url
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values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
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stringlengths 1
11.2k
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stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
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|---|---|---|---|---|---|---|---|---|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_inj | [384, 1] | [420, 18] | rw [hGxn, hGyn] | case h.intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
h : ¬InjOn f U
x : ℂ
hx : x ∈ U
y : ℂ
hy : y ∈ U
hfxy : f x = f y
hxy : x ≠ y
g : ℂ → ℂ := fun z => f z - f x
G : ι → ℂ → ℂ := fun n z => F n z - f x
hG : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (G n) U
hg : TendstoLocallyUniformlyOn G g p U
hgx : g x = 0
hgy : g y = 0
hi : ¬∀ z ∈ U, g z = 0
h1 : DifferentiableOn ℂ g U
h2 : ∀ z₀ ∈ U, ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, g z ≠ 0
u v : Set ℂ
hu : u ∈ 𝓝 x
hv : v ∈ 𝓝 y
huv : Disjoint u v
h3 : ∀ᶠ (n : ι) in p, ∃ z ∈ u ∩ U, G n z = 0
h4 : ∀ᶠ (n : ι) in p, ∃ z ∈ v ∩ U, G n z = 0
n : ι
xn : ℂ
hxn : xn ∈ u ∩ U
hGxn : F n xn = f x
yn : ℂ
hyn : yn ∈ v ∩ U
hGyn : F n yn = f x
⊢ F n xn = F n yn | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_inj | [384, 1] | [420, 18] | simp [InjOn] at h | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : ∃ᶠ (n : ι) in p, InjOn (F n) U
h : ¬InjOn f U
⊢ ∃ x ∈ U, ∃ y ∈ U, f x = f y ∧ x ≠ y | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : ∃ᶠ (n : ι) in p, InjOn (F n) U
h : ∃ x ∈ U, ∃ x_1, f x = f x_1 ∧ x_1 ∈ U ∧ ¬x = x_1
⊢ ∃ x ∈ U, ∃ y ∈ U, f x = f y ∧ x ≠ y |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_inj | [384, 1] | [420, 18] | obtain ⟨x, h1, y, h2, h3, h4⟩ := h | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : ∃ᶠ (n : ι) in p, InjOn (F n) U
h : ∃ x ∈ U, ∃ x_1, f x = f x_1 ∧ x_1 ∈ U ∧ ¬x = x_1
⊢ ∃ x ∈ U, ∃ y ∈ U, f x = f y ∧ x ≠ y | case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : ∃ᶠ (n : ι) in p, InjOn (F n) U
x : ℂ
h1 : x ∈ U
y : ℂ
h2 : f x = f y
h3 : y ∈ U
h4 : ¬x = y
⊢ ∃ x ∈ U, ∃ y ∈ U, f x = f y ∧ x ≠ y |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_inj | [384, 1] | [420, 18] | refine ⟨x, h1, y, h3, h2, h4⟩ | case intro.intro.intro.intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : ∃ᶠ (n : ι) in p, InjOn (F n) U
x : ℂ
h1 : x ∈ U
y : ℂ
h2 : f x = f y
