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 |
|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_bound | [89, 1] | [121, 92] | rw [cauchyPowerSeries_apply f c r n w, smul_eq_mul, Complex.abs.map_mul] | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
⊢ Complex.abs ((cauchyPowerSeries f c (↑r) n) fun x => w) ≤ wr | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
⊢ Complex.abs (2 * ↑π * I)⁻¹ * Complex.abs (∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • f z) ≤ wr |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_bound | [89, 1] | [121, 92] | generalize hg : (fun z ↦ (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
⊢ Complex.abs (2 * ↑π * I)⁻¹ * Complex.abs (∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • f z) ≤ wr | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
⊢ Complex.abs (2 * ↑π * I)⁻¹ * Complex.abs (circleIntegral g c ↑r) ≤ wr |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_bound | [89, 1] | [121, 92] | have gs : ∀ z ∈ sphere c r, ‖g z‖ ≤ wr * r⁻¹ := by
intro z; simp only [mem_sphere_iff_norm, Complex.norm_eq_abs]; intro zr
simp only [← hg, zr, div_pow, Algebra.id.smul_eq_mul, AbsoluteValue.map_mul, map_div₀,
Complex.abs_pow, map_inv₀]
have zb : z ∈ closedBall c r := by
simp only [Metric.mem_closedBall, ... | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
⊢ Complex.abs (2 * ↑π * I)⁻¹ * Complex.abs (circleIntegral g c ↑r) ≤ wr | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹
⊢ Complex.abs (2 * ↑π * I)⁻¹ * Complex.abs (circleIntegral g c ↑r) ≤ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_bound | [89, 1] | [121, 92] | have cn := circleIntegral.norm_integral_le_of_norm_le_const (NNReal.coe_nonneg r) gs | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹
⊢ Complex.abs (2 * ↑π * I)⁻¹ * Complex.abs (circleIntegral g c ↑r) ≤ ... | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹
cn : ‖∮ (z : ℂ) in C(c, ↑r), g z‖ ≤ 2 * π * ↑r * (wr * ↑r⁻¹)
⊢ Comple... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_bound | [89, 1] | [121, 92] | rw [Complex.norm_eq_abs] at cn | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹
cn : ‖∮ (z : ℂ) in C(c, ↑r), g z‖ ≤ 2 * π * ↑r * (wr * ↑r⁻¹)
⊢ Comple... | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹
cn : Complex.abs (∮ (z : ℂ) in C(c, ↑r), g z) ≤ 2 * π * ↑r * (wr * ↑r... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_bound | [89, 1] | [121, 92] | simp only [mul_inv_rev, Complex.inv_I, AbsoluteValue.map_neg, AbsoluteValue.map_mul,
Complex.abs_I, map_inv₀, Complex.abs_ofReal, Complex.abs_two, one_mul, div_pow,
Algebra.id.smul_eq_mul] at hg cn ⊢ | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹
cn : Complex.abs (∮ (z : ℂ) in C(c, ↑r), g z) ≤ 2 * π * ↑r * (wr * ↑r... | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹
cn : Complex.abs (∮ (z : ℂ) in C(c, ↑r), g z) ≤ 2 * π * ↑r * (wr * ↑r⁻¹)
hg : (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_bound | [89, 1] | [121, 92] | have p3 : |π| = π := abs_eq_self.mpr (by bound) | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹
cn : Complex.abs (∮ (z : ℂ) in C(c, ↑r), g z) ≤ 2 * π * ↑r * (wr * ↑r⁻¹)
hg : (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f ... | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹
cn : Complex.abs (∮ (z : ℂ) in C(c, ↑r), g z) ≤ 2 * π * ↑r * (wr * ↑r⁻¹)
hg : (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_bound | [89, 1] | [121, 92] | calc |π|⁻¹ * 2⁻¹ * abs (circleIntegral g c ↑r)
_ ≤ |π|⁻¹ * 2⁻¹ * (2 * π * r * (wr * r⁻¹)) := by bound
_ = π * |π|⁻¹ * (r * r⁻¹) * wr := by ring
_ = π * π⁻¹ * (r * r⁻¹) * wr := by rw [p3]
_ = 1 * (r * r⁻¹) * wr := by rw [mul_inv_cancel Real.pi_ne_zero]
_ = wr := by
rw [NNReal.coe_inv, mul_inv_cancel (NNRea... | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹
cn : Complex.abs (∮ (z : ℂ) in C(c, ↑r), g z) ≤ 2 * π * ↑r * (wr * ↑r⁻¹)
hg : (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_bound | [89, 1] | [121, 92] | intro z | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
⊢ ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹ | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
z : ℂ
⊢ z ∈ sphere c ↑r → ‖g z‖ ≤ wr * ↑r⁻¹ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_bound | [89, 1] | [121, 92] | simp only [mem_sphere_iff_norm, Complex.norm_eq_abs] | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
z : ℂ
⊢ z ∈ sphere c ↑r → ‖g z‖ ≤ wr * ↑r⁻¹ | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
z : ℂ
⊢ Complex.abs (z - c) = ↑r → Complex.abs (g z) ≤ wr * ↑r⁻¹ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_bound | [89, 1] | [121, 92] | intro zr | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
z : ℂ
⊢ Complex.abs (z - c) = ↑r → Complex.abs (g z) ≤ wr * ↑r⁻¹ | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
z : ℂ
zr : Complex.abs (z - c) = ↑r
⊢ Complex.abs (g z) ≤ wr * ↑r⁻¹ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_bound | [89, 1] | [121, 92] | simp only [← hg, zr, div_pow, Algebra.id.smul_eq_mul, AbsoluteValue.map_mul, map_div₀,
Complex.abs_pow, map_inv₀] | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
z : ℂ
zr : Complex.abs (z - c) = ↑r
⊢ Complex.abs (g z) ≤ wr * ↑r⁻¹ | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
z : ℂ
zr : Complex.abs (z - c) = ↑r
⊢ Complex.abs w ^ n / ↑r ^ n * ((↑r)⁻¹ * Complex.abs (f z)) ≤ wr * ↑r⁻¹ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_bound | [89, 1] | [121, 92] | have zb : z ∈ closedBall c r := by
simp only [Metric.mem_closedBall, dist_le_coe, ← NNReal.coe_le_coe, coe_nndist,
Complex.dist_eq, zr, le_refl] | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
z : ℂ
zr : Complex.abs (z - c) = ↑r
⊢ Complex.abs w ^ n / ↑r ^ n * ((↑r)⁻¹ * Complex.abs (f z)) ≤ wr * ↑r⁻¹ | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
z : ℂ
zr : Complex.abs (z - c) = ↑r
zb : z ∈ closedBall c ↑r
⊢ Complex.abs w ^ n / ↑r ^ n * ((↑r)⁻¹ * Complex.a... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_bound | [89, 1] | [121, 92] | have zs := h z zb | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
z : ℂ
zr : Complex.abs (z - c) = ↑r
zb : z ∈ closedBall c ↑r
⊢ Complex.abs w ^ n / ↑r ^ n * ((↑r)⁻¹ * Complex.a... | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
z : ℂ
zr : Complex.abs (z - c) = ↑r
zb : z ∈ closedBall c ↑r
zs : Complex.abs (f z) ≤ ↑d
⊢ Complex.abs w ^ n / ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_bound | [89, 1] | [121, 92] | calc abs w ^ n / ↑r ^ n * (r⁻¹ * abs (f z))
_ = abs w ^ n * (r⁻¹ ^ n : ℝ≥0) * (r⁻¹ * abs (f z)) := by
rw [div_eq_mul_inv, ← inv_pow, NNReal.coe_pow, NNReal.coe_inv]
_ ≤ abs w ^ n * r⁻¹ ^ n * (r⁻¹ * d) := by bound
_ = abs w ^ n * r⁻¹ ^ n * d * r⁻¹ := by ring
_ = wr * r⁻¹ := rfl | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
z : ℂ
zr : Complex.abs (z - c) = ↑r
zb : z ∈ closedBall c ↑r
zs : Complex.abs (f z) ≤ ↑d
⊢ Complex.abs w ^ n / ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_bound | [89, 1] | [121, 92] | simp only [Metric.mem_closedBall, dist_le_coe, ← NNReal.coe_le_coe, coe_nndist,
