphere c0 r)
inner_b :
∀ z0 ∈ sphere c0 r,
‖(2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1)‖ ≤ b * r⁻¹ ^ n1
⊢ ‖h.series2Coeff n0 n1‖ ≤ b * r⁻¹ ^ (n0 + n1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | series2Coeff_bound | [426, 1] | [441, 59] | have e : b * r⁻¹ ^ n1 * r⁻¹ ^ n0 = b * r⁻¹ ^ (n0 + n1) := by
rw [mul_assoc, ← pow_add, add_comm n0 _] | E : Type
inst✝² : NormedAddCommGroup E
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
n0 n1 : ℕ
inner_c :
ContinuousOn (fun z0 => (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1))
(sphere c0 r)
inner_b :
∀ z0 ∈ sphere c0 r,
‖(2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1)‖ ≤ b * r⁻¹ ^ n1
outer :
‖(2 * ↑π * I)⁻¹ •
∮ (z : ℂ) in C(c0, r),
(z - c0)⁻¹ ^ n0 •
(z - c0)⁻¹ • (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z, z1)‖ ≤
b * r⁻¹ ^ n1 * r⁻¹ ^ n0
⊢ ‖h.series2Coeff n0 n1‖ ≤ b * r⁻¹ ^ (n0 + n1) | E : Type
inst✝² : NormedAddCommGroup E
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
n0 n1 : ℕ
inner_c :
ContinuousOn (fun z0 => (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1))
(sphere c0 r)
inner_b :
∀ z0 ∈ sphere c0 r,
‖(2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1)‖ ≤ b * r⁻¹ ^ n1
outer :
‖(2 * ↑π * I)⁻¹ •
∮ (z : ℂ) in C(c0, r),
(z - c0)⁻¹ ^ n0 •
(z - c0)⁻¹ • (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z, z1)‖ ≤
b * r⁻¹ ^ n1 * r⁻¹ ^ n0
e : b * r⁻¹ ^ n1 * r⁻¹ ^ n0 = b * r⁻¹ ^ (n0 + n1)
⊢ ‖h.series2Coeff n0 n1‖ ≤ b * r⁻¹ ^ (n0 + n1) | 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
n0 n1 : ℕ
inner_c :
ContinuousOn (fun z0 => (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1))
(sphere c0 r)
inner_b :
∀ z0 ∈ sphere c0 r,
‖(2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1)‖ ≤ b * r⁻¹ ^ n1
outer :
‖(2 * ↑π * I)⁻¹ •
∮ (z : ℂ) in C(c0, r),
(z - c0)⁻¹ ^ n0 •
(z - c0)⁻¹ • (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z, z1)‖ ≤
b * r⁻¹ ^ n1 * r⁻¹ ^ n0
⊢ ‖h.series2Coeff n0 n1‖ ≤ b * r⁻¹ ^ (n0 + n1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | series2Coeff_bound | [426, 1] | [441, 59] | rw [Separate.series2Coeff] | E : Type
inst✝² : NormedAddCommGroup E
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
n0 n1 : ℕ
inner_c :
ContinuousOn (fun z0 => (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1))
(sphere c0 r)
inner_b :
∀ z0 ∈ sphere c0 r,
‖(2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1)‖ ≤ b * r⁻¹ ^ n1
outer :
‖(2 * ↑π * I)⁻¹ •
∮ (z : ℂ) in C(c0, r),
(z - c0)⁻¹ ^ n0 •
(z - c0)⁻¹ • (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z, z1)‖ ≤
b * r⁻¹ ^ n1 * r⁻¹ ^ n0
e : b * r⁻¹ ^ n1 * r⁻¹ ^ n0 = b * r⁻¹ ^ (n0 + n1)
⊢ ‖h.series2Coeff n0 n1‖ ≤ b * r⁻¹ ^ (n0 + n1) | E : Type
inst✝² : NormedAddCommGroup E
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
n0 n1 : ℕ
inner_c :
ContinuousOn (fun z0 => (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1))
(sphere c0 r)
inner_b :
∀ z0 ∈ sphere c0 r,
‖(2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1)‖ ≤ b * r⁻¹ ^ n1
outer :
‖(2 * ↑π * I)⁻¹ •
∮ (z : ℂ) in C(c0, r),
(z - c0)⁻¹ ^ n0 •
(z - c0)⁻¹ • (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z, z1)‖ ≤
b * r⁻¹ ^ n1 * r⁻¹ ^ n0
e : b * r⁻¹ ^ n1 * r⁻¹ ^ n0 = b * r⁻¹ ^ (n0 + n1)
⊢ ‖(2 * ↑π * I)⁻¹ •
∮ (z0 : ℂ) in C(c0, r),
(z0 - c0)⁻¹ ^ n0 •
(z0 - c0)⁻¹ • (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1)‖ ≤
b * r⁻¹ ^ (n0 + n1) | 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
n0 n1 : ℕ
inner_c :
ContinuousOn (fun z0 => (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1))
(sphere c0 r)
inner_b :
∀ z0 ∈ sphere c0 r,
‖(2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1)‖ ≤ b * r⁻¹ ^ n1
outer :
‖(2 * ↑π * I)⁻¹ •
∮ (z : ℂ) in C(c0, r),
(z - c0)⁻¹ ^ n0 •
(z - c0)⁻¹ • (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z, z1)‖ ≤
b * r⁻¹ ^ n1 * r⁻¹ ^ n0
e : b * r⁻¹ ^ n1 * r⁻¹ ^ n0 = b * r⁻¹ ^ (n0 + n1)
⊢ ‖h.series2Coeff n0 n1‖ ≤ b * r⁻¹ ^ (n0 + n1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | series2Coeff_bound | [426, 1] | [441, 59] | rw [e] at outer | E : Type
inst✝² : NormedAddCommGroup E
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
n0 n1 : ℕ
inner_c :
ContinuousOn (fun z0 => (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1))
(sphere c0 r)
inner_b :
∀ z0 ∈ sphere c0 r,
‖(2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1)‖ ≤ b * r⁻¹ ^ n1
outer :
‖(2 * ↑π * I)⁻¹ •
∮ (z : ℂ) in C(c0, r),
(z - c0)⁻¹ ^ n0 •
(z - c0)⁻¹ • (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z, z1)‖ ≤
b * r⁻¹ ^ n1 * r⁻¹ ^ n0
e : b * r⁻¹ ^ n1 * r⁻¹ ^ n0 = b * r⁻¹ ^ (n0 + n1)
⊢ ‖(2 * ↑π * I)⁻¹ •
∮ (z0 : ℂ) in C(c0, r),
(z0 - c0)⁻¹ ^ n0 •
(z0 - c0)⁻¹ • (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1)‖ ≤
b * r⁻¹ ^ (n0 + n1) | E : Type
inst✝² : NormedAddCommGroup E
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
n0 n1 : ℕ
inner_c :
ContinuousOn (fun z0 => (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1))
(sphere c0 r)
inner_b :
∀ z0 ∈ sphere c0 r,
‖(2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1)‖ ≤ b * r⁻¹ ^ n1
outer :
‖(2 * ↑π * I)⁻¹ •
