url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
|
|---|---|---|---|---|---|---|---|---|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.tendsto_circle_integral | [209, 1] | [231, 59] | exact β¨zβ + r, this, h _ thisβ© | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hr : 0 < r
F_cont : βαΆ (n : ΞΉ) in p, ContinuousOn (F n) (sphere zβ r)
F_conv : TendstoUniformlyOn F f p (sphere zβ r)
hβ : NeBot p
f_cont : ContinuousOn f (sphere zβ r)
Ξ΅ : β
hΞ΅ : Ξ΅ > 0
twopir_ne_zero : 2 * Real.pi * r β 0
thisβ : (2 * Real.pi * r)β»ΒΉ * Ξ΅ > 0
n : ΞΉ
h' : ContinuousOn (F n) (sphere zβ r)
h : β x β sphere zβ r, βF n x - f xβ < (2 * Real.pi * r)β»ΒΉ * Ξ΅
this : zβ + βr β sphere zβ r
β’ β x β sphere zβ r, βF n x - f xβ < (2 * Real.pi * r)β»ΒΉ * Ξ΅ | no goals |
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] | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hr : 0 < r
F_cont : βαΆ (n : ΞΉ) in p, ContinuousOn (F n) (sphere zβ r)
F_conv : TendstoUniformlyOn F f p (sphere zβ r)
hβ : NeBot p
f_cont : ContinuousOn f (sphere zβ r)
Ξ΅ : β
hΞ΅ : Ξ΅ > 0
twopir_ne_zero : 2 * Real.pi * r β 0
this : (2 * Real.pi * r)β»ΒΉ * Ξ΅ > 0
n : ΞΉ
h' : ContinuousOn (F n) (sphere zβ r)
h : β x β sphere zβ r, βF n x - f xβ < (2 * Real.pi * r)β»ΒΉ * Ξ΅
β’ zβ + βr β sphere zβ r | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2_2 | [233, 1] | [255, 65] | by_cases h : NeBot p | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere zβ r β U
hf1 : β z β sphere zβ r, f z β 0
β’ Tendsto (cindex zβ r β F) p (π (cindex zβ r f)) | case pos
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere zβ r β U
hf1 : β z β sphere zβ r, f z β 0
h : NeBot p
β’ Tendsto (cindex zβ r β F) p (π (cindex zβ r f))
case neg
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere zβ r β U
hf1 : β z β sphere zβ r, f z β 0
h : Β¬NeBot p
β’ Tendsto (cindex zβ r β F) p (π (cindex zβ r f)) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2_2 | [233, 1] | [255, 65] | case neg => simp at h; simp [h] | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere zβ r β U
hf1 : β z β sphere zβ r, f z β 0
h : Β¬NeBot p
β’ Tendsto (cindex zβ r β F) p (π (cindex zβ r f)) | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2_2 | [233, 1] | [255, 65] | simp at h | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere zβ r β U
hf1 : β z β sphere zβ r, f z β 0
h : Β¬NeBot p
β’ Tendsto (cindex zβ r β F) p (π (cindex zβ r f)) | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere zβ r β U
hf1 : β z β sphere zβ r, f z β 0
h : p = β₯
β’ Tendsto (cindex zβ r β F) p (π (cindex zβ r f)) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2_2 | [233, 1] | [255, 65] | simp [h] | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere zβ r β U
hf1 : β z β sphere zβ r, f z β 0
h : p = β₯
β’ Tendsto (cindex zβ r β F) p (π (cindex zβ r f)) | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2_2 | [233, 1] | [255, 65] | have H1 : IsCompact (sphere zβ r) := isCompact_sphere zβ r | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere zβ r β U
hf1 : β z β sphere zβ r, f z β 0
h : NeBot p
β’ Tendsto (cindex zβ r β F) p (π (cindex zβ r f)) | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere zβ r β U
hf1 : β z β sphere zβ r, f z β 0
h : NeBot p
H1 : IsCompact (sphere zβ r)
β’ Tendsto (cindex zβ r β F) p (π (cindex zβ r f)) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2_2 | [233, 1] | [255, 65] | have H2 : TendstoUniformlyOn F f p (sphere zβ r) :=
(tendstoLocallyUniformlyOn_iff_forall_isCompact hU).1 hf _ hr2 H1 | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere zβ r β U
hf1 : β z β sphere zβ r, f z β 0
h : NeBot p
H1 : IsCompact (sphere zβ r)
β’ Tendsto (cindex zβ r β F) p (π (cindex zβ r f)) | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere zβ r β U
hf1 : β z β sphere zβ r, f z β 0
h : NeBot p
H1 : IsCompact (sphere zβ r)
H2 : TendstoUniformlyOn F f p (sphere zβ r)
β’ Tendsto (cindex zβ r β F) p (π (cindex zβ r f)) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2_2 | [233, 1] | [255, 65] | have H3 : DifferentiableOn β f U := hf.differentiableOn hF hU | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere zβ r β U
hf1 : β z β sphere zβ r, f z β 0
h : NeBot p
H1 : IsCompact (sphere zβ r)
H2 : TendstoUniformlyOn F f p (sphere zβ r)
β’ Tendsto (cindex zβ r β F) p (π (cindex zβ r f)) | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere zβ r β U
hf1 : β z β sphere zβ r, f z β 0
h : NeBot p
H1 : IsCompact (sphere zβ r)
H2 : TendstoUniformlyOn F f p (sphere zβ r)
H3 : DifferentiableOn β f U
β’ Tendsto (cindex zβ r β F) p (π (cindex zβ r f)) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2_2 | [233, 1] | [255, 65] | have H4 : ContinuousOn f (sphere zβ r) := H3.continuousOn.mono hr2 | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere zβ r β U
hf1 : β z β sphere zβ r, f z β 0
h : NeBot p
H1 : IsCompact (sphere zβ r)
H2 : TendstoUniformlyOn F f p (sphere zβ r)
H3 : DifferentiableOn β f U
β’ Tendsto (cindex zβ r β F) p (π (cindex zβ r f)) | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere zβ r β U
hf1 : β z β sphere zβ r, f z β 0
h : NeBot p
H1 : IsCompact (sphere zβ r)
H2 : TendstoUniformlyOn F f p (sphere zβ r)
H3 : DifferentiableOn β f U
H4 : ContinuousOn f (sphere zβ r)
β’ Tendsto (cindex zβ r β F) p (π (cindex zβ r f)) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2_2 | [233, 1] | [255, 65] | have H5 : βαΆ n in p, ContinuousOn (F n) (sphere zβ r) := by
filter_upwards [hF] with n h using h.continuousOn.mono hr2 | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere zβ r β U
hf1 : β z β sphere zβ r, f z β 0
h : NeBot p
H1 : IsCompact (sphere zβ r)
H2 : TendstoUniformlyOn F f p (sphere zβ r)
H3 : DifferentiableOn β f U
H4 : ContinuousOn f (sphere zβ r)
β’ Tendsto (cindex zβ r β F) p (π (cindex zβ r f)) | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere zβ r β U
hf1 : β z β sphere zβ r, f z β 0
h : NeBot p
H1 : IsCompact (sphere zβ r)
H2 : TendstoUniformlyOn F f p (sphere zβ r)
H3 : DifferentiableOn β f U
H4 : ContinuousOn f (sphere zβ r)
H5 : βαΆ (n : ΞΉ) in p, ContinuousOn (F n) (sphere zβ r)
β’ Tendsto (cindex zβ r β F) p (π (cindex zβ r f)) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2_2 | [233, 1] | [255, 65] | have H6 : βαΆ n in p, ContinuousOn (deriv (F n)) (sphere zβ r) := by
filter_upwards [hF] with n h using (h.deriv hU).continuousOn.mono hr2 | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere zβ r β U
hf1 : β z β sphere zβ r, f z β 0
h : NeBot p
H1 : IsCompact (sphere zβ r)
H2 : TendstoUniformlyOn F f p (sphere zβ r)
H3 : DifferentiableOn β f U
H4 : ContinuousOn f (sphere zβ r)
H5 : βαΆ (n : ΞΉ) in p, ContinuousOn (F n) (sphere zβ r)
β’ Tendsto (cindex zβ r β F) p (π (cindex zβ r f)) | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere zβ r β U
hf1 : β z β sphere zβ r, f z β 0
h : NeBot p
H1 : IsCompact (sphere zβ r)
H2 : TendstoUniformlyOn F f p (sphere zβ r)
H3 : DifferentiableOn β f U
H4 : ContinuousOn f (sphere zβ r)
H5 : βαΆ (n : ΞΉ) in p, ContinuousOn (F n) (sphere zβ r)
H6 : βαΆ (n : ΞΉ) in p, ContinuousOn (deriv (F n)) (sphere zβ r)
β’ Tendsto (cindex zβ r β F) p (π (cindex zβ r f)) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2_2 | [233, 1] | [255, 65] | have H7 : TendstoUniformlyOn (deriv β F) (deriv f) p (sphere zβ r) :=
(tendstoLocallyUniformlyOn_iff_forall_isCompact hU).1 (hf.deriv hF hU) _ hr2 H1 | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere zβ r β U
hf1 : β z β sphere zβ r, f z β 0
h : NeBot p
H1 : IsCompact (sphere zβ r)
H2 : TendstoUniformlyOn F f p (sphere zβ r)
H3 : DifferentiableOn β f U
H4 : ContinuousOn f (sphere zβ r)
H5 : βαΆ (n : ΞΉ) in p, ContinuousOn (F n) (sphere zβ r)
H6 : βαΆ (n : ΞΉ) in p, ContinuousOn (deriv (F n)) (sphere zβ r)
β’ Tendsto (cindex zβ r β F) p (π (cindex zβ r f)) | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere zβ r β U
hf1 : β z β sphere zβ r, f z β 0
h : NeBot p
H1 : IsCompact (sphere zβ r)
H2 : TendstoUniformlyOn F f p (sphere zβ r)
H3 : DifferentiableOn β f U
H4 : ContinuousOn f (sphere zβ r)
H5 : βαΆ (n : ΞΉ) in p, ContinuousOn (F n) (sphere zβ r)
H6 : βαΆ (n : ΞΉ) in p, ContinuousOn (deriv (F n)) (sphere zβ r)
H7 : TendstoUniformlyOn (deriv β F) (deriv f) p (sphere zβ r)
β’ Tendsto (cindex zβ r β F) p (π (cindex zβ r f)) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2_2 | [233, 1] | [255, 65] | have H8 : ContinuousOn (deriv f) (sphere zβ r) :=
(H3.deriv hU).continuousOn.mono hr2 | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere zβ r β U
hf1 : β z β sphere zβ r, f z β 0
h : NeBot p
H1 : IsCompact (sphere zβ r)
H2 : TendstoUniformlyOn F f p (sphere zβ r)
H3 : DifferentiableOn β f U
H4 : ContinuousOn f (sphere zβ r)
H5 : βαΆ (n : ΞΉ) in p, ContinuousOn (F n) (sphere zβ r)
H6 : βαΆ (n : ΞΉ) in p, ContinuousOn (deriv (F n)) (sphere zβ r)
H7 : TendstoUniformlyOn (deriv β F) (deriv f) p (sphere zβ r)
β’ Tendsto (cindex zβ r β F) p (π (cindex zβ r f)) | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere zβ r β U
hf1 : β z β sphere zβ r, f z β 0
h : NeBot p
H1 : IsCompact (sphere zβ r)
H2 : TendstoUniformlyOn F f p (sphere zβ r)
H3 : DifferentiableOn β f U
H4 : ContinuousOn f (sphere zβ r)
H5 : βαΆ (n : ΞΉ) in p, ContinuousOn (F n) (sphere zβ r)
H6 : βαΆ (n : ΞΉ) in p, ContinuousOn (deriv (F n)) (sphere zβ r)
H7 : TendstoUniformlyOn (deriv β F) (deriv f) p (sphere zβ r)
H8 : ContinuousOn (deriv f) (sphere zβ r)
β’ Tendsto (cindex zβ r β F) p (π (cindex zβ r f)) |
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 ?_ ?_) | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere zβ r β U
hf1 : β z β sphere zβ r, f z β 0
h : NeBot p
H1 : IsCompact (sphere zβ r)
H2 : TendstoUniformlyOn F f p (sphere zβ r)
H3 : DifferentiableOn β f U
H4 : ContinuousOn f (sphere zβ r)
H5 : βαΆ (n : ΞΉ) in p, ContinuousOn (F n) (sphere zβ r)
H6 : βαΆ (n : ΞΉ) in p, ContinuousOn (deriv (F n)) (sphere zβ r)
