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/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | HasFPowerSeriesAt.dslope_order_eventually_ne_zero | [66, 1] | [74, 24] | refine ContinuousAt.eventually_ne ?h (hp.iterate_dslope_fslope_ne_zero h) | E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
p : FormalMultilinearSeries ℂ ℂ E
U : Set ℂ
z₀ c : ℂ
n : ℕ
hp : HasFPowerSeriesAt f p z₀
h : p ≠ 0
⊢ ∀ᶠ (z : ℂ) in 𝓝 z₀, (swap dslope z₀)^[FormalMultilinearSeries.order p] f z ≠ 0 | case h
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
p : FormalMultilinearSeries ℂ ℂ E
U : Set ℂ
z₀ c : ℂ
n : ℕ
hp : HasFPowerSeriesAt f p z₀
h : p ≠ 0
⊢ ContinuousAt (fun z => (swap dslope z₀)^[FormalMultilinearSeries.order p] f z) z₀ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | HasFPowerSeriesAt.dslope_order_eventually_ne_zero | [66, 1] | [74, 24] | obtain ⟨r, hf⟩ := hp | case h
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
p : FormalMultilinearSeries ℂ ℂ E
U : Set ℂ
z₀ c : ℂ
n : ℕ
hp : HasFPowerSeriesAt f p z₀
h : p ≠ 0
⊢ ContinuousAt (fun z => (swap dslope z₀)^[FormalMultilinearSeries.order p] f z) z₀ | case h.intro
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
p : FormalMultilinearSeries ℂ ℂ E
U : Set ℂ
z₀ c : ℂ
n : ℕ
h : p ≠ 0
r : ENNReal
hf : HasFPowerSeriesOnBall f p z₀ r
⊢ ContinuousAt (fun z => (swap dslope z₀)^[FormalMultilinearSeries.order p] f z) z₀ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | HasFPowerSeriesAt.dslope_order_eventually_ne_zero | [66, 1] | [74, 24] | have hr : 0 < r := hf.r_pos | case h.intro
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
p : FormalMultilinearSeries ℂ ℂ E
U : Set ℂ
z₀ c : ℂ
n : ℕ
h : p ≠ 0
r : ENNReal
hf : HasFPowerSeriesOnBall f p z₀ r
⊢ ContinuousAt (fun z => (swap dslope z₀)^[FormalMultilinearSeries.order p] f z) z₀ | case h.intro
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
p : FormalMultilinearSeries ℂ ℂ E
U : Set ℂ
z₀ c : ℂ
n : ℕ
h : p ≠ 0
r : ENNReal
hf : HasFPowerSeriesOnBall f p z₀ r
hr : 0 < r
⊢ ContinuousAt (fun z => (swap dslope z₀)^[FormalMultilinearSeries.order p] f z) z₀ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | HasFPowerSeriesAt.dslope_order_eventually_ne_zero | [66, 1] | [74, 24] | refine ContinuousOn.continuousAt ?h1 (EMetric.ball_mem_nhds _ hr) | case h.intro
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
p : FormalMultilinearSeries ℂ ℂ E
U : Set ℂ
z₀ c : ℂ
n : ℕ
h : p ≠ 0
r : ENNReal
hf : HasFPowerSeriesOnBall f p z₀ r
hr : 0 < r
⊢ ContinuousAt (fun z => (swap dslope z₀)^[FormalMultilinearSeries.order p] f z) z₀ | case h1
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
p : FormalMultilinearSeries ℂ ℂ E
U : Set ℂ
z₀ c : ℂ
n : ℕ
h : p ≠ 0
r : ENNReal
hf : HasFPowerSeriesOnBall f p z₀ r
hr : 0 < r
⊢ ContinuousOn (fun z => (swap dslope z₀)^[FormalMultilinearSeries.order p] f z) (EMetric.ball z₀ r) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | HasFPowerSeriesAt.dslope_order_eventually_ne_zero | [66, 1] | [74, 24] | have hh : DifferentiableOn ℂ (iterate (swap dslope z₀) p.order f) (EMetric.ball z₀ r) :=
DifferentiableOn.iterate_dslope hf.differentiableOn EMetric.isOpen_ball (EMetric.mem_ball_self hr) | case h1
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
p : FormalMultilinearSeries ℂ ℂ E
U : Set ℂ
z₀ c : ℂ
n : ℕ
h : p ≠ 0
r : ENNReal
hf : HasFPowerSeriesOnBall f p z₀ r
hr : 0 < r
⊢ ContinuousOn (fun z => (swap dslope z₀)^[FormalMultilinearSeries.order p] f z) (EMetric.ball z₀ r) | case h1
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
p : FormalMultilinearSeries ℂ ℂ E
U : Set ℂ
z₀ c : ℂ
n : ℕ
h : p ≠ 0
r : ENNReal
hf : HasFPowerSeriesOnBall f p z₀ r
hr : 0 < r
hh : DifferentiableOn ℂ ((swap dslope z₀)^[FormalMultilinearSeries.order p] f) (EMetric.ball z₀ r)
⊢ ContinuousOn (fun z => (swap dslope z₀)^[FormalMultilinearSeries.order p] f z) (EMetric.ball z₀ r) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | HasFPowerSeriesAt.dslope_order_eventually_ne_zero | [66, 1] | [74, 24] | exact hh.continuousOn | case h1
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
p : FormalMultilinearSeries ℂ ℂ E
U : Set ℂ
z₀ c : ℂ
n : ℕ
h : p ≠ 0
r : ENNReal
hf : HasFPowerSeriesOnBall f p z₀ r
hr : 0 < r
hh : DifferentiableOn ℂ ((swap dslope z₀)^[FormalMultilinearSeries.order p] f) (EMetric.ball z₀ r)
⊢ ContinuousOn (fun z => (swap dslope z₀)^[FormalMultilinearSeries.order p] f z) (EMetric.ball z₀ r) | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | deriv_div_self_eq_div_add_deriv_div_self | [80, 1] | [92, 9] | have h1 : DifferentiableAt ℂ (λ y => (y - z₀) ^ n) z :=
((differentiable_id'.sub_const z₀).pow n).differentiableAt | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hg : DifferentiableAt ℂ g z
hgz : g z ≠ 0
hfg : f =ᶠ[𝓝 z] fun w => (w - z₀) ^ n * g w
hz : z ≠ z₀
⊢ deriv f z / f z = ↑n / (z - z₀) + deriv g z / g z | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hg : DifferentiableAt ℂ g z
hgz : g z ≠ 0
hfg : f =ᶠ[𝓝 z] fun w => (w - z₀) ^ n * g w
hz : z ≠ z₀
h1 : DifferentiableAt ℂ (fun y => (y - z₀) ^ n) z
⊢ deriv f z / f z = ↑n / (z - z₀) + deriv g z / g z |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | deriv_div_self_eq_div_add_deriv_div_self | [80, 1] | [92, 9] | have h4 : DifferentiableAt ℂ (λ y => y - z₀) z := (differentiable_id'.sub_const z₀).differentiableAt | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hg : DifferentiableAt ℂ g z
hgz : g z ≠ 0
hfg : f =ᶠ[𝓝 z] fun w => (w - z₀) ^ n * g w
hz : z ≠ z₀
h1 : DifferentiableAt ℂ (fun y => (y - z₀) ^ n) z
⊢ deriv f z / f z = ↑n / (z - z₀) + deriv g z / g z | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hg : DifferentiableAt ℂ g z
hgz : g z ≠ 0
hfg : f =ᶠ[𝓝 z] fun w => (w - z₀) ^ n * g w
hz : z ≠ z₀
h1 : DifferentiableAt ℂ (fun y => (y - z₀) ^ n) z
h4 : DifferentiableAt ℂ (fun y => y - z₀) z
⊢ deriv f z / f z = ↑n / (z - z₀) + deriv g z / g z |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | deriv_div_self_eq_div_add_deriv_div_self | [80, 1] | [92, 9] | have h5 : deriv (fun y => y - z₀) z = 1 := by simp only [deriv_sub_const, deriv_id''] | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hg : DifferentiableAt ℂ g z
hgz : g z ≠ 0
hfg : f =ᶠ[𝓝 z] fun w => (w - z₀) ^ n * g w
hz : z ≠ z₀
h1 : DifferentiableAt ℂ (fun y => (y - z₀) ^ n) z
h4 : DifferentiableAt ℂ (fun y => y - z₀) z
⊢ deriv f z / f z = ↑n / (z - z₀) + deriv g z / g z | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hg : DifferentiableAt ℂ g z
hgz : g z ≠ 0
hfg : f =ᶠ[𝓝 z] fun w => (w - z₀) ^ n * g w
hz : z ≠ z₀
h1 : DifferentiableAt ℂ (fun y => (y - z₀) ^ n) z
h4 : DifferentiableAt ℂ (fun y => y - z₀) z
h5 : deriv (fun y => y - z₀) z = 1
⊢ deriv f z / f z = ↑n / (z - z₀) + deriv g z / g z |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | deriv_div_self_eq_div_add_deriv_div_self | [80, 1] | [92, 9] | simp [hfg.deriv_eq, hfg.self_of_nhds, deriv_mul h1 hg, _root_.add_div, deriv_pow'' n h4, deriv_sub_const, h5] | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hg : DifferentiableAt ℂ g z
hgz : g z ≠ 0
hfg : f =ᶠ[𝓝 z] fun w => (w - z₀) ^ n * g w
hz : z ≠ z₀
h1 : DifferentiableAt ℂ (fun y => (y - z₀) ^ n) z
h4 : DifferentiableAt ℂ (fun y => y - z₀) z
h5 : deriv (fun y => y - z₀) z = 1
⊢ deriv f z / f z = ↑n / (z - z₀) + deriv g z / g z | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hg : DifferentiableAt ℂ g z
hgz : g z ≠ 0
hfg : f =ᶠ[𝓝 z] fun w => (w - z₀) ^ n * g w
hz : z ≠ z₀
h1 : DifferentiableAt ℂ (fun y => (y - z₀) ^ n) z
h4 : DifferentiableAt ℂ (fun y => y - z₀) z
h5 : deriv (fun y => y - z₀) z = 1
⊢ ↑n * (z - z₀) ^ (n - 1) * g z / ((z - z₀) ^ n * g z) + (z - z₀) ^ n * deriv g z / ((z - z₀) ^ n * g z) =
↑n / (z - z₀) + deriv g z / g z |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | deriv_div_self_eq_div_add_deriv_div_self | [80, 1] | [92, 9] | cases n | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hg : DifferentiableAt ℂ g z
hgz : g z ≠ 0
hfg : f =ᶠ[𝓝 z] fun w => (w - z₀) ^ n * g w
hz : z ≠ z₀
h1 : DifferentiableAt ℂ (fun y => (y - z₀) ^ n) z
h4 : DifferentiableAt ℂ (fun y => y - z₀) z
h5 : deriv (fun y => y - z₀) z = 1
⊢ ↑n * (z - z₀) ^ (n - 1) * g z / ((z - z₀) ^ n * g z) + (z - z₀) ^ n * deriv g z / ((z - z₀) ^ n * g z) =
↑n / (z - z₀) + deriv g z / g z | case zero
f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
U : Set ℂ
r : ℝ
hg : DifferentiableAt ℂ g z
hgz : g z ≠ 0
hz : z ≠ z₀
h4 : DifferentiableAt ℂ (fun y => y - z₀) z
h5 : deriv (fun y => y - z₀) z = 1
hfg : f =ᶠ[𝓝 z] fun w => (w - z₀) ^ zero * g w
h1 : DifferentiableAt ℂ (fun y => (y - z₀) ^ zero) z
⊢ ↑zero * (z - z₀) ^ (zero - 1) * g z / ((z - z₀) ^ zero * g z) +
(z - z₀) ^ zero * deriv g z / ((z - z₀) ^ zero * g z) =
↑zero / (z - z₀) + deriv g z / g z
case succ
f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
U : Set ℂ
r : ℝ
hg : DifferentiableAt ℂ g z
hgz : g z ≠ 0
hz : z ≠ z₀
h4 : DifferentiableAt ℂ (fun y => y - z₀) z
h5 : deriv (fun y => y - z₀) z = 1
n✝ : ℕ
hfg : f =ᶠ[𝓝 z] fun w => (w - z₀) ^ succ n✝ * g w
h1 : DifferentiableAt ℂ (fun y => (y - z₀) ^ succ n✝) z
⊢ ↑(succ n✝) * (z - z₀) ^ (succ n✝ - 1) * g z / ((z - z₀) ^ succ n✝ * g z) +
(z - z₀) ^ succ n✝ * deriv g z / ((z - z₀) ^ succ n✝ * g z) =
↑(succ n✝) / (z - z₀) + deriv g z / g z |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | deriv_div_self_eq_div_add_deriv_div_self | [80, 1] | [92, 9] | case zero => simp | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
U : Set ℂ
r : ℝ
hg : DifferentiableAt ℂ g z
hgz : g z ≠ 0
hz : z ≠ z₀
h4 : DifferentiableAt ℂ (fun y => y - z₀) z
h5 : deriv (fun y => y - z₀) z = 1
hfg : f =ᶠ[𝓝 z] fun w => (w - z₀) ^ zero * g w
h1 : DifferentiableAt ℂ (fun y => (y - z₀) ^ zero) z
⊢ ↑zero * (z - z₀) ^ (zero - 1) * g z / ((z - z₀) ^ zero * g z) +
(z - z₀) ^ zero * deriv g z / ((z - z₀) ^ zero * g z) =
↑zero / (z - z₀) + deriv g z / g z | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | deriv_div_self_eq_div_add_deriv_div_self | [80, 1] | [92, 9] | case succ n =>
field_simp [_root_.pow_succ, sub_ne_zero.mpr hz]
ring | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
U : Set ℂ
r : ℝ
hg : DifferentiableAt ℂ g z
hgz : g z ≠ 0
hz : z ≠ z₀
h4 : DifferentiableAt ℂ (fun y => y - z₀) z