∮ (z : ℂ) in C(c0, r),
(z - c0)⁻¹ ^ n0 •
(z - c0)⁻¹ • (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z, z1)‖ ≤
b * r⁻¹ ^ (n0 + n1)
e : b * r⁻¹ ^ n1 * r⁻¹ ^ n0 = b * r⁻¹ ^ (n0 + n1)
⊢ ‖(2 * ↑π * I)⁻¹ •
∮ (z0 : ℂ) in C(c0, r),
(z0 - c0)⁻¹ ^ n0 •
(z0 - c0)⁻¹ • (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1)‖ ≤
b * r⁻¹ ^ (n0 + n1) | 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
n0 n1 : ℕ
inner_c :
ContinuousOn (fun z0 => (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1))
(sphere c0 r)
inner_b :
∀ z0 ∈ sphere c0 r,
‖(2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1)‖ ≤ b * r⁻¹ ^ n1
outer :
‖(2 * ↑π * I)⁻¹ •
∮ (z : ℂ) in C(c0, r),
(z - c0)⁻¹ ^ n0 •
(z - c0)⁻¹ • (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z, z1)‖ ≤
b * r⁻¹ ^ n1 * r⁻¹ ^ n0
e : b * r⁻¹ ^ n1 * r⁻¹ ^ n0 = b * r⁻¹ ^ (n0 + n1)
⊢ ‖(2 * ↑π * I)⁻¹ •
∮ (z0 : ℂ) in C(c0, r),
(z0 - c0)⁻¹ ^ n0 •
(z0 - c0)⁻¹ • (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1)‖ ≤
b * r⁻¹ ^ (n0 + n1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | series2Coeff_bound | [426, 1] | [441, 59] | exact outer | E : Type
inst✝² : NormedAddCommGroup E
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
n0 n1 : ℕ
inner_c :
ContinuousOn (fun z0 => (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1))
(sphere c0 r)
inner_b :
∀ z0 ∈ sphere c0 r,
‖(2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1)‖ ≤ b * r⁻¹ ^ n1
outer :
‖(2 * ↑π * I)⁻¹ •
∮ (z : ℂ) in C(c0, r),
(z - c0)⁻¹ ^ n0 •
(z - c0)⁻¹ • (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z, z1)‖ ≤
b * r⁻¹ ^ (n0 + n1)
e : b * r⁻¹ ^ n1 * r⁻¹ ^ n0 = b * r⁻¹ ^ (n0 + n1)
⊢ ‖(2 * ↑π * I)⁻¹ •
∮ (z0 : ℂ) in C(c0, r),
(z0 - c0)⁻¹ ^ n0 •
(z0 - c0)⁻¹ • (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1)‖ ≤
b * r⁻¹ ^ (n0 + n1) | 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
n0 n1 : ℕ
inner_c :
ContinuousOn (fun z0 => (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1))
(sphere c0 r)
inner_b :
∀ z0 ∈ sphere c0 r,
‖(2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1)‖ ≤ b * r⁻¹ ^ n1
outer :
‖(2 * ↑π * I)⁻¹ •
∮ (z : ℂ) in C(c0, r),
(z - c0)⁻¹ ^ n0 •
(z - c0)⁻¹ • (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z, z1)‖ ≤
b * r⁻¹ ^ (n0 + n1)
e : b * r⁻¹ ^ n1 * r⁻¹ ^ n0 = b * r⁻¹ ^ (n0 + n1)
⊢ ‖(2 * ↑π * I)⁻¹ •
∮ (z0 : ℂ) in C(c0, r),
(z0 - c0)⁻¹ ^ n0 •
(z0 - c0)⁻¹ • (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1)‖ ≤
b * r⁻¹ ^ (n0 + n1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | series2Coeff_bound | [426, 1] | [441, 59] | rw [mul_assoc, ← pow_add, add_comm n0 _] | E : Type
inst✝² : NormedAddCommGroup E
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
n0 n1 : ℕ
inner_c :
ContinuousOn (fun z0 => (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1))
(sphere c0 r)
inner_b :
∀ z0 ∈ sphere c0 r,
‖(2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1)‖ ≤ b * r⁻¹ ^ n1
outer :
‖(2 * ↑π * I)⁻¹ •
∮ (z : ℂ) in C(c0, r),
(z - c0)⁻¹ ^ n0 •
(z - c0)⁻¹ • (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z, z1)‖ ≤
b * r⁻¹ ^ n1 * r⁻¹ ^ n0
⊢ b * r⁻¹ ^ n1 * r⁻¹ ^ n0 = b * r⁻¹ ^ (n0 + n1) | 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
n0 n1 : ℕ
inner_c :
ContinuousOn (fun z0 => (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1))
(sphere c0 r)
inner_b :
∀ z0 ∈ sphere c0 r,
‖(2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1)‖ ≤ b * r⁻¹ ^ n1
outer :
‖(2 * ↑π * I)⁻¹ •
∮ (z : ℂ) in C(c0, r),
(z - c0)⁻¹ ^ n0 •
(z - c0)⁻¹ • (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z, z1)‖ ≤
b * r⁻¹ ^ n1 * r⁻¹ ^ n0
⊢ b * r⁻¹ ^ n1 * r⁻¹ ^ n0 = b * r⁻¹ ^ (n0 + n1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | series2_norm | [448, 1] | [464, 19] | rw [series2] | E : Type
inst✝² : NormedAddCommGroup E
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
⊢ ‖series2 h n‖ ≤ (↑n + 1) * 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
h : Separate f c0 c1 r b s
n : ℕ
⊢ ‖(Finset.range (n + 1)).sum fun n0 => termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖ ≤ (↑n + 1) * 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
h : Separate f c0 c1 r b s
n : ℕ
⊢ ‖series2 h n‖ ≤ (↑n + 1) * b * r⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | series2_norm | [448, 1] | [464, 19] | simp only [ge_iff_le, inv_pow] | E : Type
inst✝² : NormedAddCommGroup E
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
⊢ ‖(Finset.range (n + 1)).sum fun n0 => termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖ ≤ (↑n + 1) * 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
h : Separate f c0 c1 r b s
n : ℕ
⊢ ‖(Finset.range (n + 1)).sum fun n0 => termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖ ≤ (↑n + 1) * b * (r ^ n)⁻¹ | 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
⊢ ‖(Finset.range (n + 1)).sum fun n0 => termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖ ≤ (↑n + 1) * b * r⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | series2_norm | [448, 1] | [464, 19] | have tb : ∀ n0, n0 ∈ Finset.range (n+1) →
‖termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖ ≤ b * r⁻¹ ^ n := by