Complex.dist_eq, zr, le_refl] | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
z : ℂ
zr : Complex.abs (z - c) = ↑r
⊢ z ∈ closedBall c ↑r | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_bound | [89, 1] | [121, 92] | rw [div_eq_mul_inv, ← inv_pow, NNReal.coe_pow, NNReal.coe_inv] | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
z : ℂ
zr : Complex.abs (z - c) = ↑r
zb : z ∈ closedBall c ↑r
zs : Complex.abs (f z) ≤ ↑d
⊢ Complex.abs w ^ n / ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_bound | [89, 1] | [121, 92] | bound | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
z : ℂ
zr : Complex.abs (z - c) = ↑r
zb : z ∈ closedBall c ↑r
zs : Complex.abs (f z) ≤ ↑d
⊢ Complex.abs w ^ n * ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_bound | [89, 1] | [121, 92] | ring | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
z : ℂ
zr : Complex.abs (z - c) = ↑r
zb : z ∈ closedBall c ↑r
zs : Complex.abs (f z) ≤ ↑d
⊢ Complex.abs w ^ n * ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_bound | [89, 1] | [121, 92] | bound | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹
cn : Complex.abs (∮ (z : ℂ) in C(c, ↑r), g z) ≤ 2 * π * ↑r * (wr * ↑r⁻¹)
hg : (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_bound | [89, 1] | [121, 92] | bound | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹
cn : Complex.abs (∮ (z : ℂ) in C(c, ↑r), g z) ≤ 2 * π * ↑r * (wr * ↑r⁻¹)
hg : (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_bound | [89, 1] | [121, 92] | ring | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹
cn : Complex.abs (∮ (z : ℂ) in C(c, ↑r), g z) ≤ 2 * π * ↑r * (wr * ↑r⁻¹)
hg : (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_bound | [89, 1] | [121, 92] | rw [p3] | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹
cn : Complex.abs (∮ (z : ℂ) in C(c, ↑r), g z) ≤ 2 * π * ↑r * (wr * ↑r⁻¹)
hg : (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_bound | [89, 1] | [121, 92] | rw [mul_inv_cancel Real.pi_ne_zero] | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹
cn : Complex.abs (∮ (z : ℂ) in C(c, ↑r), g z) ≤ 2 * π * ↑r * (wr * ↑r⁻¹)
hg : (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_bound | [89, 1] | [121, 92] | rw [NNReal.coe_inv, mul_inv_cancel (NNReal.coe_ne_zero.mpr rp.ne'), one_mul, one_mul] | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹
cn : Complex.abs (∮ (z : ℂ) in C(c, ↑r), g z) ≤ 2 * π * ↑r * (wr * ↑r⁻¹)
hg : (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | circleIntegral_sub | [123, 1] | [138, 46] | rw [circleIntegral] | f g : ℂ → ℂ
c : ℂ
r : ℝ
fi : CircleIntegrable f c r
gi : CircleIntegrable g c r
⊢ circleIntegral f c r - circleIntegral g c r = circleIntegral (f - g) c r | f g : ℂ → ℂ
c : ℂ
r : ℝ
fi : CircleIntegrable f c r
gi : CircleIntegrable g c r
⊢ (∫ (θ : ℝ) in 0 ..2 * π, deriv (circleMap c r) θ • f (circleMap c r θ)) - circleIntegral g c r =
circleIntegral (f - g) c r |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | circleIntegral_sub | [123, 1] | [138, 46] | generalize hf : (fun θ : ℝ ↦ deriv (circleMap c r) θ • f (circleMap c r θ)) = fc | f g : ℂ → ℂ
c : ℂ
r : ℝ
fi : CircleIntegrable f c r
gi : CircleIntegrable g c r
⊢ (∫ (θ : ℝ) in 0 ..2 * π, deriv (circleMap c r) θ • f (circleMap c r θ)) - circleIntegral g c r =
circleIntegral (f - g) c r | f g : ℂ → ℂ
c : ℂ
r : ℝ
fi : CircleIntegrable f c r
gi : CircleIntegrable g c r
fc : ℝ → ℂ
hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc
⊢ intervalIntegral fc 0 (2 * π) volume - circleIntegral g c r = circleIntegral (f - g) c r |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | circleIntegral_sub | [123, 1] | [138, 46] | rw [circleIntegral] | f g : ℂ → ℂ
c : ℂ
r : ℝ
fi : CircleIntegrable f c r
gi : CircleIntegrable g c r
fc : ℝ → ℂ
hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc
⊢ intervalIntegral fc 0 (2 * π) volume - circleIntegral g c r = circleIntegral (f - g) c r | f g : ℂ → ℂ
c : ℂ
r : ℝ
fi : CircleIntegrable f c r
gi : CircleIntegrable g c r
fc : ℝ → ℂ
hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc
⊢ intervalIntegral fc 0 (2 * π) volume - ∫ (θ : ℝ) in 0 ..2 * π, deriv (circleMap c r) θ • g (circleMap c r θ) =
circleIntegral (f - g) c r |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | circleIntegral_sub | [123, 1] | [138, 46] | generalize hg : (fun θ : ℝ ↦ deriv (circleMap c r) θ • g (circleMap c r θ)) = gc | f g : ℂ → ℂ
c : ℂ
r : ℝ
fi : CircleIntegrable f c r
gi : CircleIntegrable g c r
fc : ℝ → ℂ
hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc
⊢ intervalIntegral fc 0 (2 * π) volume - ∫ (θ : ℝ) in 0 ..2 * π, deriv (circleMap c r) θ • g (circleMap c r θ) =
circleIntegral (f - g) c r | f g : ℂ → ℂ
c : ℂ
r : ℝ
fi : CircleIntegrable f c r
gi : CircleIntegrable g c r
fc : ℝ → ℂ
hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc
gc : ℝ → ℂ
hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc
⊢ intervalIntegral fc 0 (2 * π) volume - intervalIntegral gc 0 (2 * π) volume = circ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | circleIntegral_sub | [123, 1] | [138, 46] | rw [circleIntegral] | f g : ℂ → ℂ
c : ℂ
r : ℝ
fi : CircleIntegrable f c r
gi : CircleIntegrable g c r
fc : ℝ → ℂ
hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc
gc : ℝ → ℂ
hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc
⊢ intervalIntegral fc 0 (2 * π) volume - intervalIntegral gc 0 (2 * π) volume = circ... | f g : ℂ → ℂ
c : ℂ
r : ℝ
fi : CircleIntegrable f c r
gi : CircleIntegrable g c r
fc : ℝ → ℂ
hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc
gc : ℝ → ℂ
hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc
⊢ intervalIntegral fc 0 (2 * π) volume - intervalIntegral gc 0 (2 * π) volume =
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | circleIntegral_sub | [123, 1] | [138, 46] | generalize hfg : (fun θ : ℝ ↦ deriv (circleMap c r) θ • (f - g) (circleMap c r θ)) = fgc | f g : ℂ → ℂ
c : ℂ
r : ℝ
fi : CircleIntegrable f c r
gi : CircleIntegrable g c r
fc : ℝ → ℂ
hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc
gc : ℝ → ℂ
hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc
⊢ intervalIntegral fc 0 (2 * π) volume - intervalIntegral gc 0 (2 * π) volume =
... | f g : ℂ → ℂ
c : ℂ
r : ℝ
fi : CircleIntegrable f c r
gi : CircleIntegrable g c r
fc : ℝ → ℂ
hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc
gc : ℝ → ℂ
hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc
fgc : ℝ → ℂ
hfg : (fun θ => deriv (circleMap c r) θ • (f - g) (circleMap c r θ)) = f... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | circleIntegral_sub | [123, 1] | [138, 46] | have hs : fc - gc = fgc := by
rw [← hf, ← hg, ← hfg]; funext
simp only [deriv_circleMap, Algebra.id.smul_eq_mul, Pi.sub_apply, mul_sub_left_distrib] | f g : ℂ → ℂ
c : ℂ
r : ℝ
fi : CircleIntegrable f c r
gi : CircleIntegrable g c r
fc : ℝ → ℂ
hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc
gc : ℝ → ℂ
hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc
fgc : ℝ → ℂ