h3 : y ∈ U
h4 : ¬x = y
⊢ ∃ x ∈ U, ∃ y ∈ U, f x = f y ∧ x ≠ y | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_inj | [384, 1] | [420, 18] | filter_upwards [hF] with n hF using hF.sub (differentiableOn_const _) | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : ∃ᶠ (n : ι) in p, InjOn (F n) U
h : ¬InjOn f U
x : ℂ
hx : x ∈ U
y : ℂ
hy : y ∈ U
hfxy : f x = f y
hxy : x ≠ y
g : ℂ → ℂ := fun z => f z - f x
G : ι → ℂ → ℂ := fun n z => F n z - f x
⊢ ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (G n) U | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_inj | [384, 1] | [420, 18] | simp [g, hfxy] | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : ∃ᶠ (n : ι) in p, InjOn (F n) U
h : ¬InjOn f U
x : ℂ
hx : x ∈ U
y : ℂ
hy : y ∈ U
hfxy : f x = f y
hxy : x ≠ y
g : ℂ → ℂ := fun z => f z - f x
G : ι → ℂ → ℂ := fun n z => F n z - f x
hG : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (G n) U
hg : TendstoLocallyUniformlyOn G g p U
hgx : g x = 0
⊢ g y = 0 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_inj | [384, 1] | [420, 18] | exact ⟨f x, by simpa [sub_eq_zero, g] using this⟩ | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : ∃ᶠ (n : ι) in p, InjOn (F n) U
h : ¬InjOn f U
x : ℂ
hx : x ∈ U
y : ℂ
hy : y ∈ U
hfxy : f x = f y
hxy : x ≠ y
g : ℂ → ℂ := fun z => f z - f x
G : ι → ℂ → ℂ := fun n z => F n z - f x
hG : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (G n) U
hg : TendstoLocallyUniformlyOn G g p U
hgx : g x = 0
hgy : g y = 0
this : ∀ z ∈ U, g z = 0
⊢ ∃ w, ∀ z ∈ U, f z = w | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz_inj | [384, 1] | [420, 18] | simpa [sub_eq_zero, g] using this | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
inst✝ : NeBot p
hU : IsOpen U
hU' : IsPreconnected U
hF : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (F n) U
hf : TendstoLocallyUniformlyOn F f p U
hi : ∃ᶠ (n : ι) in p, InjOn (F n) U
h : ¬InjOn f U
x : ℂ
hx : x ∈ U
y : ℂ
hy : y ∈ U
hfxy : f x = f y
hxy : x ≠ y
g : ℂ → ℂ := fun z => f z - f x
G : ι → ℂ → ℂ := fun n z => F n z - f x
hG : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (G n) U
hg : TendstoLocallyUniformlyOn G g p U
hgx : g x = 0
hgy : g y = 0
this : ∀ z ∈ U, g z = 0
⊢ ∀ z ∈ U, f z = f x | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | IsCompact_𝓙 | [10, 1] | [32, 38] | have hU : IsOpen U := good_domain.is_open | ι : Type u_1
l : Filter ι