h5 : deriv (fun y => y - z₀) z = 1
n : ℕ
hfg : f =ᶠ[𝓝 z] fun w => (w - z₀) ^ succ n * g w
h1 : DifferentiableAt ℂ (fun y => (y - z₀) ^ succ n) z
⊢ ↑(succ n) * (z - z₀) ^ (succ n - 1) * g z / ((z - z₀) ^ succ n * g z) +
(z - z₀) ^ succ n * deriv g z / ((z - z₀) ^ succ n * g z) =
↑(succ n) / (z - z₀) + deriv g z / g z | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | deriv_div_self_eq_div_add_deriv_div_self | [80, 1] | [92, 9] | simp only [deriv_sub_const, deriv_id''] | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hg : DifferentiableAt ℂ g z
hgz : g z ≠ 0
hfg : f =ᶠ[𝓝 z] fun w => (w - z₀) ^ n * g w
hz : z ≠ z₀
h1 : DifferentiableAt ℂ (fun y => (y - z₀) ^ n) z
h4 : DifferentiableAt ℂ (fun y => y - z₀) z
⊢ deriv (fun y => y - z₀) z = 1 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | deriv_div_self_eq_div_add_deriv_div_self | [80, 1] | [92, 9] | simp | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
U : Set ℂ
r : ℝ
hg : DifferentiableAt ℂ g z
hgz : g z ≠ 0
hz : z ≠ z₀
h4 : DifferentiableAt ℂ (fun y => y - z₀) z
h5 : deriv (fun y => y - z₀) z = 1
hfg : f =ᶠ[𝓝 z] fun w => (w - z₀) ^ zero * g w
h1 : DifferentiableAt ℂ (fun y => (y - z₀) ^ zero) z
⊢ ↑zero * (z - z₀) ^ (zero - 1) * g z / ((z - z₀) ^ zero * g z) +
(z - z₀) ^ zero * deriv g z / ((z - z₀) ^ zero * g z) =
↑zero / (z - z₀) + deriv g z / g z | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | deriv_div_self_eq_div_add_deriv_div_self | [80, 1] | [92, 9] | field_simp [_root_.pow_succ, sub_ne_zero.mpr hz] | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
U : Set ℂ
r : ℝ
hg : DifferentiableAt ℂ g z
hgz : g z ≠ 0
hz : z ≠ z₀
h4 : DifferentiableAt ℂ (fun y => y - z₀) z
h5 : deriv (fun y => y - z₀) z = 1
n : ℕ
hfg : f =ᶠ[𝓝 z] fun w => (w - z₀) ^ succ n * g w
h1 : DifferentiableAt ℂ (fun y => (y - z₀) ^ succ n) z
⊢ ↑(succ n) * (z - z₀) ^ (succ n - 1) * g z / ((z - z₀) ^ succ n * g z) +
(z - z₀) ^ succ n * deriv g z / ((z - z₀) ^ succ n * g z) =
↑(succ n) / (z - z₀) + deriv g z / g z | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
U : Set ℂ
r : ℝ
hg : DifferentiableAt ℂ g z
hgz : g z ≠ 0
hz : z ≠ z₀
h4 : DifferentiableAt ℂ (fun y => y - z₀) z
h5 : deriv (fun y => y - z₀) z = 1
n : ℕ
hfg : f =ᶠ[𝓝 z] fun w => (w - z₀) ^ succ n * g w
h1 : DifferentiableAt ℂ (fun y => (y - z₀) ^ succ n) z
⊢ ((↑n + 1) * (z - z₀) ^ n * g z + (z - z₀) * (z - z₀) ^ n * deriv g z) * ((z - z₀) * g z) =
((↑n + 1) * g z + deriv g z * (z - z₀)) * ((z - z₀) * (z - z₀) ^ n * g z) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | deriv_div_self_eq_div_add_deriv_div_self | [80, 1] | [92, 9] | ring | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
U : Set ℂ
r : ℝ
hg : DifferentiableAt ℂ g z
hgz : g z ≠ 0
hz : z ≠ z₀
h4 : DifferentiableAt ℂ (fun y => y - z₀) z
h5 : deriv (fun y => y - z₀) z = 1
n : ℕ
hfg : f =ᶠ[𝓝 z] fun w => (w - z₀) ^ succ n * g w
h1 : DifferentiableAt ℂ (fun y => (y - z₀) ^ succ n) z
⊢ ((↑n + 1) * (z - z₀) ^ n * g z + (z - z₀) * (z - z₀) ^ n * deriv g z) * ((z - z₀) * g z) =
((↑n + 1) * g z + deriv g z * (z - z₀)) * ((z - z₀) * (z - z₀) ^ n * g z) | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | eventually_deriv_div_self_eq | [94, 1] | [102, 87] | intro g | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hp : HasFPowerSeriesAt f p z₀
h : p ≠ 0
⊢ let g := (swap dslope z₀)^[FormalMultilinearSeries.order p] f;
∀ᶠ (z : ℂ) in 𝓝 z₀, z ≠ z₀ → deriv f z / f z = ↑(FormalMultilinearSeries.order p) / (z - z₀) + deriv g z / g z | f g✝ : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hp : HasFPowerSeriesAt f p z₀
h : p ≠ 0
g : ℂ → ℂ := (swap dslope z₀)^[FormalMultilinearSeries.order p] f
⊢ ∀ᶠ (z : ℂ) in 𝓝 z₀, z ≠ z₀ → deriv f z / f z = ↑(FormalMultilinearSeries.order p) / (z - z₀) + deriv g z / g z |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | eventually_deriv_div_self_eq | [94, 1] | [102, 87] | obtain ⟨r, h2⟩ := hp.has_fpower_series_iterate_dslope_fslope p.order | f g✝ : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hp : HasFPowerSeriesAt f p z₀
h : p ≠ 0
g : ℂ → ℂ := (swap dslope z₀)^[FormalMultilinearSeries.order p] f
⊢ ∀ᶠ (z : ℂ) in 𝓝 z₀, z ≠ z₀ → deriv f z / f z = ↑(FormalMultilinearSeries.order p) / (z - z₀) + deriv g z / g z | case intro
f g✝ : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r✝ : ℝ
hp : HasFPowerSeriesAt f p z₀
h : p ≠ 0
g : ℂ → ℂ := (swap dslope z₀)^[FormalMultilinearSeries.order p] f
r : ENNReal
h2 :
HasFPowerSeriesOnBall ((swap dslope z₀)^[FormalMultilinearSeries.order p] f)
(FormalMultilinearSeries.fslope^[FormalMultilinearSeries.order p] p) z₀ r
⊢ ∀ᶠ (z : ℂ) in 𝓝 z₀, z ≠ z₀ → deriv f z / f z = ↑(FormalMultilinearSeries.order p) / (z - z₀) + deriv g z / g z |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | eventually_deriv_div_self_eq | [94, 1] | [102, 87] | have lh1 := h2.differentiableOn.eventually_differentiableAt (EMetric.ball_mem_nhds _ h2.r_pos) | case intro
f g✝ : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r✝ : ℝ
hp : HasFPowerSeriesAt f p z₀
h : p ≠ 0
g : ℂ → ℂ := (swap dslope z₀)^[FormalMultilinearSeries.order p] f
r : ENNReal
h2 :
HasFPowerSeriesOnBall ((swap dslope z₀)^[FormalMultilinearSeries.order p] f)
(FormalMultilinearSeries.fslope^[FormalMultilinearSeries.order p] p) z₀ r
⊢ ∀ᶠ (z : ℂ) in 𝓝 z₀, z ≠ z₀ → deriv f z / f z = ↑(FormalMultilinearSeries.order p) / (z - z₀) + deriv g z / g z | case intro
f g✝ : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r✝ : ℝ
hp : HasFPowerSeriesAt f p z₀
h : p ≠ 0
g : ℂ → ℂ := (swap dslope z₀)^[FormalMultilinearSeries.order p] f
r : ENNReal
h2 :
HasFPowerSeriesOnBall ((swap dslope z₀)^[FormalMultilinearSeries.order p] f)
(FormalMultilinearSeries.fslope^[FormalMultilinearSeries.order p] p) z₀ r
lh1 : ∀ᶠ (y : ℂ) in 𝓝 z₀, DifferentiableAt ℂ ((swap dslope z₀)^[FormalMultilinearSeries.order p] f) y
⊢ ∀ᶠ (z : ℂ) in 𝓝 z₀, z ≠ z₀ → deriv f z / f z = ↑(FormalMultilinearSeries.order p) / (z - z₀) + deriv g z / g z |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | eventually_deriv_div_self_eq | [94, 1] | [102, 87] | have lh2 := hp.dslope_order_eventually_ne_zero h | case intro
f g✝ : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r✝ : ℝ
hp : HasFPowerSeriesAt f p z₀
h : p ≠ 0
g : ℂ → ℂ := (swap dslope z₀)^[FormalMultilinearSeries.order p] f
r : ENNReal
h2 :
HasFPowerSeriesOnBall ((swap dslope z₀)^[FormalMultilinearSeries.order p] f)
(FormalMultilinearSeries.fslope^[FormalMultilinearSeries.order p] p) z₀ r
lh1 : ∀ᶠ (y : ℂ) in 𝓝 z₀, DifferentiableAt ℂ ((swap dslope z₀)^[FormalMultilinearSeries.order p] f) y
⊢ ∀ᶠ (z : ℂ) in 𝓝 z₀, z ≠ z₀ → deriv f z / f z = ↑(FormalMultilinearSeries.order p) / (z - z₀) + deriv g z / g z | case intro
f g✝ : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r✝ : ℝ
hp : HasFPowerSeriesAt f p z₀
h : p ≠ 0
g : ℂ → ℂ := (swap dslope z₀)^[FormalMultilinearSeries.order p] f
r : ENNReal
h2 :
HasFPowerSeriesOnBall ((swap dslope z₀)^[FormalMultilinearSeries.order p] f)
(FormalMultilinearSeries.fslope^[FormalMultilinearSeries.order p] p) z₀ r
lh1 : ∀ᶠ (y : ℂ) in 𝓝 z₀, DifferentiableAt ℂ ((swap dslope z₀)^[FormalMultilinearSeries.order p] f) y
lh2 : ∀ᶠ (z : ℂ) in 𝓝 z₀, (swap dslope z₀)^[FormalMultilinearSeries.order p] f z ≠ 0
⊢ ∀ᶠ (z : ℂ) in 𝓝 z₀, z ≠ z₀ → deriv f z / f z = ↑(FormalMultilinearSeries.order p) / (z - z₀) + deriv g z / g z |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | eventually_deriv_div_self_eq | [94, 1] | [102, 87] | have lh3 := eventually_eventually_nhds.mpr hp.eq_pow_order_mul_iterate_dslope | case intro
f g✝ : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r✝ : ℝ
hp : HasFPowerSeriesAt f p z₀
h : p ≠ 0
g : ℂ → ℂ := (swap dslope z₀)^[FormalMultilinearSeries.order p] f
r : ENNReal
h2 :
HasFPowerSeriesOnBall ((swap dslope z₀)^[FormalMultilinearSeries.order p] f)
(FormalMultilinearSeries.fslope^[FormalMultilinearSeries.order p] p) z₀ r
lh1 : ∀ᶠ (y : ℂ) in 𝓝 z₀, DifferentiableAt ℂ ((swap dslope z₀)^[FormalMultilinearSeries.order p] f) y
lh2 : ∀ᶠ (z : ℂ) in 𝓝 z₀, (swap dslope z₀)^[FormalMultilinearSeries.order p] f z ≠ 0
⊢ ∀ᶠ (z : ℂ) in 𝓝 z₀, z ≠ z₀ → deriv f z / f z = ↑(FormalMultilinearSeries.order p) / (z - z₀) + deriv g z / g z | case intro
f g✝ : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r✝ : ℝ
hp : HasFPowerSeriesAt f p z₀
h : p ≠ 0
g : ℂ → ℂ := (swap dslope z₀)^[FormalMultilinearSeries.order p] f
r : ENNReal
h2 :
HasFPowerSeriesOnBall ((swap dslope z₀)^[FormalMultilinearSeries.order p] f)
(FormalMultilinearSeries.fslope^[FormalMultilinearSeries.order p] p) z₀ r
lh1 : ∀ᶠ (y : ℂ) in 𝓝 z₀, DifferentiableAt ℂ ((swap dslope z₀)^[FormalMultilinearSeries.order p] f) y
lh2 : ∀ᶠ (z : ℂ) in 𝓝 z₀, (swap dslope z₀)^[FormalMultilinearSeries.order p] f z ≠ 0
lh3 :
∀ᶠ (y : ℂ) in 𝓝 z₀,
∀ᶠ (x : ℂ) in 𝓝 y,
f x = (x - z₀) ^ FormalMultilinearSeries.order p • (swap dslope z₀)^[FormalMultilinearSeries.order p] f x
⊢ ∀ᶠ (z : ℂ) in 𝓝 z₀, z ≠ z₀ → deriv f z / f z = ↑(FormalMultilinearSeries.order p) / (z - z₀) + deriv g z / g z |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | eventually_deriv_div_self_eq | [94, 1] | [102, 87] | filter_upwards [lh1, lh2, lh3] with z using deriv_div_self_eq_div_add_deriv_div_self | case intro
f g✝ : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r✝ : ℝ
hp : HasFPowerSeriesAt f p z₀
h : p ≠ 0
g : ℂ → ℂ := (swap dslope z₀)^[FormalMultilinearSeries.order p] f
r : ENNReal
h2 :
HasFPowerSeriesOnBall ((swap dslope z₀)^[FormalMultilinearSeries.order p] f)
(FormalMultilinearSeries.fslope^[FormalMultilinearSeries.order p] p) z₀ r
lh1 : ∀ᶠ (y : ℂ) in 𝓝 z₀, DifferentiableAt ℂ ((swap dslope z₀)^[FormalMultilinearSeries.order p] f) y
lh2 : ∀ᶠ (z : ℂ) in 𝓝 z₀, (swap dslope z₀)^[FormalMultilinearSeries.order p] f z ≠ 0
lh3 :
∀ᶠ (y : ℂ) in 𝓝 z₀,
∀ᶠ (x : ℂ) in 𝓝 y,
f x = (x - z₀) ^ FormalMultilinearSeries.order p • (swap dslope z₀)^[FormalMultilinearSeries.order p] f x
⊢ ∀ᶠ (z : ℂ) in 𝓝 z₀, z ≠ z₀ → deriv f z / f z = ↑(FormalMultilinearSeries.order p) / (z - z₀) + deriv g z / g z | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | cindex_eq_zero | [104, 1] | [116, 102] | simp [cindex, circle_integral_eq_zero h2 hr h3 (((f_hol.mono h1).deriv h2).div (f_hol.mono h1) h4)] | case intro.intro.intro.intro