hfg : (fun θ => deriv (circleMap c r) θ • (f - g) (circleMap c r θ)) = f... | f g : ℂ → ℂ
c : ℂ
r : ℝ
fi : CircleIntegrable f c r
gi : CircleIntegrable g c r
fc : ℝ → ℂ
hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc
gc : ℝ → ℂ
hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc
fgc : ℝ → ℂ
hfg : (fun θ => deriv (circleMap c r) θ • (f - g) (circleMap c r θ)) = f... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | circleIntegral_sub | [123, 1] | [138, 46] | rw [← hs] | f g : ℂ → ℂ
c : ℂ
r : ℝ
fi : CircleIntegrable f c r
gi : CircleIntegrable g c r
fc : ℝ → ℂ
hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc
gc : ℝ → ℂ
hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc
fgc : ℝ → ℂ
hfg : (fun θ => deriv (circleMap c r) θ • (f - g) (circleMap c r θ)) = f... | f g : ℂ → ℂ
c : ℂ
r : ℝ
fi : CircleIntegrable f c r
gi : CircleIntegrable g c r
fc : ℝ → ℂ
hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc
gc : ℝ → ℂ
hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc
fgc : ℝ → ℂ
hfg : (fun θ => deriv (circleMap c r) θ • (f - g) (circleMap c r θ)) = f... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | circleIntegral_sub | [123, 1] | [138, 46] | clear hfg hs fgc | f g : ℂ → ℂ
c : ℂ
r : ℝ
fi : CircleIntegrable f c r
gi : CircleIntegrable g c r
fc : ℝ → ℂ
hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc
gc : ℝ → ℂ
hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc
fgc : ℝ → ℂ
hfg : (fun θ => deriv (circleMap c r) θ • (f - g) (circleMap c r θ)) = f... | f g : ℂ → ℂ
c : ℂ
r : ℝ
fi : CircleIntegrable f c r
gi : CircleIntegrable g c r
fc : ℝ → ℂ
hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc
gc : ℝ → ℂ
hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc
⊢ intervalIntegral fc 0 (2 * π) volume - intervalIntegral gc 0 (2 * π) volume =
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | circleIntegral_sub | [123, 1] | [138, 46] | symm | f g : ℂ → ℂ
c : ℂ
r : ℝ
fi : CircleIntegrable f c r
gi : CircleIntegrable g c r
fc : ℝ → ℂ
hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc
gc : ℝ → ℂ
hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc
⊢ intervalIntegral fc 0 (2 * π) volume - intervalIntegral gc 0 (2 * π) volume =
... | f g : ℂ → ℂ
c : ℂ
r : ℝ
fi : CircleIntegrable f c r
gi : CircleIntegrable g c r
fc : ℝ → ℂ
hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc
gc : ℝ → ℂ
hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc
⊢ intervalIntegral (fc - gc) 0 (2 * π) volume =
intervalIntegral fc 0 (2 * π) vo... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | circleIntegral_sub | [123, 1] | [138, 46] | have fci := CircleIntegrable.out fi | f g : ℂ → ℂ
c : ℂ
r : ℝ
fi : CircleIntegrable f c r
gi : CircleIntegrable g c r
fc : ℝ → ℂ
hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc
gc : ℝ → ℂ
hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc
⊢ intervalIntegral (fc - gc) 0 (2 * π) volume =
intervalIntegral fc 0 (2 * π) vo... | f g : ℂ → ℂ
c : ℂ
r : ℝ
fi : CircleIntegrable f c r
gi : CircleIntegrable g c r
fc : ℝ → ℂ
hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc
gc : ℝ → ℂ
hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc
fci : IntervalIntegrable (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) vo... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | circleIntegral_sub | [123, 1] | [138, 46] | rw [hf] at fci | f g : ℂ → ℂ
c : ℂ
r : ℝ
fi : CircleIntegrable f c r
gi : CircleIntegrable g c r
fc : ℝ → ℂ
hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc
gc : ℝ → ℂ
hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc
fci : IntervalIntegrable (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) vo... | f g : ℂ → ℂ
c : ℂ
r : ℝ
fi : CircleIntegrable f c r
gi : CircleIntegrable g c r
fc : ℝ → ℂ
hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc
gc : ℝ → ℂ
hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc
fci : IntervalIntegrable fc volume 0 (2 * π)
⊢ intervalIntegral (fc - gc) 0 (2 * π) ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | circleIntegral_sub | [123, 1] | [138, 46] | have gci := CircleIntegrable.out gi | f g : ℂ → ℂ
c : ℂ
r : ℝ
fi : CircleIntegrable f c r
gi : CircleIntegrable g c r
fc : ℝ → ℂ
hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc
gc : ℝ → ℂ
hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc
fci : IntervalIntegrable fc volume 0 (2 * π)
⊢ intervalIntegral (fc - gc) 0 (2 * π) ... | f g : ℂ → ℂ
c : ℂ
r : ℝ
fi : CircleIntegrable f c r
gi : CircleIntegrable g c r
fc : ℝ → ℂ
hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc
gc : ℝ → ℂ
hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc
fci : IntervalIntegrable fc volume 0 (2 * π)
gci : IntervalIntegrable (fun θ => deri... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | circleIntegral_sub | [123, 1] | [138, 46] | rw [hg] at gci | f g : ℂ → ℂ
c : ℂ
r : ℝ
fi : CircleIntegrable f c r
gi : CircleIntegrable g c r
fc : ℝ → ℂ
hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc
gc : ℝ → ℂ
hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc
fci : IntervalIntegrable fc volume 0 (2 * π)
gci : IntervalIntegrable (fun θ => deri... | f g : ℂ → ℂ
c : ℂ
r : ℝ
fi : CircleIntegrable f c r
gi : CircleIntegrable g c r
fc : ℝ → ℂ
hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc
gc : ℝ → ℂ
hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc
fci : IntervalIntegrable fc volume 0 (2 * π)
gci : IntervalIntegrable gc volume 0 (2... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | circleIntegral_sub | [123, 1] | [138, 46] | exact intervalIntegral.integral_sub fci gci | f g : ℂ → ℂ
c : ℂ
r : ℝ
fi : CircleIntegrable f c r
gi : CircleIntegrable g c r
fc : ℝ → ℂ
hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc
gc : ℝ → ℂ
hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc
fci : IntervalIntegrable fc volume 0 (2 * π)
gci : IntervalIntegrable gc volume 0 (2... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | circleIntegral_sub | [123, 1] | [138, 46] | rw [← hf, ← hg, ← hfg] | f g : ℂ → ℂ
c : ℂ
r : ℝ
fi : CircleIntegrable f c r
gi : CircleIntegrable g c r
fc : ℝ → ℂ
hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc
gc : ℝ → ℂ
hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc
fgc : ℝ → ℂ
hfg : (fun θ => deriv (circleMap c r) θ • (f - g) (circleMap c r θ)) = f... | f g : ℂ → ℂ
c : ℂ
r : ℝ
fi : CircleIntegrable f c r
gi : CircleIntegrable g c r
fc : ℝ → ℂ
hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc
gc : ℝ → ℂ
hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc
fgc : ℝ → ℂ
hfg : (fun θ => deriv (circleMap c r) θ • (f - g) (circleMap c r θ)) = f... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | circleIntegral_sub | [123, 1] | [138, 46] | funext | f g : ℂ → ℂ
c : ℂ
r : ℝ
fi : CircleIntegrable f c r
gi : CircleIntegrable g c r
fc : ℝ → ℂ
hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc
gc : ℝ → ℂ
hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc
fgc : ℝ → ℂ
hfg : (fun θ => deriv (circleMap c r) θ • (f - g) (circleMap c r θ)) = f... | case h