intro n0 n0n; simp at n0n
apply le_trans (termCmmap_norm ℂ n n0 (h.series2Coeff n0 (n - n0)))
have sb := series2Coeff_bound h n0 (n - n0)
rw [← Nat.add_sub_assoc (Nat.le_of_lt_succ n0n) n0, Nat.add_sub_cancel_left] at sb
assumption | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
n : ℕ
⊢ ‖(Finset.range (n + 1)).sum fun n0 => termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖ ≤ (↑n + 1) * b * (r ^ n)⁻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
tb : ∀ n0 ∈ Finset.range (n + 1), ‖termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖ ≤ b * r⁻¹ ^ n
⊢ ‖(Finset.range (n + 1)).sum fun n0 => termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖ ≤ (↑n + 1) * b * (r ^ n)⁻¹ | 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
⊢ ‖(Finset.range (n + 1)).sum fun n0 => termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖ ≤ (↑n + 1) * b * (r ^ n)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | series2_norm | [448, 1] | [464, 19] | trans (Finset.range (n + 1)).sum fun n0 ↦ ‖termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖ | E : Type
inst✝² : NormedAddCommGroup E
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
tb : ∀ n0 ∈ Finset.range (n + 1), ‖termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖ ≤ b * r⁻¹ ^ n
⊢ ‖(Finset.range (n + 1)).sum fun n0 => termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖ ≤ (↑n + 1) * b * (r ^ n)⁻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
tb : ∀ n0 ∈ Finset.range (n + 1), ‖termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖ ≤ b * r⁻¹ ^ n
⊢ ‖(Finset.range (n + 1)).sum fun n0 => termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖ ≤
(Finset.range (n + 1)).sum fun n0 => ‖termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖
E : Type
inst✝² : NormedAddCommGroup E
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
tb : ∀ n0 ∈ Finset.range (n + 1), ‖termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖ ≤ b * r⁻¹ ^ n
⊢ ((Finset.range (n + 1)).sum fun n0 => ‖termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖) ≤ (↑n + 1) * b * (r ^ n)⁻¹ | 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
tb : ∀ n0 ∈ Finset.range (n + 1), ‖termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖ ≤ b * r⁻¹ ^ n
⊢ ‖(Finset.range (n + 1)).sum fun n0 => termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖ ≤ (↑n + 1) * b * (r ^ n)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | series2_norm | [448, 1] | [464, 19] | intro n0 n0n | E : Type
inst✝² : NormedAddCommGroup E
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
⊢ ∀ n0 ∈ Finset.range (n + 1), ‖termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖ ≤ 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
h : Separate f c0 c1 r b s
n n0 : ℕ
n0n : n0 ∈ Finset.range (n + 1)
⊢ ‖termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖ ≤ 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
h : Separate f c0 c1 r b s
n : ℕ
⊢ ∀ n0 ∈ Finset.range (n + 1), ‖termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖ ≤ b * r⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | series2_norm | [448, 1] | [464, 19] | simp at n0n | E : Type
inst✝² : NormedAddCommGroup E
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 n0 : ℕ
n0n : n0 ∈ Finset.range (n + 1)
⊢ ‖termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖ ≤ 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
h : Separate f c0 c1 r b s
n n0 : ℕ
n0n : n0 < n + 1
⊢ ‖termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖ ≤ 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
h : Separate f c0 c1 r b s
n n0 : ℕ
n0n : n0 ∈ Finset.range (n + 1)
⊢ ‖termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖ ≤ b * r⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | series2_norm | [448, 1] | [464, 19] | apply le_trans (termCmmap_norm ℂ n n0 (h.series2Coeff n0 (n - n0))) | E : Type
inst✝² : NormedAddCommGroup E
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 n0 : ℕ
n0n : n0 < n + 1
⊢ ‖termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖ ≤ 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
h : Separate f c0 c1 r b s
n n0 : ℕ
n0n : n0 < n + 1
⊢ ‖h.series2Coeff n0 (n - n0)‖ ≤ 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
h : Separate f c0 c1 r b s
n n0 : ℕ
n0n : n0 < n + 1
⊢ ‖termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖ ≤ b * r⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | series2_norm | [448, 1] | [464, 19] | have sb := series2Coeff_bound h n0 (n - n0) | E : Type
inst✝² : NormedAddCommGroup E
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 n0 : ℕ
n0n : n0 < n + 1
⊢ ‖h.series2Coeff n0 (n - n0)‖ ≤ 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
h : Separate f c0 c1 r b s
n n0 : ℕ
n0n : n0 < n + 1
sb : ‖h.series2Coeff n0 (n - n0)‖ ≤ b * r⁻¹ ^ (n0 + (n - n0))
⊢ ‖h.series2Coeff n0 (n - n0)‖ ≤ 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
h : Separate f c0 c1 r b s
n n0 : ℕ
n0n : n0 < n + 1