U : Set ℂ
z₀ : ℂ
inst✝ : good_domain U
⊢ IsCompact (𝓙 U) | ι : Type u_1
l : Filter ι
U : Set ℂ
z₀ : ℂ
inst✝ : good_domain U
hU : IsOpen U
⊢ IsCompact (𝓙 U) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | IsCompact_𝓙 | [10, 1] | [32, 38] | refine (isCompact_𝓜 hU).of_isClosed_subset ?_ (λ _ hf => hf.1) | ι : Type u_1
l : Filter ι
U : Set ℂ
z₀ : ℂ
inst✝ : good_domain U
hU : IsOpen U
⊢ IsCompact (𝓙 U) | ι : Type u_1
l : Filter ι
U : Set ℂ
z₀ : ℂ
inst✝ : good_domain U
hU : IsOpen U
⊢ IsClosed (𝓙 U) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | IsCompact_𝓙 | [10, 1] | [32, 38] | refine isClosed_iff_clusterPt.2 (λ f hf => ?_) | ι : Type u_1
l : Filter ι
U : Set ℂ
z₀ : ℂ
inst✝ : good_domain U
hU : IsOpen U
⊢ IsClosed (𝓙 U) | ι : Type u_1
l : Filter ι
U : Set ℂ
z₀ : ℂ
inst✝ : good_domain U
hU : IsOpen U
f : 𝓒 U
hf : ClusterPt f (𝓟 (𝓙 U))
⊢ f ∈ 𝓙 U |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | IsCompact_𝓙 | [10, 1] | [32, 38] | set l := 𝓝 f ⊓ 𝓟 (𝓙 U) | ι : Type u_1
l : Filter ι
U : Set ℂ
z₀ : ℂ
inst✝ : good_domain U
hU : IsOpen U
f : 𝓒 U
hf : ClusterPt f (𝓟 (𝓙 U))
⊢ f ∈ 𝓙 U | ι : Type u_1
l✝ : Filter ι
U : Set ℂ
z₀ : ℂ
inst✝ : good_domain U
hU : IsOpen U
f : 𝓒 U
hf : ClusterPt f (𝓟 (𝓙 U))
l : Filter (𝓒 U) := 𝓝 f ⊓ 𝓟 (𝓙 U)
⊢ f ∈ 𝓙 U |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | IsCompact_𝓙 | [10, 1] | [32, 38] | haveI : l.NeBot := hf | ι : Type u_1
l✝ : Filter ι
U : Set ℂ
z₀ : ℂ
inst✝ : good_domain U
hU : IsOpen U
f : 𝓒 U
hf : ClusterPt f (𝓟 (𝓙 U))
l : Filter (𝓒 U) := 𝓝 f ⊓ 𝓟 (𝓙 U)
⊢ f ∈ 𝓙 U | ι : Type u_1
l✝ : Filter ι
U : Set ℂ
z₀ : ℂ
inst✝ : good_domain U
hU : IsOpen U
f : 𝓒 U
hf : ClusterPt f (𝓟 (𝓙 U))
l : Filter (𝓒 U) := 𝓝 f ⊓ 𝓟 (𝓙 U)
this : NeBot l
⊢ f ∈ 𝓙 U |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | IsCompact_𝓙 | [10, 1] | [32, 38] | obtain ⟨h1, h2⟩ := tendsto_inf.1 (@tendsto_id _ l) | ι : Type u_1
l✝ : Filter ι
U : Set ℂ
z₀ : ℂ
inst✝ : good_domain U
hU : IsOpen U
f : 𝓒 U
hf : ClusterPt f (𝓟 (𝓙 U))
l : Filter (𝓒 U) := 𝓝 f ⊓ 𝓟 (𝓙 U)
this : NeBot l
⊢ f ∈ 𝓙 U | case intro
ι : Type u_1
l✝ : Filter ι
U : Set ℂ
z₀ : ℂ
inst✝ : good_domain U
hU : IsOpen U
f : 𝓒 U
hf : ClusterPt f (𝓟 (𝓙 U))
l : Filter (𝓒 U) := 𝓝 f ⊓ 𝓟 (𝓙 U)
this : NeBot l
h1 : Tendsto id l (𝓝 f)
h2 : Tendsto id l (𝓟 (𝓙 U))
⊢ f ∈ 𝓙 U |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | IsCompact_𝓙 | [10, 1] | [32, 38] | rw [tendsto_principal] at h2 | case intro