f g : ℂ → ℂ
c : ℂ
r : ℝ
fi : CircleIntegrable f c r
gi : CircleIntegrable g c r
fc : ℝ → ℂ
hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc
gc : ℝ → ℂ
hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc
fgc : ℝ → ℂ
hfg : (fun θ => deriv (circleMap c r) θ • (f - g) (circleMap c r ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | circleIntegral_sub | [123, 1] | [138, 46] | simp only [deriv_circleMap, Algebra.id.smul_eq_mul, Pi.sub_apply, mul_sub_left_distrib] | case h
f g : ℂ → ℂ
c : ℂ
r : ℝ
fi : CircleIntegrable f c r
gi : CircleIntegrable g c r
fc : ℝ → ℂ
hf : (fun θ => deriv (circleMap c r) θ • f (circleMap c r θ)) = fc
gc : ℝ → ℂ
hg : (fun θ => deriv (circleMap c r) θ • g (circleMap c r θ)) = gc
fgc : ℝ → ℂ
hfg : (fun θ => deriv (circleMap c r) θ • (f - g) (circleMap c r ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | circleMap_nz | [140, 1] | [142, 67] | simp only [circleMap_sub_center, Ne, circleMap_eq_center_iff, NNReal.coe_eq_zero] | c : ℂ
r : ℝ≥0
θ : ℝ
rp : r > 0
⊢ circleMap c (↑r) θ - c ≠ 0 | c : ℂ
r : ℝ≥0
θ : ℝ
rp : r > 0
⊢ ¬r = 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | circleMap_nz | [140, 1] | [142, 67] | intro h | c : ℂ
r : ℝ≥0
θ : ℝ
rp : r > 0
⊢ ¬r = 0 | c : ℂ
r : ℝ≥0
θ : ℝ
rp : r > 0
h : r = 0
⊢ False |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | circleMap_nz | [140, 1] | [142, 67] | rw [h] at rp | c : ℂ
r : ℝ≥0
θ : ℝ
rp : r > 0
h : r = 0
⊢ False | c : ℂ
r : ℝ≥0
θ : ℝ
rp : 0 > 0
h : r = 0
⊢ False |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | circleMap_nz | [140, 1] | [142, 67] | simp only [gt_iff_lt, not_lt_zero'] at rp | c : ℂ
r : ℝ≥0
θ : ℝ
rp : 0 > 0
h : r = 0
⊢ False | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_is_circleIntegrable | [144, 1] | [157, 74] | refine ContinuousOn.intervalIntegrable ?_ | f : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
h : ContinuousOn f (closedBall c ↑r)
⊢ CircleIntegrable (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) c ↑r | f : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
h : ContinuousOn f (closedBall c ↑r)
⊢ ContinuousOn (fun θ => (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) (circleMap c (↑r) θ)) (Set.uIcc 0 (2 * π)) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_is_circleIntegrable | [144, 1] | [157, 74] | refine ContinuousOn.mul ?_ ?_ | f : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
h : ContinuousOn f (closedBall c ↑r)
⊢ ContinuousOn (fun θ => (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) (circleMap c (↑r) θ)) (Set.uIcc 0 (2 * π)) | case refine_1
f : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
h : ContinuousOn f (closedBall c ↑r)
⊢ ContinuousOn (fun θ => w ^ n / (circleMap c (↑r) θ - c) ^ n) (Set.uIcc 0 (2 * π))
case refine_2
f : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
h : ContinuousOn f (closedBall c ↑r)
⊢ ContinuousOn (fun θ => (circleMap c (... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_is_circleIntegrable | [144, 1] | [157, 74] | refine ContinuousOn.mul continuousOn_const ?_ | case refine_1
f : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
h : ContinuousOn f (closedBall c ↑r)
⊢ ContinuousOn (fun θ => w ^ n / (circleMap c (↑r) θ - c) ^ n) (Set.uIcc 0 (2 * π)) | case refine_1
f : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
h : ContinuousOn f (closedBall c ↑r)
⊢ ContinuousOn (fun θ => ((circleMap c (↑r) θ - c) ^ n)⁻¹) (Set.uIcc 0 (2 * π)) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_is_circleIntegrable | [144, 1] | [157, 74] | apply Continuous.continuousOn | case refine_1
f : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
h : ContinuousOn f (closedBall c ↑r)
⊢ ContinuousOn (fun θ => ((circleMap c (↑r) θ - c) ^ n)⁻¹) (Set.uIcc 0 (2 * π)) | case refine_1.h
f : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
h : ContinuousOn f (closedBall c ↑r)
⊢ Continuous fun θ => ((circleMap c (↑r) θ - c) ^ n)⁻¹ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_is_circleIntegrable | [144, 1] | [157, 74] | refine Continuous.inv₀ ?_ fun x ↦ pow_ne_zero n (circleMap_nz rp) | case refine_1.h
f : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
h : ContinuousOn f (closedBall c ↑r)
⊢ Continuous fun θ => ((circleMap c (↑r) θ - c) ^ n)⁻¹ | case refine_1.h
f : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
h : ContinuousOn f (closedBall c ↑r)
⊢ Continuous fun θ => (circleMap c (↑r) θ - c) ^ n |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_is_circleIntegrable | [144, 1] | [157, 74] | apply Continuous.pow | case refine_1.h
f : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
h : ContinuousOn f (closedBall c ↑r)
⊢ Continuous fun θ => (circleMap c (↑r) θ - c) ^ n | case refine_1.h.h
f : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
h : ContinuousOn f (closedBall c ↑r)
⊢ Continuous fun b => circleMap c (↑r) b - c |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_is_circleIntegrable | [144, 1] | [157, 74] | continuity | case refine_1.h.h
f : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
h : ContinuousOn f (closedBall c ↑r)
⊢ Continuous fun b => circleMap c (↑r) b - c | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_is_circleIntegrable | [144, 1] | [157, 74] | refine ContinuousOn.mul ?_ ?_ | case refine_2
f : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
h : ContinuousOn f (closedBall c ↑r)
⊢ ContinuousOn (fun θ => (circleMap c (↑r) θ - c)⁻¹ * f (circleMap c (↑r) θ)) (Set.uIcc 0 (2 * π)) | case refine_2.refine_1
f : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
h : ContinuousOn f (closedBall c ↑r)
⊢ ContinuousOn (fun θ => (circleMap c (↑r) θ - c)⁻¹) (Set.uIcc 0 (2 * π))
case refine_2.refine_2
f : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
h : ContinuousOn f (closedBall c ↑r)
⊢ ContinuousOn (fun θ => f (cir... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_is_circleIntegrable | [144, 1] | [157, 74] | apply Continuous.continuousOn | case refine_2.refine_1
f : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
h : ContinuousOn f (closedBall c ↑r)
⊢ ContinuousOn (fun θ => (circleMap c (↑r) θ - c)⁻¹) (Set.uIcc 0 (2 * π)) | case refine_2.refine_1.h
f : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
h : ContinuousOn f (closedBall c ↑r)
⊢ Continuous fun θ => (circleMap c (↑r) θ - c)⁻¹ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_is_circleIntegrable | [144, 1] | [157, 74] | refine Continuous.inv₀ (by continuity) fun x ↦ circleMap_nz rp | case refine_2.refine_1.h
f : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
h : ContinuousOn f (closedBall c ↑r)
⊢ Continuous fun θ => (circleMap c (↑r) θ - c)⁻¹ | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_is_circleIntegrable | [144, 1] | [157, 74] | continuity | f : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
h : ContinuousOn f (closedBall c ↑r)
⊢ Continuous fun θ => circleMap c (↑r) θ - c | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_is_circleIntegrable | [144, 1] | [157, 74] | refine ContinuousOn.comp h (Continuous.continuousOn (by continuity)) ?_ | case refine_2.refine_2
f : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
h : ContinuousOn f (closedBall c ↑r)
⊢ ContinuousOn (fun θ => f (circleMap c (↑r) θ)) (Set.uIcc 0 (2 * π)) | case refine_2.refine_2
f : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
h : ContinuousOn f (closedBall c ↑r)
⊢ Set.MapsTo (fun θ => circleMap c (↑r) θ) (Set.uIcc 0 (2 * π)) (closedBall c ↑r) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_is_circleIntegrable | [144, 1] | [157, 74] | intro θ _ | case refine_2.refine_2
f : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
h : ContinuousOn f (closedBall c ↑r)
⊢ Set.MapsTo (fun θ => circleMap c (↑r) θ) (Set.uIcc 0 (2 * π)) (closedBall c ↑r) | case refine_2.refine_2
f : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
h : ContinuousOn f (closedBall c ↑r)
θ : ℝ
a✝ : θ ∈ Set.uIcc 0 (2 * π)
⊢ (fun θ => circleMap c (↑r) θ) θ ∈ closedBall c ↑r |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_is_circleIntegrable | [144, 1] | [157, 74] | exact circleMap_mem_closedBall c (NNReal.coe_nonneg r) θ | case refine_2.refine_2
f : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
h : ContinuousOn f (closedBall c ↑r)
θ : ℝ
a✝ : θ ∈ Set.uIcc 0 (2 * π)
⊢ (fun θ => circleMap c (↑r) θ) θ ∈ closedBall c ↑r | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_is_circleIntegrable | [144, 1] | [157, 74] | continuity | f : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
h : ContinuousOn f (closedBall c ↑r)
⊢ Continuous fun θ => circleMap c (↑r) θ | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_sub | [159, 1] | [178, 41] | rw [cauchyPowerSeries_apply f c r n w] | f g : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
cf : ContinuousOn f (closedBall c ↑r)
cg : ContinuousOn g (closedBall c ↑r)
⊢ (((cauchyPowerSeries f c (↑r) n) fun x => w) - (cauchyPowerSeries g c (↑r) n) fun x => w) =
(cauchyPowerSeries (f - g) c (↑r) n) fun x => w | f g : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
cf : ContinuousOn f (closedBall c ↑r)
cg : ContinuousOn g (closedBall c ↑r)
⊢ (((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • f z) -
(cauchyPowerSeries g c (↑r) n) fun x => w) =
(cauchyPowerSeries (f - g) c (↑r) n) fun x => w |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_sub | [159, 1] | [178, 41] | rw [cauchyPowerSeries_apply g c r n w] | f g : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
cf : ContinuousOn f (closedBall c ↑r)
cg : ContinuousOn g (closedBall c ↑r)
⊢ (((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • f z) -
(cauchyPowerSeries g c (↑r) n) fun x => w) =
(cauchyPowerSeries (f - g) c (↑r) n) fun x => w | f g : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
cf : ContinuousOn f (closedBall c ↑r)
cg : ContinuousOn g (closedBall c ↑r)
⊢ (((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • f z) -
(2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • g z) =
(cauchyPowerSeries (f -... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_sub | [159, 1] | [178, 41] | rw [cauchyPowerSeries_apply (f - g) c r n w] | f g : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
cf : ContinuousOn f (closedBall c ↑r)
cg : ContinuousOn g (closedBall c ↑r)
⊢ (((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • f z) -
(2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • g z) =
(cauchyPowerSeries (f -... | f g : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
cf : ContinuousOn f (closedBall c ↑r)
cg : ContinuousOn g (closedBall c ↑r)
⊢ (((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • f z) -
(2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • g z) =
(2 * ↑π * I)⁻¹ • ∮ (z :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_sub | [159, 1] | [178, 41] | set s : ℂ := (2 * π * I)⁻¹ | f g : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
cf : ContinuousOn f (closedBall c ↑r)
cg : ContinuousOn g (closedBall c ↑r)
⊢ (((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • f z) -
(2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • g z) =
(2 * ↑π * I)⁻¹ • ∮ (z :... | f g : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
cf : ContinuousOn f (closedBall c ↑r)
cg : ContinuousOn g (closedBall c ↑r)
s : ℂ := (2 * ↑π * I)⁻¹
⊢ ((s • ∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • f z) -
s • ∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • g z) =
s • ∮ (z : ℂ) in C(c, ↑r)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_sub | [159, 1] | [178, 41] | simp only [div_pow, Algebra.id.smul_eq_mul, Pi.sub_apply] | f g : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
cf : ContinuousOn f (closedBall c ↑r)
cg : ContinuousOn g (closedBall c ↑r)
s : ℂ := (2 * ↑π * I)⁻¹
⊢ ((s • ∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • f z) -
s • ∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • g z) =
s • ∮ (z : ℂ) in C(c, ↑r)... | f g : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
cf : ContinuousOn f (closedBall c ↑r)
cg : ContinuousOn g (closedBall c ↑r)
s : ℂ := (2 * ↑π * I)⁻¹
⊢ ((s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) -
s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) =
s * ∮ (z : ℂ) in ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_sub | [159, 1] | [178, 41] | have fi := cauchy_is_circleIntegrable n w rp cf | f g : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
cf : ContinuousOn f (closedBall c ↑r)
cg : ContinuousOn g (closedBall c ↑r)
s : ℂ := (2 * ↑π * I)⁻¹
⊢ ((s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) -
s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) =
s * ∮ (z : ℂ) in ... | f g : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
cf : ContinuousOn f (closedBall c ↑r)
cg : ContinuousOn g (closedBall c ↑r)
s : ℂ := (2 * ↑π * I)⁻¹
fi : CircleIntegrable (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) c ↑r
⊢ ((s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) -
s * ∮ (z : ℂ)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_sub | [159, 1] | [178, 41] | have gi := cauchy_is_circleIntegrable n w rp cg | f g : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
cf : ContinuousOn f (closedBall c ↑r)
cg : ContinuousOn g (closedBall c ↑r)
s : ℂ := (2 * ↑π * I)⁻¹
fi : CircleIntegrable (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) c ↑r
⊢ ((s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) -
s * ∮ (z : ℂ)... | f g : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
cf : ContinuousOn f (closedBall c ↑r)
cg : ContinuousOn g (closedBall c ↑r)
s : ℂ := (2 * ↑π * I)⁻¹
fi : CircleIntegrable (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) c ↑r
gi : CircleIntegrable (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) c ↑r
⊢ ((s * ∮ (z : ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_sub | [159, 1] | [178, 41] | have cia := circleIntegral_sub fi gi | f g : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
cf : ContinuousOn f (closedBall c ↑r)
cg : ContinuousOn g (closedBall c ↑r)
s : ℂ := (2 * ↑π * I)⁻¹
