url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.tendsto_circle_integral | [209, 1] | [231, 59] | exact ⟨z₀ + r, this, h _ this⟩ | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h✝ : NeBot p
f_cont : ContinuousOn f (sphere z₀ r)
ε : ℝ
hε : ε > 0
twopir_ne_zero : 2 * Real.pi * r ≠ 0
this✝ : (2 * Real.pi * r... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h✝ : NeBot p
f_cont : ContinuousOn f (sphere z₀ r)
ε : ℝ
hε : ε > 0
... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.tendsto_circle_integral | [209, 1] | [231, 59] | simp [hr.le, Real.norm_eq_abs] | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h✝ : NeBot p
f_cont : ContinuousOn f (sphere z₀ r)
ε : ℝ
hε : ε > 0
twopir_ne_zero : 2 * Real.pi * r ≠ 0
this : (2 * Real.pi * r)... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h✝ : NeBot p
f_cont : ContinuousOn f (sphere z₀ r)
ε : ℝ
hε : ε > 0
... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2_2 | [233, 1] | [255, 65] | by_cases h : NeBot p | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
⊢ Tendsto (cindex z₀ r ∘ F) p (𝓝 (cindex z₀ r f)) | case pos
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
h : NeBot p
⊢ Tendsto (cindex z₀ r ∘ F) p (𝓝 (cindex z₀ r f))
case ... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
⊢ Tendsto (cindex... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2_2 | [233, 1] | [255, 65] | case neg => simp at h; simp [h] | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
h : ¬NeBot p
⊢ Tendsto (cindex z₀ r ∘ F) p (𝓝 (cindex z₀ r f)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
h : ¬NeBot p
⊢ Te... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2_2 | [233, 1] | [255, 65] | simp at h | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
h : ¬NeBot p
⊢ Tendsto (cindex z₀ r ∘ F) p (𝓝 (cindex z₀ r f)) | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
h : p = ⊥
⊢ Tendsto (cindex z₀ r ∘ F) p (𝓝 (cindex z₀ r f)) | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
h : ¬NeBot p
⊢ Te... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2_2 | [233, 1] | [255, 65] | simp [h] | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
h : p = ⊥
⊢ Tendsto (cindex z₀ r ∘ F) p (𝓝 (cindex z₀ r f)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
h : p = ⊥
⊢ Tends... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2_2 | [233, 1] | [255, 65] | have H1 : IsCompact (sphere z₀ r) := isCompact_sphere z₀ r | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
h : NeBot p
⊢ Tendsto (cindex z₀ r ∘ F) p (𝓝 (cindex z₀ r f)) | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
h : NeBot p
H1 : IsCompact (sphere z₀ r)
⊢ Tendsto (cindex z₀ r ∘ F) p (𝓝 (ci... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
h : NeBot p
⊢ Ten... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2_2 | [233, 1] | [255, 65] | have H2 : TendstoUniformlyOn F f p (sphere z₀ r) :=
(tendstoLocallyUniformlyOn_iff_forall_isCompact hU).1 hf _ hr2 H1 | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
h : NeBot p
H1 : IsCompact (sphere z₀ r)
⊢ Tendsto (cindex z₀ r ∘ F) p (𝓝 (ci... | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
h : NeBot p
H1 : IsCompact (sphere z₀ r)
H2 : TendstoUniformlyOn F f p (sphere... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
h : NeBot p
H1 : ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2_2 | [233, 1] | [255, 65] | have H3 : DifferentiableOn ℂ f U := hf.differentiableOn hF hU | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
h : NeBot p
H1 : IsCompact (sphere z₀ r)
H2 : TendstoUniformlyOn F f p (sphere... | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
h : NeBot p
H1 : IsCompact (sphere z₀ r)
H2 : TendstoUniformlyOn F f p (sphere... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
h : NeBot p
H1 : ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2_2 | [233, 1] | [255, 65] | have H4 : ContinuousOn f (sphere z₀ r) := H3.continuousOn.mono hr2 | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
h : NeBot p
H1 : IsCompact (sphere z₀ r)
H2 : TendstoUniformlyOn F f p (sphere... | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
h : NeBot p
H1 : IsCompact (sphere z₀ r)
H2 : TendstoUniformlyOn F f p (sphere... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
h : NeBot p
H1 : ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2_2 | [233, 1] | [255, 65] | have H5 : ∀ᶠ n in p, ContinuousOn (F n) (sphere z₀ r) := by
filter_upwards [hF] with n h using h.continuousOn.mono hr2 | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
h : NeBot p
H1 : IsCompact (sphere z₀ r)
H2 : TendstoUniformlyOn F f p (sphere... | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
h : NeBot p
H1 : IsCompact (sphere z₀ r)
H2 : TendstoUniformlyOn F f p (sphere... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
h : NeBot p
H1 : ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2_2 | [233, 1] | [255, 65] | have H6 : ∀ᶠ n in p, ContinuousOn (deriv (F n)) (sphere z₀ r) := by
filter_upwards [hF] with n h using (h.deriv hU).continuousOn.mono hr2 | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
h : NeBot p
H1 : IsCompact (sphere z₀ r)
H2 : TendstoUniformlyOn F f p (sphere... | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
h : NeBot p
H1 : IsCompact (sphere z₀ r)
H2 : TendstoUniformlyOn F f p (sphere... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
h : NeBot p
H1 : ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2_2 | [233, 1] | [255, 65] | have H7 : TendstoUniformlyOn (deriv ∘ F) (deriv f) p (sphere z₀ r) :=
(tendstoLocallyUniformlyOn_iff_forall_isCompact hU).1 (hf.deriv hF hU) _ hr2 H1 | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
h : NeBot p
H1 : IsCompact (sphere z₀ r)
H2 : TendstoUniformlyOn F f p (sphere... | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
h : NeBot p