ι : Type u_1
l✝ : Filter ι
U : Set ℂ
z₀ : ℂ
inst✝ : good_domain U
hU : IsOpen U
f : 𝓒 U
hf : ClusterPt f (𝓟 (𝓙 U))
l : Filter (𝓒 U) := 𝓝 f ⊓ 𝓟 (𝓙 U)
this : NeBot l
h1 : Tendsto id l (𝓝 f)
h2 : Tendsto id l (𝓟 (𝓙 U))
⊢ f ∈ 𝓙 U | case intro
ι : Type u_1
l✝ : Filter ι
U : Set ℂ
z₀ : ℂ
inst✝ : good_domain U
hU : IsOpen U
f : 𝓒 U
hf : ClusterPt f (𝓟 (𝓙 U))
l : Filter (𝓒 U) := 𝓝 f ⊓ 𝓟 (𝓙 U)
this : NeBot l
h1 : Tendsto id l (𝓝 f)
h2 : ∀ᶠ (a : 𝓒 U) in l, id a ∈ 𝓙 U
⊢ f ∈ 𝓙 U |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | IsCompact_𝓙 | [10, 1] | [32, 38] | refine ⟨(IsClosed_𝓜 hU).mem_of_tendsto h1 (h2.mono (λ _ h => h.1)), ?_⟩ | case intro
ι : Type u_1
l✝ : Filter ι
U : Set ℂ
z₀ : ℂ
inst✝ : good_domain U
hU : IsOpen U
f : 𝓒 U
hf : ClusterPt f (𝓟 (𝓙 U))
l : Filter (𝓒 U) := 𝓝 f ⊓ 𝓟 (𝓙 U)
this : NeBot l
h1 : Tendsto id l (𝓝 f)
h2 : ∀ᶠ (a : 𝓒 U) in l, id a ∈ 𝓙 U
⊢ f ∈ 𝓙 U | case intro
ι : Type u_1
l✝ : Filter ι
U : Set ℂ
z₀ : ℂ
inst✝ : good_domain U
hU : IsOpen U
f : 𝓒 U
hf : ClusterPt f (𝓟 (𝓙 U))
l : Filter (𝓒 U) := 𝓝 f ⊓ 𝓟 (𝓙 U)
this : NeBot l
h1 : Tendsto id l (𝓝 f)
h2 : ∀ᶠ (a : 𝓒 U) in l, id a ∈ 𝓙 U
⊢ InjOn f U ∨ ∃ w, EqOn f (fun x => w) U |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | IsCompact_𝓙 | [10, 1] | [32, 38] | by_cases h : ∃ᶠ f in l, InjOn f U | case intro
ι : Type u_1
l✝ : Filter ι
U : Set ℂ
z₀ : ℂ
inst✝ : good_domain U
hU : IsOpen U
f : 𝓒 U
hf : ClusterPt f (𝓟 (𝓙 U))
l : Filter (𝓒 U) := 𝓝 f ⊓ 𝓟 (𝓙 U)
this : NeBot l
h1 : Tendsto id l (𝓝 f)
h2 : ∀ᶠ (a : 𝓒 U) in l, id a ∈ 𝓙 U
⊢ InjOn f U ∨ ∃ w, EqOn f (fun x => w) U | case pos
ι : Type u_1
l✝ : Filter ι
U : Set ℂ
z₀ : ℂ
inst✝ : good_domain U
hU : IsOpen U
f : 𝓒 U
hf : ClusterPt f (𝓟 (𝓙 U))
l : Filter (𝓒 U) := 𝓝 f ⊓ 𝓟 (𝓙 U)
this : NeBot l
h1 : Tendsto id l (𝓝 f)
h2 : ∀ᶠ (a : 𝓒 U) in l, id a ∈ 𝓙 U
h : ∃ᶠ (f : ℂ → ℂ) in l, InjOn f U
⊢ InjOn f U ∨ ∃ w, EqOn f (fun x => w) U
case neg
ι : Type u_1
l✝ : Filter ι
U : Set ℂ
z₀ : ℂ
inst✝ : good_domain U
hU : IsOpen U
f : 𝓒 U
hf : ClusterPt f (𝓟 (𝓙 U))
l : Filter (𝓒 U) := 𝓝 f ⊓ 𝓟 (𝓙 U)
this : NeBot l
h1 : Tendsto id l (𝓝 f)
h2 : ∀ᶠ (a : 𝓒 U) in l, id a ∈ 𝓙 U
h : ¬∃ᶠ (f : ℂ → ℂ) in l, InjOn f U
⊢ InjOn f U ∨ ∃ w, EqOn f (fun x => w) U |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | IsCompact_𝓙 | [10, 1] | [32, 38] | case pos =>
refine (hurwitz_inj hU good_domain.is_preconnected ?_ ((tendsto_𝓒_iff hU).1 h1) h).symm