fi : CircleIntegrable (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) c ↑r
gi : CircleIntegrable (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) c ↑r
⊢ ((s * ∮ (z : ... | f g : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
cf : ContinuousOn f (closedBall c ↑r)
cg : ContinuousOn g (closedBall c ↑r)
s : ℂ := (2 * ↑π * I)⁻¹
fi : CircleIntegrable (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) c ↑r
gi : CircleIntegrable (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) c ↑r
cia :
((∮ (z ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_sub | [159, 1] | [178, 41] | rw [← mul_sub_left_distrib, cia] | f g : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
cf : ContinuousOn f (closedBall c ↑r)
cg : ContinuousOn g (closedBall c ↑r)
s : ℂ := (2 * ↑π * I)⁻¹
fi : CircleIntegrable (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) c ↑r
gi : CircleIntegrable (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) c ↑r
cia :
((∮ (z ... | f g : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
cf : ContinuousOn f (closedBall c ↑r)
cg : ContinuousOn g (closedBall c ↑r)
s : ℂ := (2 * ↑π * I)⁻¹
fi : CircleIntegrable (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) c ↑r
gi : CircleIntegrable (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) c ↑r
cia :
((∮ (z ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_sub | [159, 1] | [178, 41] | clear cia fi gi cf cg rp | f g : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
cf : ContinuousOn f (closedBall c ↑r)
cg : ContinuousOn g (closedBall c ↑r)
s : ℂ := (2 * ↑π * I)⁻¹
fi : CircleIntegrable (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) c ↑r
gi : CircleIntegrable (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) c ↑r
cia :
((∮ (z ... | f g : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
s : ℂ := (2 * ↑π * I)⁻¹
⊢ s *
circleIntegral
((fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) - fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) c ↑r =
s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * (f z - g z)) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_sub | [159, 1] | [178, 41] | have flip :
((fun z ↦ w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) - fun z ↦
w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) =
fun z ↦ w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z) - w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z) :=
rfl | f g : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
s : ℂ := (2 * ↑π * I)⁻¹
⊢ s *
circleIntegral
((fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) - fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) c ↑r =
s * ∮ (z : ℂ) in C(c, ↑r), w ^ n / (z - c) ^ n * ((z - c)⁻¹ * (f z - g z)) | f g : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
s : ℂ := (2 * ↑π * I)⁻¹
flip :
((fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) - fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) = fun z =>
w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z) - w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)
⊢ s *
circleIntegral
((fun z => w ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_sub | [159, 1] | [178, 41] | simp only [flip, mul_sub_left_distrib] | f g : ℂ → ℂ
c : ℂ
r : ℝ≥0
n : ℕ
w : ℂ
s : ℂ := (2 * ↑π * I)⁻¹
flip :
((fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z)) - fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)) = fun z =>
w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f z) - w ^ n / (z - c) ^ n * ((z - c)⁻¹ * g z)
⊢ s *
circleIntegral
((fun z => w ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_dist | [180, 1] | [186, 69] | rw [Complex.dist_eq, cauchy_sub n w rp cf cg] | f g : ℂ → ℂ
c : ℂ
r d : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
cf : ContinuousOn f (closedBall c ↑r)
cg : ContinuousOn g (closedBall c ↑r)
h : ∀ z ∈ closedBall c ↑r, Complex.abs (f z - g z) ≤ ↑d
⊢ dist ((cauchyPowerSeries f c (↑r) n) fun x => w) ((cauchyPowerSeries g c (↑r) n) fun x => w) ≤
Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d | f g : ℂ → ℂ
c : ℂ
r d : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
cf : ContinuousOn f (closedBall c ↑r)
cg : ContinuousOn g (closedBall c ↑r)
h : ∀ z ∈ closedBall c ↑r, Complex.abs (f z - g z) ≤ ↑d
⊢ Complex.abs ((cauchyPowerSeries (f - g) c (↑r) n) fun x => w) ≤ Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_dist | [180, 1] | [186, 69] | refine cauchy_bound rp ?_ | f g : ℂ → ℂ
c : ℂ
r d : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
cf : ContinuousOn f (closedBall c ↑r)
cg : ContinuousOn g (closedBall c ↑r)
h : ∀ z ∈ closedBall c ↑r, Complex.abs (f z - g z) ≤ ↑d
⊢ Complex.abs ((cauchyPowerSeries (f - g) c (↑r) n) fun x => w) ≤ Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d | f g : ℂ → ℂ
c : ℂ
r d : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
cf : ContinuousOn f (closedBall c ↑r)
cg : ContinuousOn g (closedBall c ↑r)
h : ∀ z ∈ closedBall c ↑r, Complex.abs (f z - g z) ≤ ↑d
⊢ ∀ w ∈ closedBall c ↑r, Complex.abs ((f - g) w) ≤ ↑d |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_dist | [180, 1] | [186, 69] | intro z zr | f g : ℂ → ℂ
c : ℂ
r d : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
cf : ContinuousOn f (closedBall c ↑r)
cg : ContinuousOn g (closedBall c ↑r)
h : ∀ z ∈ closedBall c ↑r, Complex.abs (f z - g z) ≤ ↑d
⊢ ∀ w ∈ closedBall c ↑r, Complex.abs ((f - g) w) ≤ ↑d | f g : ℂ → ℂ
c : ℂ
r d : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
cf : ContinuousOn f (closedBall c ↑r)
cg : ContinuousOn g (closedBall c ↑r)
h : ∀ z ∈ closedBall c ↑r, Complex.abs (f z - g z) ≤ ↑d
z : ℂ
zr : z ∈ closedBall c ↑r
⊢ Complex.abs ((f - g) z) ≤ ↑d |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_dist | [180, 1] | [186, 69] | simp at h zr | f g : ℂ → ℂ
c : ℂ
r d : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
cf : ContinuousOn f (closedBall c ↑r)
cg : ContinuousOn g (closedBall c ↑r)
h : ∀ z ∈ closedBall c ↑r, Complex.abs (f z - g z) ≤ ↑d
z : ℂ
zr : z ∈ closedBall c ↑r
⊢ Complex.abs ((f - g) z) ≤ ↑d | f g : ℂ → ℂ
c : ℂ
r d : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
cf : ContinuousOn f (closedBall c ↑r)
cg : ContinuousOn g (closedBall c ↑r)
z : ℂ
h : ∀ (z : ℂ), nndist z c ≤ r → Complex.abs (f z - g z) ≤ ↑d
zr : nndist z c ≤ r
⊢ Complex.abs ((f - g) z) ≤ ↑d |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_dist | [180, 1] | [186, 69] | refine h z zr | f g : ℂ → ℂ
c : ℂ
r d : ℝ≥0
n : ℕ
w : ℂ
rp : r > 0
cf : ContinuousOn f (closedBall c ↑r)
cg : ContinuousOn g (closedBall c ↑r)
z : ℂ
h : ∀ (z : ℂ), nndist z c ≤ r → Complex.abs (f z - g z) ≤ ↑d
zr : nndist z c ≤ r
⊢ Complex.abs ((f - g) z) ≤ ↑d | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | intro c hc | I : Type
inst✝¹ : Lattice I
inst✝ : Nonempty I
f : I → ℂ → ℂ
g : ℂ → ℂ
s : Set ℂ
o : IsOpen s
h : ∀ (n : I), AnalyticOn ℂ (f n) s
u : TendstoUniformlyOn f g atTop s
⊢ AnalyticOn ℂ g s | I : Type
inst✝¹ : Lattice I
inst✝ : Nonempty I
f : I → ℂ → ℂ
g : ℂ → ℂ
s : Set ℂ