⊢ ‖h.series2Coeff n0 (n - n0)‖ ≤ b * r⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | series2_norm | [448, 1] | [464, 19] | rw [← Nat.add_sub_assoc (Nat.le_of_lt_succ n0n) n0, Nat.add_sub_cancel_left] at 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
h : Separate f c0 c1 r b s
n n0 : ℕ
n0n : n0 < n + 1
sb : ‖h.series2Coeff n0 (n - n0)‖ ≤ b * r⁻¹ ^ (n0 + (n - n0))
⊢ ‖h.series2Coeff n0 (n - n0)‖ ≤ 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
h : Separate f c0 c1 r b s
n n0 : ℕ
n0n : n0 < n + 1
sb : ‖h.series2Coeff n0 (n - n0)‖ ≤ b * r⁻¹ ^ n
⊢ ‖h.series2Coeff n0 (n - n0)‖ ≤ 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
h : Separate f c0 c1 r b s
n n0 : ℕ
n0n : n0 < n + 1
sb : ‖h.series2Coeff n0 (n - n0)‖ ≤ b * r⁻¹ ^ (n0 + (n - n0))
⊢ ‖h.series2Coeff n0 (n - n0)‖ ≤ b * r⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | series2_norm | [448, 1] | [464, 19] | assumption | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
n n0 : ℕ
n0n : n0 < n + 1
sb : ‖h.series2Coeff n0 (n - n0)‖ ≤ b * r⁻¹ ^ n
⊢ ‖h.series2Coeff n0 (n - n0)‖ ≤ 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 n0 : ℕ
n0n : n0 < n + 1
sb : ‖h.series2Coeff n0 (n - n0)‖ ≤ b * r⁻¹ ^ n
⊢ ‖h.series2Coeff n0 (n - n0)‖ ≤ b * r⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | series2_norm | [448, 1] | [464, 19] | 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
tb : ∀ n0 ∈ Finset.range (n + 1), ‖termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖ ≤ b * r⁻¹ ^ n
⊢ ‖(Finset.range (n + 1)).sum fun n0 => termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖ ≤
(Finset.range (n + 1)).sum fun n0 => ‖termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖ | 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
tb : ∀ n0 ∈ Finset.range (n + 1), ‖termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖ ≤ b * r⁻¹ ^ n
⊢ ‖(Finset.range (n + 1)).sum fun n0 => termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖ ≤
(Finset.range (n + 1)).sum fun n0 => ‖termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | series2_norm | [448, 1] | [464, 19] | trans (Finset.range (n + 1)).sum fun _ ↦ 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
h : Separate f c0 c1 r b s
n : ℕ
tb : ∀ n0 ∈ Finset.range (n + 1), ‖termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖ ≤ b * r⁻¹ ^ n
⊢ ((Finset.range (n + 1)).sum fun n0 => ‖termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖) ≤ (↑n + 1) * b * (r ^ n)⁻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
tb : ∀ n0 ∈ Finset.range (n + 1), ‖termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖ ≤ b * r⁻¹ ^ n
⊢ ((Finset.range (n + 1)).sum fun n0 => ‖termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖) ≤
(Finset.range (n + 1)).sum fun x => 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
h : Separate f c0 c1 r b s
n : ℕ
tb : ∀ n0 ∈ Finset.range (n + 1), ‖termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖ ≤ b * r⁻¹ ^ n
⊢ ((Finset.range (n + 1)).sum fun x => b * r⁻¹ ^ n) ≤ (↑n + 1) * b * (r ^ n)⁻¹ | 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
tb : ∀ n0 ∈ Finset.range (n + 1), ‖termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖ ≤ b * r⁻¹ ^ n
⊢ ((Finset.range (n + 1)).sum fun n0 => ‖termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖) ≤ (↑n + 1) * b * (r ^ n)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | series2_norm | [448, 1] | [464, 19] | 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
tb : ∀ n0 ∈ Finset.range (n + 1), ‖termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖ ≤ b * r⁻¹ ^ n
⊢ ((Finset.range (n + 1)).sum fun n0 => ‖termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖) ≤
(Finset.range (n + 1)).sum fun x => 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
tb : ∀ n0 ∈ Finset.range (n + 1), ‖termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖ ≤ b * r⁻¹ ^ n
⊢ ((Finset.range (n + 1)).sum fun n0 => ‖termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖) ≤
(Finset.range (n + 1)).sum fun x => b * r⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | series2_norm | [448, 1] | [464, 19] | clear tb | E : Type
inst✝² : NormedAddCommGroup E
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
tb : ∀ n0 ∈ Finset.range (n + 1), ‖termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖ ≤ b * r⁻¹ ^ n
⊢ ((Finset.range (n + 1)).sum fun x => b * r⁻¹ ^ n) ≤ (↑n + 1) * b * (r ^ n)⁻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
⊢ ((Finset.range (n + 1)).sum fun x => b * r⁻¹ ^ n) ≤ (↑n + 1) * b * (r ^ n)⁻¹ | 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
tb : ∀ n0 ∈ Finset.range (n + 1), ‖termCmmap ℂ n n0 (h.series2Coeff n0 (n - n0))‖ ≤ b * r⁻¹ ^ n
⊢ ((Finset.range (n + 1)).sum fun x => b * r⁻¹ ^ n) ≤ (↑n + 1) * b * (r ^ n)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | series2_norm | [448, 1] | [464, 19] | rw [Finset.sum_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
h : Separate f c0 c1 r b s
n : ℕ
⊢ ((Finset.range (n + 1)).sum fun x => b * r⁻¹ ^ n) ≤ (↑n + 1) * b * (r ^ n)⁻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