H1 : IsCompact (sphere z₀ r)
H2 : TendstoUniformlyOn F f p (sphere... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
h : NeBot p
H1 : ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2_2 | [233, 1] | [255, 65] | have H8 : ContinuousOn (deriv f) (sphere z₀ r) :=
(H3.deriv hU).continuousOn.mono hr2 | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
h : NeBot p
H1 : IsCompact (sphere z₀ r)
H2 : TendstoUniformlyOn F f p (sphere... | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
h : NeBot p
H1 : IsCompact (sphere z₀ r)
H2 : TendstoUniformlyOn F f p (sphere... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
h : NeBot p
H1 : ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2_2 | [233, 1] | [255, 65] | refine Tendsto.const_mul _ (TendstoUniformlyOn.tendsto_circle_integral hr1 ?_ ?_) | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
h : NeBot p
H1 : IsCompact (sphere z₀ r)
H2 : TendstoUniformlyOn F f p (sphere... | case refine_1
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
h : NeBot p
H1 : IsCompact (sphere z₀ r)
H2 : TendstoUniformlyOn... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
h : NeBot p
H1 : ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2_2 | [233, 1] | [255, 65] | filter_upwards [hF] with n h using h.continuousOn.mono hr2 | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
h : NeBot p
H1 : IsCompact (sphere z₀ r)
H2 : TendstoUniformlyOn F f p (sphere... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
h : NeBot p
H1 : ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2_2 | [233, 1] | [255, 65] | filter_upwards [hF] with n h using (h.deriv hU).continuousOn.mono hr2 | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
h : NeBot p
H1 : IsCompact (sphere z₀ r)
H2 : TendstoUniformlyOn F f p (sphere... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
h : NeBot p
H1 : ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2_2 | [233, 1] | [255, 65] | filter_upwards [hurwitz2_1 H1 H2 H4 hf1, H6, H5] with n hn H6 H5 using ContinuousOn.div H6 H5 hn | case refine_1
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
h : NeBot p
H1 : IsCompact (sphere z₀ r)
H2 : TendstoUniformlyOn... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
h :... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2_2 | [233, 1] | [255, 65] | exact TendstoUniformlyOn.div_of_compact H7 H2 H8 H4 hf1 H1 | case refine_2
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
h : NeBot p
H1 : IsCompact (sphere z₀ r)
H2 : TendstoUniformlyOn... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
h :... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2 | [257, 1] | [281, 71] | by_cases h : NeBot p | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
hf2 : cindex z₀ r f ≠ 0
⊢ ∀ᶠ (n : ι) in p, ∃ z ∈ ball z₀ r, F n z = 0 | case pos
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
hf2 : cindex z₀ r f ≠ 0
h : NeBot p
⊢ ∀ᶠ (n : ι) in p, ∃ z ∈ ball... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
hf2 : cindex ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2 | [257, 1] | [281, 71] | case neg => simp at h; simp [h] | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
hf2 : cindex z₀ r f ≠ 0
h : ¬NeBot p
⊢ ∀ᶠ (n : ι) in p, ∃ z ∈ ball z₀ r, F... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
hf2 : cindex ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2 | [257, 1] | [281, 71] | case pos =>
have H1 : IsCompact (sphere z₀ r) := isCompact_sphere z₀ r
have H2 : sphere z₀ r ⊆ U := sphere_subset_closedBall.trans hr2
have H3 : TendstoUniformlyOn F f p (sphere z₀ r) :=
(tendstoLocallyUniformlyOn_iff_forall_isCompact hU).1 hf _ H2 H1
have H4 : ContinuousOn f (sphere z₀ r) :=
(hf.differ... | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
hf2 : cindex z₀ r f ≠ 0
h : NeBot p
⊢ ∀ᶠ (n : ι) in p, ∃ z ∈ ball z₀ r, F ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
hf2 : cindex ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2 | [257, 1] | [281, 71] | simp at h | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
hf2 : cindex z₀ r f ≠ 0
h : ¬NeBot p
⊢ ∀ᶠ (n : ι) in p, ∃ z ∈ ball z₀ r, F... | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
hf2 : cindex z₀ r f ≠ 0
h : p = ⊥
⊢ ∀ᶠ (n : ι) in p, ∃ z ∈ ball z₀ r, F n ... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
hf2 : cindex ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2 | [257, 1] | [281, 71] | simp [h] | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
hf2 : cindex z₀ r f ≠ 0
h : p = ⊥
⊢ ∀ᶠ (n : ι) in p, ∃ z ∈ ball z₀ r, F n ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
hf2 : cindex ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2 | [257, 1] | [281, 71] | have H1 : IsCompact (sphere z₀ r) := isCompact_sphere z₀ r | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
hf2 : cindex z₀ r f ≠ 0
h : NeBot p
⊢ ∀ᶠ (n : ι) in p, ∃ z ∈ ball z₀ r, F ... | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
hf2 : cindex z₀ r f ≠ 0
h : NeBot p
H1 : IsCompact (sphere z₀ r)
⊢ ∀ᶠ (n :... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
hf2 : cindex ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2 | [257, 1] | [281, 71] | have H2 : sphere z₀ r ⊆ U := sphere_subset_closedBall.trans hr2 | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
hf2 : cindex z₀ r f ≠ 0
h : NeBot p
H1 : IsCompact (sphere z₀ r)
⊢ ∀ᶠ (n :... | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
hf2 : cindex z₀ r f ≠ 0
h : NeBot p
H1 : IsCompact (sphere z₀ r)
H2 : sphe... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
hf2 : cindex ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2 | [257, 1] | [281, 71] | have H3 : TendstoUniformlyOn F f p (sphere z₀ r) :=
(tendstoLocallyUniformlyOn_iff_forall_isCompact hU).1 hf _ H2 H1 | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
hf2 : cindex z₀ r f ≠ 0