H7 : TendstoUniformlyOn (deriv β F) (deriv f) p (sphere zβ r)
H8 : ContinuousOn (deriv f) (sphere zβ r)
β’ Tendsto (cindex zβ r β F) p (π (cindex zβ r f)) | case refine_1
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere zβ r β U
hf1 : β z β sphere zβ r, f z β 0
h : NeBot p
H1 : IsCompact (sphere zβ r)
H2 : TendstoUniformlyOn F f p (sphere zβ r)
H3 : DifferentiableOn β f U
H4 : ContinuousOn f (sphere zβ r)
H5 : βαΆ (n : ΞΉ) in p, ContinuousOn (F n) (sphere zβ r)
H6 : βαΆ (n : ΞΉ) in p, ContinuousOn (deriv (F n)) (sphere zβ r)
H7 : TendstoUniformlyOn (deriv β F) (deriv f) p (sphere zβ r)
H8 : ContinuousOn (deriv f) (sphere zβ r)
β’ βαΆ (n : ΞΉ) in p, ContinuousOn (fun z => deriv (F n) z / F n z) (sphere zβ r)
case refine_2
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere zβ r β U
hf1 : β z β sphere zβ r, f z β 0
h : NeBot p
H1 : IsCompact (sphere zβ r)
H2 : TendstoUniformlyOn F f p (sphere zβ r)
H3 : DifferentiableOn β f U
H4 : ContinuousOn f (sphere zβ r)
H5 : βαΆ (n : ΞΉ) in p, ContinuousOn (F n) (sphere zβ r)
H6 : βαΆ (n : ΞΉ) in p, ContinuousOn (deriv (F n)) (sphere zβ r)
H7 : TendstoUniformlyOn (deriv β F) (deriv f) p (sphere zβ r)
H8 : ContinuousOn (deriv f) (sphere zβ r)
β’ TendstoUniformlyOn (fun k z => deriv (F k) z / F k z) (fun z => deriv f z / f z) p (sphere zβ r) |
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 | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere zβ r β U
hf1 : β z β sphere zβ r, f z β 0
h : NeBot p
H1 : IsCompact (sphere zβ r)
H2 : TendstoUniformlyOn F f p (sphere zβ r)
H3 : DifferentiableOn β f U
H4 : ContinuousOn f (sphere zβ r)
β’ βαΆ (n : ΞΉ) in p, ContinuousOn (F n) (sphere zβ r) | no goals |
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 | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere zβ r β U
hf1 : β z β sphere zβ r, f z β 0
h : NeBot p
H1 : IsCompact (sphere zβ r)
H2 : TendstoUniformlyOn F f p (sphere zβ r)
H3 : DifferentiableOn β f U
H4 : ContinuousOn f (sphere zβ r)
H5 : βαΆ (n : ΞΉ) in p, ContinuousOn (F n) (sphere zβ r)
β’ βαΆ (n : ΞΉ) in p, ContinuousOn (deriv (F n)) (sphere zβ r) | no goals |
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
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere zβ r β U
hf1 : β z β sphere zβ r, f z β 0
h : NeBot p
H1 : IsCompact (sphere zβ r)
H2 : TendstoUniformlyOn F f p (sphere zβ r)
H3 : DifferentiableOn β f U
H4 : ContinuousOn f (sphere zβ r)
H5 : βαΆ (n : ΞΉ) in p, ContinuousOn (F n) (sphere zβ r)
H6 : βαΆ (n : ΞΉ) in p, ContinuousOn (deriv (F n)) (sphere zβ r)
H7 : TendstoUniformlyOn (deriv β F) (deriv f) p (sphere zβ r)
H8 : ContinuousOn (deriv f) (sphere zβ r)
β’ βαΆ (n : ΞΉ) in p, ContinuousOn (fun z => deriv (F n) z / F n z) (sphere zβ r) | no goals |
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
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : sphere zβ r β U
hf1 : β z β sphere zβ r, f z β 0
h : NeBot p
H1 : IsCompact (sphere zβ r)
H2 : TendstoUniformlyOn F f p (sphere zβ r)
H3 : DifferentiableOn β f U
H4 : ContinuousOn f (sphere zβ r)
H5 : βαΆ (n : ΞΉ) in p, ContinuousOn (F n) (sphere zβ r)
H6 : βαΆ (n : ΞΉ) in p, ContinuousOn (deriv (F n)) (sphere zβ r)
H7 : TendstoUniformlyOn (deriv β F) (deriv f) p (sphere zβ r)
H8 : ContinuousOn (deriv f) (sphere zβ r)
β’ TendstoUniformlyOn (fun k z => deriv (F k) z / F k z) (fun z => deriv f z / f z) p (sphere zβ r) | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2 | [257, 1] | [281, 71] | by_cases h : NeBot p | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall zβ r β U
hf1 : β z β sphere zβ r, f z β 0
hf2 : cindex zβ r f β 0
β’ βαΆ (n : ΞΉ) in p, β z β ball zβ r, F n z = 0 | case pos
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall zβ r β U
hf1 : β z β sphere zβ r, f z β 0
hf2 : cindex zβ r f β 0
h : NeBot p
β’ βαΆ (n : ΞΉ) in p, β z β ball zβ r, F n z = 0
case neg
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall zβ r β U
hf1 : β z β sphere zβ r, f z β 0
hf2 : cindex zβ r f β 0
h : Β¬NeBot p
β’ βαΆ (n : ΞΉ) in p, β z β ball zβ r, F n z = 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2 | [257, 1] | [281, 71] | case neg => simp at h; simp [h] | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall zβ r β U
hf1 : β z β sphere zβ r, f z β 0
hf2 : cindex zβ r f β 0
h : Β¬NeBot p
β’ βαΆ (n : ΞΉ) in p, β z β ball zβ r, F n z = 0 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2 | [257, 1] | [281, 71] | case pos =>
have H1 : IsCompact (sphere zβ r) := isCompact_sphere zβ r
have H2 : sphere zβ r β U := sphere_subset_closedBall.trans hr2
have H3 : TendstoUniformlyOn F f p (sphere zβ r) :=
(tendstoLocallyUniformlyOn_iff_forall_isCompact hU).1 hf _ H2 H1
have H4 : ContinuousOn f (sphere zβ r) :=
(hf.differentiableOn hF hU).continuousOn.mono H2
have H5 : βαΆ n in p, β z β sphere zβ r, F n z β 0 := hurwitz2_1 H1 H3 H4 hf1
filter_upwards [(hurwitz2_2 hU hF hf hr1 H2 hf1).eventually_ne hf2, H5, hF] with n h h' hF
contrapose! h
have : β (z : β), z β ball zβ r βͺ sphere zβ r β F n z β 0 := Ξ» z hz => hz.casesOn (h z) (h' z)
refine cindex_eq_zero hU hr1 hr2 hF (by rwa [β ball_union_sphere]) | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall zβ r β U
hf1 : β z β sphere zβ r, f z β 0
hf2 : cindex zβ r f β 0
h : NeBot p
β’ βαΆ (n : ΞΉ) in p, β z β ball zβ r, F n z = 0 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2 | [257, 1] | [281, 71] | simp at h | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall zβ r β U
hf1 : β z β sphere zβ r, f z β 0
hf2 : cindex zβ r f β 0
h : Β¬NeBot p
β’ βαΆ (n : ΞΉ) in p, β z β ball zβ r, F n z = 0 | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall zβ r β U
hf1 : β z β sphere zβ r, f z β 0
hf2 : cindex zβ r f β 0
h : p = β₯
β’ βαΆ (n : ΞΉ) in p, β z β ball zβ r, F n z = 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2 | [257, 1] | [281, 71] | simp [h] | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall zβ r β U
hf1 : β z β sphere zβ r, f z β 0
hf2 : cindex zβ r f β 0
h : p = β₯
β’ βαΆ (n : ΞΉ) in p, β z β ball zβ r, F n z = 0 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2 | [257, 1] | [281, 71] | have H1 : IsCompact (sphere zβ r) := isCompact_sphere zβ r | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall zβ r β U
hf1 : β z β sphere zβ r, f z β 0
hf2 : cindex zβ r f β 0
h : NeBot p
β’ βαΆ (n : ΞΉ) in p, β z β ball zβ r, F n z = 0 | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall zβ r β U
hf1 : β z β sphere zβ r, f z β 0
hf2 : cindex zβ r f β 0
h : NeBot p
H1 : IsCompact (sphere zβ r)
β’ βαΆ (n : ΞΉ) in p, β z β ball zβ r, F n z = 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2 | [257, 1] | [281, 71] | have H2 : sphere zβ r β U := sphere_subset_closedBall.trans hr2 | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall zβ r β U
hf1 : β z β sphere zβ r, f z β 0
hf2 : cindex zβ r f β 0
h : NeBot p
H1 : IsCompact (sphere zβ r)
β’ βαΆ (n : ΞΉ) in p, β z β ball zβ r, F n z = 0 | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall zβ r β U
hf1 : β z β sphere zβ r, f z β 0
hf2 : cindex zβ r f β 0
h : NeBot p
H1 : IsCompact (sphere zβ r)
H2 : sphere zβ r β U
β’ βαΆ (n : ΞΉ) in p, β z β ball zβ r, F n z = 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2 | [257, 1] | [281, 71] | have H3 : TendstoUniformlyOn F f p (sphere zβ r) :=
(tendstoLocallyUniformlyOn_iff_forall_isCompact hU).1 hf _ H2 H1 | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall zβ r β U
hf1 : β z β sphere zβ r, f z β 0
hf2 : cindex zβ r f β 0
h : NeBot p
H1 : IsCompact (sphere zβ r)
H2 : sphere zβ r β U
β’ βαΆ (n : ΞΉ) in p, β z β ball zβ r, F n z = 0 | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall zβ r β U
hf1 : β z β sphere zβ r, f z β 0
hf2 : cindex zβ r f β 0
h : NeBot p
H1 : IsCompact (sphere zβ r)
H2 : sphere zβ r β U
H3 : TendstoUniformlyOn F f p (sphere zβ r)
β’ βαΆ (n : ΞΉ) in p, β z β ball zβ r, F n z = 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2 | [257, 1] | [281, 71] | have H4 : ContinuousOn f (sphere zβ r) :=
(hf.differentiableOn hF hU).continuousOn.mono H2 | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall zβ r β U
hf1 : β z β sphere zβ r, f z β 0
hf2 : cindex zβ r f β 0
h : NeBot p
H1 : IsCompact (sphere zβ r)
H2 : sphere zβ r β U
H3 : TendstoUniformlyOn F f p (sphere zβ r)
β’ βαΆ (n : ΞΉ) in p, β z β ball zβ r, F n z = 0 | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall zβ r β U
hf1 : β z β sphere zβ r, f z β 0
hf2 : cindex zβ r f β 0
h : NeBot p
H1 : IsCompact (sphere zβ r)
H2 : sphere zβ r β U
H3 : TendstoUniformlyOn F f p (sphere zβ r)
H4 : ContinuousOn f (sphere zβ r)
β’ βαΆ (n : ΞΉ) in p, β z β ball zβ r, F n z = 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2 | [257, 1] | [281, 71] | have H5 : βαΆ n in p, β z β sphere zβ r, F n z β 0 := hurwitz2_1 H1 H3 H4 hf1 | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall zβ r β U
hf1 : β z β sphere zβ r, f z β 0
hf2 : cindex zβ r f β 0
h : NeBot p
H1 : IsCompact (sphere zβ r)
H2 : sphere zβ r β U
H3 : TendstoUniformlyOn F f p (sphere zβ r)
H4 : ContinuousOn f (sphere zβ r)
β’ βαΆ (n : ΞΉ) in p, β z β ball zβ r, F n z = 0 | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall zβ r β U
hf1 : β z β sphere zβ r, f z β 0
hf2 : cindex zβ r f β 0
h : NeBot p
H1 : IsCompact (sphere zβ r)
H2 : sphere zβ r β U
H3 : TendstoUniformlyOn F f p (sphere zβ r)
H4 : ContinuousOn f (sphere zβ r)
H5 : βαΆ (n : ΞΉ) in p, β z β sphere zβ r, F n z β 0
β’ βαΆ (n : ΞΉ) in p, β z β ball zβ r, F n z = 0 |
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 | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall zβ r β U
hf1 : β z β sphere zβ r, f z β 0
hf2 : cindex zβ r f β 0
h : NeBot p
H1 : IsCompact (sphere zβ r)
H2 : sphere zβ r β U
H3 : TendstoUniformlyOn F f p (sphere zβ r)
H4 : ContinuousOn f (sphere zβ r)
H5 : βαΆ (n : ΞΉ) in p, β z β sphere zβ r, F n z β 0