f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
hcr : closedBall c r ⊆ U
f_hol : DifferentiableOn ℂ f U
hf : ∀ z ∈ closedBall c r, f z ≠ 0
V : Set ℂ
h1 : V ⊆ U
h2 : IsOpen V
h3 : closedBall c r ⊆ V
h4 : ∀ z ∈ V, f z ≠ 0
⊢ cindex c r f = 0 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | cindex_eq_zero | [104, 1] | [116, 102] | set s : Set ℂ := { z ∈ U | f z ≠ 0 } | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
hcr : closedBall c r ⊆ U
f_hol : DifferentiableOn ℂ f U
hf : ∀ z ∈ closedBall c r, f z ≠ 0
⊢ ∃ V ⊆ U, IsOpen V ∧ closedBall c r ⊆ V ∧ ∀ z ∈ V, f z ≠ 0 | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
hcr : closedBall c r ⊆ U
f_hol : DifferentiableOn ℂ f U
hf : ∀ z ∈ closedBall c r, f z ≠ 0
s : Set ℂ := {z | z ∈ U ∧ f z ≠ 0}
⊢ ∃ V ⊆ U, IsOpen V ∧ closedBall c r ⊆ V ∧ ∀ z ∈ V, f z ≠ 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | cindex_eq_zero | [104, 1] | [116, 102] | have e1 : IsCompact (closedBall c r) := isCompact_closedBall _ _ | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
hcr : closedBall c r ⊆ U
f_hol : DifferentiableOn ℂ f U
hf : ∀ z ∈ closedBall c r, f z ≠ 0
s : Set ℂ := {z | z ∈ U ∧ f z ≠ 0}
⊢ ∃ V ⊆ U, IsOpen V ∧ closedBall c r ⊆ V ∧ ∀ z ∈ V, f z ≠ 0 | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
hcr : closedBall c r ⊆ U
f_hol : DifferentiableOn ℂ f U
hf : ∀ z ∈ closedBall c r, f z ≠ 0
s : Set ℂ := {z | z ∈ U ∧ f z ≠ 0}
e1 : IsCompact (closedBall c r)
⊢ ∃ V ⊆ U, IsOpen V ∧ closedBall c r ⊆ V ∧ ∀ z ∈ V, f z ≠ 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | cindex_eq_zero | [104, 1] | [116, 102] | have e2 : IsOpen s := f_hol.continuousOn.isOpen_inter_preimage hU isOpen_compl_singleton | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
hcr : closedBall c r ⊆ U
f_hol : DifferentiableOn ℂ f U
hf : ∀ z ∈ closedBall c r, f z ≠ 0
s : Set ℂ := {z | z ∈ U ∧ f z ≠ 0}
e1 : IsCompact (closedBall c r)
⊢ ∃ V ⊆ U, IsOpen V ∧ closedBall c r ⊆ V ∧ ∀ z ∈ V, f z ≠ 0 | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
hcr : closedBall c r ⊆ U
f_hol : DifferentiableOn ℂ f U
hf : ∀ z ∈ closedBall c r, f z ≠ 0
s : Set ℂ := {z | z ∈ U ∧ f z ≠ 0}
e1 : IsCompact (closedBall c r)
e2 : IsOpen s
⊢ ∃ V ⊆ U, IsOpen V ∧ closedBall c r ⊆ V ∧ ∀ z ∈ V, f z ≠ 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | cindex_eq_zero | [104, 1] | [116, 102] | have e3 : closedBall c r ⊆ s := λ z hz => ⟨hcr hz, hf z hz⟩ | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
hcr : closedBall c r ⊆ U
f_hol : DifferentiableOn ℂ f U
hf : ∀ z ∈ closedBall c r, f z ≠ 0
s : Set ℂ := {z | z ∈ U ∧ f z ≠ 0}
e1 : IsCompact (closedBall c r)
e2 : IsOpen s
⊢ ∃ V ⊆ U, IsOpen V ∧ closedBall c r ⊆ V ∧ ∀ z ∈ V, f z ≠ 0 | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
hcr : closedBall c r ⊆ U
f_hol : DifferentiableOn ℂ f U
hf : ∀ z ∈ closedBall c r, f z ≠ 0
s : Set ℂ := {z | z ∈ U ∧ f z ≠ 0}
e1 : IsCompact (closedBall c r)
e2 : IsOpen s
e3 : closedBall c r ⊆ s
⊢ ∃ V ⊆ U, IsOpen V ∧ closedBall c r ⊆ V ∧ ∀ z ∈ V, f z ≠ 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | cindex_eq_zero | [104, 1] | [116, 102] | obtain ⟨δ, e4, e5⟩ := e1.exists_thickening_subset_open e2 e3 | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
hcr : closedBall c r ⊆ U
f_hol : DifferentiableOn ℂ f U
hf : ∀ z ∈ closedBall c r, f z ≠ 0
s : Set ℂ := {z | z ∈ U ∧ f z ≠ 0}
e1 : IsCompact (closedBall c r)
e2 : IsOpen s
e3 : closedBall c r ⊆ s
⊢ ∃ V ⊆ U, IsOpen V ∧ closedBall c r ⊆ V ∧ ∀ z ∈ V, f z ≠ 0 | case intro.intro
f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
hcr : closedBall c r ⊆ U
f_hol : DifferentiableOn ℂ f U
hf : ∀ z ∈ closedBall c r, f z ≠ 0
s : Set ℂ := {z | z ∈ U ∧ f z ≠ 0}
e1 : IsCompact (closedBall c r)
e2 : IsOpen s
e3 : closedBall c r ⊆ s
δ : ℝ
e4 : 0 < δ
e5 : thickening δ (closedBall c r) ⊆ s
⊢ ∃ V ⊆ U, IsOpen V ∧ closedBall c r ⊆ V ∧ ∀ z ∈ V, f z ≠ 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | cindex_eq_zero | [104, 1] | [116, 102] | refine ⟨thickening δ (closedBall c r), ?_, isOpen_thickening, self_subset_thickening e4 _, ?_⟩ | case intro.intro
f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
hcr : closedBall c r ⊆ U
f_hol : DifferentiableOn ℂ f U
hf : ∀ z ∈ closedBall c r, f z ≠ 0
s : Set ℂ := {z | z ∈ U ∧ f z ≠ 0}
e1 : IsCompact (closedBall c r)
e2 : IsOpen s
e3 : closedBall c r ⊆ s
δ : ℝ
e4 : 0 < δ
e5 : thickening δ (closedBall c r) ⊆ s
⊢ ∃ V ⊆ U, IsOpen V ∧ closedBall c r ⊆ V ∧ ∀ z ∈ V, f z ≠ 0 | case intro.intro.refine_1
f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
hcr : closedBall c r ⊆ U
f_hol : DifferentiableOn ℂ f U
hf : ∀ z ∈ closedBall c r, f z ≠ 0
s : Set ℂ := {z | z ∈ U ∧ f z ≠ 0}
e1 : IsCompact (closedBall c r)
e2 : IsOpen s
e3 : closedBall c r ⊆ s
δ : ℝ
e4 : 0 < δ
e5 : thickening δ (closedBall c r) ⊆ s
⊢ thickening δ (closedBall c r) ⊆ U
case intro.intro.refine_2
f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
hcr : closedBall c r ⊆ U
f_hol : DifferentiableOn ℂ f U
hf : ∀ z ∈ closedBall c r, f z ≠ 0
s : Set ℂ := {z | z ∈ U ∧ f z ≠ 0}
e1 : IsCompact (closedBall c r)
e2 : IsOpen s
e3 : closedBall c r ⊆ s
δ : ℝ
e4 : 0 < δ
e5 : thickening δ (closedBall c r) ⊆ s
⊢ ∀ z ∈ thickening δ (closedBall c r), f z ≠ 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | cindex_eq_zero | [104, 1] | [116, 102] | exact (e5.trans $ Set.sep_subset _ _) | case intro.intro.refine_1
f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
hcr : closedBall c r ⊆ U
f_hol : DifferentiableOn ℂ f U
hf : ∀ z ∈ closedBall c r, f z ≠ 0
s : Set ℂ := {z | z ∈ U ∧ f z ≠ 0}
e1 : IsCompact (closedBall c r)
e2 : IsOpen s
e3 : closedBall c r ⊆ s
δ : ℝ
e4 : 0 < δ
e5 : thickening δ (closedBall c r) ⊆ s
⊢ thickening δ (closedBall c r) ⊆ U | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | cindex_eq_zero | [104, 1] | [116, 102] | exact λ z hz => (e5 hz).2 | case intro.intro.refine_2
f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
hcr : closedBall c r ⊆ U
f_hol : DifferentiableOn ℂ f U
hf : ∀ z ∈ closedBall c r, f z ≠ 0
s : Set ℂ := {z | z ∈ U ∧ f z ≠ 0}
e1 : IsCompact (closedBall c r)
e2 : IsOpen s
e3 : closedBall c r ⊆ s
δ : ℝ
e4 : 0 < δ
e5 : thickening δ (closedBall c r) ⊆ s
⊢ ∀ z ∈ thickening δ (closedBall c r), f z ≠ 0 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | cindex_eq_order_aux | [120, 1] | [144, 7] | have e2 : sphere z₀ r ⊆ U := sphere_subset_closedBall.trans h0 | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
h0 : closedBall z₀ r ⊆ U
h1 : DifferentiableOn ℂ g U
h2 : ∀ z ∈ closedBall z₀ r, g z ≠ 0
h3 : ∀ {z : ℂ}, z ∈ sphere z₀ r → deriv f z / f z = c / (z - z₀) + deriv g z / g z
⊢ cindex z₀ r f = c | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
h0 : closedBall z₀ r ⊆ U
h1 : DifferentiableOn ℂ g U
h2 : ∀ z ∈ closedBall z₀ r, g z ≠ 0
h3 : ∀ {z : ℂ}, z ∈ sphere z₀ r → deriv f z / f z = c / (z - z₀) + deriv g z / g z
e2 : sphere z₀ r ⊆ U
⊢ cindex z₀ r f = c |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | cindex_eq_order_aux | [120, 1] | [144, 7] | have e4 : (∮ z in C(z₀,r), deriv f z / f z) = ∮ z in C(z₀,r), c / (z - z₀) + deriv g z / g z :=
circleIntegral.integral_congr hr.le (λ z hz => h3 hz) | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
h0 : closedBall z₀ r ⊆ U
h1 : DifferentiableOn ℂ g U
h2 : ∀ z ∈ closedBall z₀ r, g z ≠ 0
h3 : ∀ {z : ℂ}, z ∈ sphere z₀ r → deriv f z / f z = c / (z - z₀) + deriv g z / g z
e2 : sphere z₀ r ⊆ U
⊢ cindex z₀ r f = c | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
h0 : closedBall z₀ r ⊆ U
h1 : DifferentiableOn ℂ g U
h2 : ∀ z ∈ closedBall z₀ r, g z ≠ 0
h3 : ∀ {z : ℂ}, z ∈ sphere z₀ r → deriv f z / f z = c / (z - z₀) + deriv g z / g z
e2 : sphere z₀ r ⊆ U
e4 : (∮ (z : ℂ) in C(z₀, r), deriv f z / f z) = ∮ (z : ℂ) in C(z₀, r), c / (z - z₀) + deriv g z / g z
⊢ cindex z₀ r f = c |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | cindex_eq_order_aux | [120, 1] | [144, 7] | have e6 : (∮ z in C(z₀, r), deriv g z / g z) = 0 := by
have := cindex_eq_zero hU hr h0 h1 h2
simpa [cindex, Real.pi_ne_zero, I_ne_zero] using this | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
h0 : closedBall z₀ r ⊆ U
h1 : DifferentiableOn ℂ g U
h2 : ∀ z ∈ closedBall z₀ r, g z ≠ 0
h3 : ∀ {z : ℂ}, z ∈ sphere z₀ r → deriv f z / f z = c / (z - z₀) + deriv g z / g z
e2 : sphere z₀ r ⊆ U
e4 : (∮ (z : ℂ) in C(z₀, r), deriv f z / f z) = ∮ (z : ℂ) in C(z₀, r), c / (z - z₀) + deriv g z / g z
e5 :
(∮ (z : ℂ) in C(z₀, r), c / (z - z₀) + deriv g z / g z) =
(∮ (z : ℂ) in C(z₀, r), c / (z - z₀)) + ∮ (z : ℂ) in C(z₀, r), deriv g z / g z
⊢ cindex z₀ r f = c | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
h0 : closedBall z₀ r ⊆ U
h1 : DifferentiableOn ℂ g U
h2 : ∀ z ∈ closedBall z₀ r, g z ≠ 0
h3 : ∀ {z : ℂ}, z ∈ sphere z₀ r → deriv f z / f z = c / (z - z₀) + deriv g z / g z
e2 : sphere z₀ r ⊆ U
e4 : (∮ (z : ℂ) in C(z₀, r), deriv f z / f z) = ∮ (z : ℂ) in C(z₀, r), c / (z - z₀) + deriv g z / g z
e5 :
(∮ (z : ℂ) in C(z₀, r), c / (z - z₀) + deriv g z / g z) =
(∮ (z : ℂ) in C(z₀, r), c / (z - z₀)) + ∮ (z : ℂ) in C(z₀, r), deriv g z / g z
e6 : (∮ (z : ℂ) in C(z₀, r), deriv g z / g z) = 0
⊢ cindex z₀ r f = c |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | cindex_eq_order_aux | [120, 1] | [144, 7] | have e7 : (∮ z in C(z₀, r), c / (z - z₀)) = 2 * π * I * c := by
simpa [div_eq_mul_inv, mul_comm _ _⁻¹] using circle_integral_sub_center_inv_smul hr | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
h0 : closedBall z₀ r ⊆ U
h1 : DifferentiableOn ℂ g U
h2 : ∀ z ∈ closedBall z₀ r, g z ≠ 0
h3 : ∀ {z : ℂ}, z ∈ sphere z₀ r → deriv f z / f z = c / (z - z₀) + deriv g z / g z
e2 : sphere z₀ r ⊆ U
e4 : (∮ (z : ℂ) in C(z₀, r), deriv f z / f z) = ∮ (z : ℂ) in C(z₀, r), c / (z - z₀) + deriv g z / g z
e5 :
(∮ (z : ℂ) in C(z₀, r), c / (z - z₀) + deriv g z / g z) =
(∮ (z : ℂ) in C(z₀, r), c / (z - z₀)) + ∮ (z : ℂ) in C(z₀, r), deriv g z / g z
e6 : (∮ (z : ℂ) in C(z₀, r), deriv g z / g z) = 0
⊢ cindex z₀ r f = c | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
h0 : closedBall z₀ r ⊆ U
h1 : DifferentiableOn ℂ g U
h2 : ∀ z ∈ closedBall z₀ r, g z ≠ 0
h3 : ∀ {z : ℂ}, z ∈ sphere z₀ r → deriv f z / f z = c / (z - z₀) + deriv g z / g z
e2 : sphere z₀ r ⊆ U
e4 : (∮ (z : ℂ) in C(z₀, r), deriv f z / f z) = ∮ (z : ℂ) in C(z₀, r), c / (z - z₀) + deriv g z / g z