h : NeBot p
H1 : IsCompact (sphere z₀ r)
H2 : sphe... | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
hf2 : cindex z₀ r f ≠ 0
h : NeBot p
H1 : IsCompact (sphere z₀ r)
H2 : sphe... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
hf2 : cindex ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2 | [257, 1] | [281, 71] | have H4 : ContinuousOn f (sphere z₀ r) :=
(hf.differentiableOn hF hU).continuousOn.mono H2 | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
hf2 : cindex z₀ r f ≠ 0
h : NeBot p
H1 : IsCompact (sphere z₀ r)
H2 : sphe... | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
hf2 : cindex z₀ r f ≠ 0
h : NeBot p
H1 : IsCompact (sphere z₀ r)
H2 : sphe... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
hf2 : cindex ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2 | [257, 1] | [281, 71] | have H5 : ∀ᶠ n in p, ∀ z ∈ sphere z₀ r, F n z ≠ 0 := hurwitz2_1 H1 H3 H4 hf1 | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
hf2 : cindex z₀ r f ≠ 0
h : NeBot p
H1 : IsCompact (sphere z₀ r)
H2 : sphe... | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
hf2 : cindex z₀ r f ≠ 0
h : NeBot p
H1 : IsCompact (sphere z₀ r)
H2 : sphe... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
hf2 : cindex ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2 | [257, 1] | [281, 71] | filter_upwards [(hurwitz2_2 hU hF hf hr1 H2 hf1).eventually_ne hf2, H5, hF] with n h h' hF | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
hf2 : cindex z₀ r f ≠ 0
h : NeBot p
H1 : IsCompact (sphere z₀ r)
H2 : sphe... | case h
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF✝ : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
hf2 : cindex z₀ r f ≠ 0
h✝ : NeBot p
H1 : IsCompact (sphere z₀ r)
... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
hf2 : cindex ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2 | [257, 1] | [281, 71] | contrapose! h | case h
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF✝ : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
hf2 : cindex z₀ r f ≠ 0
h✝ : NeBot p
H1 : IsCompact (sphere z₀ r)
... | case h
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF✝ : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
hf2 : cindex z₀ r f ≠ 0
h✝ : NeBot p
H1 : IsCompact (sphere z₀ r)
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF✝ : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
hf2 :... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2 | [257, 1] | [281, 71] | have : ∀ (z : ℂ), z ∈ ball z₀ r ∪ sphere z₀ r → F n z ≠ 0 := λ z hz => hz.casesOn (h z) (h' z) | case h
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF✝ : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
hf2 : cindex z₀ r f ≠ 0
h✝ : NeBot p
H1 : IsCompact (sphere z₀ r)
... | case h
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF✝ : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
hf2 : cindex z₀ r f ≠ 0
h✝ : NeBot p
H1 : IsCompact (sphere z₀ r)
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF✝ : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
hf2 :... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2 | [257, 1] | [281, 71] | refine cindex_eq_zero hU hr1 hr2 hF (by rwa [← ball_union_sphere]) | case h
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF✝ : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
hf2 : cindex z₀ r f ≠ 0
h✝ : NeBot p
H1 : IsCompact (sphere z₀ r)
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF✝ : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
hf2 :... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2 | [257, 1] | [281, 71] | rwa [← ball_union_sphere] | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF✝ : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
hf2 : cindex z₀ r f ≠ 0
h✝ : NeBot p
H1 : IsCompact (sphere z₀ r)
H2 : sp... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hF✝ : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall z₀ r ⊆ U
hf1 : ∀ z ∈ sphere z₀ r, f z ≠ 0
hf2 : cindex... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz3 | [283, 1] | [310, 53] | by_cases h : NeBot p | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
hs : s ∈ 𝓝 z₀
⊢ ∀ᶠ (n : ι) in p, ∃ z ∈ s, F n z = 0 | case pos
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
hs : s ∈ 𝓝 z₀
h : NeBot p
⊢ ∀ᶠ (n : ι) in p, ∃ z ∈ s, F n z = 0
case... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
hs : s ∈ 𝓝 z₀
⊢ ∀... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz3 | [283, 1] | [310, 53] | case neg => simp at h; simp [h] | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
hs : s ∈ 𝓝 z₀
h : ¬NeBot p
⊢ ∀ᶠ (n : ι) in p, ∃ z ∈ s, F n z = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
hs : s ∈ 𝓝 z₀
h :... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz3 | [283, 1] | [310, 53] | case pos =>
have H1 := (hf.differentiableOn hF hU).analyticAt (hU.mem_nhds hz₀)
have H5 := cindex_pos H1 h1 h2
rw [eventually_nhdsWithin_iff] at h2
have h3 := eventually_nhds_iff_eventually_closed_ball.1 h2
have h4 : ∀ᶠ r in 𝓝[>] 0, closedBall z₀ r ⊆ U :=
(eventually_closedBall_subset (hU.mem_nhds hz₀)).... | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
hs : s ∈ 𝓝 z₀
h : NeBot p
⊢ ∀ᶠ (n : ι) in p, ∃ z ∈ s, F n z = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
hs : s ∈ 𝓝 z₀
h :... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz3 | [283, 1] | [310, 53] | simp at h | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
hs : s ∈ 𝓝 z₀
h : ¬NeBot p
⊢ ∀ᶠ (n : ι) in p, ∃ z ∈ s, F n z = 0 | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
hs : s ∈ 𝓝 z₀
h : p = ⊥
⊢ ∀ᶠ (n : ι) in p, ∃ z ∈ s, F n z = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
hs : s ∈ 𝓝 z₀
h :... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz3 | [283, 1] | [310, 53] | simp [h] | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