filter_upwards [h2] with g hg using hg.1.1 | ι : Type u_1
l✝ : Filter ι
U : Set ℂ
z₀ : ℂ
inst✝ : good_domain U
hU : IsOpen U
f : 𝓒 U
hf : ClusterPt f (𝓟 (𝓙 U))
l : Filter (𝓒 U) := 𝓝 f ⊓ 𝓟 (𝓙 U)
this : NeBot l
h1 : Tendsto id l (𝓝 f)
h2 : ∀ᶠ (a : 𝓒 U) in l, id a ∈ 𝓙 U
h : ∃ᶠ (f : ℂ → ℂ) in l, InjOn f U
⊢ InjOn f U ∨ ∃ w, EqOn f (fun x => w) U | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | IsCompact_𝓙 | [10, 1] | [32, 38] | case neg =>
obtain ⟨z₀, hz₀⟩ : U.Nonempty := good_domain.is_nonempty
have : ∀ z ∈ U, Tendsto (eval z) l (𝓝 (f z)) := by
refine λ z hz => (map_mono inf_le_left).trans ?_
exact ((UniformOnFun.uniformContinuous_eval_of_mem ℂ (compacts U)
(mem_singleton z) ⟨singleton_subset_iff.2 hz, isCompact_singleton⟩).continuous).tendsto f
refine Or.inr ⟨f z₀, λ z hz => tendsto_nhds_unique ((this z hz).congr' ?_) (this z₀ hz₀)⟩
filter_upwards [not_frequently.1 h, h2] with f hf1 hf2
obtain ⟨w, hw⟩ := hf2.2.resolve_left hf1
exact (hw hz).trans (hw hz₀).symm | ι : Type u_1
l✝ : Filter ι
U : Set ℂ
z₀ : ℂ
inst✝ : good_domain U
hU : IsOpen U
f : 𝓒 U
hf : ClusterPt f (𝓟 (𝓙 U))
l : Filter (𝓒 U) := 𝓝 f ⊓ 𝓟 (𝓙 U)
this : NeBot l
h1 : Tendsto id l (𝓝 f)
h2 : ∀ᶠ (a : 𝓒 U) in l, id a ∈ 𝓙 U
h : ¬∃ᶠ (f : ℂ → ℂ) in l, InjOn f U
⊢ InjOn f U ∨ ∃ w, EqOn f (fun x => w) U | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | IsCompact_𝓙 | [10, 1] | [32, 38] | refine (hurwitz_inj hU good_domain.is_preconnected ?_ ((tendsto_𝓒_iff hU).1 h1) h).symm | ι : Type u_1
l✝ : Filter ι
U : Set ℂ
z₀ : ℂ
inst✝ : good_domain U
hU : IsOpen U
f : 𝓒 U
hf : ClusterPt f (𝓟 (𝓙 U))
l : Filter (𝓒 U) := 𝓝 f ⊓ 𝓟 (𝓙 U)
this : NeBot l
h1 : Tendsto id l (𝓝 f)
h2 : ∀ᶠ (a : 𝓒 U) in l, id a ∈ 𝓙 U
h : ∃ᶠ (f : ℂ → ℂ) in l, InjOn f U
⊢ InjOn f U ∨ ∃ w, EqOn f (fun x => w) U | ι : Type u_1
l✝ : Filter ι
U : Set ℂ
z₀ : ℂ
inst✝ : good_domain U
hU : IsOpen U
f : 𝓒 U
hf : ClusterPt f (𝓟 (𝓙 U))
l : Filter (𝓒 U) := 𝓝 f ⊓ 𝓟 (𝓙 U)
this : NeBot l
h1 : Tendsto id l (𝓝 f)
h2 : ∀ᶠ (a : 𝓒 U) in l, id a ∈ 𝓙 U
h : ∃ᶠ (f : ℂ → ℂ) in l, InjOn f U
⊢ ∀ᶠ (n : 𝓒 U) in l, DifferentiableOn ℂ (id n) U |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | IsCompact_𝓙 | [10, 1] | [32, 38] | filter_upwards [h2] with g hg using hg.1.1 | ι : Type u_1
l✝ : Filter ι
U : Set ℂ
z₀ : ℂ
inst✝ : good_domain U
hU : IsOpen U
f : 𝓒 U
hf : ClusterPt f (𝓟 (𝓙 U))
l : Filter (𝓒 U) := 𝓝 f ⊓ 𝓟 (𝓙 U)
this : NeBot l
h1 : Tendsto id l (𝓝 f)