e5 :
(∮ (z : ℂ) in C(z₀, r), c / (z - z₀) + deriv g z / g z) =
(∮ (z : ℂ) in C(z₀, r), c / (z - z₀)) + ∮ (z : ℂ) in C(z₀, r), deriv g z / g z
e6 : (∮ (z : ℂ) in C(z₀, r), deriv g z / g z) = 0
e7 : (∮ (z : ℂ) in C(z₀, r), c / (z - z₀)) = 2 * ↑π * I * c
⊢ cindex z₀ r f = c |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | cindex_eq_order_aux | [120, 1] | [144, 7] | field_simp [cindex, e4, e5, e6, e7, Real.pi_ne_zero, I_ne_zero, two_ne_zero] | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
h0 : closedBall z₀ r ⊆ U
h1 : DifferentiableOn ℂ g U
h2 : ∀ z ∈ closedBall z₀ r, g z ≠ 0
h3 : ∀ {z : ℂ}, z ∈ sphere z₀ r → deriv f z / f z = c / (z - z₀) + deriv g z / g z
e2 : sphere z₀ r ⊆ U
e4 : (∮ (z : ℂ) in C(z₀, r), deriv f z / f z) = ∮ (z : ℂ) in C(z₀, r), c / (z - z₀) + deriv g z / g z
e5 :
(∮ (z : ℂ) in C(z₀, r), c / (z - z₀) + deriv g z / g z) =
(∮ (z : ℂ) in C(z₀, r), c / (z - z₀)) + ∮ (z : ℂ) in C(z₀, r), deriv g z / g z
e6 : (∮ (z : ℂ) in C(z₀, r), deriv g z / g z) = 0
e7 : (∮ (z : ℂ) in C(z₀, r), c / (z - z₀)) = 2 * ↑π * I * c
⊢ cindex z₀ r f = c | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
h0 : closedBall z₀ r ⊆ U
h1 : DifferentiableOn ℂ g U
h2 : ∀ z ∈ closedBall z₀ r, g z ≠ 0
h3 : ∀ {z : ℂ}, z ∈ sphere z₀ r → deriv f z / f z = c / (z - z₀) + deriv g z / g z
e2 : sphere z₀ r ⊆ U
e4 : (∮ (z : ℂ) in C(z₀, r), deriv f z / f z) = ∮ (z : ℂ) in C(z₀, r), c / (z - z₀) + deriv g z / g z
e5 :
(∮ (z : ℂ) in C(z₀, r), c / (z - z₀) + deriv g z / g z) =
(∮ (z : ℂ) in C(z₀, r), c / (z - z₀)) + ∮ (z : ℂ) in C(z₀, r), deriv g z / g z
e6 : (∮ (z : ℂ) in C(z₀, r), deriv g z / g z) = 0
e7 : (∮ (z : ℂ) in C(z₀, r), c / (z - z₀)) = 2 * ↑π * I * c
⊢ 2 * ↑π * I * c = c * (2 * ↑π * I) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | cindex_eq_order_aux | [120, 1] | [144, 7] | ring | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
h0 : closedBall z₀ r ⊆ U
h1 : DifferentiableOn ℂ g U
h2 : ∀ z ∈ closedBall z₀ r, g z ≠ 0
h3 : ∀ {z : ℂ}, z ∈ sphere z₀ r → deriv f z / f z = c / (z - z₀) + deriv g z / g z
e2 : sphere z₀ r ⊆ U
e4 : (∮ (z : ℂ) in C(z₀, r), deriv f z / f z) = ∮ (z : ℂ) in C(z₀, r), c / (z - z₀) + deriv g z / g z
e5 :
(∮ (z : ℂ) in C(z₀, r), c / (z - z₀) + deriv g z / g z) =
(∮ (z : ℂ) in C(z₀, r), c / (z - z₀)) + ∮ (z : ℂ) in C(z₀, r), deriv g z / g z
e6 : (∮ (z : ℂ) in C(z₀, r), deriv g z / g z) = 0
e7 : (∮ (z : ℂ) in C(z₀, r), c / (z - z₀)) = 2 * ↑π * I * c
⊢ 2 * ↑π * I * c = c * (2 * ↑π * I) | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | cindex_eq_order_aux | [120, 1] | [144, 7] | apply circleIntegral.integral_add | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
h0 : closedBall z₀ r ⊆ U
h1 : DifferentiableOn ℂ g U
h2 : ∀ z ∈ closedBall z₀ r, g z ≠ 0
h3 : ∀ {z : ℂ}, z ∈ sphere z₀ r → deriv f z / f z = c / (z - z₀) + deriv g z / g z
e2 : sphere z₀ r ⊆ U
e4 : (∮ (z : ℂ) in C(z₀, r), deriv f z / f z) = ∮ (z : ℂ) in C(z₀, r), c / (z - z₀) + deriv g z / g z
⊢ (∮ (z : ℂ) in C(z₀, r), c / (z - z₀) + deriv g z / g z) =
(∮ (z : ℂ) in C(z₀, r), c / (z - z₀)) + ∮ (z : ℂ) in C(z₀, r), deriv g z / g z | case hf
f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
h0 : closedBall z₀ r ⊆ U
h1 : DifferentiableOn ℂ g U
h2 : ∀ z ∈ closedBall z₀ r, g z ≠ 0
h3 : ∀ {z : ℂ}, z ∈ sphere z₀ r → deriv f z / f z = c / (z - z₀) + deriv g z / g z
e2 : sphere z₀ r ⊆ U
e4 : (∮ (z : ℂ) in C(z₀, r), deriv f z / f z) = ∮ (z : ℂ) in C(z₀, r), c / (z - z₀) + deriv g z / g z
⊢ CircleIntegrable (fun z => c / (z - z₀)) z₀ r
case hg
f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
h0 : closedBall z₀ r ⊆ U
h1 : DifferentiableOn ℂ g U
h2 : ∀ z ∈ closedBall z₀ r, g z ≠ 0
h3 : ∀ {z : ℂ}, z ∈ sphere z₀ r → deriv f z / f z = c / (z - z₀) + deriv g z / g z
e2 : sphere z₀ r ⊆ U
e4 : (∮ (z : ℂ) in C(z₀, r), deriv f z / f z) = ∮ (z : ℂ) in C(z₀, r), c / (z - z₀) + deriv g z / g z
⊢ CircleIntegrable (fun z => deriv g z / g z) z₀ r |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | cindex_eq_order_aux | [120, 1] | [144, 7] | apply ContinuousOn.circleIntegrable hr.le | case hf
f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
h0 : closedBall z₀ r ⊆ U
h1 : DifferentiableOn ℂ g U
h2 : ∀ z ∈ closedBall z₀ r, g z ≠ 0
h3 : ∀ {z : ℂ}, z ∈ sphere z₀ r → deriv f z / f z = c / (z - z₀) + deriv g z / g z
e2 : sphere z₀ r ⊆ U
e4 : (∮ (z : ℂ) in C(z₀, r), deriv f z / f z) = ∮ (z : ℂ) in C(z₀, r), c / (z - z₀) + deriv g z / g z
⊢ CircleIntegrable (fun z => c / (z - z₀)) z₀ r | case hf
f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
h0 : closedBall z₀ r ⊆ U
h1 : DifferentiableOn ℂ g U
h2 : ∀ z ∈ closedBall z₀ r, g z ≠ 0
h3 : ∀ {z : ℂ}, z ∈ sphere z₀ r → deriv f z / f z = c / (z - z₀) + deriv g z / g z
e2 : sphere z₀ r ⊆ U
e4 : (∮ (z : ℂ) in C(z₀, r), deriv f z / f z) = ∮ (z : ℂ) in C(z₀, r), c / (z - z₀) + deriv g z / g z
⊢ ContinuousOn (fun z => c / (z - z₀)) (sphere z₀ r) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | cindex_eq_order_aux | [120, 1] | [144, 7] | apply ContinuousOn.div continuousOn_const (continuousOn_id.sub continuousOn_const) | case hf
f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
h0 : closedBall z₀ r ⊆ U
h1 : DifferentiableOn ℂ g U
h2 : ∀ z ∈ closedBall z₀ r, g z ≠ 0
h3 : ∀ {z : ℂ}, z ∈ sphere z₀ r → deriv f z / f z = c / (z - z₀) + deriv g z / g z
e2 : sphere z₀ r ⊆ U
e4 : (∮ (z : ℂ) in C(z₀, r), deriv f z / f z) = ∮ (z : ℂ) in C(z₀, r), c / (z - z₀) + deriv g z / g z
⊢ ContinuousOn (fun z => c / (z - z₀)) (sphere z₀ r) | case hf
f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
h0 : closedBall z₀ r ⊆ U
h1 : DifferentiableOn ℂ g U
h2 : ∀ z ∈ closedBall z₀ r, g z ≠ 0
h3 : ∀ {z : ℂ}, z ∈ sphere z₀ r → deriv f z / f z = c / (z - z₀) + deriv g z / g z
e2 : sphere z₀ r ⊆ U
e4 : (∮ (z : ℂ) in C(z₀, r), deriv f z / f z) = ∮ (z : ℂ) in C(z₀, r), c / (z - z₀) + deriv g z / g z
⊢ ∀ x ∈ sphere z₀ r, id x - z₀ ≠ 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | cindex_eq_order_aux | [120, 1] | [144, 7] | exact λ z hz => sub_ne_zero.mpr (ne_of_mem_sphere hz hr.ne.symm) | case hf
f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
h0 : closedBall z₀ r ⊆ U
h1 : DifferentiableOn ℂ g U
h2 : ∀ z ∈ closedBall z₀ r, g z ≠ 0
h3 : ∀ {z : ℂ}, z ∈ sphere z₀ r → deriv f z / f z = c / (z - z₀) + deriv g z / g z
e2 : sphere z₀ r ⊆ U
e4 : (∮ (z : ℂ) in C(z₀, r), deriv f z / f z) = ∮ (z : ℂ) in C(z₀, r), c / (z - z₀) + deriv g z / g z
⊢ ∀ x ∈ sphere z₀ r, id x - z₀ ≠ 0 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | cindex_eq_order_aux | [120, 1] | [144, 7] | apply ContinuousOn.circleIntegrable hr.le | case hg
f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
h0 : closedBall z₀ r ⊆ U
h1 : DifferentiableOn ℂ g U
h2 : ∀ z ∈ closedBall z₀ r, g z ≠ 0
h3 : ∀ {z : ℂ}, z ∈ sphere z₀ r → deriv f z / f z = c / (z - z₀) + deriv g z / g z
e2 : sphere z₀ r ⊆ U
e4 : (∮ (z : ℂ) in C(z₀, r), deriv f z / f z) = ∮ (z : ℂ) in C(z₀, r), c / (z - z₀) + deriv g z / g z
⊢ CircleIntegrable (fun z => deriv g z / g z) z₀ r | case hg
f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
h0 : closedBall z₀ r ⊆ U
h1 : DifferentiableOn ℂ g U
h2 : ∀ z ∈ closedBall z₀ r, g z ≠ 0
h3 : ∀ {z : ℂ}, z ∈ sphere z₀ r → deriv f z / f z = c / (z - z₀) + deriv g z / g z
e2 : sphere z₀ r ⊆ U
e4 : (∮ (z : ℂ) in C(z₀, r), deriv f z / f z) = ∮ (z : ℂ) in C(z₀, r), c / (z - z₀) + deriv g z / g z
⊢ ContinuousOn (fun z => deriv g z / g z) (sphere z₀ r) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | cindex_eq_order_aux | [120, 1] | [144, 7] | refine ContinuousOn.div ?_ (h1.continuousOn.mono e2) (λ z hz => h2 _ (sphere_subset_closedBall hz)) | case hg
f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
h0 : closedBall z₀ r ⊆ U
h1 : DifferentiableOn ℂ g U
h2 : ∀ z ∈ closedBall z₀ r, g z ≠ 0
h3 : ∀ {z : ℂ}, z ∈ sphere z₀ r → deriv f z / f z = c / (z - z₀) + deriv g z / g z
e2 : sphere z₀ r ⊆ U
e4 : (∮ (z : ℂ) in C(z₀, r), deriv f z / f z) = ∮ (z : ℂ) in C(z₀, r), c / (z - z₀) + deriv g z / g z
⊢ ContinuousOn (fun z => deriv g z / g z) (sphere z₀ r) | case hg
f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
h0 : closedBall z₀ r ⊆ U
h1 : DifferentiableOn ℂ g U
h2 : ∀ z ∈ closedBall z₀ r, g z ≠ 0
h3 : ∀ {z : ℂ}, z ∈ sphere z₀ r → deriv f z / f z = c / (z - z₀) + deriv g z / g z
e2 : sphere z₀ r ⊆ U
e4 : (∮ (z : ℂ) in C(z₀, r), deriv f z / f z) = ∮ (z : ℂ) in C(z₀, r), c / (z - z₀) + deriv g z / g z
⊢ ContinuousOn (fun z => deriv g z) (sphere z₀ r) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | cindex_eq_order_aux | [120, 1] | [144, 7] | refine ContinuousOn.mono ?_ e2 | case hg
f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
h0 : closedBall z₀ r ⊆ U
h1 : DifferentiableOn ℂ g U
h2 : ∀ z ∈ closedBall z₀ r, g z ≠ 0
h3 : ∀ {z : ℂ}, z ∈ sphere z₀ r → deriv f z / f z = c / (z - z₀) + deriv g z / g z
e2 : sphere z₀ r ⊆ U
e4 : (∮ (z : ℂ) in C(z₀, r), deriv f z / f z) = ∮ (z : ℂ) in C(z₀, r), c / (z - z₀) + deriv g z / g z
⊢ ContinuousOn (fun z => deriv g z) (sphere z₀ r) | case hg
f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
h0 : closedBall z₀ r ⊆ U
h1 : DifferentiableOn ℂ g U
h2 : ∀ z ∈ closedBall z₀ r, g z ≠ 0
h3 : ∀ {z : ℂ}, z ∈ sphere z₀ r → deriv f z / f z = c / (z - z₀) + deriv g z / g z
e2 : sphere z₀ r ⊆ U
e4 : (∮ (z : ℂ) in C(z₀, r), deriv f z / f z) = ∮ (z : ℂ) in C(z₀, r), c / (z - z₀) + deriv g z / g z
⊢ ContinuousOn (fun z => deriv g z) U |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | cindex_eq_order_aux | [120, 1] | [144, 7] | apply ContDiffOn.continuousOn_deriv_of_isOpen ?_ hU le_rfl | case hg
f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
h0 : closedBall z₀ r ⊆ U
h1 : DifferentiableOn ℂ g U
h2 : ∀ z ∈ closedBall z₀ r, g z ≠ 0
h3 : ∀ {z : ℂ}, z ∈ sphere z₀ r → deriv f z / f z = c / (z - z₀) + deriv g z / g z
e2 : sphere z₀ r ⊆ U
e4 : (∮ (z : ℂ) in C(z₀, r), deriv f z / f z) = ∮ (z : ℂ) in C(z₀, r), c / (z - z₀) + deriv g z / g z
⊢ ContinuousOn (fun z => deriv g z) U | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
h0 : closedBall z₀ r ⊆ U
h1 : DifferentiableOn ℂ g U
h2 : ∀ z ∈ closedBall z₀ r, g z ≠ 0
h3 : ∀ {z : ℂ}, z ∈ sphere z₀ r → deriv f z / f z = c / (z - z₀) + deriv g z / g z
e2 : sphere z₀ r ⊆ U