β’ βαΆ (n : ΞΉ) in p, β z β ball zβ r, F n z = 0 | case h
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hFβ : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall zβ r β U
hf1 : β z β sphere zβ r, f z β 0
hf2 : cindex zβ r f β 0
hβ : NeBot p
H1 : IsCompact (sphere zβ r)
H2 : sphere zβ r β U
H3 : TendstoUniformlyOn F f p (sphere zβ r)
H4 : ContinuousOn f (sphere zβ r)
H5 : βαΆ (n : ΞΉ) in p, β z β sphere zβ r, F n z β 0
n : ΞΉ
h : (cindex zβ r β fun n => F n) n β 0
h' : β z β sphere zβ r, F n z β 0
hF : DifferentiableOn β (F n) U
β’ β z β ball zβ r, F n z = 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2 | [257, 1] | [281, 71] | contrapose! h | case h
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hFβ : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall zβ r β U
hf1 : β z β sphere zβ r, f z β 0
hf2 : cindex zβ r f β 0
hβ : NeBot p
H1 : IsCompact (sphere zβ r)
H2 : sphere zβ r β U
H3 : TendstoUniformlyOn F f p (sphere zβ r)
H4 : ContinuousOn f (sphere zβ r)
H5 : βαΆ (n : ΞΉ) in p, β z β sphere zβ r, F n z β 0
n : ΞΉ
h : (cindex zβ r β fun n => F n) n β 0
h' : β z β sphere zβ r, F n z β 0
hF : DifferentiableOn β (F n) U
β’ β z β ball zβ r, F n z = 0 | case h
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hFβ : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall zβ r β U
hf1 : β z β sphere zβ r, f z β 0
hf2 : cindex zβ r f β 0
hβ : NeBot p
H1 : IsCompact (sphere zβ r)
H2 : sphere zβ r β U
H3 : TendstoUniformlyOn F f p (sphere zβ r)
H4 : ContinuousOn f (sphere zβ r)
H5 : βαΆ (n : ΞΉ) in p, β z β sphere zβ r, F n z β 0
n : ΞΉ
h' : β z β sphere zβ r, F n z β 0
hF : DifferentiableOn β (F n) U
h : β z β ball zβ r, F n z β 0
β’ (cindex zβ r β fun n => F n) n = 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2 | [257, 1] | [281, 71] | have : β (z : β), z β ball zβ r βͺ sphere zβ r β F n z β 0 := Ξ» z hz => hz.casesOn (h z) (h' z) | case h
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hFβ : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall zβ r β U
hf1 : β z β sphere zβ r, f z β 0
hf2 : cindex zβ r f β 0
hβ : NeBot p
H1 : IsCompact (sphere zβ r)
H2 : sphere zβ r β U
H3 : TendstoUniformlyOn F f p (sphere zβ r)
H4 : ContinuousOn f (sphere zβ r)
H5 : βαΆ (n : ΞΉ) in p, β z β sphere zβ r, F n z β 0
n : ΞΉ
h' : β z β sphere zβ r, F n z β 0
hF : DifferentiableOn β (F n) U
h : β z β ball zβ r, F n z β 0
β’ (cindex zβ r β fun n => F n) n = 0 | case h
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hFβ : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall zβ r β U
hf1 : β z β sphere zβ r, f z β 0
hf2 : cindex zβ r f β 0
hβ : NeBot p
H1 : IsCompact (sphere zβ r)
H2 : sphere zβ r β U
H3 : TendstoUniformlyOn F f p (sphere zβ r)
H4 : ContinuousOn f (sphere zβ r)
H5 : βαΆ (n : ΞΉ) in p, β z β sphere zβ r, F n z β 0
n : ΞΉ
h' : β z β sphere zβ r, F n z β 0
hF : DifferentiableOn β (F n) U
h : β z β ball zβ r, F n z β 0
this : β z β ball zβ r βͺ sphere zβ r, F n z β 0
β’ (cindex zβ r β fun n => F n) n = 0 |
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 [β ball_union_sphere]) | case h
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hFβ : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall zβ r β U
hf1 : β z β sphere zβ r, f z β 0
hf2 : cindex zβ r f β 0
hβ : NeBot p
H1 : IsCompact (sphere zβ r)
H2 : sphere zβ r β U
H3 : TendstoUniformlyOn F f p (sphere zβ r)
H4 : ContinuousOn f (sphere zβ r)
H5 : βαΆ (n : ΞΉ) in p, β z β sphere zβ r, F n z β 0
n : ΞΉ
h' : β z β sphere zβ r, F n z β 0
hF : DifferentiableOn β (F n) U
h : β z β ball zβ r, F n z β 0
this : β z β ball zβ r βͺ sphere zβ r, F n z β 0
β’ (cindex zβ r β fun n => F n) n = 0 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2 | [257, 1] | [281, 71] | rwa [β ball_union_sphere] | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hFβ : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hr1 : 0 < r
hr2 : closedBall zβ r β U
hf1 : β z β sphere zβ r, f z β 0
hf2 : cindex zβ r f β 0
hβ : NeBot p
H1 : IsCompact (sphere zβ r)
H2 : sphere zβ r β U
H3 : TendstoUniformlyOn F f p (sphere zβ r)
H4 : ContinuousOn f (sphere zβ r)
H5 : βαΆ (n : ΞΉ) in p, β z β sphere zβ r, F n z β 0
n : ΞΉ
h' : β z β sphere zβ r, F n z β 0
hF : DifferentiableOn β (F n) U
h : β z β ball zβ r, F n z β 0
this : β z β ball zβ r βͺ sphere zβ r, F n z β 0
β’ β z β closedBall zβ r, (fun n => F n) n z β 0 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz3 | [283, 1] | [310, 53] | by_cases h : NeBot p | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U s : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
h1 : f zβ = 0
h2 : βαΆ (z : β) in π[β ] zβ, f z β 0
hs : s β π zβ
β’ βαΆ (n : ΞΉ) in p, β z β s, F n z = 0 | case pos
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U s : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
h1 : f zβ = 0
h2 : βαΆ (z : β) in π[β ] zβ, f z β 0
hs : s β π zβ
h : NeBot p
β’ βαΆ (n : ΞΉ) in p, β z β s, F n z = 0
case neg
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U s : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
h1 : f zβ = 0
h2 : βαΆ (z : β) in π[β ] zβ, f z β 0
hs : s β π zβ
h : Β¬NeBot p
β’ βαΆ (n : ΞΉ) in p, β z β s, F n z = 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz3 | [283, 1] | [310, 53] | case neg => simp at h; simp [h] | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U s : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
h1 : f zβ = 0
h2 : βαΆ (z : β) in π[β ] zβ, f z β 0
hs : s β π zβ
h : Β¬NeBot p
β’ βαΆ (n : ΞΉ) in p, β z β s, F n z = 0 | no goals |
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 hzβ)
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 : βαΆ r in π[>] 0, closedBall zβ r β U :=
(eventually_closedBall_subset (hU.mem_nhds hzβ)).filter_mono nhdsWithin_le_nhds
have h4' : βαΆ r in π[>] 0, closedBall zβ r β s :=
(eventually_closedBall_subset hs).filter_mono nhdsWithin_le_nhds
obtain β¨r, hr, h5, h6, h7, h9β© := (h3.and (h4.and (H5.and h4'))).exists'
have h8 : β z β sphere zβ r, f z β 0 := by
exact Ξ» z hz => h5 z (sphere_subset_closedBall hz) (ne_of_mem_sphere hz hr.lt.ne.symm)
refine (hurwitz2 hU hF hf hr h6 h8 h7).mono ?_
rintro n β¨z, hz, hFnzβ©
refine β¨z, h9 (ball_subset_closedBall hz), hFnzβ© | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U s : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
h1 : f zβ = 0
h2 : βαΆ (z : β) in π[β ] zβ, f z β 0
hs : s β π zβ
h : NeBot p
β’ βαΆ (n : ΞΉ) in p, β z β s, F n z = 0 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz3 | [283, 1] | [310, 53] | simp at h | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U s : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
h1 : f zβ = 0
h2 : βαΆ (z : β) in π[β ] zβ, f z β 0
hs : s β π zβ
h : Β¬NeBot p
β’ βαΆ (n : ΞΉ) in p, β z β s, F n z = 0 | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U s : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
h1 : f zβ = 0
h2 : βαΆ (z : β) in π[β ] zβ, f z β 0
hs : s β π zβ
h : p = β₯
β’ βαΆ (n : ΞΉ) in p, β z β s, F n z = 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz3 | [283, 1] | [310, 53] | simp [h] | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U s : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
h1 : f zβ = 0
h2 : βαΆ (z : β) in π[β ] zβ, f z β 0
hs : s β π zβ
h : p = β₯
β’ βαΆ (n : ΞΉ) in p, β z β s, F n z = 0 | no goals |
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 hzβ) | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U s : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
h1 : f zβ = 0
h2 : βαΆ (z : β) in π[β ] zβ, f z β 0
hs : s β π zβ
h : NeBot p
β’ βαΆ (n : ΞΉ) in p, β z β s, F n z = 0 | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U s : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
h1 : f zβ = 0
h2 : βαΆ (z : β) in π[β ] zβ, f z β 0
hs : s β π zβ
h : NeBot p
H1 : AnalyticAt β f zβ
β’ βαΆ (n : ΞΉ) in p, β z β s, F n z = 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz3 | [283, 1] | [310, 53] | have H5 := cindex_pos H1 h1 h2 | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U s : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
h1 : f zβ = 0
h2 : βαΆ (z : β) in π[β ] zβ, f z β 0
hs : s β π zβ
h : NeBot p
H1 : AnalyticAt β f zβ
β’ βαΆ (n : ΞΉ) in p, β z β s, F n z = 0 | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U s : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
h1 : f zβ = 0
h2 : βαΆ (z : β) in π[β ] zβ, f z β 0
hs : s β π zβ
h : NeBot p
H1 : AnalyticAt β f zβ
H5 : βαΆ (r : β) in π[>] 0, cindex zβ r f β 0
β’ βαΆ (n : ΞΉ) in p, β z β s, F n z = 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz3 | [283, 1] | [310, 53] | rw [eventually_nhdsWithin_iff] at h2 | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U s : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
h1 : f zβ = 0
h2 : βαΆ (z : β) in π[β ] zβ, f z β 0
hs : s β π zβ
h : NeBot p
H1 : AnalyticAt β f zβ
H5 : βαΆ (r : β) in π[>] 0, cindex zβ r f β 0
β’ βαΆ (n : ΞΉ) in p, β z β s, F n z = 0 | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U s : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
h1 : f zβ = 0
h2 : βαΆ (x : β) in π zβ, x β {zβ}αΆ β f x β 0
hs : s β π zβ
h : NeBot p
H1 : AnalyticAt β f zβ
H5 : βαΆ (r : β) in π[>] 0, cindex zβ r f β 0
β’ βαΆ (n : ΞΉ) in p, β z β s, F n z = 0 |
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 | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U s : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
h1 : f zβ = 0
h2 : βαΆ (x : β) in π zβ, x β {zβ}αΆ β f x β 0
hs : s β π zβ
h : NeBot p
H1 : AnalyticAt β f zβ
H5 : βαΆ (r : β) in π[>] 0, cindex zβ r f β 0
β’ βαΆ (n : ΞΉ) in p, β z β s, F n z = 0 | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U s : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
h1 : f zβ = 0
h2 : βαΆ (x : β) in π zβ, x β {zβ}αΆ β f x β 0
hs : s β π zβ
h : NeBot p
H1 : AnalyticAt β f zβ
H5 : βαΆ (r : β) in π[>] 0, cindex zβ r f β 0