h2 : ∀ᶠ (a : 𝓒 U) in l, id a ∈ 𝓙 U
h : ∃ᶠ (f : ℂ → ℂ) in l, InjOn f U
⊢ ∀ᶠ (n : 𝓒 U) in l, DifferentiableOn ℂ (id n) U | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | IsCompact_𝓙 | [10, 1] | [32, 38] | obtain ⟨z₀, hz₀⟩ : U.Nonempty := good_domain.is_nonempty | ι : Type u_1
l✝ : Filter ι
U : Set ℂ
z₀ : ℂ
inst✝ : good_domain U
hU : IsOpen U
f : 𝓒 U
hf : ClusterPt f (𝓟 (𝓙 U))
l : Filter (𝓒 U) := 𝓝 f ⊓ 𝓟 (𝓙 U)
this : NeBot l
h1 : Tendsto id l (𝓝 f)
h2 : ∀ᶠ (a : 𝓒 U) in l, id a ∈ 𝓙 U
h : ¬∃ᶠ (f : ℂ → ℂ) in l, InjOn f U
⊢ InjOn f U ∨ ∃ w, EqOn f (fun x => w) U | case intro
ι : Type u_1
l✝ : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
hU : IsOpen U
f : 𝓒 U
hf : ClusterPt f (𝓟 (𝓙 U))
l : Filter (𝓒 U) := 𝓝 f ⊓ 𝓟 (𝓙 U)
this : NeBot l
h1 : Tendsto id l (𝓝 f)
h2 : ∀ᶠ (a : 𝓒 U) in l, id a ∈ 𝓙 U
h : ¬∃ᶠ (f : ℂ → ℂ) in l, InjOn f U
z₀ : ℂ
hz₀ : z₀ ∈ U
⊢ InjOn f U ∨ ∃ w, EqOn f (fun x => w) U |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | IsCompact_𝓙 | [10, 1] | [32, 38] | have : ∀ z ∈ U, Tendsto (eval z) l (𝓝 (f z)) := by
refine λ z hz => (map_mono inf_le_left).trans ?_
exact ((UniformOnFun.uniformContinuous_eval_of_mem ℂ (compacts U)
(mem_singleton z) ⟨singleton_subset_iff.2 hz, isCompact_singleton⟩).continuous).tendsto f | case intro
ι : Type u_1
l✝ : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
hU : IsOpen U
f : 𝓒 U
hf : ClusterPt f (𝓟 (𝓙 U))
l : Filter (𝓒 U) := 𝓝 f ⊓ 𝓟 (𝓙 U)
this : NeBot l
h1 : Tendsto id l (𝓝 f)
h2 : ∀ᶠ (a : 𝓒 U) in l, id a ∈ 𝓙 U
h : ¬∃ᶠ (f : ℂ → ℂ) in l, InjOn f U
z₀ : ℂ
hz₀ : z₀ ∈ U
⊢ InjOn f U ∨ ∃ w, EqOn f (fun x => w) U | case intro
ι : Type u_1
l✝ : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
hU : IsOpen U
f : 𝓒 U
hf : ClusterPt f (𝓟 (𝓙 U))
l : Filter (𝓒 U) := 𝓝 f ⊓ 𝓟 (𝓙 U)
this✝ : NeBot l
h1 : Tendsto id l (𝓝 f)
h2 : ∀ᶠ (a : 𝓒 U) in l, id a ∈ 𝓙 U
h : ¬∃ᶠ (f : ℂ → ℂ) in l, InjOn f U
z₀ : ℂ
hz₀ : z₀ ∈ U
this : ∀ z ∈ U, Tendsto (eval z) l (𝓝 (f z))
⊢ InjOn f U ∨ ∃ w, EqOn f (fun x => w) U |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | IsCompact_𝓙 | [10, 1] | [32, 38] | refine Or.inr ⟨f z₀, λ z hz => tendsto_nhds_unique ((this z hz).congr' ?_) (this z₀ hz₀)⟩ | case intro
ι : Type u_1
l✝ : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
hU : IsOpen U
f : 𝓒 U
hf : ClusterPt f (𝓟 (𝓙 U))
l : Filter (𝓒 U) := 𝓝 f ⊓ 𝓟 (𝓙 U)
this✝ : NeBot l
h1 : Tendsto id l (𝓝 f)
h2 : ∀ᶠ (a : 𝓒 U) in l, id a ∈ 𝓙 U