hs : s ∈ 𝓝 z₀
h : p = ⊥
⊢ ∀ᶠ (n : ι) in p, ∃ z ∈ s, F n z = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
hs : s ∈ 𝓝 z₀
h :... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz3 | [283, 1] | [310, 53] | have H1 := (hf.differentiableOn hF hU).analyticAt (hU.mem_nhds hz₀) | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
hs : s ∈ 𝓝 z₀
h : NeBot p
⊢ ∀ᶠ (n : ι) in p, ∃ z ∈ s, F n z = 0 | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
hs : s ∈ 𝓝 z₀
h : NeBot p
H1 : AnalyticAt ℂ f z₀
⊢ ∀ᶠ (n : ι) in p, ∃ z ∈ s, F... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
hs : s ∈ 𝓝 z₀
h :... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz3 | [283, 1] | [310, 53] | have H5 := cindex_pos H1 h1 h2 | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
hs : s ∈ 𝓝 z₀
h : NeBot p
H1 : AnalyticAt ℂ f z₀
⊢ ∀ᶠ (n : ι) in p, ∃ z ∈ s, F... | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
hs : s ∈ 𝓝 z₀
h : NeBot p
H1 : AnalyticAt ℂ f z₀
H5 : ∀ᶠ (r : ℝ) in 𝓝[>] 0, c... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
hs : s ∈ 𝓝 z₀
h :... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz3 | [283, 1] | [310, 53] | rw [eventually_nhdsWithin_iff] at h2 | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
hs : s ∈ 𝓝 z₀
h : NeBot p
H1 : AnalyticAt ℂ f z₀
H5 : ∀ᶠ (r : ℝ) in 𝓝[>] 0, c... | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (x : ℂ) in 𝓝 z₀, x ∈ {z₀}ᶜ → f x ≠ 0
hs : s ∈ 𝓝 z₀
h : NeBot p
H1 : AnalyticAt ℂ f z₀
H5 : ∀ᶠ (r : ℝ) in �... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
hs : s ∈ 𝓝 z₀
h :... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz3 | [283, 1] | [310, 53] | have h3 := eventually_nhds_iff_eventually_closed_ball.1 h2 | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (x : ℂ) in 𝓝 z₀, x ∈ {z₀}ᶜ → f x ≠ 0
hs : s ∈ 𝓝 z₀
h : NeBot p
H1 : AnalyticAt ℂ f z₀
H5 : ∀ᶠ (r : ℝ) in �... | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (x : ℂ) in 𝓝 z₀, x ∈ {z₀}ᶜ → f x ≠ 0
hs : s ∈ 𝓝 z₀
h : NeBot p
H1 : AnalyticAt ℂ f z₀
H5 : ∀ᶠ (r : ℝ) in �... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (x : ℂ) in 𝓝 z₀, x ∈ {z₀}ᶜ → f x ≠ 0
hs : s ∈ ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz3 | [283, 1] | [310, 53] | have h4 : ∀ᶠ r in 𝓝[>] 0, closedBall z₀ r ⊆ U :=
(eventually_closedBall_subset (hU.mem_nhds hz₀)).filter_mono nhdsWithin_le_nhds | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (x : ℂ) in 𝓝 z₀, x ∈ {z₀}ᶜ → f x ≠ 0
hs : s ∈ 𝓝 z₀
h : NeBot p
H1 : AnalyticAt ℂ f z₀
H5 : ∀ᶠ (r : ℝ) in �... | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (x : ℂ) in 𝓝 z₀, x ∈ {z₀}ᶜ → f x ≠ 0
hs : s ∈ 𝓝 z₀
h : NeBot p
H1 : AnalyticAt ℂ f z₀
H5 : ∀ᶠ (r : ℝ) in �... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (x : ℂ) in 𝓝 z₀, x ∈ {z₀}ᶜ → f x ≠ 0
hs : s ∈ ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz3 | [283, 1] | [310, 53] | have h4' : ∀ᶠ r in 𝓝[>] 0, closedBall z₀ r ⊆ s :=
(eventually_closedBall_subset hs).filter_mono nhdsWithin_le_nhds | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (x : ℂ) in 𝓝 z₀, x ∈ {z₀}ᶜ → f x ≠ 0
hs : s ∈ 𝓝 z₀
h : NeBot p
H1 : AnalyticAt ℂ f z₀
H5 : ∀ᶠ (r : ℝ) in �... | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (x : ℂ) in 𝓝 z₀, x ∈ {z₀}ᶜ → f x ≠ 0
hs : s ∈ 𝓝 z₀
h : NeBot p
H1 : AnalyticAt ℂ f z₀
H5 : ∀ᶠ (r : ℝ) in �... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (x : ℂ) in 𝓝 z₀, x ∈ {z₀}ᶜ → f x ≠ 0
hs : s ∈ ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz3 | [283, 1] | [310, 53] | obtain ⟨r, hr, h5, h6, h7, h9⟩ := (h3.and (h4.and (H5.and h4'))).exists' | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (x : ℂ) in 𝓝 z₀, x ∈ {z₀}ᶜ → f x ≠ 0
hs : s ∈ 𝓝 z₀
h : NeBot p
H1 : AnalyticAt ℂ f z₀
H5 : ∀ᶠ (r : ℝ) in �... | case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r✝ : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (x : ℂ) in 𝓝 z₀, x ∈ {z₀}ᶜ → f x ≠ 0
hs : s ∈ 𝓝 z₀
h : NeBot p
H1 : An... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (x : ℂ) in 𝓝 z₀, x ∈ {z₀}ᶜ → f x ≠ 0
hs : s ∈ ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz3 | [283, 1] | [310, 53] | have h8 : ∀ z ∈ sphere z₀ r, f z ≠ 0 := by
exact λ z hz => h5 z (sphere_subset_closedBall hz) (ne_of_mem_sphere hz hr.lt.ne.symm) | case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r✝ : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (x : ℂ) in 𝓝 z₀, x ∈ {z₀}ᶜ → f x ≠ 0
hs : s ∈ 𝓝 z₀
h : NeBot p
H1 : An... | case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r✝ : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (x : ℂ) in 𝓝 z₀, x ∈ {z₀}ᶜ → f x ≠ 0
hs : s ∈ 𝓝 z₀
h : NeBot p
H1 : An... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r✝ : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (x : ℂ) in ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz3 | [283, 1] | [310, 53] | refine (hurwitz2 hU hF hf hr h6 h8 h7).mono ?_ | case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r✝ : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (x : ℂ) in 𝓝 z₀, x ∈ {z₀}ᶜ → f x ≠ 0
hs : s ∈ 𝓝 z₀
h : NeBot p
H1 : An... | case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r✝ : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (x : ℂ) in 𝓝 z₀, x ∈ {z₀}ᶜ → f x ≠ 0
hs : s ∈ 𝓝 z₀
h : NeBot p
H1 : An... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r✝ : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (x : ℂ) in ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz3 | [283, 1] | [310, 53] | rintro n ⟨z, hz, hFnz⟩ | case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r✝ : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (x : ℂ) in 𝓝 z₀, x ∈ {z₀}ᶜ → f x ≠ 0
hs : s ∈ 𝓝 z₀
h : NeBot p
H1 : An... | case intro.intro.intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r✝ : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (x : ℂ) in 𝓝 z₀, x ∈ {z₀}ᶜ → f x ≠ 0
hs : s ∈ 𝓝 z₀