e4 : (∮ (z : ℂ) in C(z₀, r), deriv f z / f z) = ∮ (z : ℂ) in C(z₀, r), c / (z - z₀) + deriv g z / g z
⊢ ContDiffOn ℂ 1 g U |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | cindex_eq_order_aux | [120, 1] | [144, 7] | exact h1.contDiffOn hU | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
h0 : closedBall z₀ r ⊆ U
h1 : DifferentiableOn ℂ g U
h2 : ∀ z ∈ closedBall z₀ r, g z ≠ 0
h3 : ∀ {z : ℂ}, z ∈ sphere z₀ r → deriv f z / f z = c / (z - z₀) + deriv g z / g z
e2 : sphere z₀ r ⊆ U
e4 : (∮ (z : ℂ) in C(z₀, r), deriv f z / f z) = ∮ (z : ℂ) in C(z₀, r), c / (z - z₀) + deriv g z / g z
⊢ ContDiffOn ℂ 1 g U | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | cindex_eq_order_aux | [120, 1] | [144, 7] | have := cindex_eq_zero hU hr h0 h1 h2 | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
h0 : closedBall z₀ r ⊆ U
h1 : DifferentiableOn ℂ g U
h2 : ∀ z ∈ closedBall z₀ r, g z ≠ 0
h3 : ∀ {z : ℂ}, z ∈ sphere z₀ r → deriv f z / f z = c / (z - z₀) + deriv g z / g z
e2 : sphere z₀ r ⊆ U
e4 : (∮ (z : ℂ) in C(z₀, r), deriv f z / f z) = ∮ (z : ℂ) in C(z₀, r), c / (z - z₀) + deriv g z / g z
e5 :
(∮ (z : ℂ) in C(z₀, r), c / (z - z₀) + deriv g z / g z) =
(∮ (z : ℂ) in C(z₀, r), c / (z - z₀)) + ∮ (z : ℂ) in C(z₀, r), deriv g z / g z
⊢ (∮ (z : ℂ) in C(z₀, r), deriv g z / g z) = 0 | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
h0 : closedBall z₀ r ⊆ U
h1 : DifferentiableOn ℂ g U
h2 : ∀ z ∈ closedBall z₀ r, g z ≠ 0
h3 : ∀ {z : ℂ}, z ∈ sphere z₀ r → deriv f z / f z = c / (z - z₀) + deriv g z / g z
e2 : sphere z₀ r ⊆ U
e4 : (∮ (z : ℂ) in C(z₀, r), deriv f z / f z) = ∮ (z : ℂ) in C(z₀, r), c / (z - z₀) + deriv g z / g z
e5 :
(∮ (z : ℂ) in C(z₀, r), c / (z - z₀) + deriv g z / g z) =
(∮ (z : ℂ) in C(z₀, r), c / (z - z₀)) + ∮ (z : ℂ) in C(z₀, r), deriv g z / g z
this : cindex z₀ r g = 0
⊢ (∮ (z : ℂ) in C(z₀, r), deriv g z / g z) = 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | cindex_eq_order_aux | [120, 1] | [144, 7] | simpa [cindex, Real.pi_ne_zero, I_ne_zero] using this | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
h0 : closedBall z₀ r ⊆ U
h1 : DifferentiableOn ℂ g U
h2 : ∀ z ∈ closedBall z₀ r, g z ≠ 0
h3 : ∀ {z : ℂ}, z ∈ sphere z₀ r → deriv f z / f z = c / (z - z₀) + deriv g z / g z
e2 : sphere z₀ r ⊆ U
e4 : (∮ (z : ℂ) in C(z₀, r), deriv f z / f z) = ∮ (z : ℂ) in C(z₀, r), c / (z - z₀) + deriv g z / g z
e5 :
(∮ (z : ℂ) in C(z₀, r), c / (z - z₀) + deriv g z / g z) =
(∮ (z : ℂ) in C(z₀, r), c / (z - z₀)) + ∮ (z : ℂ) in C(z₀, r), deriv g z / g z
this : cindex z₀ r g = 0
⊢ (∮ (z : ℂ) in C(z₀, r), deriv g z / g z) = 0 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | cindex_eq_order_aux | [120, 1] | [144, 7] | simpa [div_eq_mul_inv, mul_comm _ _⁻¹] using circle_integral_sub_center_inv_smul hr | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hU : IsOpen U
hr : 0 < r
h0 : closedBall z₀ r ⊆ U
h1 : DifferentiableOn ℂ g U
h2 : ∀ z ∈ closedBall z₀ r, g z ≠ 0
h3 : ∀ {z : ℂ}, z ∈ sphere z₀ r → deriv f z / f z = c / (z - z₀) + deriv g z / g z
e2 : sphere z₀ r ⊆ U
e4 : (∮ (z : ℂ) in C(z₀, r), deriv f z / f z) = ∮ (z : ℂ) in C(z₀, r), c / (z - z₀) + deriv g z / g z
e5 :
(∮ (z : ℂ) in C(z₀, r), c / (z - z₀) + deriv g z / g z) =
(∮ (z : ℂ) in C(z₀, r), c / (z - z₀)) + ∮ (z : ℂ) in C(z₀, r), deriv g z / g z
e6 : (∮ (z : ℂ) in C(z₀, r), deriv g z / g z) = 0
⊢ (∮ (z : ℂ) in C(z₀, r), c / (z - z₀)) = 2 * ↑π * I * c | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | exists_cindex_eq_order' | [146, 1] | [161, 41] | set g : ℂ → ℂ := iterate (swap dslope z₀) p.order f | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hp : HasFPowerSeriesAt f p z₀
h : p ≠ 0
⊢ ∃ R > 0, ∀ r ∈ Set.Ioo 0 R, cindex z₀ r f = ↑(FormalMultilinearSeries.order p) | f g✝ : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hp : HasFPowerSeriesAt f p z₀
h : p ≠ 0
g : ℂ → ℂ := (swap dslope z₀)^[FormalMultilinearSeries.order p] f
⊢ ∃ R > 0, ∀ r ∈ Set.Ioo 0 R, cindex z₀ r f = ↑(FormalMultilinearSeries.order p) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | exists_cindex_eq_order' | [146, 1] | [161, 41] | have lh1 : ∀ᶠ z in 𝓝 z₀, g z ≠ 0 := hp.dslope_order_eventually_ne_zero h | f g✝ : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hp : HasFPowerSeriesAt f p z₀
h : p ≠ 0
g : ℂ → ℂ := (swap dslope z₀)^[FormalMultilinearSeries.order p] f
⊢ ∃ R > 0, ∀ r ∈ Set.Ioo 0 R, cindex z₀ r f = ↑(FormalMultilinearSeries.order p) | f g✝ : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hp : HasFPowerSeriesAt f p z₀
h : p ≠ 0
g : ℂ → ℂ := (swap dslope z₀)^[FormalMultilinearSeries.order p] f
lh1 : ∀ᶠ (z : ℂ) in 𝓝 z₀, g z ≠ 0
⊢ ∃ R > 0, ∀ r ∈ Set.Ioo 0 R, cindex z₀ r f = ↑(FormalMultilinearSeries.order p) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | exists_cindex_eq_order' | [146, 1] | [161, 41] | have lh2 : ∀ᶠ z in 𝓝 z₀, z ≠ z₀ → deriv f z / f z = p.order / (z - z₀) + deriv g z / g z :=
eventually_deriv_div_self_eq hp h | f g✝ : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hp : HasFPowerSeriesAt f p z₀
h : p ≠ 0
g : ℂ → ℂ := (swap dslope z₀)^[FormalMultilinearSeries.order p] f
lh1 : ∀ᶠ (z : ℂ) in 𝓝 z₀, g z ≠ 0
⊢ ∃ R > 0, ∀ r ∈ Set.Ioo 0 R, cindex z₀ r f = ↑(FormalMultilinearSeries.order p) | f g✝ : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hp : HasFPowerSeriesAt f p z₀
h : p ≠ 0
g : ℂ → ℂ := (swap dslope z₀)^[FormalMultilinearSeries.order p] f
lh1 : ∀ᶠ (z : ℂ) in 𝓝 z₀, g z ≠ 0
lh2 : ∀ᶠ (z : ℂ) in 𝓝 z₀, z ≠ z₀ → deriv f z / f z = ↑(FormalMultilinearSeries.order p) / (z - z₀) + deriv g z / g z
⊢ ∃ R > 0, ∀ r ∈ Set.Ioo 0 R, cindex z₀ r f = ↑(FormalMultilinearSeries.order p) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | exists_cindex_eq_order' | [146, 1] | [161, 41] | have lh3 : ∀ᶠ z in 𝓝 z₀, DifferentiableAt ℂ g z :=
(hp.has_fpower_series_iterate_dslope_fslope p.order).eventually_differentiable_at | f g✝ : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hp : HasFPowerSeriesAt f p z₀
h : p ≠ 0
g : ℂ → ℂ := (swap dslope z₀)^[FormalMultilinearSeries.order p] f
lh1 : ∀ᶠ (z : ℂ) in 𝓝 z₀, g z ≠ 0
lh2 : ∀ᶠ (z : ℂ) in 𝓝 z₀, z ≠ z₀ → deriv f z / f z = ↑(FormalMultilinearSeries.order p) / (z - z₀) + deriv g z / g z
⊢ ∃ R > 0, ∀ r ∈ Set.Ioo 0 R, cindex z₀ r f = ↑(FormalMultilinearSeries.order p) | f g✝ : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hp : HasFPowerSeriesAt f p z₀
h : p ≠ 0
g : ℂ → ℂ := (swap dslope z₀)^[FormalMultilinearSeries.order p] f
lh1 : ∀ᶠ (z : ℂ) in 𝓝 z₀, g z ≠ 0
lh2 : ∀ᶠ (z : ℂ) in 𝓝 z₀, z ≠ z₀ → deriv f z / f z = ↑(FormalMultilinearSeries.order p) / (z - z₀) + deriv g z / g z
lh3 : ∀ᶠ (z : ℂ) in 𝓝 z₀, DifferentiableAt ℂ g z
⊢ ∃ R > 0, ∀ r ∈ Set.Ioo 0 R, cindex z₀ r f = ↑(FormalMultilinearSeries.order p) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | exists_cindex_eq_order' | [146, 1] | [161, 41] | obtain ⟨R, hR₁, hh⟩ := Metric.mem_nhds_iff.mp (lh1.and (lh2.and lh3)) | f g✝ : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hp : HasFPowerSeriesAt f p z₀
h : p ≠ 0
g : ℂ → ℂ := (swap dslope z₀)^[FormalMultilinearSeries.order p] f
lh1 : ∀ᶠ (z : ℂ) in 𝓝 z₀, g z ≠ 0
lh2 : ∀ᶠ (z : ℂ) in 𝓝 z₀, z ≠ z₀ → deriv f z / f z = ↑(FormalMultilinearSeries.order p) / (z - z₀) + deriv g z / g z
lh3 : ∀ᶠ (z : ℂ) in 𝓝 z₀, DifferentiableAt ℂ g z
⊢ ∃ R > 0, ∀ r ∈ Set.Ioo 0 R, cindex z₀ r f = ↑(FormalMultilinearSeries.order p) | case intro.intro
f g✝ : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hp : HasFPowerSeriesAt f p z₀
h : p ≠ 0
g : ℂ → ℂ := (swap dslope z₀)^[FormalMultilinearSeries.order p] f
lh1 : ∀ᶠ (z : ℂ) in 𝓝 z₀, g z ≠ 0
lh2 : ∀ᶠ (z : ℂ) in 𝓝 z₀, z ≠ z₀ → deriv f z / f z = ↑(FormalMultilinearSeries.order p) / (z - z₀) + deriv g z / g z
lh3 : ∀ᶠ (z : ℂ) in 𝓝 z₀, DifferentiableAt ℂ g z
R : ℝ
hR₁ : R > 0
hh :
ball z₀ R ⊆
{x |
(fun x =>
g x ≠ 0 ∧
(x ≠ z₀ → deriv f x / f x = ↑(FormalMultilinearSeries.order p) / (x - z₀) + deriv g x / g x) ∧
DifferentiableAt ℂ g x)
x}
⊢ ∃ R > 0, ∀ r ∈ Set.Ioo 0 R, cindex z₀ r f = ↑(FormalMultilinearSeries.order p) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | exists_cindex_eq_order' | [146, 1] | [161, 41] | refine ⟨R, hR₁, λ r hr => ?_⟩ | case intro.intro
f g✝ : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hp : HasFPowerSeriesAt f p z₀
h : p ≠ 0
g : ℂ → ℂ := (swap dslope z₀)^[FormalMultilinearSeries.order p] f
lh1 : ∀ᶠ (z : ℂ) in 𝓝 z₀, g z ≠ 0
lh2 : ∀ᶠ (z : ℂ) in 𝓝 z₀, z ≠ z₀ → deriv f z / f z = ↑(FormalMultilinearSeries.order p) / (z - z₀) + deriv g z / g z
lh3 : ∀ᶠ (z : ℂ) in 𝓝 z₀, DifferentiableAt ℂ g z
R : ℝ
hR₁ : R > 0
hh :
ball z₀ R ⊆
{x |
(fun x =>
g x ≠ 0 ∧
(x ≠ z₀ → deriv f x / f x = ↑(FormalMultilinearSeries.order p) / (x - z₀) + deriv g x / g x) ∧
DifferentiableAt ℂ g x)
x}
⊢ ∃ R > 0, ∀ r ∈ Set.Ioo 0 R, cindex z₀ r f = ↑(FormalMultilinearSeries.order p) | case intro.intro
f g✝ : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r✝ : ℝ
hp : HasFPowerSeriesAt f p z₀
h : p ≠ 0
g : ℂ → ℂ := (swap dslope z₀)^[FormalMultilinearSeries.order p] f
lh1 : ∀ᶠ (z : ℂ) in 𝓝 z₀, g z ≠ 0
lh2 : ∀ᶠ (z : ℂ) in 𝓝 z₀, z ≠ z₀ → deriv f z / f z = ↑(FormalMultilinearSeries.order p) / (z - z₀) + deriv g z / g z
lh3 : ∀ᶠ (z : ℂ) in 𝓝 z₀, DifferentiableAt ℂ g z
R : ℝ
hR₁ : R > 0
hh :
ball z₀ R ⊆
{x |
(fun x =>
g x ≠ 0 ∧
(x ≠ z₀ → deriv f x / f x = ↑(FormalMultilinearSeries.order p) / (x - z₀) + deriv g x / g x) ∧
DifferentiableAt ℂ g x)
x}
r : ℝ
hr : r ∈ Set.Ioo 0 R
⊢ cindex z₀ r f = ↑(FormalMultilinearSeries.order p) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | exists_cindex_eq_order' | [146, 1] | [161, 41] | refine cindex_eq_order_aux isOpen_ball hr.1 (closedBall_subset_ball hr.2)
(λ z hz => (hh hz).2.2.differentiableWithinAt)
(λ z hz => (hh (closedBall_subset_ball hr.2 hz)).1)
(λ hz => ?_) | case intro.intro
f g✝ : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r✝ : ℝ
hp : HasFPowerSeriesAt f p z₀
h : p ≠ 0
g : ℂ → ℂ := (swap dslope z₀)^[FormalMultilinearSeries.order p] f
lh1 : ∀ᶠ (z : ℂ) in 𝓝 z₀, g z ≠ 0
lh2 : ∀ᶠ (z : ℂ) in 𝓝 z₀, z ≠ z₀ → deriv f z / f z = ↑(FormalMultilinearSeries.order p) / (z - z₀) + deriv g z / g z
lh3 : ∀ᶠ (z : ℂ) in 𝓝 z₀, DifferentiableAt ℂ g z
R : ℝ
hR₁ : R > 0
hh :
ball z₀ R ⊆
{x |
(fun x =>
g x ≠ 0 ∧
(x ≠ z₀ → deriv f x / f x = ↑(FormalMultilinearSeries.order p) / (x - z₀) + deriv g x / g x) ∧