h3 : βαΆ (r : β) in π[>] 0, β z β closedBall zβ r, z β {zβ}αΆ β f z β 0
β’ βαΆ (n : ΞΉ) in p, β z β s, F n z = 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz3 | [283, 1] | [310, 53] | have h4 : βαΆ r in π[>] 0, closedBall zβ r β U :=
(eventually_closedBall_subset (hU.mem_nhds hzβ)).filter_mono nhdsWithin_le_nhds | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U s : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
h1 : f zβ = 0
h2 : βαΆ (x : β) in π zβ, x β {zβ}αΆ β f x β 0
hs : s β π zβ
h : NeBot p
H1 : AnalyticAt β f zβ
H5 : βαΆ (r : β) in π[>] 0, cindex zβ r f β 0
h3 : βαΆ (r : β) in π[>] 0, β z β closedBall zβ r, z β {zβ}αΆ β f z β 0
β’ βαΆ (n : ΞΉ) in p, β z β s, F n z = 0 | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U s : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
h1 : f zβ = 0
h2 : βαΆ (x : β) in π zβ, x β {zβ}αΆ β f x β 0
hs : s β π zβ
h : NeBot p
H1 : AnalyticAt β f zβ
H5 : βαΆ (r : β) in π[>] 0, cindex zβ r f β 0
h3 : βαΆ (r : β) in π[>] 0, β z β closedBall zβ r, z β {zβ}αΆ β f z β 0
h4 : βαΆ (r : β) in π[>] 0, closedBall zβ r β U
β’ βαΆ (n : ΞΉ) in p, β z β s, F n z = 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz3 | [283, 1] | [310, 53] | have h4' : βαΆ r in π[>] 0, closedBall zβ r β s :=
(eventually_closedBall_subset hs).filter_mono nhdsWithin_le_nhds | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U s : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
h1 : f zβ = 0
h2 : βαΆ (x : β) in π zβ, x β {zβ}αΆ β f x β 0
hs : s β π zβ
h : NeBot p
H1 : AnalyticAt β f zβ
H5 : βαΆ (r : β) in π[>] 0, cindex zβ r f β 0
h3 : βαΆ (r : β) in π[>] 0, β z β closedBall zβ r, z β {zβ}αΆ β f z β 0
h4 : βαΆ (r : β) in π[>] 0, closedBall zβ r β U
β’ βαΆ (n : ΞΉ) in p, β z β s, F n z = 0 | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U s : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
h1 : f zβ = 0
h2 : βαΆ (x : β) in π zβ, x β {zβ}αΆ β f x β 0
hs : s β π zβ
h : NeBot p
H1 : AnalyticAt β f zβ
H5 : βαΆ (r : β) in π[>] 0, cindex zβ r f β 0
h3 : βαΆ (r : β) in π[>] 0, β z β closedBall zβ r, z β {zβ}αΆ β f z β 0
h4 : βαΆ (r : β) in π[>] 0, closedBall zβ r β U
h4' : βαΆ (r : β) in π[>] 0, closedBall zβ r β s
β’ βαΆ (n : ΞΉ) in p, β z β s, F n z = 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz3 | [283, 1] | [310, 53] | obtain β¨r, hr, h5, h6, h7, h9β© := (h3.and (h4.and (H5.and h4'))).exists' | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U s : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
h1 : f zβ = 0
h2 : βαΆ (x : β) in π zβ, x β {zβ}αΆ β f x β 0
hs : s β π zβ
h : NeBot p
H1 : AnalyticAt β f zβ
H5 : βαΆ (r : β) in π[>] 0, cindex zβ r f β 0
h3 : βαΆ (r : β) in π[>] 0, β z β closedBall zβ r, z β {zβ}αΆ β f z β 0
h4 : βαΆ (r : β) in π[>] 0, closedBall zβ r β U
h4' : βαΆ (r : β) in π[>] 0, closedBall zβ r β s
β’ βαΆ (n : ΞΉ) in p, β z β s, F n z = 0 | case intro.intro.intro.intro.intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
rβ : β
U s : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
h1 : f zβ = 0
h2 : βαΆ (x : β) in π zβ, x β {zβ}αΆ β f x β 0
hs : s β π zβ
h : NeBot p
H1 : AnalyticAt β f zβ
H5 : βαΆ (r : β) in π[>] 0, cindex zβ r f β 0
h3 : βαΆ (r : β) in π[>] 0, β z β closedBall zβ r, z β {zβ}αΆ β f z β 0
h4 : βαΆ (r : β) in π[>] 0, closedBall zβ r β U
h4' : βαΆ (r : β) in π[>] 0, closedBall zβ r β s
r : β
hr : r > 0
h5 : β z β closedBall zβ r, z β {zβ}αΆ β f z β 0
h6 : closedBall zβ r β U
h7 : cindex zβ r f β 0
h9 : closedBall zβ r β s
β’ βαΆ (n : ΞΉ) in p, β z β s, F n z = 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz3 | [283, 1] | [310, 53] | have h8 : β z β sphere zβ r, f z β 0 := by
exact Ξ» z hz => h5 z (sphere_subset_closedBall hz) (ne_of_mem_sphere hz hr.lt.ne.symm) | case intro.intro.intro.intro.intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
rβ : β
U s : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
h1 : f zβ = 0
h2 : βαΆ (x : β) in π zβ, x β {zβ}αΆ β f x β 0
hs : s β π zβ
h : NeBot p
H1 : AnalyticAt β f zβ
H5 : βαΆ (r : β) in π[>] 0, cindex zβ r f β 0
h3 : βαΆ (r : β) in π[>] 0, β z β closedBall zβ r, z β {zβ}αΆ β f z β 0
h4 : βαΆ (r : β) in π[>] 0, closedBall zβ r β U
h4' : βαΆ (r : β) in π[>] 0, closedBall zβ r β s
r : β
hr : r > 0
h5 : β z β closedBall zβ r, z β {zβ}αΆ β f z β 0
h6 : closedBall zβ r β U
h7 : cindex zβ r f β 0
h9 : closedBall zβ r β s
β’ βαΆ (n : ΞΉ) in p, β z β s, F n z = 0 | case intro.intro.intro.intro.intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
rβ : β
U s : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
h1 : f zβ = 0
h2 : βαΆ (x : β) in π zβ, x β {zβ}αΆ β f x β 0
hs : s β π zβ
h : NeBot p
H1 : AnalyticAt β f zβ
H5 : βαΆ (r : β) in π[>] 0, cindex zβ r f β 0
h3 : βαΆ (r : β) in π[>] 0, β z β closedBall zβ r, z β {zβ}αΆ β f z β 0
h4 : βαΆ (r : β) in π[>] 0, closedBall zβ r β U
h4' : βαΆ (r : β) in π[>] 0, closedBall zβ r β s
r : β
hr : r > 0
h5 : β z β closedBall zβ r, z β {zβ}αΆ β f z β 0
h6 : closedBall zβ r β U
h7 : cindex zβ r f β 0
h9 : closedBall zβ r β s
h8 : β z β sphere zβ r, f z β 0
β’ βαΆ (n : ΞΉ) in p, β z β s, F n z = 0 |
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
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
rβ : β
U s : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
h1 : f zβ = 0
h2 : βαΆ (x : β) in π zβ, x β {zβ}αΆ β f x β 0
hs : s β π zβ
h : NeBot p
H1 : AnalyticAt β f zβ
H5 : βαΆ (r : β) in π[>] 0, cindex zβ r f β 0
h3 : βαΆ (r : β) in π[>] 0, β z β closedBall zβ r, z β {zβ}αΆ β f z β 0
h4 : βαΆ (r : β) in π[>] 0, closedBall zβ r β U
h4' : βαΆ (r : β) in π[>] 0, closedBall zβ r β s
r : β
hr : r > 0
h5 : β z β closedBall zβ r, z β {zβ}αΆ β f z β 0
h6 : closedBall zβ r β U
h7 : cindex zβ r f β 0
h9 : closedBall zβ r β s
h8 : β z β sphere zβ r, f z β 0
β’ βαΆ (n : ΞΉ) in p, β z β s, F n z = 0 | case intro.intro.intro.intro.intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
rβ : β
U s : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
h1 : f zβ = 0
h2 : βαΆ (x : β) in π zβ, x β {zβ}αΆ β f x β 0
hs : s β π zβ
h : NeBot p
H1 : AnalyticAt β f zβ
H5 : βαΆ (r : β) in π[>] 0, cindex zβ r f β 0
h3 : βαΆ (r : β) in π[>] 0, β z β closedBall zβ r, z β {zβ}αΆ β f z β 0
h4 : βαΆ (r : β) in π[>] 0, closedBall zβ r β U
h4' : βαΆ (r : β) in π[>] 0, closedBall zβ r β s
r : β
hr : r > 0
h5 : β z β closedBall zβ r, z β {zβ}αΆ β f z β 0
h6 : closedBall zβ r β U
h7 : cindex zβ r f β 0
h9 : closedBall zβ r β s
h8 : β z β sphere zβ r, f z β 0
β’ β (x : ΞΉ), (β z β ball zβ r, F x z = 0) β β z β s, F x z = 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz3 | [283, 1] | [310, 53] | rintro n β¨z, hz, hFnzβ© | case intro.intro.intro.intro.intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
rβ : β
U s : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
h1 : f zβ = 0
h2 : βαΆ (x : β) in π zβ, x β {zβ}αΆ β f x β 0
hs : s β π zβ
h : NeBot p
H1 : AnalyticAt β f zβ
H5 : βαΆ (r : β) in π[>] 0, cindex zβ r f β 0
h3 : βαΆ (r : β) in π[>] 0, β z β closedBall zβ r, z β {zβ}αΆ β f z β 0
h4 : βαΆ (r : β) in π[>] 0, closedBall zβ r β U
h4' : βαΆ (r : β) in π[>] 0, closedBall zβ r β s
r : β
hr : r > 0
h5 : β z β closedBall zβ r, z β {zβ}αΆ β f z β 0
h6 : closedBall zβ r β U
h7 : cindex zβ r f β 0
h9 : closedBall zβ r β s
h8 : β z β sphere zβ r, f z β 0
β’ β (x : ΞΉ), (β z β ball zβ r, F x z = 0) β β z β s, F x z = 0 | case intro.intro.intro.intro.intro.intro.intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
rβ : β
U s : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
h1 : f zβ = 0
h2 : βαΆ (x : β) in π zβ, x β {zβ}αΆ β f x β 0
hs : s β π zβ
h : NeBot p
H1 : AnalyticAt β f zβ
H5 : βαΆ (r : β) in π[>] 0, cindex zβ r f β 0
h3 : βαΆ (r : β) in π[>] 0, β z β closedBall zβ r, z β {zβ}αΆ β f z β 0
h4 : βαΆ (r : β) in π[>] 0, closedBall zβ r β U
h4' : βαΆ (r : β) in π[>] 0, closedBall zβ r β s
r : β
hr : r > 0
h5 : β z β closedBall zβ r, z β {zβ}αΆ β f z β 0
h6 : closedBall zβ r β U
h7 : cindex zβ r f β 0
h9 : closedBall zβ r β s
h8 : β z β sphere zβ r, f z β 0
n : ΞΉ
z : β
hz : z β ball zβ r
hFnz : F n z = 0
β’ β z β s, F n z = 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz3 | [283, 1] | [310, 53] | refine β¨z, h9 (ball_subset_closedBall hz), hFnzβ© | case intro.intro.intro.intro.intro.intro.intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
rβ : β
U s : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
h1 : f zβ = 0
h2 : βαΆ (x : β) in π zβ, x β {zβ}αΆ β f x β 0
hs : s β π zβ
h : NeBot p
H1 : AnalyticAt β f zβ
H5 : βαΆ (r : β) in π[>] 0, cindex zβ r f β 0
h3 : βαΆ (r : β) in π[>] 0, β z β closedBall zβ r, z β {zβ}αΆ β f z β 0
h4 : βαΆ (r : β) in π[>] 0, closedBall zβ r β U
h4' : βαΆ (r : β) in π[>] 0, closedBall zβ r β s
r : β
hr : r > 0
h5 : β z β closedBall zβ r, z β {zβ}αΆ β f z β 0
h6 : closedBall zβ r β U
h7 : cindex zβ r f β 0
h9 : closedBall zβ r β s
h8 : β z β sphere zβ r, f z β 0
n : ΞΉ
z : β
hz : z β ball zβ r
hFnz : F n z = 0
β’ β z β s, F n z = 0 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz3 | [283, 1] | [310, 53] | exact Ξ» z hz => h5 z (sphere_subset_closedBall hz) (ne_of_mem_sphere hz hr.lt.ne.symm) | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
rβ : β
U s : Set β
hU : IsOpen U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
h1 : f zβ = 0
h2 : βαΆ (x : β) in π zβ, x β {zβ}αΆ β f x β 0
hs : s β π zβ
h : NeBot p
H1 : AnalyticAt β f zβ
H5 : βαΆ (r : β) in π[>] 0, cindex zβ r f β 0
h3 : βαΆ (r : β) in π[>] 0, β z β closedBall zβ r, z β {zβ}αΆ β f z β 0
h4 : βαΆ (r : β) in π[>] 0, closedBall zβ r β U
h4' : βαΆ (r : β) in π[>] 0, closedBall zβ r β s
r : β
hr : r > 0
h5 : β z β closedBall zβ r, z β {zβ}αΆ β f z β 0
h6 : closedBall zβ r β U
h7 : cindex zβ r f β 0
h9 : closedBall zβ r β s
β’ β z β sphere zβ r, f z β 0 | no goals |
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 hzβ) | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