h : NeB... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r✝ : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (x : ℂ) in ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz3 | [283, 1] | [310, 53] | refine ⟨z, h9 (ball_subset_closedBall hz), hFnz⟩ | case intro.intro.intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r✝ : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (x : ℂ) in 𝓝 z₀, x ∈ {z₀}ᶜ → f x ≠ 0
hs : s ∈ 𝓝 z₀
h : NeB... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r✝ : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz3 | [283, 1] | [310, 53] | exact λ z hz => h5 z (sphere_subset_closedBall hz) (ne_of_mem_sphere hz hr.lt.ne.symm) | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r✝ : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (x : ℂ) in 𝓝 z₀, x ∈ {z₀}ᶜ → f x ≠ 0
hs : s ∈ 𝓝 z₀
h : NeBot p
H1 : AnalyticAt ℂ f z₀
H5 : ∀ᶠ (r : ℝ) in ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r✝ : ℝ
U s : Set ℂ
hU : IsOpen U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
h1 : f z₀ = 0
h2 : ∀ᶠ (x : ℂ) in 𝓝 z₀, x ∈ {z₀}ᶜ → f x ≠ 0
hs : s ∈... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | local_hurwitz | [314, 1] | [342, 71] | have H1 := (F_conv.differentiableOn F_holo hU).analyticAt (hU.mem_nhds hz₀) | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f z₀ = 0
⊢ ∀ᶠ (z : ℂ) in 𝓝 z₀, f z = 0 | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f z₀ = 0
H1 : AnalyticAt ℂ f z₀
⊢ ∀ᶠ (z : ℂ) in 𝓝 z₀, f z = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | local_hurwitz | [314, 1] | [342, 71] | cases H1.eventually_eq_zero_or_eventually_ne_zero | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f z₀ = 0
H1 : AnalyticAt ℂ f z₀
⊢ ∀ᶠ (z : ℂ) in 𝓝 z₀, f z = 0 | case inl
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f z₀ = 0
H1 : AnalyticAt ℂ f z₀
h✝ : ∀ᶠ (z : ℂ) in 𝓝 ... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | local_hurwitz | [314, 1] | [342, 71] | case inl => assumption | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f z₀ = 0
H1 : AnalyticAt ℂ f z₀
h✝ : ∀ᶠ (z : ℂ) in 𝓝 z₀, f z =... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | local_hurwitz | [314, 1] | [342, 71] | assumption | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f z₀ = 0
H1 : AnalyticAt ℂ f z₀
h✝ : ∀ᶠ (z : ℂ) in 𝓝 z₀, f z =... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | local_hurwitz | [314, 1] | [342, 71] | obtain ⟨pf, hp⟩ := H1 | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f z₀ = 0
H1 : AnalyticAt ℂ f z₀
h : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z... | case intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f z₀ = 0
h : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
pf : Fo... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | local_hurwitz | [314, 1] | [342, 71] | by_contra hh | case intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f z₀ = 0
h : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
pf : Fo... | case intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f z₀ = 0
h : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
pf : Fo... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | local_hurwitz | [314, 1] | [342, 71] | rw [Filter.not_eventually] at hh | case intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f z₀ = 0
h : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
pf : Fo... | case intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f z₀ = 0
h : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
pf : Fo... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | local_hurwitz | [314, 1] | [342, 71] | have h1 := (order_pos_iff hp hfz₀).2 hh | case intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f z₀ = 0
h : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
pf : Fo... | case intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f z₀ = 0
h : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
pf : Fo... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | local_hurwitz | [314, 1] | [342, 71] | obtain ⟨n, z, h5, h6⟩ := (hurwitz2 hU F_holo F_conv h1 h2 h3 h4).exists | case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r✝ : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f z₀ = 0
h : ∀ᶠ (z : ℂ) in ... | case intro.intro.intro.intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r✝ : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f z₀ = 0
... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r✝ : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUnifor... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | local_hurwitz | [314, 1] | [342, 71] | cases F_noz n z (h2 (ball_subset_closedBall (mem_ball.mpr h5))) h6 | case intro.intro.intro.intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r✝ : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f z₀ = 0
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r✝ : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : Te... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | local_hurwitz | [314, 1] | [342, 71] | rw [eventually_nhdsWithin_iff, eventually_nhds_iff_eventually_closed_ball] at h | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f z₀ = 0
h : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
pf : FormalMultili... | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f z₀ = 0
h : ∀ᶠ (r : ℝ) in 𝓝[>] 0, ∀ z ∈ closedBall z₀ r, z ∈ ... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | local_hurwitz | [314, 1] | [342, 71] | have h4 := cindex_eventually_eq_order hp | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f z₀ = 0
h : ∀ᶠ (r : ℝ) in 𝓝[>] 0, ∀ z ∈ closedBall z₀ r, z ∈ ... | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f z₀ = 0
h : ∀ᶠ (r : ℝ) in 𝓝[>] 0, ∀ z ∈ closedBall z₀ r, z ∈ ... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | local_hurwitz | [314, 1] | [342, 71] | have h5 : ∀ᶠ r in 𝓝[>] 0, closedBall z₀ r ⊆ U :=