o : IsOpen s
h : ∀ (n : I), AnalyticOn ℂ (f n) s
u : TendstoUniformlyOn f g atTop s
c : ℂ
hc : c ∈ s
⊢ AnalyticAt ℂ g c |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | rcases Metric.nhds_basis_closedBall.mem_iff.mp (o.mem_nhds hc) with ⟨r, rp, cb⟩ | I : Type
inst✝¹ : Lattice I
inst✝ : Nonempty I
f : I → ℂ → ℂ
g : ℂ → ℂ
s : Set ℂ
o : IsOpen s
h : ∀ (n : I), AnalyticOn ℂ (f n) s
u : TendstoUniformlyOn f g atTop s
c : ℂ
hc : c ∈ s
⊢ AnalyticAt ℂ g c | case intro.intro
I : Type
inst✝¹ : Lattice I
inst✝ : Nonempty I
f : I → ℂ → ℂ
g : ℂ → ℂ
s : Set ℂ
o : IsOpen s
h : ∀ (n : I), AnalyticOn ℂ (f n) s
u : TendstoUniformlyOn f g atTop s
c : ℂ
hc : c ∈ s
r : ℝ
rp : 0 < r
cb : closedBall c r ⊆ s
⊢ AnalyticAt ℂ g c |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | lift r to ℝ≥0 using rp.le | case intro.intro
I : Type
inst✝¹ : Lattice I
inst✝ : Nonempty I
f : I → ℂ → ℂ
g : ℂ → ℂ
s : Set ℂ
o : IsOpen s
h : ∀ (n : I), AnalyticOn ℂ (f n) s
u : TendstoUniformlyOn f g atTop s
c : ℂ
hc : c ∈ s
r : ℝ
rp : 0 < r
cb : closedBall c r ⊆ s
⊢ AnalyticAt ℂ g c | case intro.intro.intro
I : Type
inst✝¹ : Lattice I
inst✝ : Nonempty I
f : I → ℂ → ℂ
g : ℂ → ℂ
s : Set ℂ
o : IsOpen s
h : ∀ (n : I), AnalyticOn ℂ (f n) s
u : TendstoUniformlyOn f g atTop s
c : ℂ
hc : c ∈ s
r : ℝ≥0
rp : 0 < ↑r
cb : closedBall c ↑r ⊆ s
⊢ AnalyticAt ℂ g c |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | simp only [NNReal.coe_pos] at rp | case intro.intro.intro
I : Type
inst✝¹ : Lattice I
inst✝ : Nonempty I
f : I → ℂ → ℂ
g : ℂ → ℂ
s : Set ℂ
o : IsOpen s
h : ∀ (n : I), AnalyticOn ℂ (f n) s
u : TendstoUniformlyOn f g atTop s
c : ℂ
hc : c ∈ s
r : ℝ≥0
rp : 0 < ↑r
cb : closedBall c ↑r ⊆ s
⊢ AnalyticAt ℂ g c | case intro.intro.intro
I : Type
inst✝¹ : Lattice I
inst✝ : Nonempty I
f : I → ℂ → ℂ
g : ℂ → ℂ
s : Set ℂ
o : IsOpen s
h : ∀ (n : I), AnalyticOn ℂ (f n) s
u : TendstoUniformlyOn f g atTop s
c : ℂ
hc : c ∈ s
r : ℝ≥0
cb : closedBall c ↑r ⊆ s
rp : 0 < r
⊢ AnalyticAt ℂ g c |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | have hb : ∀ n, AnalyticOn ℂ (f n) (closedBall c r) := fun n ↦ (h n).mono cb | case intro.intro.intro
I : Type
inst✝¹ : Lattice I
inst✝ : Nonempty I
f : I → ℂ → ℂ
g : ℂ → ℂ
s : Set ℂ
o : IsOpen s
h : ∀ (n : I), AnalyticOn ℂ (f n) s
u : TendstoUniformlyOn f g atTop s
c : ℂ
hc : c ∈ s
r : ℝ≥0
cb : closedBall c ↑r ⊆ s
rp : 0 < r
⊢ AnalyticAt ℂ g c | case intro.intro.intro
I : Type
inst✝¹ : Lattice I
inst✝ : Nonempty I
f : I → ℂ → ℂ
g : ℂ → ℂ
s : Set ℂ
o : IsOpen s
h : ∀ (n : I), AnalyticOn ℂ (f n) s
u : TendstoUniformlyOn f g atTop s
c : ℂ
hc : c ∈ s
r : ℝ≥0
cb : closedBall c ↑r ⊆ s
rp : 0 < r
hb : ∀ (n : I), AnalyticOn ℂ (f n) (closedBall c ↑r)
⊢ AnalyticAt ℂ g c |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | set pr := fun n ↦ cauchyPowerSeries (f n) c r | case intro.intro.intro
I : Type
inst✝¹ : Lattice I
inst✝ : Nonempty I
f : I → ℂ → ℂ
g : ℂ → ℂ
s : Set ℂ
o : IsOpen s
h : ∀ (n : I), AnalyticOn ℂ (f n) s
u : TendstoUniformlyOn f g atTop s
c : ℂ
hc : c ∈ s
r : ℝ≥0
cb : closedBall c ↑r ⊆ s
rp : 0 < r
hb : ∀ (n : I), AnalyticOn ℂ (f n) (closedBall c ↑r)
⊢ AnalyticAt ℂ g c | case intro.intro.intro
I : Type
inst✝¹ : Lattice I
inst✝ : Nonempty I
f : I → ℂ → ℂ
g : ℂ → ℂ
s : Set ℂ
o : IsOpen s
h : ∀ (n : I), AnalyticOn ℂ (f n) s
u : TendstoUniformlyOn f g atTop s
c : ℂ
hc : c ∈ s
r : ℝ≥0
cb : closedBall c ↑r ⊆ s
rp : 0 < r
hb : ∀ (n : I), AnalyticOn ℂ (f n) (closedBall c ↑r)
pr : I → FormalMul... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | have hpf : ∀ n, HasFPowerSeriesOnBall (f n) (pr n) c r := by
intro n
have cs := cauchy_on_cball_radius rp (hb n)
have pn : pr n = cauchyPowerSeries (f n) c r := rfl
rw [← pn] at cs; exact cs | case intro.intro.intro
I : Type
inst✝¹ : Lattice I
inst✝ : Nonempty I
f : I → ℂ → ℂ
g : ℂ → ℂ
s : Set ℂ
o : IsOpen s
h : ∀ (n : I), AnalyticOn ℂ (f n) s
u : TendstoUniformlyOn f g atTop s
c : ℂ
hc : c ∈ s
r : ℝ≥0
cb : closedBall c ↑r ⊆ s
rp : 0 < r
hb : ∀ (n : I), AnalyticOn ℂ (f n) (closedBall c ↑r)
pr : I → FormalMul... | case intro.intro.intro
I : Type
inst✝¹ : Lattice I
inst✝ : Nonempty I
f : I → ℂ → ℂ
g : ℂ → ℂ
s : Set ℂ
o : IsOpen s
h : ∀ (n : I), AnalyticOn ℂ (f n) s
u : TendstoUniformlyOn f g atTop s
c : ℂ
hc : c ∈ s
r : ℝ≥0
cb : closedBall c ↑r ⊆ s
rp : 0 < r
hb : ∀ (n : I), AnalyticOn ℂ (f n) (closedBall c ↑r)
pr : I → FormalMul... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | have cfs : ∀ n, ContinuousOn (f n) s := fun n ↦ AnalyticOn.continuousOn (h n) | case intro.intro.intro
I : Type
inst✝¹ : Lattice I
inst✝ : Nonempty I
f : I → ℂ → ℂ
g : ℂ → ℂ
s : Set ℂ
o : IsOpen s
h : ∀ (n : I), AnalyticOn ℂ (f n) s
u : TendstoUniformlyOn f g atTop s
c : ℂ
hc : c ∈ s
r : ℝ≥0
cb : closedBall c ↑r ⊆ s
rp : 0 < r
hb : ∀ (n : I), AnalyticOn ℂ (f n) (closedBall c ↑r)
pr : I → FormalMul... | case intro.intro.intro
I : Type
inst✝¹ : Lattice I
inst✝ : Nonempty I
f : I → ℂ → ℂ
g : ℂ → ℂ
s : Set ℂ
o : IsOpen s
h : ∀ (n : I), AnalyticOn ℂ (f n) s
u : TendstoUniformlyOn f g atTop s
c : ℂ
hc : c ∈ s
r : ℝ≥0
cb : closedBall c ↑r ⊆ s
rp : 0 < r
hb : ∀ (n : I), AnalyticOn ℂ (f n) (closedBall c ↑r)
pr : I → FormalMul... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | have cf : ∀ n, ContinuousOn (f n) (closedBall c r) := fun n ↦ ContinuousOn.mono (cfs n) cb | case intro.intro.intro
I : Type
inst✝¹ : Lattice I
inst✝ : Nonempty I
f : I → ℂ → ℂ
g : ℂ → ℂ
s : Set ℂ
o : IsOpen s
h : ∀ (n : I), AnalyticOn ℂ (f n) s
u : TendstoUniformlyOn f g atTop s
c : ℂ
hc : c ∈ s
r : ℝ≥0
cb : closedBall c ↑r ⊆ s
rp : 0 < r
hb : ∀ (n : I), AnalyticOn ℂ (f n) (closedBall c ↑r)
pr : I → FormalMul... | case intro.intro.intro
I : Type
inst✝¹ : Lattice I
inst✝ : Nonempty I
f : I → ℂ → ℂ
g : ℂ → ℂ
s : Set ℂ
o : IsOpen s
h : ∀ (n : I), AnalyticOn ℂ (f n) s
u : TendstoUniformlyOn f g atTop s
c : ℂ
hc : c ∈ s
r : ℝ≥0
cb : closedBall c ↑r ⊆ s
rp : 0 < r
hb : ∀ (n : I), AnalyticOn ℂ (f n) (closedBall c ↑r)
pr : I → FormalMul... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | have cg : ContinuousOn g (closedBall c r) :=
ContinuousOn.mono (TendstoUniformlyOn.continuousOn u (Filter.eventually_of_forall cfs)) cb | case intro.intro.intro
I : Type
inst✝¹ : Lattice I
inst✝ : Nonempty I
f : I → ℂ → ℂ
g : ℂ → ℂ
s : Set ℂ
o : IsOpen s
h : ∀ (n : I), AnalyticOn ℂ (f n) s
u : TendstoUniformlyOn f g atTop s
c : ℂ
hc : c ∈ s
r : ℝ≥0
cb : closedBall c ↑r ⊆ s
rp : 0 < r
hb : ∀ (n : I), AnalyticOn ℂ (f n) (closedBall c ↑r)
pr : I → FormalMul... | case intro.intro.intro
I : Type
inst✝¹ : Lattice I
inst✝ : Nonempty I
f : I → ℂ → ℂ
g : ℂ → ℂ
s : Set ℂ
o : IsOpen s
h : ∀ (n : I), AnalyticOn ℂ (f n) s
u : TendstoUniformlyOn f g atTop s
c : ℂ
hc : c ∈ s
r : ℝ≥0
cb : closedBall c ↑r ⊆ s
rp : 0 < r
hb : ∀ (n : I), AnalyticOn ℂ (f n) (closedBall c ↑r)
pr : I → FormalMul... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | clear h hb hc o cfs | case intro.intro.intro
I : Type
inst✝¹ : Lattice I
inst✝ : Nonempty I
f : I → ℂ → ℂ
g : ℂ → ℂ
s : Set ℂ
o : IsOpen s
h : ∀ (n : I), AnalyticOn ℂ (f n) s
u : TendstoUniformlyOn f g atTop s