⊢ (Finset.range (n + 1)).card • (b * r⁻¹ ^ n) ≤ (↑n + 1) * b * (r ^ n)⁻¹ | 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
⊢ ((Finset.range (n + 1)).sum fun x => b * r⁻¹ ^ n) ≤ (↑n + 1) * b * (r ^ n)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | series2_norm | [448, 1] | [464, 19] | simp only [Finset.card_range, inv_pow, nsmul_eq_mul, Nat.cast_add, Nat.cast_one] | E : Type
inst✝² : NormedAddCommGroup E
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
⊢ (Finset.range (n + 1)).card • (b * r⁻¹ ^ n) ≤ (↑n + 1) * b * (r ^ n)⁻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
⊢ (↑n + 1) * (b * (r ^ n)⁻¹) ≤ (↑n + 1) * b * (r ^ n)⁻¹ | 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
⊢ (Finset.range (n + 1)).card • (b * r⁻¹ ^ n) ≤ (↑n + 1) * b * (r ^ n)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | series2_norm | [448, 1] | [464, 19] | 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
⊢ (↑n + 1) * (b * (r ^ n)⁻¹) ≤ (↑n + 1) * b * (r ^ n)⁻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
⊢ ↑n * b * r⁻¹ ^ n + b * r⁻¹ ^ n ≤ ↑n * b * r⁻¹ ^ n + 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
h : Separate f c0 c1 r b s
n : ℕ
⊢ (↑n + 1) * (b * (r ^ n)⁻¹) ≤ (↑n + 1) * b * (r ^ n)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | series2_norm | [448, 1] | [464, 19] | rfl | E : Type
inst✝² : NormedAddCommGroup E
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
⊢ ↑n * b * r⁻¹ ^ n + b * r⁻¹ ^ n ≤ ↑n * 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
h : Separate f c0 c1 r b s
n : ℕ
⊢ ↑n * b * r⁻¹ ^ n + b * r⁻¹ ^ n ≤ ↑n * b * r⁻¹ ^ n + b * r⁻¹ ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_radius | [467, 1] | [484, 51] | apply ENNReal.le_of_forall_nnreal_lt | E : Type
inst✝² : NormedAddCommGroup E
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
⊢ ENNReal.ofReal r ≤ (series2 h).radius | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
⊢ ∀ (r_1 : ℝ≥0), ↑r_1 < ENNReal.ofReal r → ↑r_1 ≤ (series2 h).radius | 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
⊢ ENNReal.ofReal r ≤ (series2 h).radius
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_radius | [467, 1] | [484, 51] | intro t tr | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
⊢ ∀ (r_1 : ℝ≥0), ↑r_1 < ENNReal.ofReal r → ↑r_1 ≤ (series2 h).radius | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
t : ℝ≥0
tr : ↑t < ENNReal.ofReal r
⊢ ↑t ≤ (series2 h).radius | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
⊢ ∀ (r_1 : ℝ≥0), ↑r_1 < ENNReal.ofReal r → ↑r_1 ≤ (series2 h).radius
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_radius | [467, 1] | [484, 51] | rw [←ENNReal.toReal_lt_toReal (@ENNReal.coe_ne_top t) (@ENNReal.ofReal_ne_top r)] at tr | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
t : ℝ≥0
tr : ↑t < ENNReal.ofReal r
⊢ ↑t ≤ (series2 h).radius | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
t : ℝ≥0
tr : (↑t).toReal < (ENNReal.ofReal r).toReal
⊢ ↑t ≤ (series2 h).radius | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
t : ℝ≥0
tr : ↑t < ENNReal.ofReal r
⊢ ↑t ≤ (series2 h).radius
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_radius | [467, 1] | [484, 51] | rw [ENNReal.coe_toReal, ENNReal.toReal_ofReal h.rp.le] at tr | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
t : ℝ≥0
tr : (↑t).toReal < (ENNReal.ofReal r).toReal
⊢ ↑t ≤ (series2 h).radius | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
t : ℝ≥0
tr : ↑t < r
⊢ ↑t ≤ (series2 h).radius | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
t : ℝ≥0
tr : (↑t).toReal < (ENNReal.ofReal r).toReal
⊢ ↑t ≤ (series2 h).radius
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_radius | [467, 1] | [484, 51] | apply FormalMultilinearSeries.le_radius_of_summable_nnnorm | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
t : ℝ≥0
tr : ↑t < r
⊢ ↑t ≤ (series2 h).radius | case h.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
t : ℝ≥0
tr : ↑t < r
⊢ Summable fun n => ‖series2 h n‖₊ * t ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
t : ℝ≥0
tr : ↑t < r
⊢ ↑t ≤ (series2 h).radius
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_radius | [467, 1] | [484, 51] | simp_rw [← norm_toNNReal, ← NNReal.summable_coe] | case h.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
t : ℝ≥0
tr : ↑t < r
⊢ Summable fun n => ‖series2 h n‖₊ * t ^ n | case h.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
t : ℝ≥0
tr : ↑t < r
⊢ Summable fun a => ↑(‖series2 h a‖.toNNReal * t ^ a) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.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
t : ℝ≥0
tr : ↑t < r
⊢ Summable fun n => ‖series2 h n‖₊ * t ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_radius | [467, 1] | [484, 51] | simp | case h.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
t : ℝ≥0
tr : ↑t < r
⊢ Summable fun a => ↑(‖series2 h a‖.toNNReal * t ^ a) | case h.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
t : ℝ≥0
tr : ↑t < r
⊢ Summable fun a => ‖series2 h a‖ * ↑t ^ a | Please generate a tactic in lean4 to solve the state.