hfzβ : f zβ = 0
β’ βαΆ (z : β) in π zβ, f z = 0 | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
hfzβ : f zβ = 0
H1 : AnalyticAt β f zβ
β’ βαΆ (z : β) in π zβ, f z = 0 |
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 | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
hfzβ : f zβ = 0
H1 : AnalyticAt β f zβ
β’ βαΆ (z : β) in π zβ, f z = 0 | case inl
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
hfzβ : f zβ = 0
H1 : AnalyticAt β f zβ
hβ : βαΆ (z : β) in π zβ, f z = 0
β’ βαΆ (z : β) in π zβ, f z = 0
case inr
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
hfzβ : f zβ = 0
H1 : AnalyticAt β f zβ
hβ : βαΆ (z : β) in π[β ] zβ, f z β 0
β’ βαΆ (z : β) in π zβ, f z = 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | local_hurwitz | [314, 1] | [342, 71] | case inl => assumption | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
hfzβ : f zβ = 0
H1 : AnalyticAt β f zβ
hβ : βαΆ (z : β) in π zβ, f z = 0
β’ βαΆ (z : β) in π zβ, f z = 0 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | local_hurwitz | [314, 1] | [342, 71] | assumption | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
hfzβ : f zβ = 0
H1 : AnalyticAt β f zβ
hβ : βαΆ (z : β) in π zβ, f z = 0
β’ βαΆ (z : β) in π zβ, f z = 0 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | local_hurwitz | [314, 1] | [342, 71] | obtain β¨pf, hpβ© := H1 | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
hfzβ : f zβ = 0
H1 : AnalyticAt β f zβ
h : βαΆ (z : β) in π[β ] zβ, f z β 0
β’ βαΆ (z : β) in π zβ, f z = 0 | case intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
hfzβ : f zβ = 0
h : βαΆ (z : β) in π[β ] zβ, f z β 0
pf : FormalMultilinearSeries β β β
hp : HasFPowerSeriesAt f pf zβ
β’ βαΆ (z : β) in π zβ, f z = 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | local_hurwitz | [314, 1] | [342, 71] | by_contra hh | case intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
hfzβ : f zβ = 0
h : βαΆ (z : β) in π[β ] zβ, f z β 0
pf : FormalMultilinearSeries β β β
hp : HasFPowerSeriesAt f pf zβ
β’ βαΆ (z : β) in π zβ, f z = 0 | case intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
hfzβ : f zβ = 0
h : βαΆ (z : β) in π[β ] zβ, f z β 0
pf : FormalMultilinearSeries β β β
hp : HasFPowerSeriesAt f pf zβ
hh : Β¬βαΆ (z : β) in π zβ, f z = 0
β’ False |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | local_hurwitz | [314, 1] | [342, 71] | rw [Filter.not_eventually] at hh | case intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
hfzβ : f zβ = 0
h : βαΆ (z : β) in π[β ] zβ, f z β 0
pf : FormalMultilinearSeries β β β
hp : HasFPowerSeriesAt f pf zβ
hh : Β¬βαΆ (z : β) in π zβ, f z = 0
β’ False | case intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
hfzβ : f zβ = 0
h : βαΆ (z : β) in π[β ] zβ, f z β 0
pf : FormalMultilinearSeries β β β
hp : HasFPowerSeriesAt f pf zβ
hh : βαΆ (x : β) in π zβ, Β¬f x = 0
β’ False |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | local_hurwitz | [314, 1] | [342, 71] | have h1 := (order_pos_iff hp hfzβ).2 hh | case intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
hfzβ : f zβ = 0
h : βαΆ (z : β) in π[β ] zβ, f z β 0
pf : FormalMultilinearSeries β β β
hp : HasFPowerSeriesAt f pf zβ
hh : βαΆ (x : β) in π zβ, Β¬f x = 0
β’ False | case intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
hfzβ : f zβ = 0
h : βαΆ (z : β) in π[β ] zβ, f z β 0
pf : FormalMultilinearSeries β β β
hp : HasFPowerSeriesAt f pf zβ
hh : βαΆ (x : β) in π zβ, Β¬f x = 0
h1 : 0 < FormalMultilinearSeries.order pf
β’ False |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | local_hurwitz | [314, 1] | [342, 71] | obtain β¨n, z, h5, h6β© := (hurwitz2 hU F_holo F_conv h1 h2 h3 h4).exists | case intro.intro.intro.intro.intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
rβ : β
U : Set β
instβ : NeBot p
hU : IsOpen U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
hfzβ : f zβ = 0
h : βαΆ (z : β) in π[β ] zβ, f z β 0
pf : FormalMultilinearSeries β β β
hp : HasFPowerSeriesAt f pf zβ
hh : βαΆ (x : β) in π zβ, Β¬f x = 0
h1β : 0 < FormalMultilinearSeries.order pf
r : β
h1 : r > 0
h2 : closedBall zβ r β U
h3 : β z β sphere zβ r, f z β 0
h4 : cindex zβ r f β 0
β’ False | case intro.intro.intro.intro.intro.intro.intro.intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
rβ : β
U : Set β
instβ : NeBot p
hU : IsOpen U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
hfzβ : f zβ = 0
h : βαΆ (z : β) in π[β ] zβ, f z β 0
pf : FormalMultilinearSeries β β β
hp : HasFPowerSeriesAt f pf zβ
hh : βαΆ (x : β) in π zβ, Β¬f x = 0
h1β : 0 < FormalMultilinearSeries.order pf
r : β
h1 : r > 0
h2 : closedBall zβ r β U
h3 : β z β sphere zβ r, f z β 0
h4 : cindex zβ r f β 0
n : ΞΉ
z : β
h5 : z β ball zβ r
h6 : F n z = 0
β’ False |
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
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
rβ : β
U : Set β
instβ : NeBot p
hU : IsOpen U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
hfzβ : f zβ = 0
h : βαΆ (z : β) in π[β ] zβ, f z β 0
pf : FormalMultilinearSeries β β β
hp : HasFPowerSeriesAt f pf zβ
hh : βαΆ (x : β) in π zβ, Β¬f x = 0
h1β : 0 < FormalMultilinearSeries.order pf
r : β
h1 : r > 0
h2 : closedBall zβ r β U
h3 : β z β sphere zβ r, f z β 0
h4 : cindex zβ r f β 0
n : ΞΉ
z : β
h5 : z β ball zβ r
h6 : F n z = 0
β’ False | no goals |
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 | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
hfzβ : f zβ = 0
h : βαΆ (z : β) in π[β ] zβ, f z β 0
pf : FormalMultilinearSeries β β β
hp : HasFPowerSeriesAt f pf zβ
hh : βαΆ (x : β) in π zβ, Β¬f x = 0
h1 : 0 < FormalMultilinearSeries.order pf
β’ β r > 0, closedBall zβ r β U β§ (β z β sphere zβ r, f z β 0) β§ cindex zβ r f β 0 | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
hfzβ : f zβ = 0
h : βαΆ (r : β) in π[>] 0, β z β closedBall zβ r, z β {zβ}αΆ β f z β 0
pf : FormalMultilinearSeries β β β
hp : HasFPowerSeriesAt f pf zβ
hh : βαΆ (x : β) in π zβ, Β¬f x = 0
h1 : 0 < FormalMultilinearSeries.order pf
β’ β r > 0, closedBall zβ r β U β§ (β z β sphere zβ r, f z β 0) β§ cindex zβ r f β 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | local_hurwitz | [314, 1] | [342, 71] | have h4 := cindex_eventually_eq_order hp | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
hfzβ : f zβ = 0
h : βαΆ (r : β) in π[>] 0, β z β closedBall zβ r, z β {zβ}αΆ β f z β 0
pf : FormalMultilinearSeries β β β
hp : HasFPowerSeriesAt f pf zβ
hh : βαΆ (x : β) in π zβ, Β¬f x = 0
h1 : 0 < FormalMultilinearSeries.order pf
β’ β r > 0, closedBall zβ r β U β§ (β z β sphere zβ r, f z β 0) β§ cindex zβ r f β 0 | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
hfzβ : f zβ = 0
h : βαΆ (r : β) in π[>] 0, β z β closedBall zβ r, z β {zβ}αΆ β f z β 0
pf : FormalMultilinearSeries β β β
hp : HasFPowerSeriesAt f pf zβ
hh : βαΆ (x : β) in π zβ, Β¬f x = 0
h1 : 0 < FormalMultilinearSeries.order pf
h4 : βαΆ (r : β) in π[>] 0, cindex zβ r f = β(FormalMultilinearSeries.order pf)
β’ β r > 0, closedBall zβ r β U β§ (β z β sphere zβ r, f z β 0) β§ cindex zβ r f β 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | local_hurwitz | [314, 1] | [342, 71] | have h5 : βαΆ r in π[>] 0, closedBall zβ r β U :=
(eventually_closedBall_subset (hU.mem_nhds hzβ)).filter_mono nhdsWithin_le_nhds | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
hfzβ : f zβ = 0
h : βαΆ (r : β) in π[>] 0, β z β closedBall zβ r, z β {zβ}αΆ β f z β 0
pf : FormalMultilinearSeries β β β
hp : HasFPowerSeriesAt f pf zβ
hh : βαΆ (x : β) in π zβ, Β¬f x = 0
h1 : 0 < FormalMultilinearSeries.order pf
h4 : βαΆ (r : β) in π[>] 0, cindex zβ r f = β(FormalMultilinearSeries.order pf)
β’ β r > 0, closedBall zβ r β U β§ (β z β sphere zβ r, f z β 0) β§ cindex zβ r f β 0 | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
hfzβ : f zβ = 0
h : βαΆ (r : β) in π[>] 0, β z β closedBall zβ r, z β {zβ}αΆ β f z β 0
pf : FormalMultilinearSeries β β β
hp : HasFPowerSeriesAt f pf zβ
hh : βαΆ (x : β) in π zβ, Β¬f x = 0
h1 : 0 < FormalMultilinearSeries.order pf
h4 : βαΆ (r : β) in π[>] 0, cindex zβ r f = β(FormalMultilinearSeries.order pf)
h5 : βαΆ (r : β) in π[>] 0, closedBall zβ r β U
β’ β r > 0, closedBall zβ r β U β§ (β z β sphere zβ r, f z β 0) β§ cindex zβ r f β 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | local_hurwitz | [314, 1] | [342, 71] | obtain β¨r, h6, h7, h8, h9β© := (h.and (h4.and h5)).exists' | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
hfzβ : f zβ = 0
h : βαΆ (r : β) in π[>] 0, β z β closedBall zβ r, z β {zβ}αΆ β f z β 0
pf : FormalMultilinearSeries β β β
hp : HasFPowerSeriesAt f pf zβ
hh : βαΆ (x : β) in π zβ, Β¬f x = 0
h1 : 0 < FormalMultilinearSeries.order pf
h4 : βαΆ (r : β) in π[>] 0, cindex zβ r f = β(FormalMultilinearSeries.order pf)
h5 : βαΆ (r : β) in π[>] 0, closedBall zβ r β U
β’ β r > 0, closedBall zβ r β U β§ (β z β sphere zβ r, f z β 0) β§ cindex zβ r f β 0 | case intro.intro.intro.intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
rβ : β
U : Set β
instβ : NeBot p
hU : IsOpen U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
hfzβ : f zβ = 0
h : βαΆ (r : β) in π[>] 0, β z β closedBall zβ r, z β {zβ}αΆ β f z β 0
pf : FormalMultilinearSeries β β β
hp : HasFPowerSeriesAt f pf zβ
hh : βαΆ (x : β) in π zβ, Β¬f x = 0
h1 : 0 < FormalMultilinearSeries.order pf
h4 : βαΆ (r : β) in π[>] 0, cindex zβ r f = β(FormalMultilinearSeries.order pf)
h5 : βαΆ (r : β) in π[>] 0, closedBall zβ r β U
r : β
h6 : r > 0
h7 : β z β closedBall zβ r, z β {zβ}αΆ β f z β 0
h8 : cindex zβ r f = β(FormalMultilinearSeries.order pf)
h9 : closedBall zβ r β U
β’ β r > 0, closedBall zβ r β U β§ (β z β sphere zβ r, f z β 0) β§ cindex zβ r f β 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | local_hurwitz | [314, 1] | [342, 71] | refine β¨r, h6, h9, ?_, ?_β© | case intro.intro.intro.intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
rβ : β
U : Set β
instβ : NeBot p