DifferentiableAt ℂ g x)
x}
r : ℝ
hr : r ∈ Set.Ioo 0 R
⊢ cindex z₀ r f = ↑(FormalMultilinearSeries.order p) | case intro.intro
f g✝ : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r✝ : ℝ
hp : HasFPowerSeriesAt f p z₀
h : p ≠ 0
g : ℂ → ℂ := (swap dslope z₀)^[FormalMultilinearSeries.order p] f
lh1 : ∀ᶠ (z : ℂ) in 𝓝 z₀, g z ≠ 0
lh2 : ∀ᶠ (z : ℂ) in 𝓝 z₀, z ≠ z₀ → deriv f z / f z = ↑(FormalMultilinearSeries.order p) / (z - z₀) + deriv g z / g z
lh3 : ∀ᶠ (z : ℂ) in 𝓝 z₀, DifferentiableAt ℂ g z
R : ℝ
hR₁ : R > 0
hh :
ball z₀ R ⊆
{x |
(fun x =>
g x ≠ 0 ∧
(x ≠ z₀ → deriv f x / f x = ↑(FormalMultilinearSeries.order p) / (x - z₀) + deriv g x / g x) ∧
DifferentiableAt ℂ g x)
x}
r : ℝ
hr : r ∈ Set.Ioo 0 R
z✝ : ℂ
hz : z✝ ∈ sphere z₀ r
⊢ deriv f z✝ / f z✝ = ↑(FormalMultilinearSeries.order p) / (z✝ - z₀) + deriv g z✝ / g z✝ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | exists_cindex_eq_order' | [146, 1] | [161, 41] | refine (hh (sphere_subset_closedBall.trans (closedBall_subset_ball hr.2) hz)).2.1 ?_ | case intro.intro
f g✝ : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r✝ : ℝ
hp : HasFPowerSeriesAt f p z₀
h : p ≠ 0
g : ℂ → ℂ := (swap dslope z₀)^[FormalMultilinearSeries.order p] f
lh1 : ∀ᶠ (z : ℂ) in 𝓝 z₀, g z ≠ 0
lh2 : ∀ᶠ (z : ℂ) in 𝓝 z₀, z ≠ z₀ → deriv f z / f z = ↑(FormalMultilinearSeries.order p) / (z - z₀) + deriv g z / g z
lh3 : ∀ᶠ (z : ℂ) in 𝓝 z₀, DifferentiableAt ℂ g z
R : ℝ
hR₁ : R > 0
hh :
ball z₀ R ⊆
{x |
(fun x =>
g x ≠ 0 ∧
(x ≠ z₀ → deriv f x / f x = ↑(FormalMultilinearSeries.order p) / (x - z₀) + deriv g x / g x) ∧
DifferentiableAt ℂ g x)
x}
r : ℝ
hr : r ∈ Set.Ioo 0 R
z✝ : ℂ
hz : z✝ ∈ sphere z₀ r
⊢ deriv f z✝ / f z✝ = ↑(FormalMultilinearSeries.order p) / (z✝ - z₀) + deriv g z✝ / g z✝ | case intro.intro
f g✝ : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r✝ : ℝ
hp : HasFPowerSeriesAt f p z₀
h : p ≠ 0
g : ℂ → ℂ := (swap dslope z₀)^[FormalMultilinearSeries.order p] f
lh1 : ∀ᶠ (z : ℂ) in 𝓝 z₀, g z ≠ 0
lh2 : ∀ᶠ (z : ℂ) in 𝓝 z₀, z ≠ z₀ → deriv f z / f z = ↑(FormalMultilinearSeries.order p) / (z - z₀) + deriv g z / g z
lh3 : ∀ᶠ (z : ℂ) in 𝓝 z₀, DifferentiableAt ℂ g z
R : ℝ
hR₁ : R > 0
hh :
ball z₀ R ⊆
{x |
(fun x =>
g x ≠ 0 ∧
(x ≠ z₀ → deriv f x / f x = ↑(FormalMultilinearSeries.order p) / (x - z₀) + deriv g x / g x) ∧
DifferentiableAt ℂ g x)
x}
r : ℝ
hr : r ∈ Set.Ioo 0 R
z✝ : ℂ
hz : z✝ ∈ sphere z₀ r
⊢ z✝ ≠ z₀ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | exists_cindex_eq_order' | [146, 1] | [161, 41] | exact ne_of_mem_sphere hz hr.1.ne.symm | case intro.intro
f g✝ : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r✝ : ℝ
hp : HasFPowerSeriesAt f p z₀
h : p ≠ 0
g : ℂ → ℂ := (swap dslope z₀)^[FormalMultilinearSeries.order p] f
lh1 : ∀ᶠ (z : ℂ) in 𝓝 z₀, g z ≠ 0
lh2 : ∀ᶠ (z : ℂ) in 𝓝 z₀, z ≠ z₀ → deriv f z / f z = ↑(FormalMultilinearSeries.order p) / (z - z₀) + deriv g z / g z
lh3 : ∀ᶠ (z : ℂ) in 𝓝 z₀, DifferentiableAt ℂ g z
R : ℝ
hR₁ : R > 0
hh :
ball z₀ R ⊆
{x |
(fun x =>
g x ≠ 0 ∧
(x ≠ z₀ → deriv f x / f x = ↑(FormalMultilinearSeries.order p) / (x - z₀) + deriv g x / g x) ∧
DifferentiableAt ℂ g x)
x}
r : ℝ
hr : r ∈ Set.Ioo 0 R
z✝ : ℂ
hz : z✝ ∈ sphere z₀ r
⊢ z✝ ≠ z₀ | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | exists_cindex_eq_order | [163, 1] | [180, 26] | by_cases h : p = 0 | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hp : HasFPowerSeriesAt f p z₀
⊢ ∃ R > 0, ∀ r ∈ Set.Ioo 0 R, cindex z₀ r f = ↑(FormalMultilinearSeries.order p) | case pos
f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hp : HasFPowerSeriesAt f p z₀
h : p = 0
⊢ ∃ R > 0, ∀ r ∈ Set.Ioo 0 R, cindex z₀ r f = ↑(FormalMultilinearSeries.order p)
case neg
f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hp : HasFPowerSeriesAt f p z₀
h : ¬p = 0
⊢ ∃ R > 0, ∀ r ∈ Set.Ioo 0 R, cindex z₀ r f = ↑(FormalMultilinearSeries.order p) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | exists_cindex_eq_order | [163, 1] | [180, 26] | case neg => exact exists_cindex_eq_order' hp h | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hp : HasFPowerSeriesAt f p z₀
h : ¬p = 0
⊢ ∃ R > 0, ∀ r ∈ Set.Ioo 0 R, cindex z₀ r f = ↑(FormalMultilinearSeries.order p) | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | exists_cindex_eq_order | [163, 1] | [180, 26] | case pos =>
subst_vars
obtain ⟨R, hR, hf⟩ := Metric.eventually_nhds_iff.mp (hp.locally_zero_iff.mpr rfl)
refine ⟨R, hR, λ r hr => ?_⟩
simp [cindex, Real.pi_ne_zero, Complex.I_ne_zero]
have : Set.EqOn (λ z => deriv f z / f z) 0 (sphere z₀ r) := by
intro z hz
simp
right
apply hf
rw [hz.symm.symm]
exact hr.2
rw [circleIntegral.integral_congr hr.1.le this]
simp [circleIntegral] | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hp : HasFPowerSeriesAt f p z₀
h : p = 0
⊢ ∃ R > 0, ∀ r ∈ Set.Ioo 0 R, cindex z₀ r f = ↑(FormalMultilinearSeries.order p) | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | exists_cindex_eq_order | [163, 1] | [180, 26] | exact exists_cindex_eq_order' hp h | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hp : HasFPowerSeriesAt f p z₀
h : ¬p = 0
⊢ ∃ R > 0, ∀ r ∈ Set.Ioo 0 R, cindex z₀ r f = ↑(FormalMultilinearSeries.order p) | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | exists_cindex_eq_order | [163, 1] | [180, 26] | subst_vars | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hp : HasFPowerSeriesAt f p z₀
h : p = 0
⊢ ∃ R > 0, ∀ r ∈ Set.Ioo 0 R, cindex z₀ r f = ↑(FormalMultilinearSeries.order p) | f g : ℂ → ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hp : HasFPowerSeriesAt f 0 z₀
⊢ ∃ R > 0, ∀ r ∈ Set.Ioo 0 R, cindex z₀ r f = ↑(FormalMultilinearSeries.order 0) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | exists_cindex_eq_order | [163, 1] | [180, 26] | obtain ⟨R, hR, hf⟩ := Metric.eventually_nhds_iff.mp (hp.locally_zero_iff.mpr rfl) | f g : ℂ → ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hp : HasFPowerSeriesAt f 0 z₀
⊢ ∃ R > 0, ∀ r ∈ Set.Ioo 0 R, cindex z₀ r f = ↑(FormalMultilinearSeries.order 0) | case intro.intro
f g : ℂ → ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hp : HasFPowerSeriesAt f 0 z₀
R : ℝ
hR : R > 0
hf : ∀ ⦃y : ℂ⦄, dist y z₀ < R → f y = 0
⊢ ∃ R > 0, ∀ r ∈ Set.Ioo 0 R, cindex z₀ r f = ↑(FormalMultilinearSeries.order 0) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | exists_cindex_eq_order | [163, 1] | [180, 26] | refine ⟨R, hR, λ r hr => ?_⟩ | case intro.intro
f g : ℂ → ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hp : HasFPowerSeriesAt f 0 z₀
R : ℝ
hR : R > 0
hf : ∀ ⦃y : ℂ⦄, dist y z₀ < R → f y = 0
⊢ ∃ R > 0, ∀ r ∈ Set.Ioo 0 R, cindex z₀ r f = ↑(FormalMultilinearSeries.order 0) | case intro.intro
f g : ℂ → ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r✝ : ℝ
hp : HasFPowerSeriesAt f 0 z₀
R : ℝ
hR : R > 0
hf : ∀ ⦃y : ℂ⦄, dist y z₀ < R → f y = 0
r : ℝ
hr : r ∈ Set.Ioo 0 R
⊢ cindex z₀ r f = ↑(FormalMultilinearSeries.order 0) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | exists_cindex_eq_order | [163, 1] | [180, 26] | simp [cindex, Real.pi_ne_zero, Complex.I_ne_zero] | case intro.intro
f g : ℂ → ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r✝ : ℝ
hp : HasFPowerSeriesAt f 0 z₀
R : ℝ
hR : R > 0
hf : ∀ ⦃y : ℂ⦄, dist y z₀ < R → f y = 0
r : ℝ
hr : r ∈ Set.Ioo 0 R
⊢ cindex z₀ r f = ↑(FormalMultilinearSeries.order 0) | case intro.intro
f g : ℂ → ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r✝ : ℝ
hp : HasFPowerSeriesAt f 0 z₀
R : ℝ
hR : R > 0
hf : ∀ ⦃y : ℂ⦄, dist y z₀ < R → f y = 0
r : ℝ
hr : r ∈ Set.Ioo 0 R
⊢ (∮ (z : ℂ) in C(z₀, r), deriv f z / f z) = 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | exists_cindex_eq_order | [163, 1] | [180, 26] | have : Set.EqOn (λ z => deriv f z / f z) 0 (sphere z₀ r) := by
intro z hz
simp
right
apply hf
rw [hz.symm.symm]
exact hr.2 | case intro.intro
f g : ℂ → ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r✝ : ℝ
hp : HasFPowerSeriesAt f 0 z₀
R : ℝ
hR : R > 0
hf : ∀ ⦃y : ℂ⦄, dist y z₀ < R → f y = 0
r : ℝ
hr : r ∈ Set.Ioo 0 R
⊢ (∮ (z : ℂ) in C(z₀, r), deriv f z / f z) = 0 | case intro.intro
f g : ℂ → ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r✝ : ℝ
hp : HasFPowerSeriesAt f 0 z₀
R : ℝ
hR : R > 0
hf : ∀ ⦃y : ℂ⦄, dist y z₀ < R → f y = 0
r : ℝ
hr : r ∈ Set.Ioo 0 R
this : Set.EqOn (fun z => deriv f z / f z) 0 (sphere z₀ r)
⊢ (∮ (z : ℂ) in C(z₀, r), deriv f z / f z) = 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | exists_cindex_eq_order | [163, 1] | [180, 26] | rw [circleIntegral.integral_congr hr.1.le this] | case intro.intro
f g : ℂ → ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r✝ : ℝ
hp : HasFPowerSeriesAt f 0 z₀
R : ℝ
hR : R > 0
hf : ∀ ⦃y : ℂ⦄, dist y z₀ < R → f y = 0
r : ℝ
hr : r ∈ Set.Ioo 0 R
this : Set.EqOn (fun z => deriv f z / f z) 0 (sphere z₀ r)
⊢ (∮ (z : ℂ) in C(z₀, r), deriv f z / f z) = 0 | case intro.intro
f g : ℂ → ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r✝ : ℝ
hp : HasFPowerSeriesAt f 0 z₀
R : ℝ
hR : R > 0
hf : ∀ ⦃y : ℂ⦄, dist y z₀ < R → f y = 0
r : ℝ
hr : r ∈ Set.Ioo 0 R
this : Set.EqOn (fun z => deriv f z / f z) 0 (sphere z₀ r)
⊢ (∮ (z : ℂ) in C(z₀, r), 0 z) = 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | exists_cindex_eq_order | [163, 1] | [180, 26] | simp [circleIntegral] | case intro.intro
f g : ℂ → ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r✝ : ℝ
hp : HasFPowerSeriesAt f 0 z₀
R : ℝ
hR : R > 0
hf : ∀ ⦃y : ℂ⦄, dist y z₀ < R → f y = 0
r : ℝ
hr : r ∈ Set.Ioo 0 R
this : Set.EqOn (fun z => deriv f z / f z) 0 (sphere z₀ r)
⊢ (∮ (z : ℂ) in C(z₀, r), 0 z) = 0 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | exists_cindex_eq_order | [163, 1] | [180, 26] | intro z hz | f g : ℂ → ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r✝ : ℝ
hp : HasFPowerSeriesAt f 0 z₀
R : ℝ
hR : R > 0
hf : ∀ ⦃y : ℂ⦄, dist y z₀ < R → f y = 0
r : ℝ
hr : r ∈ Set.Ioo 0 R
⊢ Set.EqOn (fun z => deriv f z / f z) 0 (sphere z₀ r) | f g : ℂ → ℂ
z✝ z₀ c : ℂ
n : ℕ
U : Set ℂ
r✝ : ℝ
hp : HasFPowerSeriesAt f 0 z₀
R : ℝ
hR : R > 0
hf : ∀ ⦃y : ℂ⦄, dist y z₀ < R → f y = 0
r : ℝ
hr : r ∈ Set.Ioo 0 R
z : ℂ
hz : z ∈ sphere z₀ r
⊢ (fun z => deriv f z / f z) z = 0 z |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | exists_cindex_eq_order | [163, 1] | [180, 26] | simp | f g : ℂ → ℂ
z✝ z₀ c : ℂ
n : ℕ