(eventually_closedBall_subset (hU.mem_nhds hz₀)).filter_mono nhdsWithin_le_nhds | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f z₀ = 0
h : ∀ᶠ (r : ℝ) in 𝓝[>] 0, ∀ z ∈ closedBall z₀ r, z ∈ ... | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f z₀ = 0
h : ∀ᶠ (r : ℝ) in 𝓝[>] 0, ∀ z ∈ closedBall z₀ r, z ∈ ... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | local_hurwitz | [314, 1] | [342, 71] | obtain ⟨r, h6, h7, h8, h9⟩ := (h.and (h4.and h5)).exists' | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f z₀ = 0
h : ∀ᶠ (r : ℝ) in 𝓝[>] 0, ∀ z ∈ closedBall z₀ r, z ∈ ... | case intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r✝ : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f z₀ = 0
h : ∀ᶠ (r : ℝ) in 𝓝[>] ... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | local_hurwitz | [314, 1] | [342, 71] | refine ⟨r, h6, h9, ?_, ?_⟩ | case intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r✝ : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f z₀ = 0
h : ∀ᶠ (r : ℝ) in 𝓝[>] ... | case intro.intro.intro.intro.refine_1
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r✝ : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f z₀ = 0
h : ∀ᶠ (r : ℝ) ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r✝ : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | local_hurwitz | [314, 1] | [342, 71] | exact λ z hz => h7 z (sphere_subset_closedBall hz) (ne_of_mem_sphere hz h6.lt.ne.symm) | case intro.intro.intro.intro.refine_1
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r✝ : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f z₀ = 0
h : ∀ᶠ (r : ℝ) ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.refine_1
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r✝ : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUni... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | local_hurwitz | [314, 1] | [342, 71] | simp [h8, h1.ne.symm] | case intro.intro.intro.intro.refine_2
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r✝ : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f z₀ = 0
h : ∀ᶠ (r : ℝ) ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.refine_2
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r✝ : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUni... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz | [344, 1] | [357, 71] | have := local_hurwitz hU F_holo F_noz F_conv hz₀ hfz₀ | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f z₀ = 0
⊢ ∀ z ∈ U, f z = 0 | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f z₀ = 0
this : ∀ᶠ (z : ℂ) in 𝓝 z₀, f z... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz | [344, 1] | [357, 71] | have h1 : DifferentiableOn ℂ f U := F_conv.differentiableOn F_holo hU | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f z₀ = 0
this : ∀ᶠ (z : ℂ) in 𝓝 z₀, f z... | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f z₀ = 0
this : ∀ᶠ (z : ℂ) in 𝓝 z₀, f z... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz | [344, 1] | [357, 71] | have h2 := h1.analyticOn hU | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f z₀ = 0
this : ∀ᶠ (z : ℂ) in 𝓝 z₀, f z... | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f z₀ = 0
this : ∀ᶠ (z : ℂ) in 𝓝 z₀, f z... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz | [344, 1] | [357, 71] | exact h2.eqOn_zero_of_preconnected_of_eventuallyEq_zero hU' hz₀ this | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
hz₀ : z₀ ∈ U
hfz₀ : f z₀ = 0
this : ∀ᶠ (z : ℂ) in 𝓝 z₀, f z... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz' | [359, 1] | [370, 49] | refine or_iff_not_imp_left.mpr (λ h => ?_) | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
⊢ (∀ z ∈ U, f z ≠ 0) ∨ ∀ z ∈ U, f z = 0 | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
h : ¬∀ z ∈ U, f z ≠ 0
⊢ ∀ z ∈ U, f z = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz' | [359, 1] | [370, 49] | push_neg at h | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
h : ¬∀ z ∈ U, f z ≠ 0
⊢ ∀ z ∈ U, f z = 0 | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
h : ∃ z ∈ U, f z = 0
⊢ ∀ z ∈ U, f z = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz' | [359, 1] | [370, 49] | obtain ⟨z₀, h1, h2⟩ := h | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
h : ∃ z ∈ U, f z = 0
⊢ ∀ z ∈ U, f z = 0 | case intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀✝ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
z₀ : ℂ
h1 : z₀ ∈ U
h2 : f z₀ = 0
⊢ ∀ z ∈ U... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz' | [359, 1] | [370, 49] | exact hurwitz hU hU' F_holo F_noz F_conv h1 h2 | case intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀✝ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyUniformlyOn F f p U
z₀ : ℂ
h1 : z₀ ∈ U
h2 : f z₀ = 0
⊢ ∀ z ∈ U... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀✝ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
F_holo : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
F_noz : ∀ (n : ι), ∀ z ∈ U, F n z ≠ 0
F_conv : TendstoLocallyU... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_1 | [372, 1] | [376, 85] | refine or_iff_not_imp_right.2 (λ h => ?_) | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hU' : IsPreconnected U
hf : DifferentiableOn ℂ f U
⊢ EqOn f 0 U ∨ ∀ z₀ ∈ U, ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0 | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hU' : IsPreconnected U
hf : DifferentiableOn ℂ f U
h : ¬∀ z₀ ∈ U, ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
⊢ EqOn f 0 U | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hU' : IsPreconnected U
hf : DifferentiableOn ℂ f U
⊢ EqOn f 0 U ∨ ∀ z₀ ∈ U, ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_1 | [372, 1] | [376, 85] | obtain ⟨z₀, h1, h2⟩ : ∃ z₀ ∈ U, ∃ᶠ z in 𝓝[≠] z₀, f z = 0 := by simpa [not_forall] using h | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hU' : IsPreconnected U