h : ¬∃ᶠ (f : ℂ → ℂ) in l, InjOn f U
z₀ : ℂ
hz₀ : z₀ ∈ U
this : ∀ z ∈ U, Tendsto (eval z) l (𝓝 (f z))
⊢ InjOn f U ∨ ∃ w, EqOn f (fun x => w) U | case intro
ι : Type u_1
l✝ : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
hU : IsOpen U
f : 𝓒 U
hf : ClusterPt f (𝓟 (𝓙 U))
l : Filter (𝓒 U) := 𝓝 f ⊓ 𝓟 (𝓙 U)
this✝ : NeBot l
h1 : Tendsto id l (𝓝 f)
h2 : ∀ᶠ (a : 𝓒 U) in l, id a ∈ 𝓙 U
h : ¬∃ᶠ (f : ℂ → ℂ) in l, InjOn f U
z₀ : ℂ
hz₀ : z₀ ∈ U
this : ∀ z ∈ U, Tendsto (eval z) l (𝓝 (f z))
z : ℂ
hz : z ∈ U
⊢ eval z =ᶠ[l] eval z₀ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | IsCompact_𝓙 | [10, 1] | [32, 38] | filter_upwards [not_frequently.1 h, h2] with f hf1 hf2 | case intro
ι : Type u_1
l✝ : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
hU : IsOpen U
f : 𝓒 U
hf : ClusterPt f (𝓟 (𝓙 U))
l : Filter (𝓒 U) := 𝓝 f ⊓ 𝓟 (𝓙 U)
this✝ : NeBot l
h1 : Tendsto id l (𝓝 f)
h2 : ∀ᶠ (a : 𝓒 U) in l, id a ∈ 𝓙 U
h : ¬∃ᶠ (f : ℂ → ℂ) in l, InjOn f U
z₀ : ℂ
hz₀ : z₀ ∈ U
this : ∀ z ∈ U, Tendsto (eval z) l (𝓝 (f z))
z : ℂ
hz : z ∈ U
⊢ eval z =ᶠ[l] eval z₀ | case h
ι : Type u_1
l✝ : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
hU : IsOpen U
f✝ : 𝓒 U
hf : ClusterPt f✝ (𝓟 (𝓙 U))
l : Filter (𝓒 U) := 𝓝 f✝ ⊓ 𝓟 (𝓙 U)
this✝ : NeBot l
h1 : Tendsto id l (𝓝 f✝)
h2 : ∀ᶠ (a : 𝓒 U) in l, id a ∈ 𝓙 U
h : ¬∃ᶠ (f : ℂ → ℂ) in l, InjOn f U
z₀ : ℂ
hz₀ : z₀ ∈ U
this : ∀ z ∈ U, Tendsto (eval z) l (𝓝 (f✝ z))
z : ℂ
hz : z ∈ U
f : ℂ → ℂ
hf1 : ¬InjOn f U
hf2 : f ∈ {x | id x ∈ 𝓙 U}
⊢ eval z f = eval z₀ f |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | IsCompact_𝓙 | [10, 1] | [32, 38] | obtain ⟨w, hw⟩ := hf2.2.resolve_left hf1 | case h
ι : Type u_1
l✝ : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
hU : IsOpen U
f✝ : 𝓒 U
hf : ClusterPt f✝ (𝓟 (𝓙 U))
l : Filter (𝓒 U) := 𝓝 f✝ ⊓ 𝓟 (𝓙 U)
this✝ : NeBot l
h1 : Tendsto id l (𝓝 f✝)
h2 : ∀ᶠ (a : 𝓒 U) in l, id a ∈ 𝓙 U
h : ¬∃ᶠ (f : ℂ → ℂ) in l, InjOn f U
z₀ : ℂ
hz₀ : z₀ ∈ U
this : ∀ z ∈ U, Tendsto (eval z) l (𝓝 (f✝ z))
z : ℂ
hz : z ∈ U
f : ℂ → ℂ
hf1 : ¬InjOn f U
hf2 : f ∈ {x | id x ∈ 𝓙 U}
⊢ eval z f = eval z₀ f | case h.intro
ι : Type u_1
l✝ : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
hU : IsOpen U
f✝ : 𝓒 U
hf : ClusterPt f✝ (𝓟 (𝓙 U))
l : Filter (𝓒 U) := 𝓝 f✝ ⊓ 𝓟 (𝓙 U)
this✝ : NeBot l
h1 : Tendsto id l (𝓝 f✝)
h2 : ∀ᶠ (a : 𝓒 U) in l, id a ∈ 𝓙 U
h : ¬∃ᶠ (f : ℂ → ℂ) in l, InjOn f U
z₀ : ℂ
hz₀ : z₀ ∈ U
this : ∀ z ∈ U, Tendsto (eval z) l (𝓝 (f✝ z))