U : Set ℂ
r✝ : ℝ
hp : HasFPowerSeriesAt f 0 z₀
R : ℝ
hR : R > 0
hf : ∀ ⦃y : ℂ⦄, dist y z₀ < R → f y = 0
r : ℝ
hr : r ∈ Set.Ioo 0 R
z : ℂ
hz : z ∈ sphere z₀ r
⊢ (fun z => deriv f z / f z) z = 0 z | f g : ℂ → ℂ
z✝ z₀ c : ℂ
n : ℕ
U : Set ℂ
r✝ : ℝ
hp : HasFPowerSeriesAt f 0 z₀
R : ℝ
hR : R > 0
hf : ∀ ⦃y : ℂ⦄, dist y z₀ < R → f y = 0
r : ℝ
hr : r ∈ Set.Ioo 0 R
z : ℂ
hz : z ∈ sphere z₀ r
⊢ deriv f z = 0 ∨ f z = 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | exists_cindex_eq_order | [163, 1] | [180, 26] | right | f g : ℂ → ℂ
z✝ z₀ c : ℂ
n : ℕ
U : Set ℂ
r✝ : ℝ
hp : HasFPowerSeriesAt f 0 z₀
R : ℝ
hR : R > 0
hf : ∀ ⦃y : ℂ⦄, dist y z₀ < R → f y = 0
r : ℝ
hr : r ∈ Set.Ioo 0 R
z : ℂ
hz : z ∈ sphere z₀ r
⊢ deriv f z = 0 ∨ f z = 0 | case h
f g : ℂ → ℂ
z✝ z₀ c : ℂ
n : ℕ
U : Set ℂ
r✝ : ℝ
hp : HasFPowerSeriesAt f 0 z₀
R : ℝ
hR : R > 0
hf : ∀ ⦃y : ℂ⦄, dist y z₀ < R → f y = 0
r : ℝ
hr : r ∈ Set.Ioo 0 R
z : ℂ
hz : z ∈ sphere z₀ r
⊢ f z = 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | exists_cindex_eq_order | [163, 1] | [180, 26] | apply hf | case h
f g : ℂ → ℂ
z✝ z₀ c : ℂ
n : ℕ
U : Set ℂ
r✝ : ℝ
hp : HasFPowerSeriesAt f 0 z₀
R : ℝ
hR : R > 0
hf : ∀ ⦃y : ℂ⦄, dist y z₀ < R → f y = 0
r : ℝ
hr : r ∈ Set.Ioo 0 R
z : ℂ
hz : z ∈ sphere z₀ r
⊢ f z = 0 | case h.a
f g : ℂ → ℂ
z✝ z₀ c : ℂ
n : ℕ
U : Set ℂ
r✝ : ℝ
hp : HasFPowerSeriesAt f 0 z₀
R : ℝ
hR : R > 0
hf : ∀ ⦃y : ℂ⦄, dist y z₀ < R → f y = 0
r : ℝ
hr : r ∈ Set.Ioo 0 R
z : ℂ
hz : z ∈ sphere z₀ r
⊢ dist z z₀ < R |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | exists_cindex_eq_order | [163, 1] | [180, 26] | rw [hz.symm.symm] | case h.a
f g : ℂ → ℂ
z✝ z₀ c : ℂ
n : ℕ
U : Set ℂ
r✝ : ℝ
hp : HasFPowerSeriesAt f 0 z₀
R : ℝ
hR : R > 0
hf : ∀ ⦃y : ℂ⦄, dist y z₀ < R → f y = 0
r : ℝ
hr : r ∈ Set.Ioo 0 R
z : ℂ
hz : z ∈ sphere z₀ r
⊢ dist z z₀ < R | case h.a
f g : ℂ → ℂ
z✝ z₀ c : ℂ
n : ℕ
U : Set ℂ
r✝ : ℝ
hp : HasFPowerSeriesAt f 0 z₀
R : ℝ
hR : R > 0
hf : ∀ ⦃y : ℂ⦄, dist y z₀ < R → f y = 0
r : ℝ
hr : r ∈ Set.Ioo 0 R
z : ℂ
hz : z ∈ sphere z₀ r
⊢ r < R |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | exists_cindex_eq_order | [163, 1] | [180, 26] | exact hr.2 | case h.a
f g : ℂ → ℂ
z✝ z₀ c : ℂ
n : ℕ
U : Set ℂ
r✝ : ℝ
hp : HasFPowerSeriesAt f 0 z₀
R : ℝ
hR : R > 0
hf : ∀ ⦃y : ℂ⦄, dist y z₀ < R → f y = 0
r : ℝ
hr : r ∈ Set.Ioo 0 R
z : ℂ
hz : z ∈ sphere z₀ r
⊢ r < R | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | cindex_eventually_eq_order | [182, 1] | [186, 76] | rw [eventually_nhdsWithin_iff, Metric.eventually_nhds_iff] | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hp : HasFPowerSeriesAt f p z₀
⊢ ∀ᶠ (r : ℝ) in 𝓝[>] 0, cindex z₀ r f = ↑(FormalMultilinearSeries.order p) | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hp : HasFPowerSeriesAt f p z₀
⊢ ∃ ε > 0, ∀ ⦃y : ℝ⦄, dist y 0 < ε → y ∈ Set.Ioi 0 → cindex z₀ y f = ↑(FormalMultilinearSeries.order p) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | cindex_eventually_eq_order | [182, 1] | [186, 76] | obtain ⟨R, hR, hf⟩ := exists_cindex_eq_order hp | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hp : HasFPowerSeriesAt f p z₀
⊢ ∃ ε > 0, ∀ ⦃y : ℝ⦄, dist y 0 < ε → y ∈ Set.Ioi 0 → cindex z₀ y f = ↑(FormalMultilinearSeries.order p) | case intro.intro
f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hp : HasFPowerSeriesAt f p z₀
R : ℝ
hR : R > 0
hf : ∀ r ∈ Set.Ioo 0 R, cindex z₀ r f = ↑(FormalMultilinearSeries.order p)
⊢ ∃ ε > 0, ∀ ⦃y : ℝ⦄, dist y 0 < ε → y ∈ Set.Ioi 0 → cindex z₀ y f = ↑(FormalMultilinearSeries.order p) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | cindex_eventually_eq_order | [182, 1] | [186, 76] | exact ⟨R, hR, λ r hr1 hr2 => hf r ⟨hr2, by simpa using lt_of_abs_lt hr1⟩⟩ | case intro.intro
f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r : ℝ
hp : HasFPowerSeriesAt f p z₀
R : ℝ
hR : R > 0
hf : ∀ r ∈ Set.Ioo 0 R, cindex z₀ r f = ↑(FormalMultilinearSeries.order p)
⊢ ∃ ε > 0, ∀ ⦃y : ℝ⦄, dist y 0 < ε → y ∈ Set.Ioi 0 → cindex z₀ y f = ↑(FormalMultilinearSeries.order p) | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | cindex_eventually_eq_order | [182, 1] | [186, 76] | simpa using lt_of_abs_lt hr1 | f g : ℂ → ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
z z₀ c : ℂ
n : ℕ
U : Set ℂ
r✝ : ℝ
hp : HasFPowerSeriesAt f p z₀
R : ℝ
hR : R > 0
hf : ∀ r ∈ Set.Ioo 0 R, cindex z₀ r f = ↑(FormalMultilinearSeries.order p)
r : ℝ
hr1 : dist r 0 < R
hr2 : r ∈ Set.Ioi 0
⊢ r < R | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.deriv | [14, 1] | [30, 32] | rintro K ⟨hK1, hK2⟩ | ι : Type u_1
U K : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
⊢ UniformlyBoundedOn (_root_.deriv ∘ F) U | case intro
ι : Type u_1
U K✝ : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
K : Set ℂ
hK1 : K ⊆ U
hK2 : IsCompact K
⊢ ∃ Q, IsCompact Q ∧ ∀ (i : ι), MapsTo ((_root_.deriv ∘ F) i) K Q |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.deriv | [14, 1] | [30, 32] | obtain ⟨δ, hδ, h⟩ := hK2.exists_cthickening_subset_open hU hK1 | case intro
ι : Type u_1
U K✝ : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
K : Set ℂ
hK1 : K ⊆ U
hK2 : IsCompact K
⊢ ∃ Q, IsCompact Q ∧ ∀ (i : ι), MapsTo ((_root_.deriv ∘ F) i) K Q | case intro.intro.intro
ι : Type u_1
U K✝ : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
K : Set ℂ
hK1 : K ⊆ U
hK2 : IsCompact K
δ : ℝ
hδ : 0 < δ
h : cthickening δ K ⊆ U
⊢ ∃ Q, IsCompact Q ∧ ∀ (i : ι), MapsTo ((_root_.deriv ∘ F) i) K Q |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.deriv | [14, 1] | [30, 32] | have e1 : cthickening δ K ∈ compacts U :=
⟨h, isCompact_of_isClosed_isBounded isClosed_cthickening hK2.isBounded.cthickening⟩ | case intro.intro.intro
ι : Type u_1
U K✝ : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
K : Set ℂ
hK1 : K ⊆ U
hK2 : IsCompact K
δ : ℝ
hδ : 0 < δ
h : cthickening δ K ⊆ U
⊢ ∃ Q, IsCompact Q ∧ ∀ (i : ι), MapsTo ((_root_.deriv ∘ F) i) K Q | case intro.intro.intro
ι : Type u_1
U K✝ : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
K : Set ℂ
hK1 : K ⊆ U
hK2 : IsCompact K
δ : ℝ
hδ : 0 < δ
h : cthickening δ K ⊆ U
e1 : cthickening δ K ∈ compacts U
⊢ ∃ Q, IsCompact Q ∧ ∀ (i : ι), MapsTo ((_root_.deriv ∘ F) i) K Q |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.deriv | [14, 1] | [30, 32] | obtain ⟨Q, hQ1, hQ2⟩ := h1 _ e1 | case intro.intro.intro
ι : Type u_1
U K✝ : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
K : Set ℂ
hK1 : K ⊆ U
hK2 : IsCompact K
δ : ℝ
hδ : 0 < δ
h : cthickening δ K ⊆ U
e1 : cthickening δ K ∈ compacts U
⊢ ∃ Q, IsCompact Q ∧ ∀ (i : ι), MapsTo ((_root_.deriv ∘ F) i) K Q | case intro.intro.intro.intro.intro
ι : Type u_1
U K✝ : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q✝ : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
K : Set ℂ
hK1 : K ⊆ U
hK2 : IsCompact K
δ : ℝ
hδ : 0 < δ
h : cthickening δ K ⊆ U
e1 : cthickening δ K ∈ compacts U
Q : Set ℂ
hQ1 : IsCompact Q
hQ2 : ∀ (i : ι), MapsTo (F i) (cthickening δ K) Q
⊢ ∃ Q, IsCompact Q ∧ ∀ (i : ι), MapsTo ((_root_.deriv ∘ F) i) K Q |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.deriv | [14, 1] | [30, 32] | obtain ⟨M, hM⟩ := hQ1.isBounded.subset_closedBall 0 | case intro.intro.intro.intro.intro
ι : Type u_1
U K✝ : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q✝ : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
K : Set ℂ
hK1 : K ⊆ U
hK2 : IsCompact K
δ : ℝ
hδ : 0 < δ
h : cthickening δ K ⊆ U
e1 : cthickening δ K ∈ compacts U
Q : Set ℂ
hQ1 : IsCompact Q
hQ2 : ∀ (i : ι), MapsTo (F i) (cthickening δ K) Q
⊢ ∃ Q, IsCompact Q ∧ ∀ (i : ι), MapsTo ((_root_.deriv ∘ F) i) K Q | case intro.intro.intro.intro.intro.intro
ι : Type u_1
U K✝ : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q✝ : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
K : Set ℂ
hK1 : K ⊆ U
hK2 : IsCompact K
δ : ℝ
hδ : 0 < δ
h : cthickening δ K ⊆ U
e1 : cthickening δ K ∈ compacts U
Q : Set ℂ
hQ1 : IsCompact Q
hQ2 : ∀ (i : ι), MapsTo (F i) (cthickening δ K) Q
M : ℝ
hM : Q ⊆ closedBall 0 M
⊢ ∃ Q, IsCompact Q ∧ ∀ (i : ι), MapsTo ((_root_.deriv ∘ F) i) K Q |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.deriv | [14, 1] | [30, 32] | refine ⟨closedBall 0 (M / δ), isCompact_closedBall _ _, ?_⟩ | case intro.intro.intro.intro.intro.intro
ι : Type u_1
U K✝ : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q✝ : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
K : Set ℂ
hK1 : K ⊆ U
hK2 : IsCompact K
δ : ℝ
hδ : 0 < δ
h : cthickening δ K ⊆ U
e1 : cthickening δ K ∈ compacts U
Q : Set ℂ
hQ1 : IsCompact Q
hQ2 : ∀ (i : ι), MapsTo (F i) (cthickening δ K) Q
M : ℝ
hM : Q ⊆ closedBall 0 M
⊢ ∃ Q, IsCompact Q ∧ ∀ (i : ι), MapsTo ((_root_.deriv ∘ F) i) K Q | case intro.intro.intro.intro.intro.intro
ι : Type u_1
U K✝ : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q✝ : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
K : Set ℂ
hK1 : K ⊆ U
hK2 : IsCompact K
δ : ℝ
hδ : 0 < δ
h : cthickening δ K ⊆ U
e1 : cthickening δ K ∈ compacts U
Q : Set ℂ
hQ1 : IsCompact Q
hQ2 : ∀ (i : ι), MapsTo (F i) (cthickening δ K) Q
M : ℝ
hM : Q ⊆ closedBall 0 M
⊢ ∀ (i : ι), MapsTo ((_root_.deriv ∘ F) i) K (closedBall 0 (M / δ)) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.deriv | [14, 1] | [30, 32] | intro i x hx | case intro.intro.intro.intro.intro.intro
ι : Type u_1
U K✝ : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q✝ : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
K : Set ℂ
hK1 : K ⊆ U
hK2 : IsCompact K
δ : ℝ
hδ : 0 < δ
h : cthickening δ K ⊆ U
e1 : cthickening δ K ∈ compacts U
Q : Set ℂ
hQ1 : IsCompact Q
hQ2 : ∀ (i : ι), MapsTo (F i) (cthickening δ K) Q
M : ℝ
hM : Q ⊆ closedBall 0 M
⊢ ∀ (i : ι), MapsTo ((_root_.deriv ∘ F) i) K (closedBall 0 (M / δ)) | case intro.intro.intro.intro.intro.intro
ι : Type u_1
U K✝ : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q✝ : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
K : Set ℂ
hK1 : K ⊆ U
hK2 : IsCompact K
δ : ℝ
hδ : 0 < δ
h : cthickening δ K ⊆ U
e1 : cthickening δ K ∈ compacts U
Q : Set ℂ
hQ1 : IsCompact Q
hQ2 : ∀ (i : ι), MapsTo (F i) (cthickening δ K) Q
M : ℝ
hM : Q ⊆ closedBall 0 M
i : ι
x : ℂ
hx : x ∈ K
⊢ (_root_.deriv ∘ F) i x ∈ closedBall 0 (M / δ) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.deriv | [14, 1] | [30, 32] | simp only [mem_closedBall_zero_iff] | case intro.intro.intro.intro.intro.intro