STATE:
case h.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
t : ℝ≥0
tr : ↑t < r
⊢ Summable fun a => ↑(‖series2 h a‖.toNNReal * t ^ a)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_radius | [467, 1] | [484, 51] | have lo : ∀ n : ℕ, 0 ≤ ‖series2 h n‖ * (t:ℝ)^n := by intro; bound | case h.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
t : ℝ≥0
tr : ↑t < r
⊢ Summable fun a => ‖series2 h a‖ * ↑t ^ a | case h.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
t : ℝ≥0
tr : ↑t < r
lo : ∀ (n : ℕ), 0 ≤ ‖series2 h n‖ * ↑t ^ n
⊢ Summable fun a => ‖series2 h a‖ * ↑t ^ a | Please generate a tactic in lean4 to solve the state.
STATE:
case h.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
t : ℝ≥0
tr : ↑t < r
⊢ Summable fun a => ‖series2 h a‖ * ↑t ^ a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_radius | [467, 1] | [484, 51] | refine .of_nonneg_of_le lo hi ?_ | case h.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
t : ℝ≥0
tr : ↑t < r
lo : ∀ (n : ℕ), 0 ≤ ‖series2 h n‖ * ↑t ^ n
hi : ∀ (n : ℕ), ‖series2 h n‖ * ↑t ^ n ≤ (↑n + 1) * b * (↑t / r) ^ n
⊢ Summable fun a => ‖series2 h a‖ * ↑t ^ a | case h.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
t : ℝ≥0
tr : ↑t < r
lo : ∀ (n : ℕ), 0 ≤ ‖series2 h n‖ * ↑t ^ n
hi : ∀ (n : ℕ), ‖series2 h n‖ * ↑t ^ n ≤ (↑n + 1) * b * (↑t / r) ^ n
⊢ Summable fun b_1 => (↑b_1 + 1) * b * (↑t / r) ^ b_1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.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
t : ℝ≥0
tr : ↑t < r
lo : ∀ (n : ℕ), 0 ≤ ‖series2 h n‖ * ↑t ^ n
hi : ∀ (n : ℕ), ‖series2 h n‖ * ↑t ^ n ≤ (↑n + 1) * b * (↑t / r) ^ n
⊢ Summable fun a => ‖series2 h a‖ * ↑t ^ a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_radius | [467, 1] | [484, 51] | simp_rw [mul_comm _ b, mul_assoc b _ _] | case h.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
t : ℝ≥0
tr : ↑t < r
lo : ∀ (n : ℕ), 0 ≤ ‖series2 h n‖ * ↑t ^ n
hi : ∀ (n : ℕ), ‖series2 h n‖ * ↑t ^ n ≤ (↑n + 1) * b * (↑t / r) ^ n
⊢ Summable fun b_1 => (↑b_1 + 1) * b * (↑t / r) ^ b_1 | case h.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
t : ℝ≥0
tr : ↑t < r
lo : ∀ (n : ℕ), 0 ≤ ‖series2 h n‖ * ↑t ^ n
hi : ∀ (n : ℕ), ‖series2 h n‖ * ↑t ^ n ≤ (↑n + 1) * b * (↑t / r) ^ n
⊢ Summable fun b_1 => b * ((↑b_1 + 1) * (↑t / r) ^ b_1) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.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
t : ℝ≥0
tr : ↑t < r
lo : ∀ (n : ℕ), 0 ≤ ‖series2 h n‖ * ↑t ^ n
hi : ∀ (n : ℕ), ‖series2 h n‖ * ↑t ^ n ≤ (↑n + 1) * b * (↑t / r) ^ n
⊢ Summable fun b_1 => (↑b_1 + 1) * b * (↑t / r) ^ b_1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_radius | [467, 1] | [484, 51] | apply Summable.mul_left b | case h.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
t : ℝ≥0
tr : ↑t < r
lo : ∀ (n : ℕ), 0 ≤ ‖series2 h n‖ * ↑t ^ n
hi : ∀ (n : ℕ), ‖series2 h n‖ * ↑t ^ n ≤ (↑n + 1) * b * (↑t / r) ^ n
⊢ Summable fun b_1 => b * ((↑b_1 + 1) * (↑t / r) ^ b_1) | case h.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
t : ℝ≥0
tr : ↑t < r
lo : ∀ (n : ℕ), 0 ≤ ‖series2 h n‖ * ↑t ^ n
hi : ∀ (n : ℕ), ‖series2 h n‖ * ↑t ^ n ≤ (↑n + 1) * b * (↑t / r) ^ n
⊢ Summable fun i => (↑i + 1) * (↑t / r) ^ i | Please generate a tactic in lean4 to solve the state.
STATE:
case h.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
t : ℝ≥0
tr : ↑t < r
lo : ∀ (n : ℕ), 0 ≤ ‖series2 h n‖ * ↑t ^ n
hi : ∀ (n : ℕ), ‖series2 h n‖ * ↑t ^ n ≤ (↑n + 1) * b * (↑t / r) ^ n
⊢ Summable fun b_1 => b * ((↑b_1 + 1) * (↑t / r) ^ b_1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_radius | [467, 1] | [484, 51] | have trn : ‖↑t / r‖ < 1 := by simp; rw [abs_of_pos h.rp, div_lt_one h.rp]; assumption | case h.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
t : ℝ≥0
tr : ↑t < r
lo : ∀ (n : ℕ), 0 ≤ ‖series2 h n‖ * ↑t ^ n
hi : ∀ (n : ℕ), ‖series2 h n‖ * ↑t ^ n ≤ (↑n + 1) * b * (↑t / r) ^ n
⊢ Summable fun i => (↑i + 1) * (↑t / r) ^ i | case h.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
t : ℝ≥0
tr : ↑t < r
lo : ∀ (n : ℕ), 0 ≤ ‖series2 h n‖ * ↑t ^ n
hi : ∀ (n : ℕ), ‖series2 h n‖ * ↑t ^ n ≤ (↑n + 1) * b * (↑t / r) ^ n
trn : ‖↑t / r‖ < 1
⊢ Summable fun i => (↑i + 1) * (↑t / r) ^ i | Please generate a tactic in lean4 to solve the state.