hf : DifferentiableOn ℂ f U
h : ¬∀ z₀ ∈ U, ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
⊢ EqOn f 0 U | case intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀✝ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hU' : IsPreconnected U
hf : DifferentiableOn ℂ f U
h : ¬∀ z₀ ∈ U, ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
z₀ : ℂ
h1 : z₀ ∈ U
h2 : ∃ᶠ (z : ℂ) in 𝓝[≠] z₀, f z = 0
⊢ EqOn f 0 U | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hU' : IsPreconnected U
hf : DifferentiableOn ℂ f U
h : ¬∀ z₀ ∈ U, ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
⊢ EqOn f 0 U
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_1 | [372, 1] | [376, 85] | exact (hf.analyticOn hU).eqOn_zero_of_preconnected_of_frequently_eq_zero hU' h1 h2 | case intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀✝ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hU' : IsPreconnected U
hf : DifferentiableOn ℂ f U
h : ¬∀ z₀ ∈ U, ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
z₀ : ℂ
h1 : z₀ ∈ U
h2 : ∃ᶠ (z : ℂ) in 𝓝[≠] z₀, f z = 0
⊢ EqOn f 0 U | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀✝ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hU' : IsPreconnected U
hf : DifferentiableOn ℂ f U
h : ¬∀ z₀ ∈ U, ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
z₀ : ℂ
h1 : z₀ ∈ U
h2 : ∃ᶠ (z : ℂ) in 𝓝[≠] z₀, f z = 0
... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_1 | [372, 1] | [376, 85] | simpa [not_forall] using h | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hU' : IsPreconnected U
hf : DifferentiableOn ℂ f U
h : ¬∀ z₀ ∈ U, ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
⊢ ∃ z₀ ∈ U, ∃ᶠ (z : ℂ) in 𝓝[≠] z₀, f z = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hU : IsOpen U
hU' : IsPreconnected U
hf : DifferentiableOn ℂ f U
h : ¬∀ z₀ ∈ U, ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
⊢ ∃ z₀ ∈ U, ∃ᶠ (z : ℂ) in 𝓝[≠] z₀, f z = 0
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_inj | [384, 1] | [420, 18] | refine or_iff_not_imp_right.2 (λ h => ?_) | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : ∃ᶠ (n : ι) in p, InjOn (F n) U
⊢ (∃ w, ∀ z ∈ U, f z = w) ∨ InjOn f U | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : ∃ᶠ (n : ι) in p, InjOn (F n) U
h : ¬InjOn f U
⊢ ∃ w, ∀ z ∈ U, f z = w | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : ∃ᶠ (n : ι) in p, InjOn (F n) U
⊢ (∃ w, ∀... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_inj | [384, 1] | [420, 18] | obtain ⟨x, hx, y, hy, hfxy, hxy⟩ : ∃ x ∈ U, ∃ y ∈ U, f x = f y ∧ x ≠ y := by
simp [InjOn] at h
obtain ⟨x, h1, y, h2, h3, h4⟩ := h
refine ⟨x, h1, y, h3, h2, h4⟩ | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : ∃ᶠ (n : ι) in p, InjOn (F n) U
h : ¬InjOn f U
⊢ ∃ w, ∀ z ∈ U, f z = w | case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : ∃ᶠ (n : ι) in p, InjOn (F n) U
h : ¬InjOn f U
x : ℂ
hx : x ∈ U
y :... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : ∃ᶠ (n : ι) in p, InjOn (F n) U
h : ¬InjO... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_inj | [384, 1] | [420, 18] | set g : ℂ → ℂ := λ z => f z - f x | case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : ∃ᶠ (n : ι) in p, InjOn (F n) U
h : ¬InjOn f U
x : ℂ
hx : x ∈ U
y :... | case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : ∃ᶠ (n : ι) in p, InjOn (F n) U
h : ¬InjOn f U
x : ℂ
hx : x ∈ U
y :... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : ∃ᶠ (n... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_inj | [384, 1] | [420, 18] | set G : ι → ℂ → ℂ := λ n z => F n z - f x | case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : ∃ᶠ (n : ι) in p, InjOn (F n) U
h : ¬InjOn f U
x : ℂ
hx : x ∈ U
y :... | case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : ∃ᶠ (n : ι) in p, InjOn (F n) U
h : ¬InjOn f U
x : ℂ
hx : x ∈ U
y :... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : ∃ᶠ (n... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_inj | [384, 1] | [420, 18] | have hG : ∀ᶠ n in p, DifferentiableOn ℂ (G n) U := by
filter_upwards [hF] with n hF using hF.sub (differentiableOn_const _) | case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : ∃ᶠ (n : ι) in p, InjOn (F n) U
h : ¬InjOn f U
x : ℂ
hx : x ∈ U
y :... | case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : ∃ᶠ (n : ι) in p, InjOn (F n) U
h : ¬InjOn f U
x : ℂ
hx : x ∈ U
y :... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : ∃ᶠ (n... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_inj | [384, 1] | [420, 18] | have hg : TendstoLocallyUniformlyOn G g p U :=
hurwitz4 hf (uniformContinuous_id.sub uniformContinuous_const) | case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : ∃ᶠ (n : ι) in p, InjOn (F n) U
h : ¬InjOn f U
x : ℂ
hx : x ∈ U
y :... | case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : ∃ᶠ (n : ι) in p, InjOn (F n) U
h : ¬InjOn f U
x : ℂ
hx : x ∈ U
y :... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : ∃ᶠ (n... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_inj | [384, 1] | [420, 18] | have hgx : g x = 0 := sub_self _ | case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : ∃ᶠ (n : ι) in p, InjOn (F n) U
h : ¬InjOn f U
x : ℂ
hx : x ∈ U
y :... | case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : ∃ᶠ (n : ι) in p, InjOn (F n) U
h : ¬InjOn f U
x : ℂ
hx : x ∈ U
y :... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : ∃ᶠ (n... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_inj | [384, 1] | [420, 18] | have hgy : g y = 0 := by simp [g, hfxy] | case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : ∃ᶠ (n : ι) in p, InjOn (F n) U
h : ¬InjOn f U
x : ℂ
hx : x ∈ U
y :... | case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : ∃ᶠ (n : ι) in p, InjOn (F n) U
h : ¬InjOn f U
x : ℂ
hx : x ∈ U