hU : IsOpen U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
hfzβ : f zβ = 0
h : βαΆ (r : β) in π[>] 0, β z β closedBall zβ r, z β {zβ}αΆ β f z β 0
pf : FormalMultilinearSeries β β β
hp : HasFPowerSeriesAt f pf zβ
hh : βαΆ (x : β) in π zβ, Β¬f x = 0
h1 : 0 < FormalMultilinearSeries.order pf
h4 : βαΆ (r : β) in π[>] 0, cindex zβ r f = β(FormalMultilinearSeries.order pf)
h5 : βαΆ (r : β) in π[>] 0, closedBall zβ r β U
r : β
h6 : r > 0
h7 : β z β closedBall zβ r, z β {zβ}αΆ β f z β 0
h8 : cindex zβ r f = β(FormalMultilinearSeries.order pf)
h9 : closedBall zβ r β U
β’ β r > 0, closedBall zβ r β U β§ (β z β sphere zβ r, f z β 0) β§ cindex zβ r f β 0 | case intro.intro.intro.intro.refine_1
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
rβ : β
U : Set β
instβ : NeBot p
hU : IsOpen U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
hfzβ : f zβ = 0
h : βαΆ (r : β) in π[>] 0, β z β closedBall zβ r, z β {zβ}αΆ β f z β 0
pf : FormalMultilinearSeries β β β
hp : HasFPowerSeriesAt f pf zβ
hh : βαΆ (x : β) in π zβ, Β¬f x = 0
h1 : 0 < FormalMultilinearSeries.order pf
h4 : βαΆ (r : β) in π[>] 0, cindex zβ r f = β(FormalMultilinearSeries.order pf)
h5 : βαΆ (r : β) in π[>] 0, closedBall zβ r β U
r : β
h6 : r > 0
h7 : β z β closedBall zβ r, z β {zβ}αΆ β f z β 0
h8 : cindex zβ r f = β(FormalMultilinearSeries.order pf)
h9 : closedBall zβ r β U
β’ β z β sphere zβ r, f z β 0
case intro.intro.intro.intro.refine_2
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
rβ : β
U : Set β
instβ : NeBot p
hU : IsOpen U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
hfzβ : f zβ = 0
h : βαΆ (r : β) in π[>] 0, β z β closedBall zβ r, z β {zβ}αΆ β f z β 0
pf : FormalMultilinearSeries β β β
hp : HasFPowerSeriesAt f pf zβ
hh : βαΆ (x : β) in π zβ, Β¬f x = 0
h1 : 0 < FormalMultilinearSeries.order pf
h4 : βαΆ (r : β) in π[>] 0, cindex zβ r f = β(FormalMultilinearSeries.order pf)
h5 : βαΆ (r : β) in π[>] 0, closedBall zβ r β U
r : β
h6 : r > 0
h7 : β z β closedBall zβ r, z β {zβ}αΆ β f z β 0
h8 : cindex zβ r f = β(FormalMultilinearSeries.order pf)
h9 : closedBall zβ r β U
β’ cindex zβ r f β 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | local_hurwitz | [314, 1] | [342, 71] | exact Ξ» z hz => h7 z (sphere_subset_closedBall hz) (ne_of_mem_sphere hz h6.lt.ne.symm) | case intro.intro.intro.intro.refine_1
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
rβ : β
U : Set β
instβ : NeBot p
hU : IsOpen U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
hfzβ : f zβ = 0
h : βαΆ (r : β) in π[>] 0, β z β closedBall zβ r, z β {zβ}αΆ β f z β 0
pf : FormalMultilinearSeries β β β
hp : HasFPowerSeriesAt f pf zβ
hh : βαΆ (x : β) in π zβ, Β¬f x = 0
h1 : 0 < FormalMultilinearSeries.order pf
h4 : βαΆ (r : β) in π[>] 0, cindex zβ r f = β(FormalMultilinearSeries.order pf)
h5 : βαΆ (r : β) in π[>] 0, closedBall zβ r β U
r : β
h6 : r > 0
h7 : β z β closedBall zβ r, z β {zβ}αΆ β f z β 0
h8 : cindex zβ r f = β(FormalMultilinearSeries.order pf)
h9 : closedBall zβ r β U
β’ β z β sphere zβ r, f z β 0 | no goals |
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
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
rβ : β
U : Set β
instβ : NeBot p
hU : IsOpen U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
hfzβ : f zβ = 0
h : βαΆ (r : β) in π[>] 0, β z β closedBall zβ r, z β {zβ}αΆ β f z β 0
pf : FormalMultilinearSeries β β β
hp : HasFPowerSeriesAt f pf zβ
hh : βαΆ (x : β) in π zβ, Β¬f x = 0
h1 : 0 < FormalMultilinearSeries.order pf
h4 : βαΆ (r : β) in π[>] 0, cindex zβ r f = β(FormalMultilinearSeries.order pf)
h5 : βαΆ (r : β) in π[>] 0, closedBall zβ r β U
r : β
h6 : r > 0
h7 : β z β closedBall zβ r, z β {zβ}αΆ β f z β 0
h8 : cindex zβ r f = β(FormalMultilinearSeries.order pf)
h9 : closedBall zβ r β U
β’ cindex zβ r f β 0 | no goals |
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 hzβ hfzβ | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
hfzβ : f zβ = 0
β’ β z β U, f z = 0 | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
hfzβ : f zβ = 0
this : βαΆ (z : β) in π zβ, f z = 0
β’ β z β U, f z = 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz | [344, 1] | [357, 71] | have h1 : DifferentiableOn β f U := F_conv.differentiableOn F_holo hU | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
hfzβ : f zβ = 0
this : βαΆ (z : β) in π zβ, f z = 0
β’ β z β U, f z = 0 | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
hfzβ : f zβ = 0
this : βαΆ (z : β) in π zβ, f z = 0
h1 : DifferentiableOn β f U
β’ β z β U, f z = 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz | [344, 1] | [357, 71] | have h2 := h1.analyticOn hU | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
hfzβ : f zβ = 0
this : βαΆ (z : β) in π zβ, f z = 0
h1 : DifferentiableOn β f U
β’ β z β U, f z = 0 | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
hfzβ : f zβ = 0
this : βαΆ (z : β) in π zβ, f z = 0
h1 : DifferentiableOn β f U
h2 : AnalyticOn β f U
β’ β z β U, f z = 0 |
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' hzβ this | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
hzβ : zβ β U
hfzβ : f zβ = 0
this : βαΆ (z : β) in π zβ, f z = 0
h1 : DifferentiableOn β f U
h2 : AnalyticOn β f U
β’ β z β U, f z = 0 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz' | [359, 1] | [370, 49] | refine or_iff_not_imp_left.mpr (Ξ» h => ?_) | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
β’ (β z β U, f z β 0) β¨ β z β U, f z = 0 | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
h : Β¬β z β U, f z β 0
β’ β z β U, f z = 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz' | [359, 1] | [370, 49] | push_neg at h | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
h : Β¬β z β U, f z β 0
β’ β z β U, f z = 0 | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
h : β z β U, f z = 0
β’ β z β U, f z = 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz' | [359, 1] | [370, 49] | obtain β¨zβ, h1, h2β© := h | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
h : β z β U, f z = 0
β’ β z β U, f z = 0 | case intro.intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zββ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
zβ : β
h1 : zβ β U
h2 : f zβ = 0
β’ β z β U, f z = 0 |
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
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zββ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
F_holo : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
F_noz : β (n : ΞΉ), β z β U, F n z β 0
F_conv : TendstoLocallyUniformlyOn F f p U
zβ : β
h1 : zβ β U
h2 : f zβ = 0
β’ β z β U, f z = 0 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_1 | [372, 1] | [376, 85] | refine or_iff_not_imp_right.2 (Ξ» h => ?_) | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hU' : IsPreconnected U
hf : DifferentiableOn β f U
β’ EqOn f 0 U β¨ β zβ β U, βαΆ (z : β) in π[β ] zβ, f z β 0 | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hU' : IsPreconnected U
hf : DifferentiableOn β f U
h : Β¬β zβ β U, βαΆ (z : β) in π[β ] zβ, f z β 0
β’ EqOn f 0 U |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_1 | [372, 1] | [376, 85] | obtain β¨zβ, h1, h2β© : β zβ β U, βαΆ z in π[β ] zβ, f z = 0 := by simpa [not_forall] using h | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hU' : IsPreconnected U
hf : DifferentiableOn β f U
h : Β¬β zβ β U, βαΆ (z : β) in π[β ] zβ, f z β 0
β’ EqOn f 0 U | case intro.intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zββ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hU' : IsPreconnected U
hf : DifferentiableOn β f U
h : Β¬β zβ β U, βαΆ (z : β) in π[β ] zβ, f z β 0
zβ : β
h1 : zβ β U
h2 : βαΆ (z : β) in π[β ] zβ, f z = 0
β’ EqOn f 0 U |
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
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zββ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hU' : IsPreconnected U
hf : DifferentiableOn β f U
h : Β¬β zβ β U, βαΆ (z : β) in π[β ] zβ, f z β 0
zβ : β
h1 : zβ β U
h2 : βαΆ (z : β) in π[β ] zβ, f z = 0
β’ EqOn f 0 U | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_1 | [372, 1] | [376, 85] | simpa [not_forall] using h | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
hU : IsOpen U
hU' : IsPreconnected U
hf : DifferentiableOn β f U
h : Β¬β zβ β U, βαΆ (z : β) in π[β ] zβ, f z β 0
β’ β zβ β U, βαΆ (z : β) in π[β ] zβ, f z = 0 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_inj | [384, 1] | [420, 18] | refine or_iff_not_imp_right.2 (Ξ» h => ?_) | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : βαΆ (n : ΞΉ) in p, InjOn (F n) U
β’ (β w, β z β U, f z = w) β¨ InjOn f U | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : βαΆ (n : ΞΉ) in p, InjOn (F n) U
h : Β¬InjOn f U
β’ β w, β z β U, f z = w |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_inj | [384, 1] | [420, 18] | obtain β¨x, hx, y, hy, hfxy, hxyβ© : β x β U, β y β U, f x = f y β§ x β y := by
simp [InjOn] at h
obtain β¨x, h1, y, h2, h3, h4β© := h
refine β¨x, h1, y, h3, h2, h4β© | ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : βαΆ (n : ΞΉ) in p, InjOn (F n) U
h : Β¬InjOn f U
β’ β w, β z β U, f z = w | case intro.intro.intro.intro.intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : βαΆ (n : ΞΉ) in p, InjOn (F n) U
h : Β¬InjOn f U
x : β
hx : x β U
y : β
hy : y β U
hfxy : f x = f y
hxy : x β y
β’ β w, β z β U, f z = w |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_inj | [384, 1] | [420, 18] | set g : β β β := Ξ» z => f z - f x | case intro.intro.intro.intro.intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : βαΆ (n : ΞΉ) in p, InjOn (F n) U
h : Β¬InjOn f U
x : β
hx : x β U
y : β
hy : y β U
hfxy : f x = f y
hxy : x β y
β’ β w, β z β U, f z = w | case intro.intro.intro.intro.intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : βαΆ (n : ΞΉ) in p, InjOn (F n) U
h : Β¬InjOn f U
x : β
hx : x β U
y : β
hy : y β U
hfxy : f x = f y
hxy : x β y
g : β β β := fun z => f z - f x