STATE:
case h.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
t : ℝ≥0
tr : ↑t < r
lo : ∀ (n : ℕ), 0 ≤ ‖series2 h n‖ * ↑t ^ n
hi : ∀ (n : ℕ), ‖series2 h n‖ * ↑t ^ n ≤ (↑n + 1) * b * (↑t / r) ^ n
⊢ Summable fun i => (↑i + 1) * (↑t / r) ^ i
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_radius | [467, 1] | [484, 51] | simp_rw [right_distrib _ _ _, one_mul] | case h.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
t : ℝ≥0
tr : ↑t < r
lo : ∀ (n : ℕ), 0 ≤ ‖series2 h n‖ * ↑t ^ n
hi : ∀ (n : ℕ), ‖series2 h n‖ * ↑t ^ n ≤ (↑n + 1) * b * (↑t / r) ^ n
trn : ‖↑t / r‖ < 1
⊢ Summable fun i => (↑i + 1) * (↑t / r) ^ i | case h.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
t : ℝ≥0
tr : ↑t < r
lo : ∀ (n : ℕ), 0 ≤ ‖series2 h n‖ * ↑t ^ n
hi : ∀ (n : ℕ), ‖series2 h n‖ * ↑t ^ n ≤ (↑n + 1) * b * (↑t / r) ^ n
trn : ‖↑t / r‖ < 1
⊢ Summable fun i => ↑i * (↑t / r) ^ i + (↑t / r) ^ i | Please generate a tactic in lean4 to solve the state.
STATE:
case h.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
t : ℝ≥0
tr : ↑t < r
lo : ∀ (n : ℕ), 0 ≤ ‖series2 h n‖ * ↑t ^ n
hi : ∀ (n : ℕ), ‖series2 h n‖ * ↑t ^ n ≤ (↑n + 1) * b * (↑t / r) ^ n
trn : ‖↑t / r‖ < 1
⊢ Summable fun i => (↑i + 1) * (↑t / r) ^ i
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_radius | [467, 1] | [484, 51] | exact Summable.add (hasSum_coe_mul_geometric_of_norm_lt_one trn).summable
(hasSum_geometric_of_norm_lt_one trn).summable | case h.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
t : ℝ≥0
tr : ↑t < r
lo : ∀ (n : ℕ), 0 ≤ ‖series2 h n‖ * ↑t ^ n
hi : ∀ (n : ℕ), ‖series2 h n‖ * ↑t ^ n ≤ (↑n + 1) * b * (↑t / r) ^ n
trn : ‖↑t / r‖ < 1
⊢ Summable fun i => ↑i * (↑t / r) ^ i + (↑t / r) ^ i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.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
t : ℝ≥0
tr : ↑t < r
lo : ∀ (n : ℕ), 0 ≤ ‖series2 h n‖ * ↑t ^ n
hi : ∀ (n : ℕ), ‖series2 h n‖ * ↑t ^ n ≤ (↑n + 1) * b * (↑t / r) ^ n
trn : ‖↑t / r‖ < 1
⊢ Summable fun i => ↑i * (↑t / r) ^ i + (↑t / r) ^ i
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_radius | [467, 1] | [484, 51] | intro | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
t : ℝ≥0
tr : ↑t < r
⊢ ∀ (n : ℕ), 0 ≤ ‖series2 h n‖ * ↑t ^ 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
t : ℝ≥0
tr : ↑t < r
n✝ : ℕ
⊢ 0 ≤ ‖series2 h n✝‖ * ↑t ^ 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
t : ℝ≥0
tr : ↑t < r
⊢ ∀ (n : ℕ), 0 ≤ ‖series2 h n‖ * ↑t ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_radius | [467, 1] | [484, 51] | 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
t : ℝ≥0
tr : ↑t < r
n✝ : ℕ
⊢ 0 ≤ ‖series2 h n✝‖ * ↑t ^ 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
t : ℝ≥0
tr : ↑t < r
n✝ : ℕ
⊢ 0 ≤ ‖series2 h n✝‖ * ↑t ^ n✝
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_radius | [467, 1] | [484, 51] | intro 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
t : ℝ≥0
tr : ↑t < r
lo : ∀ (n : ℕ), 0 ≤ ‖series2 h n‖ * ↑t ^ n
⊢ ∀ (n : ℕ), ‖series2 h n‖ * ↑t ^ n ≤ (↑n + 1) * b * (↑t / 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
t : ℝ≥0
tr : ↑t < r
lo : ∀ (n : ℕ), 0 ≤ ‖series2 h n‖ * ↑t ^ n
n : ℕ
⊢ ‖series2 h n‖ * ↑t ^ n ≤ (↑n + 1) * b * (↑t / 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
t : ℝ≥0
tr : ↑t < r
lo : ∀ (n : ℕ), 0 ≤ ‖series2 h n‖ * ↑t ^ n
⊢ ∀ (n : ℕ), ‖series2 h n‖ * ↑t ^ n ≤ (↑n + 1) * b * (↑t / r) ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_radius | [467, 1] | [484, 51] | trans (↑n + 1) * b * r⁻¹ ^ n * (t: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
t : ℝ≥0
tr : ↑t < r
lo : ∀ (n : ℕ), 0 ≤ ‖series2 h n‖ * ↑t ^ n
n : ℕ
⊢ ‖series2 h n‖ * ↑t ^ n ≤ (↑n + 1) * b * (↑t / 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
t : ℝ≥0
tr : ↑t < r
lo : ∀ (n : ℕ), 0 ≤ ‖series2 h n‖ * ↑t ^ n
n : ℕ
⊢ ‖series2 h n‖ * ↑t ^ n ≤ (↑n + 1) * b * r⁻¹ ^ n * ↑t ^ 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
t : ℝ≥0
tr : ↑t < r
lo : ∀ (n : ℕ), 0 ≤ ‖series2 h n‖ * ↑t ^ n
n : ℕ
⊢ (↑n + 1) * b * r⁻¹ ^ n * ↑t ^ n ≤ (↑n + 1) * b * (↑t / 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
t : ℝ≥0
tr : ↑t < r
lo : ∀ (n : ℕ), 0 ≤ ‖series2 h n‖ * ↑t ^ n
n : ℕ