c : ℂ
hc : c ∈ s
r : ℝ≥0
cb : closedBall c ↑r ⊆ s
rp : 0 < r
hb : ∀ (n : I), AnalyticOn ℂ (f n) (closedBall c ↑r)
pr : I → FormalMul... | case intro.intro.intro
I : Type
inst✝¹ : Lattice I
inst✝ : Nonempty I
f : I → ℂ → ℂ
g : ℂ → ℂ
s : Set ℂ
u : TendstoUniformlyOn f g atTop s
c : ℂ
r : ℝ≥0
cb : closedBall c ↑r ⊆ s
rp : 0 < r
pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r
hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | set p := cauchyPowerSeries g c r | case intro.intro.intro
I : Type
inst✝¹ : Lattice I
inst✝ : Nonempty I
f : I → ℂ → ℂ
g : ℂ → ℂ
s : Set ℂ
u : TendstoUniformlyOn f g atTop s
c : ℂ
r : ℝ≥0
cb : closedBall c ↑r ⊆ s
rp : 0 < r
pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r
hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n)... | case intro.intro.intro
I : Type
inst✝¹ : Lattice I
inst✝ : Nonempty I
f : I → ℂ → ℂ
g : ℂ → ℂ
s : Set ℂ
u : TendstoUniformlyOn f g atTop s
c : ℂ
r : ℝ≥0
cb : closedBall c ↑r ⊆ s
rp : 0 < r
pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r
hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | refine
HasFPowerSeriesOnBall.analyticAt
{ r_le := le_radius_cauchyPowerSeries g c r
r_pos := ENNReal.coe_pos.mpr rp
hasSum := ?_ } | case intro.intro.intro
I : Type
inst✝¹ : Lattice I
inst✝ : Nonempty I
f : I → ℂ → ℂ
g : ℂ → ℂ
s : Set ℂ
u : TendstoUniformlyOn f g atTop s
c : ℂ
r : ℝ≥0
cb : closedBall c ↑r ⊆ s
rp : 0 < r
pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r
hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n)... | case intro.intro.intro
I : Type
inst✝¹ : Lattice I
inst✝ : Nonempty I
f : I → ℂ → ℂ
g : ℂ → ℂ
s : Set ℂ
u : TendstoUniformlyOn f g atTop s
c : ℂ
r : ℝ≥0
cb : closedBall c ↑r ⊆ s
rp : 0 < r
pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r
hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | intro y yb | case intro.intro.intro
I : Type
inst✝¹ : Lattice I
inst✝ : Nonempty I
f : I → ℂ → ℂ
g : ℂ → ℂ
s : Set ℂ
u : TendstoUniformlyOn f g atTop s
c : ℂ
r : ℝ≥0
cb : closedBall c ↑r ⊆ s
rp : 0 < r
pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r
hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n)... | case intro.intro.intro
I : Type
inst✝¹ : Lattice I
inst✝ : Nonempty I
f : I → ℂ → ℂ
g : ℂ → ℂ
s : Set ℂ
u : TendstoUniformlyOn f g atTop s
c : ℂ
r : ℝ≥0
cb : closedBall c ↑r ⊆ s
rp : 0 < r
pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r
hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | have yr := yb | case intro.intro.intro
I : Type
inst✝¹ : Lattice I
inst✝ : Nonempty I
f : I → ℂ → ℂ
g : ℂ → ℂ
s : Set ℂ
u : TendstoUniformlyOn f g atTop s
c : ℂ
r : ℝ≥0
cb : closedBall c ↑r ⊆ s
rp : 0 < r
pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r
hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n)... | case intro.intro.intro
I : Type
inst✝¹ : Lattice I
inst✝ : Nonempty I
f : I → ℂ → ℂ
g : ℂ → ℂ
s : Set ℂ
u : TendstoUniformlyOn f g atTop s
c : ℂ
r : ℝ≥0
cb : closedBall c ↑r ⊆ s
rp : 0 < r
pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r
hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | simp at yr | case intro.intro.intro
I : Type
inst✝¹ : Lattice I
inst✝ : Nonempty I
f : I → ℂ → ℂ
g : ℂ → ℂ
s : Set ℂ
u : TendstoUniformlyOn f g atTop s
c : ℂ
r : ℝ≥0
cb : closedBall c ↑r ⊆ s
rp : 0 < r
pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r
hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n)... | case intro.intro.intro
I : Type
inst✝¹ : Lattice I
inst✝ : Nonempty I
f : I → ℂ → ℂ
g : ℂ → ℂ
s : Set ℂ
u : TendstoUniformlyOn f g atTop s
c : ℂ
r : ℝ≥0
cb : closedBall c ↑r ⊆ s
rp : 0 < r
pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r
hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | set a := abs y / r | case intro.intro.intro
I : Type
inst✝¹ : Lattice I
inst✝ : Nonempty I
f : I → ℂ → ℂ
g : ℂ → ℂ
s : Set ℂ
u : TendstoUniformlyOn f g atTop s
c : ℂ
r : ℝ≥0
cb : closedBall c ↑r ⊆ s
rp : 0 < r
pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r
hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n)... | case intro.intro.intro
I : Type
inst✝¹ : Lattice I
inst✝ : Nonempty I
f : I → ℂ → ℂ
g : ℂ → ℂ
s : Set ℂ
u : TendstoUniformlyOn f g atTop s
c : ℂ
r : ℝ≥0
cb : closedBall c ↑r ⊆ s
rp : 0 < r
pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r
hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | have a0 : a ≥ 0 := by bound | case intro.intro.intro
I : Type
inst✝¹ : Lattice I
inst✝ : Nonempty I
f : I → ℂ → ℂ
g : ℂ → ℂ
s : Set ℂ
u : TendstoUniformlyOn f g atTop s
c : ℂ
r : ℝ≥0
cb : closedBall c ↑r ⊆ s
rp : 0 < r
pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r
hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n)... | case intro.intro.intro
I : Type
inst✝¹ : Lattice I
inst✝ : Nonempty I
f : I → ℂ → ℂ
g : ℂ → ℂ
s : Set ℂ
u : TendstoUniformlyOn f g atTop s
c : ℂ
r : ℝ≥0
cb : closedBall c ↑r ⊆ s
rp : 0 < r
pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r
hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | have a1 : a < 1 := (div_lt_one (NNReal.coe_pos.mpr rp)).mpr yr | case intro.intro.intro
I : Type
inst✝¹ : Lattice I
inst✝ : Nonempty I
f : I → ℂ → ℂ
g : ℂ → ℂ
s : Set ℂ
u : TendstoUniformlyOn f g atTop s
c : ℂ
r : ℝ≥0
cb : closedBall c ↑r ⊆ s
rp : 0 < r
pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r
hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n)... | case intro.intro.intro
I : Type
inst✝¹ : Lattice I
inst✝ : Nonempty I
f : I → ℂ → ℂ
g : ℂ → ℂ
s : Set ℂ
u : TendstoUniformlyOn f g atTop s
c : ℂ
r : ℝ≥0
cb : closedBall c ↑r ⊆ s
rp : 0 < r
pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r
hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | have a1p : 1 - a > 0 := by bound | case intro.intro.intro
I : Type
inst✝¹ : Lattice I
inst✝ : Nonempty I
f : I → ℂ → ℂ
g : ℂ → ℂ
s : Set ℂ
u : TendstoUniformlyOn f g atTop s
c : ℂ
r : ℝ≥0
cb : closedBall c ↑r ⊆ s
rp : 0 < r
pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r
hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n)... | case intro.intro.intro
I : Type
inst✝¹ : Lattice I
inst✝ : Nonempty I
f : I → ℂ → ℂ
g : ℂ → ℂ
s : Set ℂ
u : TendstoUniformlyOn f g atTop s
c : ℂ
r : ℝ≥0
cb : closedBall c ↑r ⊆ s
rp : 0 < r
pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r
hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | rw [HasSum, Metric.tendsto_atTop] | case intro.intro.intro
I : Type
inst✝¹ : Lattice I
inst✝ : Nonempty I
f : I → ℂ → ℂ
g : ℂ → ℂ
s : Set ℂ
u : TendstoUniformlyOn f g atTop s
c : ℂ
r : ℝ≥0
cb : closedBall c ↑r ⊆ s
rp : 0 < r
pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r
hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n)... | case intro.intro.intro
I : Type
inst✝¹ : Lattice I
inst✝ : Nonempty I
f : I → ℂ → ℂ
g : ℂ → ℂ
s : Set ℂ
u : TendstoUniformlyOn f g atTop s
c : ℂ
r : ℝ≥0
cb : closedBall c ↑r ⊆ s
rp : 0 < r
pr : I → FormalMultilinearSeries ℂ ℂ ℂ := fun n => cauchyPowerSeries (f n) c ↑r
hpf : ∀ (n : I), HasFPowerSeriesOnBall (f n) (pr n)... |
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