β’ β w, β z β U, f z = w |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_inj | [384, 1] | [420, 18] | set G : ΞΉ β β β β := Ξ» n z => F n z - f x | case intro.intro.intro.intro.intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : βαΆ (n : ΞΉ) in p, InjOn (F n) U
h : Β¬InjOn f U
x : β
hx : x β U
y : β
hy : y β U
hfxy : f x = f y
hxy : x β y
g : β β β := fun z => f z - f x
β’ β w, β z β U, f z = w | case intro.intro.intro.intro.intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : βαΆ (n : ΞΉ) in p, InjOn (F n) U
h : Β¬InjOn f U
x : β
hx : x β U
y : β
hy : y β U
hfxy : f x = f y
hxy : x β y
g : β β β := fun z => f z - f x
G : ΞΉ β β β β := fun n z => F n z - f x
β’ β w, β z β U, f z = w |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_inj | [384, 1] | [420, 18] | have hG : βαΆ n in p, DifferentiableOn β (G n) U := by
filter_upwards [hF] with n hF using hF.sub (differentiableOn_const _) | case intro.intro.intro.intro.intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : βαΆ (n : ΞΉ) in p, InjOn (F n) U
h : Β¬InjOn f U
x : β
hx : x β U
y : β
hy : y β U
hfxy : f x = f y
hxy : x β y
g : β β β := fun z => f z - f x
G : ΞΉ β β β β := fun n z => F n z - f x
β’ β w, β z β U, f z = w | case intro.intro.intro.intro.intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : βαΆ (n : ΞΉ) in p, InjOn (F n) U
h : Β¬InjOn f U
x : β
hx : x β U
y : β
hy : y β U
hfxy : f x = f y
hxy : x β y
g : β β β := fun z => f z - f x
G : ΞΉ β β β β := fun n z => F n z - f x
hG : βαΆ (n : ΞΉ) in p, DifferentiableOn β (G n) U
β’ β w, β z β U, f z = w |
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
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : βαΆ (n : ΞΉ) in p, InjOn (F n) U
h : Β¬InjOn f U
x : β
hx : x β U
y : β
hy : y β U
hfxy : f x = f y
hxy : x β y
g : β β β := fun z => f z - f x
G : ΞΉ β β β β := fun n z => F n z - f x
hG : βαΆ (n : ΞΉ) in p, DifferentiableOn β (G n) U
β’ β w, β z β U, f z = w | case intro.intro.intro.intro.intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : βαΆ (n : ΞΉ) in p, InjOn (F n) U
h : Β¬InjOn f U
x : β
hx : x β U
y : β
hy : y β U
hfxy : f x = f y
hxy : x β y
g : β β β := fun z => f z - f x
G : ΞΉ β β β β := fun n z => F n z - f x
hG : βαΆ (n : ΞΉ) in p, DifferentiableOn β (G n) U
hg : TendstoLocallyUniformlyOn G g p U
β’ β w, β z β U, f z = w |
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
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : βαΆ (n : ΞΉ) in p, InjOn (F n) U
h : Β¬InjOn f U
x : β
hx : x β U
y : β
hy : y β U
hfxy : f x = f y
hxy : x β y
g : β β β := fun z => f z - f x
G : ΞΉ β β β β := fun n z => F n z - f x
hG : βαΆ (n : ΞΉ) in p, DifferentiableOn β (G n) U
hg : TendstoLocallyUniformlyOn G g p U
β’ β w, β z β U, f z = w | case intro.intro.intro.intro.intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : βαΆ (n : ΞΉ) in p, InjOn (F n) U
h : Β¬InjOn f U
x : β
hx : x β U
y : β
hy : y β U
hfxy : f x = f y
hxy : x β y
g : β β β := fun z => f z - f x
G : ΞΉ β β β β := fun n z => F n z - f x
hG : βαΆ (n : ΞΉ) in p, DifferentiableOn β (G n) U
hg : TendstoLocallyUniformlyOn G g p U
hgx : g x = 0
β’ β w, β z β U, f z = w |
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
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : βαΆ (n : ΞΉ) in p, InjOn (F n) U
h : Β¬InjOn f U
x : β
hx : x β U
y : β
hy : y β U
hfxy : f x = f y
hxy : x β y
g : β β β := fun z => f z - f x
G : ΞΉ β β β β := fun n z => F n z - f x
hG : βαΆ (n : ΞΉ) in p, DifferentiableOn β (G n) U
hg : TendstoLocallyUniformlyOn G g p U
hgx : g x = 0
β’ β w, β z β U, f z = w | case intro.intro.intro.intro.intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : βαΆ (n : ΞΉ) in p, InjOn (F n) U
h : Β¬InjOn f U
x : β
hx : x β U
y : β
hy : y β U
hfxy : f x = f y
hxy : x β y
g : β β β := fun z => f z - f x
G : ΞΉ β β β β := fun n z => F n z - f x
hG : βαΆ (n : ΞΉ) in p, DifferentiableOn β (G n) U
hg : TendstoLocallyUniformlyOn G g p U
hgx : g x = 0
hgy : g y = 0
β’ β w, β z β U, f z = w |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_inj | [384, 1] | [420, 18] | suffices this : β z β U, g z = 0
by exact β¨f x, by simpa [sub_eq_zero, g] using thisβ© | case intro.intro.intro.intro.intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : βαΆ (n : ΞΉ) in p, InjOn (F n) U
h : Β¬InjOn f U
x : β
hx : x β U
y : β
hy : y β U
hfxy : f x = f y
hxy : x β y
g : β β β := fun z => f z - f x
G : ΞΉ β β β β := fun n z => F n z - f x
hG : βαΆ (n : ΞΉ) in p, DifferentiableOn β (G n) U
hg : TendstoLocallyUniformlyOn G g p U
hgx : g x = 0
hgy : g y = 0
β’ β w, β z β U, f z = w | case intro.intro.intro.intro.intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : βαΆ (n : ΞΉ) in p, InjOn (F n) U
h : Β¬InjOn f U
x : β
hx : x β U
y : β
hy : y β U
hfxy : f x = f y
hxy : x β y
g : β β β := fun z => f z - f x
G : ΞΉ β β β β := fun n z => F n z - f x
hG : βαΆ (n : ΞΉ) in p, DifferentiableOn β (G n) U
hg : TendstoLocallyUniformlyOn G g p U
hgx : g x = 0
hgy : g y = 0
β’ β z β U, g z = 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_inj | [384, 1] | [420, 18] | contrapose hi | case intro.intro.intro.intro.intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : βαΆ (n : ΞΉ) in p, InjOn (F n) U
h : Β¬InjOn f U
x : β
hx : x β U
y : β
hy : y β U
hfxy : f x = f y
hxy : x β y
g : β β β := fun z => f z - f x
G : ΞΉ β β β β := fun n z => F n z - f x
hG : βαΆ (n : ΞΉ) in p, DifferentiableOn β (G n) U
hg : TendstoLocallyUniformlyOn G g p U
hgx : g x = 0
hgy : g y = 0
β’ β z β U, g z = 0 | case intro.intro.intro.intro.intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : Β¬InjOn f U
x : β
hx : x β U
y : β
hy : y β U
hfxy : f x = f y
hxy : x β y
g : β β β := fun z => f z - f x
G : ΞΉ β β β β := fun n z => F n z - f x
hG : βαΆ (n : ΞΉ) in p, DifferentiableOn β (G n) U
hg : TendstoLocallyUniformlyOn G g p U
hgx : g x = 0
hgy : g y = 0
hi : Β¬β z β U, g z = 0
β’ Β¬βαΆ (n : ΞΉ) in p, InjOn (F n) U |
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
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : Β¬InjOn f U
x : β
hx : x β U
y : β
hy : y β U
hfxy : f x = f y
hxy : x β y
g : β β β := fun z => f z - f x
G : ΞΉ β β β β := fun n z => F n z - f x
hG : βαΆ (n : ΞΉ) in p, DifferentiableOn β (G n) U
hg : TendstoLocallyUniformlyOn G g p U
hgx : g x = 0
hgy : g y = 0
hi : Β¬β z β U, g z = 0
β’ Β¬βαΆ (n : ΞΉ) in p, InjOn (F n) U | case intro.intro.intro.intro.intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : Β¬InjOn f U
x : β
hx : x β U
y : β
hy : y β U
hfxy : f x = f y
hxy : x β y
g : β β β := fun z => f z - f x
G : ΞΉ β β β β := fun n z => F n z - f x
hG : βαΆ (n : ΞΉ) in p, DifferentiableOn β (G n) U
hg : TendstoLocallyUniformlyOn G g p U
hgx : g x = 0
hgy : g y = 0
hi : Β¬β z β U, g z = 0
β’ βαΆ (x : ΞΉ) in p, β x_1, β (_ : x_1 β U), β x_2, β (_ : x_2 β U) (_ : F x x_1 = F x x_2), Β¬x_1 = x_2 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_inj | [384, 1] | [420, 18] | have h1 : DifferentiableOn β g U := hg.differentiableOn hG hU | case intro.intro.intro.intro.intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : Β¬InjOn f U
x : β
hx : x β U
y : β
hy : y β U
hfxy : f x = f y
hxy : x β y
g : β β β := fun z => f z - f x
G : ΞΉ β β β β := fun n z => F n z - f x
hG : βαΆ (n : ΞΉ) in p, DifferentiableOn β (G n) U
hg : TendstoLocallyUniformlyOn G g p U
hgx : g x = 0
hgy : g y = 0
hi : Β¬β z β U, g z = 0
β’ βαΆ (x : ΞΉ) in p, β x_1, β (_ : x_1 β U), β x_2, β (_ : x_2 β U) (_ : F x x_1 = F x x_2), Β¬x_1 = x_2 | case intro.intro.intro.intro.intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : Β¬InjOn f U
x : β
hx : x β U
y : β
hy : y β U
hfxy : f x = f y
hxy : x β y
g : β β β := fun z => f z - f x
G : ΞΉ β β β β := fun n z => F n z - f x
hG : βαΆ (n : ΞΉ) in p, DifferentiableOn β (G n) U
hg : TendstoLocallyUniformlyOn G g p U
hgx : g x = 0
hgy : g y = 0
hi : Β¬β z β U, g z = 0
h1 : DifferentiableOn β g U
β’ βαΆ (x : ΞΉ) in p, β x_1, β (_ : x_1 β U), β x_2, β (_ : x_2 β U) (_ : F x x_1 = F x x_2), Β¬x_1 = x_2 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_inj | [384, 1] | [420, 18] | have h2 : β zβ β U, βαΆ z in π[β ] zβ, g z β 0 := (hurwitz_1 hU hU' h1).resolve_left hi | case intro.intro.intro.intro.intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : Β¬InjOn f U
x : β
hx : x β U
y : β
hy : y β U
hfxy : f x = f y
hxy : x β y
g : β β β := fun z => f z - f x
G : ΞΉ β β β β := fun n z => F n z - f x
hG : βαΆ (n : ΞΉ) in p, DifferentiableOn β (G n) U
hg : TendstoLocallyUniformlyOn G g p U
hgx : g x = 0
hgy : g y = 0
hi : Β¬β z β U, g z = 0
h1 : DifferentiableOn β g U
β’ βαΆ (x : ΞΉ) in p, β x_1, β (_ : x_1 β U), β x_2, β (_ : x_2 β U) (_ : F x x_1 = F x x_2), Β¬x_1 = x_2 | case intro.intro.intro.intro.intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : Β¬InjOn f U
x : β
hx : x β U
y : β
hy : y β U
hfxy : f x = f y
hxy : x β y
g : β β β := fun z => f z - f x
G : ΞΉ β β β β := fun n z => F n z - f x
hG : βαΆ (n : ΞΉ) in p, DifferentiableOn β (G n) U
hg : TendstoLocallyUniformlyOn G g p U
hgx : g x = 0
hgy : g y = 0
hi : Β¬β z β U, g z = 0
h1 : DifferentiableOn β g U
h2 : β zβ β U, βαΆ (z : β) in π[β ] zβ, g z β 0
β’ βαΆ (x : ΞΉ) in p, β x_1, β (_ : x_1 β U), β x_2, β (_ : x_2 β U) (_ : F x x_1 = F x x_2), Β¬x_1 = x_2 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_inj | [384, 1] | [420, 18] | obtain β¨u, v, hu, hv, huvβ© := t2_separation_nhds hxy | case intro.intro.intro.intro.intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : Β¬InjOn f U
x : β
hx : x β U
y : β
hy : y β U
hfxy : f x = f y
hxy : x β y
g : β β β := fun z => f z - f x
G : ΞΉ β β β β := fun n z => F n z - f x
hG : βαΆ (n : ΞΉ) in p, DifferentiableOn β (G n) U
hg : TendstoLocallyUniformlyOn G g p U
hgx : g x = 0
hgy : g y = 0
hi : Β¬β z β U, g z = 0
h1 : DifferentiableOn β g U
h2 : β zβ β U, βαΆ (z : β) in π[β ] zβ, g z β 0