ι : Type u_1
U K✝ : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q✝ : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
K : Set ℂ
hK1 : K ⊆ U
hK2 : IsCompact K
δ : ℝ
hδ : 0 < δ
h : cthickening δ K ⊆ U
e1 : cthickening δ K ∈ compacts U
Q : Set ℂ
hQ1 : IsCompact Q
hQ2 : ∀ (i : ι), MapsTo (F i) (cthickening δ K) Q
M : ℝ
hM : Q ⊆ closedBall 0 M
i : ι
x : ℂ
hx : x ∈ K
⊢ (_root_.deriv ∘ F) i x ∈ closedBall 0 (M / δ) | case intro.intro.intro.intro.intro.intro
ι : Type u_1
U K✝ : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q✝ : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
K : Set ℂ
hK1 : K ⊆ U
hK2 : IsCompact K
δ : ℝ
hδ : 0 < δ
h : cthickening δ K ⊆ U
e1 : cthickening δ K ∈ compacts U
Q : Set ℂ
hQ1 : IsCompact Q
hQ2 : ∀ (i : ι), MapsTo (F i) (cthickening δ K) Q
M : ℝ
hM : Q ⊆ closedBall 0 M
i : ι
x : ℂ
hx : x ∈ K
⊢ ‖(_root_.deriv ∘ F) i x‖ ≤ M / δ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.deriv | [14, 1] | [30, 32] | refine norm_deriv_le_aux hδ ?_ ?_ | case intro.intro.intro.intro.intro.intro
ι : Type u_1
U K✝ : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q✝ : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
K : Set ℂ
hK1 : K ⊆ U
hK2 : IsCompact K
δ : ℝ
hδ : 0 < δ
h : cthickening δ K ⊆ U
e1 : cthickening δ K ∈ compacts U
Q : Set ℂ
hQ1 : IsCompact Q
hQ2 : ∀ (i : ι), MapsTo (F i) (cthickening δ K) Q
M : ℝ
hM : Q ⊆ closedBall 0 M
i : ι
x : ℂ
hx : x ∈ K
⊢ ‖(_root_.deriv ∘ F) i x‖ ≤ M / δ | case intro.intro.intro.intro.intro.intro.refine_1
ι : Type u_1
U K✝ : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q✝ : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
K : Set ℂ
hK1 : K ⊆ U
hK2 : IsCompact K
δ : ℝ
hδ : 0 < δ
h : cthickening δ K ⊆ U
e1 : cthickening δ K ∈ compacts U
Q : Set ℂ
hQ1 : IsCompact Q
hQ2 : ∀ (i : ι), MapsTo (F i) (cthickening δ K) Q
M : ℝ
hM : Q ⊆ closedBall 0 M
i : ι
x : ℂ
hx : x ∈ K
⊢ DiffContOnCl ℂ (F i) (ball x δ)
case intro.intro.intro.intro.intro.intro.refine_2
ι : Type u_1
U K✝ : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q✝ : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
K : Set ℂ
hK1 : K ⊆ U
hK2 : IsCompact K
δ : ℝ
hδ : 0 < δ
h : cthickening δ K ⊆ U
e1 : cthickening δ K ∈ compacts U
Q : Set ℂ
hQ1 : IsCompact Q
hQ2 : ∀ (i : ι), MapsTo (F i) (cthickening δ K) Q
M : ℝ
hM : Q ⊆ closedBall 0 M
i : ι
x : ℂ
hx : x ∈ K
⊢ ∀ z ∈ sphere x δ, ‖F i z‖ ≤ M |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.deriv | [14, 1] | [30, 32] | exact (h2 i).diffContOnCl_ball ((closedBall_subset_cthickening hx δ).trans h) | case intro.intro.intro.intro.intro.intro.refine_1
ι : Type u_1
U K✝ : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q✝ : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
K : Set ℂ
hK1 : K ⊆ U
hK2 : IsCompact K
δ : ℝ
hδ : 0 < δ
h : cthickening δ K ⊆ U
e1 : cthickening δ K ∈ compacts U
Q : Set ℂ
hQ1 : IsCompact Q
hQ2 : ∀ (i : ι), MapsTo (F i) (cthickening δ K) Q
M : ℝ
hM : Q ⊆ closedBall 0 M
i : ι
x : ℂ
hx : x ∈ K
⊢ DiffContOnCl ℂ (F i) (ball x δ) | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.deriv | [14, 1] | [30, 32] | rintro z hz | case intro.intro.intro.intro.intro.intro.refine_2
ι : Type u_1
U K✝ : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q✝ : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
K : Set ℂ
hK1 : K ⊆ U
hK2 : IsCompact K
δ : ℝ
hδ : 0 < δ
h : cthickening δ K ⊆ U
e1 : cthickening δ K ∈ compacts U
Q : Set ℂ
hQ1 : IsCompact Q
hQ2 : ∀ (i : ι), MapsTo (F i) (cthickening δ K) Q
M : ℝ
hM : Q ⊆ closedBall 0 M
i : ι
x : ℂ
hx : x ∈ K
⊢ ∀ z ∈ sphere x δ, ‖F i z‖ ≤ M | case intro.intro.intro.intro.intro.intro.refine_2
ι : Type u_1
U K✝ : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q✝ : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
K : Set ℂ
hK1 : K ⊆ U
hK2 : IsCompact K
δ : ℝ
hδ : 0 < δ
h : cthickening δ K ⊆ U
e1 : cthickening δ K ∈ compacts U
Q : Set ℂ
hQ1 : IsCompact Q
hQ2 : ∀ (i : ι), MapsTo (F i) (cthickening δ K) Q
M : ℝ
hM : Q ⊆ closedBall 0 M
i : ι
x : ℂ
hx : x ∈ K
z : ℂ
hz : z ∈ sphere x δ
⊢ ‖F i z‖ ≤ M |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.deriv | [14, 1] | [30, 32] | have : z ∈ cthickening δ K :=
sphere_subset_closedBall.trans (closedBall_subset_cthickening hx δ) hz | case intro.intro.intro.intro.intro.intro.refine_2
ι : Type u_1
U K✝ : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q✝ : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
K : Set ℂ
hK1 : K ⊆ U
hK2 : IsCompact K
δ : ℝ
hδ : 0 < δ
h : cthickening δ K ⊆ U
e1 : cthickening δ K ∈ compacts U
Q : Set ℂ
hQ1 : IsCompact Q
hQ2 : ∀ (i : ι), MapsTo (F i) (cthickening δ K) Q
M : ℝ
hM : Q ⊆ closedBall 0 M
i : ι
x : ℂ
hx : x ∈ K
z : ℂ
hz : z ∈ sphere x δ
⊢ ‖F i z‖ ≤ M | case intro.intro.intro.intro.intro.intro.refine_2
ι : Type u_1
U K✝ : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q✝ : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
K : Set ℂ
hK1 : K ⊆ U
hK2 : IsCompact K
δ : ℝ
hδ : 0 < δ
h : cthickening δ K ⊆ U
e1 : cthickening δ K ∈ compacts U
Q : Set ℂ
hQ1 : IsCompact Q
hQ2 : ∀ (i : ι), MapsTo (F i) (cthickening δ K) Q
M : ℝ
hM : Q ⊆ closedBall 0 M
i : ι
x : ℂ
hx : x ∈ K
z : ℂ
hz : z ∈ sphere x δ
this : z ∈ cthickening δ K
⊢ ‖F i z‖ ≤ M |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.deriv | [14, 1] | [30, 32] | simpa using hM (hQ2 i this) | case intro.intro.intro.intro.intro.intro.refine_2
ι : Type u_1
U K✝ : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q✝ : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
K : Set ℂ
hK1 : K ⊆ U
hK2 : IsCompact K
δ : ℝ
hδ : 0 < δ
h : cthickening δ K ⊆ U
e1 : cthickening δ K ∈ compacts U
Q : Set ℂ
hQ1 : IsCompact Q
hQ2 : ∀ (i : ι), MapsTo (F i) (cthickening δ K) Q
M : ℝ
hM : Q ⊆ closedBall 0 M
i : ι
x : ℂ
hx : x ∈ K
z : ℂ
hz : z ∈ sphere x δ
this : z ∈ cthickening δ K
⊢ ‖F i z‖ ≤ M | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.equicontinuousOn | [32, 1] | [58, 40] | apply (equicontinuous_restrict_iff _).mp | ι : Type u_1
U K : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
⊢ EquicontinuousOn F K | ι : Type u_1
U K : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
⊢ Equicontinuous (restrict K ∘ F) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.equicontinuousOn | [32, 1] | [58, 40] | rintro ⟨z, hz⟩ | ι : Type u_1
U K : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
⊢ Equicontinuous (restrict K ∘ F) | case mk
ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
⊢ EquicontinuousAt (restrict K ∘ F) { val := z, property := hz } |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.equicontinuousOn | [32, 1] | [58, 40] | obtain ⟨δ, hδ, h⟩ := nhds_basis_closedBall.mem_iff.1 (hU.mem_nhds (hK.1 hz)) | case mk
ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
⊢ EquicontinuousAt (restrict K ∘ F) { val := z, property := hz } | case mk.intro.intro
ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
δ : ℝ
hδ : 0 < δ
h : closedBall z δ ⊆ U
⊢ EquicontinuousAt (restrict K ∘ F) { val := z, property := hz } |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.equicontinuousOn | [32, 1] | [58, 40] | have : ∃ M > 0, ∀ i, MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M) := by
obtain ⟨Q, hQ1, hQ2⟩ := h1.deriv hU h2 (closedBall z δ) ⟨h, isCompact_closedBall _ _⟩
obtain ⟨M, hM⟩ := hQ1.isBounded.subset_closedBall 0
refine ⟨M ⊔ 1, by simp, fun i => ?_⟩
exact ((hQ2 i).mono_right hM).mono_right <| closedBall_subset_closedBall le_sup_left | case mk.intro.intro
ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
δ : ℝ
hδ : 0 < δ
h : closedBall z δ ⊆ U
⊢ EquicontinuousAt (restrict K ∘ F) { val := z, property := hz } | case mk.intro.intro
ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
δ : ℝ
hδ : 0 < δ
h : closedBall z δ ⊆ U
this : ∃ M > 0, ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M)
⊢ EquicontinuousAt (restrict K ∘ F) { val := z, property := hz } |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.equicontinuousOn | [32, 1] | [58, 40] | obtain ⟨M, hMp, hM⟩ := this | case mk.intro.intro
ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
δ : ℝ
hδ : 0 < δ
h : closedBall z δ ⊆ U
this : ∃ M > 0, ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M)
⊢ EquicontinuousAt (restrict K ∘ F) { val := z, property := hz } | case mk.intro.intro.intro.intro
ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
δ : ℝ
hδ : 0 < δ
h : closedBall z δ ⊆ U
M : ℝ
hMp : M > 0
hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M)
⊢ EquicontinuousAt (restrict K ∘ F) { val := z, property := hz } |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.equicontinuousOn | [32, 1] | [58, 40] | rw [equicontinuousAt_iff] | case mk.intro.intro.intro.intro
ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
δ : ℝ
hδ : 0 < δ
h : closedBall z δ ⊆ U
M : ℝ
hMp : M > 0
hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M)
⊢ EquicontinuousAt (restrict K ∘ F) { val := z, property := hz } | case mk.intro.intro.intro.intro
ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
δ : ℝ
hδ : 0 < δ
h : closedBall z δ ⊆ U
M : ℝ
hMp : M > 0
hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M)
⊢ ∀ ε > 0,
∃ δ > 0,
∀ (x : ↑K),
dist x { val := z, property := hz } < δ →
∀ (i : ι), dist ((restrict K ∘ F) i { val := z, property := hz }) ((restrict K ∘ F) i x) < ε |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.equicontinuousOn | [32, 1] | [58, 40] | rintro ε hε | case mk.intro.intro.intro.intro
ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
δ : ℝ
hδ : 0 < δ
h : closedBall z δ ⊆ U
M : ℝ
hMp : M > 0
hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M)
⊢ ∀ ε > 0,
∃ δ > 0,
∀ (x : ↑K),
dist x { val := z, property := hz } < δ →
∀ (i : ι), dist ((restrict K ∘ F) i { val := z, property := hz }) ((restrict K ∘ F) i x) < ε | case mk.intro.intro.intro.intro
ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
δ : ℝ
hδ : 0 < δ
h : closedBall z δ ⊆ U
M : ℝ
hMp : M > 0
hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M)
ε : ℝ
hε : ε > 0
⊢ ∃ δ > 0,
∀ (x : ↑K),
dist x { val := z, property := hz } < δ →
∀ (i : ι), dist ((restrict K ∘ F) i { val := z, property := hz }) ((restrict K ∘ F) i x) < ε |
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