y :... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : ∃ᶠ (n... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_inj | [384, 1] | [420, 18] | suffices this : ∀ z ∈ U, g z = 0
by exact ⟨f x, by simpa [sub_eq_zero, g] using this⟩ | case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : ∃ᶠ (n : ι) in p, InjOn (F n) U
h : ¬InjOn f U
x : ℂ
hx : x ∈ U
y :... | case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : ∃ᶠ (n : ι) in p, InjOn (F n) U
h : ¬InjOn f U
x : ℂ
hx : x ∈ U
y :... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : ∃ᶠ (n... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_inj | [384, 1] | [420, 18] | contrapose hi | case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : ∃ᶠ (n : ι) in p, InjOn (F n) U
h : ¬InjOn f U
x : ℂ
hx : x ∈ U
y :... | case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : ¬InjOn f U
x : ℂ
hx : x ∈ U
y : ℂ
hy : y ∈ U
hfxy : f x = f y
hxy :... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : ∃ᶠ (n... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_inj | [384, 1] | [420, 18] | simp only [not_frequently, InjOn, not_forall] | case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : ¬InjOn f U
x : ℂ
hx : x ∈ U
y : ℂ
hy : y ∈ U
hfxy : f x = f y
hxy :... | case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : ¬InjOn f U
x : ℂ
hx : x ∈ U
y : ℂ
hy : y ∈ U
hfxy : f x = f y
hxy :... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : ¬InjOn... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_inj | [384, 1] | [420, 18] | have h1 : DifferentiableOn ℂ g U := hg.differentiableOn hG hU | case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : ¬InjOn f U
x : ℂ
hx : x ∈ U
y : ℂ
hy : y ∈ U
hfxy : f x = f y
hxy :... | case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : ¬InjOn f U
x : ℂ
hx : x ∈ U
y : ℂ
hy : y ∈ U
hfxy : f x = f y
hxy :... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : ¬InjOn... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_inj | [384, 1] | [420, 18] | have h2 : ∀ z₀ ∈ U, ∀ᶠ z in 𝓝[≠] z₀, g z ≠ 0 := (hurwitz_1 hU hU' h1).resolve_left hi | case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : ¬InjOn f U
x : ℂ
hx : x ∈ U
y : ℂ
hy : y ∈ U
hfxy : f x = f y
hxy :... | case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : ¬InjOn f U
x : ℂ
hx : x ∈ U
y : ℂ
hy : y ∈ U
hfxy : f x = f y
hxy :... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : ¬InjOn... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_inj | [384, 1] | [420, 18] | obtain ⟨u, v, hu, hv, huv⟩ := t2_separation_nhds hxy | case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : ¬InjOn f U
x : ℂ
hx : x ∈ U
y : ℂ
hy : y ∈ U
hfxy : f x = f y
hxy :... | case intro.intro.intro.intro.intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : ¬InjOn f U
x : ℂ
hx : x ∈ U
y : ℂ
hy : y ∈ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : ¬InjOn... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_inj | [384, 1] | [420, 18] | have h3 := hurwitz3 hU hG hg hx hgx (h2 x hx) (inter_mem hu (hU.mem_nhds hx)) | case intro.intro.intro.intro.intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : ¬InjOn f U
x : ℂ
hx : x ∈ U
y : ℂ
hy : y ∈ ... | case intro.intro.intro.intro.intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : ¬InjOn f U
x : ℂ
hx : x ∈ U
y : ℂ
hy : y ∈ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUnifor... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_inj | [384, 1] | [420, 18] | have h4 := hurwitz3 hU hG hg hy hgy (h2 y hy) (inter_mem hv (hU.mem_nhds hy)) | case intro.intro.intro.intro.intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : ¬InjOn f U
x : ℂ
hx : x ∈ U
y : ℂ
hy : y ∈ ... | case intro.intro.intro.intro.intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : ¬InjOn f U
x : ℂ
hx : x ∈ U
y : ℂ
hy : y ∈ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUnifor... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_inj | [384, 1] | [420, 18] | filter_upwards [h3.and h4] with n hn | case intro.intro.intro.intro.intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : ¬InjOn f U
x : ℂ
hx : x ∈ U
y : ℂ
hy : y ∈ ... | case h
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : ¬InjOn f U
x : ℂ
hx : x ∈ U
y : ℂ
hy : y ∈ U
hfxy : f x = f y
hxy : x ≠ y
g : ℂ → ℂ := fun z =>... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUnifor... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_inj | [384, 1] | [420, 18] | obtain ⟨⟨xn, hxn, hGxn⟩, ⟨yn, hyn, hGyn⟩⟩ := hn | case h
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : ¬InjOn f U
x : ℂ
hx : x ∈ U
y : ℂ
hy : y ∈ U
hfxy : f x = f y
hxy : x ≠ y
g : ℂ → ℂ := fun z =>... | case h.intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : ¬InjOn f U
x : ℂ
hx : x ∈ U
y : ℂ
hy : y ∈ U
hfxy : f x = f y
hxy... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : ¬InjOn f U
x : ℂ
hx : x ∈ U
y : ℂ
... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_inj | [384, 1] | [420, 18] | refine ⟨xn, hxn.2, yn, hyn.2, ?_, huv.ne_of_mem hxn.1 hyn.1⟩ | case h.intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : ¬InjOn f U
x : ℂ
hx : x ∈ U
y : ℂ
hy : y ∈ U
hfxy : f x = f y
hxy... | case h.intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : ¬InjOn f U
x : ℂ
hx : x ∈ U
y : ℂ
hy : y ∈ U
hfxy : f x = f y
hxy... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : ¬Inj... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_inj | [384, 1] | [420, 18] | rw [sub_eq_zero] at hGxn hGyn | case h.intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : ¬InjOn f U
x : ℂ
hx : x ∈ U
y : ℂ
hy : y ∈ U
hfxy : f x = f y
hxy... | case h.intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : ¬InjOn f U
x : ℂ
hx : x ∈ U
y : ℂ
hy : y ∈ U
hfxy : f x = f y
hxy... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : ¬Inj... |
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