β’ βαΆ (x : ΞΉ) in p, β x_1, β (_ : x_1 β U), β x_2, β (_ : x_2 β U) (_ : F x x_1 = F x x_2), Β¬x_1 = x_2 | case intro.intro.intro.intro.intro.intro.intro.intro.intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : Β¬InjOn f U
x : β
hx : x β U
y : β
hy : y β U
hfxy : f x = f y
hxy : x β y
g : β β β := fun z => f z - f x
G : ΞΉ β β β β := fun n z => F n z - f x
hG : βαΆ (n : ΞΉ) in p, DifferentiableOn β (G n) U
hg : TendstoLocallyUniformlyOn G g p U
hgx : g x = 0
hgy : g y = 0
hi : Β¬β z β U, g z = 0
h1 : DifferentiableOn β g U
h2 : β zβ β U, βαΆ (z : β) in π[β ] zβ, g z β 0
u v : Set β
hu : u β π x
hv : v β π y
huv : Disjoint u v
β’ βαΆ (x : ΞΉ) in p, β x_1, β (_ : x_1 β U), β x_2, β (_ : x_2 β U) (_ : F x x_1 = F x x_2), Β¬x_1 = x_2 |
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
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : Β¬InjOn f U
x : β
hx : x β U
y : β
hy : y β U
hfxy : f x = f y
hxy : x β y
g : β β β := fun z => f z - f x
G : ΞΉ β β β β := fun n z => F n z - f x
hG : βαΆ (n : ΞΉ) in p, DifferentiableOn β (G n) U
hg : TendstoLocallyUniformlyOn G g p U
hgx : g x = 0
hgy : g y = 0
hi : Β¬β z β U, g z = 0
h1 : DifferentiableOn β g U
h2 : β zβ β U, βαΆ (z : β) in π[β ] zβ, g z β 0
u v : Set β
hu : u β π x
hv : v β π y
huv : Disjoint u v
β’ βαΆ (x : ΞΉ) in p, β x_1, β (_ : x_1 β U), β x_2, β (_ : x_2 β U) (_ : F x x_1 = F x x_2), Β¬x_1 = x_2 | case intro.intro.intro.intro.intro.intro.intro.intro.intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : Β¬InjOn f U
x : β
hx : x β U
y : β
hy : y β U
hfxy : f x = f y
hxy : x β y
g : β β β := fun z => f z - f x
G : ΞΉ β β β β := fun n z => F n z - f x
hG : βαΆ (n : ΞΉ) in p, DifferentiableOn β (G n) U
hg : TendstoLocallyUniformlyOn G g p U
hgx : g x = 0
hgy : g y = 0
hi : Β¬β z β U, g z = 0
h1 : DifferentiableOn β g U
h2 : β zβ β U, βαΆ (z : β) in π[β ] zβ, g z β 0
u v : Set β
hu : u β π x
hv : v β π y
huv : Disjoint u v
h3 : βαΆ (n : ΞΉ) in p, β z β u β© U, G n z = 0
β’ βαΆ (x : ΞΉ) in p, β x_1, β (_ : x_1 β U), β x_2, β (_ : x_2 β U) (_ : F x x_1 = F x x_2), Β¬x_1 = x_2 |
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
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : Β¬InjOn f U
x : β
hx : x β U
y : β
hy : y β U
hfxy : f x = f y
hxy : x β y
g : β β β := fun z => f z - f x
G : ΞΉ β β β β := fun n z => F n z - f x
hG : βαΆ (n : ΞΉ) in p, DifferentiableOn β (G n) U
hg : TendstoLocallyUniformlyOn G g p U
hgx : g x = 0
hgy : g y = 0
hi : Β¬β z β U, g z = 0
h1 : DifferentiableOn β g U
h2 : β zβ β U, βαΆ (z : β) in π[β ] zβ, g z β 0
u v : Set β
hu : u β π x
hv : v β π y
huv : Disjoint u v
h3 : βαΆ (n : ΞΉ) in p, β z β u β© U, G n z = 0
β’ βαΆ (x : ΞΉ) in p, β x_1, β (_ : x_1 β U), β x_2, β (_ : x_2 β U) (_ : F x x_1 = F x x_2), Β¬x_1 = x_2 | case intro.intro.intro.intro.intro.intro.intro.intro.intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : Β¬InjOn f U
x : β
hx : x β U
y : β
hy : y β U
hfxy : f x = f y
hxy : x β y
g : β β β := fun z => f z - f x
G : ΞΉ β β β β := fun n z => F n z - f x
hG : βαΆ (n : ΞΉ) in p, DifferentiableOn β (G n) U
hg : TendstoLocallyUniformlyOn G g p U
hgx : g x = 0
hgy : g y = 0
hi : Β¬β z β U, g z = 0
h1 : DifferentiableOn β g U
h2 : β zβ β U, βαΆ (z : β) in π[β ] zβ, g z β 0
u v : Set β
hu : u β π x
hv : v β π y
huv : Disjoint u v
h3 : βαΆ (n : ΞΉ) in p, β z β u β© U, G n z = 0
h4 : βαΆ (n : ΞΉ) in p, β z β v β© U, G n z = 0
β’ βαΆ (x : ΞΉ) in p, β x_1, β (_ : x_1 β U), β x_2, β (_ : x_2 β U) (_ : F x x_1 = F x x_2), Β¬x_1 = x_2 |
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
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : Β¬InjOn f U
x : β
hx : x β U
y : β
hy : y β U
hfxy : f x = f y
hxy : x β y
g : β β β := fun z => f z - f x
G : ΞΉ β β β β := fun n z => F n z - f x
hG : βαΆ (n : ΞΉ) in p, DifferentiableOn β (G n) U
hg : TendstoLocallyUniformlyOn G g p U
hgx : g x = 0
hgy : g y = 0
hi : Β¬β z β U, g z = 0
h1 : DifferentiableOn β g U
h2 : β zβ β U, βαΆ (z : β) in π[β ] zβ, g z β 0
u v : Set β
hu : u β π x
hv : v β π y
huv : Disjoint u v
h3 : βαΆ (n : ΞΉ) in p, β z β u β© U, G n z = 0
h4 : βαΆ (n : ΞΉ) in p, β z β v β© U, G n z = 0
β’ βαΆ (x : ΞΉ) in p, β x_1, β (_ : x_1 β U), β x_2, β (_ : x_2 β U) (_ : F x x_1 = F x x_2), Β¬x_1 = x_2 | case h
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : Β¬InjOn f U
x : β
hx : x β U
y : β
hy : y β U
hfxy : f x = f y
hxy : x β y
g : β β β := fun z => f z - f x
G : ΞΉ β β β β := fun n z => F n z - f x
hG : βαΆ (n : ΞΉ) in p, DifferentiableOn β (G n) U
hg : TendstoLocallyUniformlyOn G g p U
hgx : g x = 0
hgy : g y = 0
hi : Β¬β z β U, g z = 0
h1 : DifferentiableOn β g U
h2 : β zβ β U, βαΆ (z : β) in π[β ] zβ, g z β 0
u v : Set β
hu : u β π x
hv : v β π y
huv : Disjoint u v
h3 : βαΆ (n : ΞΉ) in p, β z β u β© U, G n z = 0
h4 : βαΆ (n : ΞΉ) in p, β z β v β© U, G n z = 0
n : ΞΉ
hn : (β z β u β© U, G n z = 0) β§ β z β v β© U, G n z = 0
β’ β x, β (_ : x β U), β x_1, β (_ : x_1 β U) (_ : F n x = F n x_1), Β¬x = x_1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_inj | [384, 1] | [420, 18] | obtain β¨β¨xn, hxn, hGxnβ©, β¨yn, hyn, hGynβ©β© := hn | case h
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : Β¬InjOn f U
x : β
hx : x β U
y : β
hy : y β U
hfxy : f x = f y
hxy : x β y
g : β β β := fun z => f z - f x
G : ΞΉ β β β β := fun n z => F n z - f x
hG : βαΆ (n : ΞΉ) in p, DifferentiableOn β (G n) U
hg : TendstoLocallyUniformlyOn G g p U
hgx : g x = 0
hgy : g y = 0
hi : Β¬β z β U, g z = 0
h1 : DifferentiableOn β g U
h2 : β zβ β U, βαΆ (z : β) in π[β ] zβ, g z β 0
u v : Set β
hu : u β π x
hv : v β π y
huv : Disjoint u v
h3 : βαΆ (n : ΞΉ) in p, β z β u β© U, G n z = 0
h4 : βαΆ (n : ΞΉ) in p, β z β v β© U, G n z = 0
n : ΞΉ
hn : (β z β u β© U, G n z = 0) β§ β z β v β© U, G n z = 0
β’ β x, β (_ : x β U), β x_1, β (_ : x_1 β U) (_ : F n x = F n x_1), Β¬x = x_1 | case h.intro.intro.intro.intro.intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : Β¬InjOn f U
x : β
hx : x β U
y : β
hy : y β U
hfxy : f x = f y
hxy : x β y
g : β β β := fun z => f z - f x
G : ΞΉ β β β β := fun n z => F n z - f x
hG : βαΆ (n : ΞΉ) in p, DifferentiableOn β (G n) U
hg : TendstoLocallyUniformlyOn G g p U
hgx : g x = 0
hgy : g y = 0
hi : Β¬β z β U, g z = 0
h1 : DifferentiableOn β g U
h2 : β zβ β U, βαΆ (z : β) in π[β ] zβ, g z β 0
u v : Set β
hu : u β π x
hv : v β π y
huv : Disjoint u v
h3 : βαΆ (n : ΞΉ) in p, β z β u β© U, G n z = 0
h4 : βαΆ (n : ΞΉ) in p, β z β v β© U, G n z = 0
n : ΞΉ
xn : β
hxn : xn β u β© U
hGxn : G n xn = 0
yn : β
hyn : yn β v β© U
hGyn : G n yn = 0
β’ β x, β (_ : x β U), β x_1, β (_ : x_1 β U) (_ : F n x = F n x_1), Β¬x = x_1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_inj | [384, 1] | [420, 18] | refine β¨xn, hxn.2, yn, hyn.2, ?_, huv.ne_of_mem hxn.1 hyn.1β© | case h.intro.intro.intro.intro.intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : Β¬InjOn f U
x : β
hx : x β U
y : β
hy : y β U
hfxy : f x = f y
hxy : x β y
g : β β β := fun z => f z - f x
G : ΞΉ β β β β := fun n z => F n z - f x
hG : βαΆ (n : ΞΉ) in p, DifferentiableOn β (G n) U
hg : TendstoLocallyUniformlyOn G g p U
hgx : g x = 0
hgy : g y = 0
hi : Β¬β z β U, g z = 0
h1 : DifferentiableOn β g U
h2 : β zβ β U, βαΆ (z : β) in π[β ] zβ, g z β 0
u v : Set β
hu : u β π x
hv : v β π y
huv : Disjoint u v
h3 : βαΆ (n : ΞΉ) in p, β z β u β© U, G n z = 0
h4 : βαΆ (n : ΞΉ) in p, β z β v β© U, G n z = 0
n : ΞΉ
xn : β
hxn : xn β u β© U
hGxn : G n xn = 0
yn : β
hyn : yn β v β© U
hGyn : G n yn = 0
β’ β x, β (_ : x β U), β x_1, β (_ : x_1 β U) (_ : F n x = F n x_1), Β¬x = x_1 | case h.intro.intro.intro.intro.intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : Β¬InjOn f U
x : β
hx : x β U
y : β
hy : y β U
hfxy : f x = f y
hxy : x β y
g : β β β := fun z => f z - f x
G : ΞΉ β β β β := fun n z => F n z - f x
hG : βαΆ (n : ΞΉ) in p, DifferentiableOn β (G n) U
hg : TendstoLocallyUniformlyOn G g p U
hgx : g x = 0
hgy : g y = 0
hi : Β¬β z β U, g z = 0
h1 : DifferentiableOn β g U
h2 : β zβ β U, βαΆ (z : β) in π[β ] zβ, g z β 0
u v : Set β
hu : u β π x
hv : v β π y
huv : Disjoint u v
h3 : βαΆ (n : ΞΉ) in p, β z β u β© U, G n z = 0
h4 : βαΆ (n : ΞΉ) in p, β z β v β© U, G n z = 0
n : ΞΉ
xn : β
hxn : xn β u β© U
hGxn : G n xn = 0
yn : β
hyn : yn β v β© U
hGyn : G n yn = 0
β’ F n xn = F n yn |
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
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : Β¬InjOn f U
x : β
hx : x β U
y : β
hy : y β U
hfxy : f x = f y
hxy : x β y
g : β β β := fun z => f z - f x
G : ΞΉ β β β β := fun n z => F n z - f x
hG : βαΆ (n : ΞΉ) in p, DifferentiableOn β (G n) U
hg : TendstoLocallyUniformlyOn G g p U
hgx : g x = 0
hgy : g y = 0
hi : Β¬β z β U, g z = 0
h1 : DifferentiableOn β g U
h2 : β zβ β U, βαΆ (z : β) in π[β ] zβ, g z β 0
u v : Set β
hu : u β π x
hv : v β π y
huv : Disjoint u v
h3 : βαΆ (n : ΞΉ) in p, β z β u β© U, G n z = 0
h4 : βαΆ (n : ΞΉ) in p, β z β v β© U, G n z = 0
n : ΞΉ
xn : β
hxn : xn β u β© U
hGxn : G n xn = 0
yn : β
hyn : yn β v β© U
hGyn : G n yn = 0
β’ F n xn = F n yn | case h.intro.intro.intro.intro.intro
ΞΉ : Type u_1
F : ΞΉ β β β β
f : β β β
zβ : β
p : Filter ΞΉ
r : β
U : Set β
instβ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : βαΆ (n : ΞΉ) in p, DifferentiableOn β (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : Β¬InjOn f U
x : β
hx : x β U
y : β
hy : y β U
hfxy : f x = f y
hxy : x β y
g : β β β := fun z => f z - f x
G : ΞΉ β β β β := fun n z => F n z - f x
hG : βαΆ (n : ΞΉ) in p, DifferentiableOn β (G n) U
hg : TendstoLocallyUniformlyOn G g p U
hgx : g x = 0
hgy : g y = 0
hi : Β¬β z β U, g z = 0
h1 : DifferentiableOn β g U
h2 : β zβ β U, βαΆ (z : β) in π[β ] zβ, g z β 0
u v : Set β
hu : u β π x
hv : v β π y
huv : Disjoint u v
h3 : βαΆ (n : ΞΉ) in p, β z β u β© U, G n z = 0
h4 : βαΆ (n : ΞΉ) in p, β z β v β© U, G n z = 0
n : ΞΉ
xn : β
hxn : xn β u β© U
hGxn : F n xn = f x
yn : β
hyn : yn β v β© U
hGyn : F n yn = f x
β’ F n xn = F n yn |
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