⊢ ‖series2 h n‖ * ↑t ^ n ≤ (↑n + 1) * b * (↑t / r) ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_radius | [467, 1] | [484, 51] | bound [series2_norm h 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
t : ℝ≥0
tr : ↑t < r
lo : ∀ (n : ℕ), 0 ≤ ‖series2 h n‖ * ↑t ^ n
n : ℕ
⊢ ‖series2 h n‖ * ↑t ^ n ≤ (↑n + 1) * b * r⁻¹ ^ n * ↑t ^ 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
t : ℝ≥0
tr : ↑t < r
lo : ∀ (n : ℕ), 0 ≤ ‖series2 h n‖ * ↑t ^ n
n : ℕ
⊢ ‖series2 h n‖ * ↑t ^ n ≤ (↑n + 1) * b * r⁻¹ ^ n * ↑t ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_radius | [467, 1] | [484, 51] | rw [mul_assoc ((↑n + 1) * b) _ _, ← mul_pow, inv_mul_eq_div] | E : Type
inst✝² : NormedAddCommGroup E
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
t : ℝ≥0
tr : ↑t < r
lo : ∀ (n : ℕ), 0 ≤ ‖series2 h n‖ * ↑t ^ n
n : ℕ
⊢ (↑n + 1) * b * r⁻¹ ^ n * ↑t ^ n ≤ (↑n + 1) * b * (↑t / 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
t : ℝ≥0
tr : ↑t < r
lo : ∀ (n : ℕ), 0 ≤ ‖series2 h n‖ * ↑t ^ n
n : ℕ
⊢ (↑n + 1) * b * r⁻¹ ^ n * ↑t ^ n ≤ (↑n + 1) * b * (↑t / r) ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_radius | [467, 1] | [484, 51] | 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
h : Separate f c0 c1 r b s
t : ℝ≥0
tr : ↑t < r
lo : ∀ (n : ℕ), 0 ≤ ‖series2 h n‖ * ↑t ^ n
hi : ∀ (n : ℕ), ‖series2 h n‖ * ↑t ^ n ≤ (↑n + 1) * b * (↑t / r) ^ n
⊢ ‖↑t / r‖ < 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
t : ℝ≥0
tr : ↑t < r
lo : ∀ (n : ℕ), 0 ≤ ‖series2 h n‖ * ↑t ^ n
hi : ∀ (n : ℕ), ‖series2 h n‖ * ↑t ^ n ≤ (↑n + 1) * b * (↑t / r) ^ n
⊢ ↑t / |r| < 1 | 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
t : ℝ≥0
tr : ↑t < r
lo : ∀ (n : ℕ), 0 ≤ ‖series2 h n‖ * ↑t ^ n
hi : ∀ (n : ℕ), ‖series2 h n‖ * ↑t ^ n ≤ (↑n + 1) * b * (↑t / r) ^ n
⊢ ‖↑t / r‖ < 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_radius | [467, 1] | [484, 51] | rw [abs_of_pos h.rp, div_lt_one h.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
h : Separate f c0 c1 r b s
t : ℝ≥0
tr : ↑t < r
lo : ∀ (n : ℕ), 0 ≤ ‖series2 h n‖ * ↑t ^ n
hi : ∀ (n : ℕ), ‖series2 h n‖ * ↑t ^ n ≤ (↑n + 1) * b * (↑t / r) ^ n
⊢ ↑t / |r| < 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
t : ℝ≥0
tr : ↑t < r
lo : ∀ (n : ℕ), 0 ≤ ‖series2 h n‖ * ↑t ^ n
hi : ∀ (n : ℕ), ‖series2 h n‖ * ↑t ^ n ≤ (↑n + 1) * b * (↑t / r) ^ n
⊢ ↑t < r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
t : ℝ≥0
tr : ↑t < r
lo : ∀ (n : ℕ), 0 ≤ ‖series2 h n‖ * ↑t ^ n
hi : ∀ (n : ℕ), ‖series2 h n‖ * ↑t ^ n ≤ (↑n + 1) * b * (↑t / r) ^ n
⊢ ↑t / |r| < 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_radius | [467, 1] | [484, 51] | assumption | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
t : ℝ≥0
tr : ↑t < r
lo : ∀ (n : ℕ), 0 ≤ ‖series2 h n‖ * ↑t ^ n
hi : ∀ (n : ℕ), ‖series2 h n‖ * ↑t ^ n ≤ (↑n + 1) * b * (↑t / r) ^ n
⊢ ↑t < 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
t : ℝ≥0
tr : ↑t < r
lo : ∀ (n : ℕ), 0 ≤ ‖series2 h n‖ * ↑t ^ n
hi : ∀ (n : ℕ), ‖series2 h n‖ * ↑t ^ n ≤ (↑n + 1) * b * (↑t / r) ^ n
⊢ ↑t < r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | generalize ha : f (c0 + w0, c1 + w1) = a | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
⊢ HasSum (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.1) (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
a : E
ha : f (c0 + w0, c1 + w1) = a
⊢ HasSum (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.1) a | 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 n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.1) (f (c0 + w0, c1 + w1))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | generalize hf : (fun n : ℕ × ℕ ↦ w0 ^ n.snd • w1 ^ n.fst • h.series2Coeff n.snd n.fst) = f | E : Type
inst✝² : NormedAddCommGroup E
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
a : E
ha : f (c0 + w0, c1 + w1) = a
⊢ HasSum (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.1) a | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.1) = f
⊢ HasSum f a | 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
a : E
ha : f (c0 + w0, c1 + w1) = a
⊢ HasSum (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.1) a
TACTIC:
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.