z : ℂ
hz : z ∈ U
f : ℂ → ℂ
hf1 : ¬InjOn f U
hf2 : f ∈ {x | id x ∈ 𝓙 U}
w : ℂ
hw : EqOn (id f) (fun x => w) U
⊢ eval z f = eval z₀ f |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | IsCompact_𝓙 | [10, 1] | [32, 38] | exact (hw hz).trans (hw hz₀).symm | case h.intro
ι : Type u_1
l✝ : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
hU : IsOpen U
f✝ : 𝓒 U
hf : ClusterPt f✝ (𝓟 (𝓙 U))
l : Filter (𝓒 U) := 𝓝 f✝ ⊓ 𝓟 (𝓙 U)
this✝ : NeBot l
h1 : Tendsto id l (𝓝 f✝)
h2 : ∀ᶠ (a : 𝓒 U) in l, id a ∈ 𝓙 U
h : ¬∃ᶠ (f : ℂ → ℂ) in l, InjOn f U
z₀ : ℂ
hz₀ : z₀ ∈ U
this : ∀ z ∈ U, Tendsto (eval z) l (𝓝 (f✝ z))
z : ℂ
hz : z ∈ U
f : ℂ → ℂ
hf1 : ¬InjOn f U
hf2 : f ∈ {x | id x ∈ 𝓙 U}
w : ℂ
hw : EqOn (id f) (fun x => w) U
⊢ eval z f = eval z₀ f | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | IsCompact_𝓙 | [10, 1] | [32, 38] | refine λ z hz => (map_mono inf_le_left).trans ?_ | ι : Type u_1
l✝ : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
hU : IsOpen U
f : 𝓒 U
hf : ClusterPt f (𝓟 (𝓙 U))
l : Filter (𝓒 U) := 𝓝 f ⊓ 𝓟 (𝓙 U)
this : NeBot l
h1 : Tendsto id l (𝓝 f)
h2 : ∀ᶠ (a : 𝓒 U) in l, id a ∈ 𝓙 U
h : ¬∃ᶠ (f : ℂ → ℂ) in l, InjOn f U
z₀ : ℂ
hz₀ : z₀ ∈ U
⊢ ∀ z ∈ U, Tendsto (eval z) l (𝓝 (f z)) | ι : Type u_1
l✝ : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
hU : IsOpen U
f : 𝓒 U
hf : ClusterPt f (𝓟 (𝓙 U))
l : Filter (𝓒 U) := 𝓝 f ⊓ 𝓟 (𝓙 U)
this : NeBot l
h1 : Tendsto id l (𝓝 f)
h2 : ∀ᶠ (a : 𝓒 U) in l, id a ∈ 𝓙 U
h : ¬∃ᶠ (f : ℂ → ℂ) in l, InjOn f U
z₀ : ℂ
hz₀ : z₀ ∈ U
z : ℂ
hz : z ∈ U
⊢ map (eval z) (𝓝 f) ≤ 𝓝 (f z) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | IsCompact_𝓙 | [10, 1] | [32, 38] | exact ((UniformOnFun.uniformContinuous_eval_of_mem ℂ (compacts U)
(mem_singleton z) ⟨singleton_subset_iff.2 hz, isCompact_singleton⟩).continuous).tendsto f | ι : Type u_1
l✝ : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
hU : IsOpen U
f : 𝓒 U
hf : ClusterPt f (𝓟 (𝓙 U))
l : Filter (𝓒 U) := 𝓝 f ⊓ 𝓟 (𝓙 U)
this : NeBot l
h1 : Tendsto id l (𝓝 f)
h2 : ∀ᶠ (a : 𝓒 U) in l, id a ∈ 𝓙 U
h : ¬∃ᶠ (f : ℂ → ℂ) in l, InjOn f U
z₀ : ℂ
hz₀ : z₀ ∈ U
z : ℂ
hz : z ∈ U
⊢ map (eval z) (𝓝 f) ≤ 𝓝 (f z) | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | ContinuousOn_obs | [38, 1] | [43, 29] | have e1 : z₀ ∈ {z₀} := mem_singleton _ | ι : Type u_1
l : Filter ι
U : Set ℂ
z₀ : ℂ
hU : IsOpen U
hz₀ : z₀ ∈ U
⊢ ContinuousOn (obs z₀) (𝓗 U) | ι : Type u_1
l : Filter ι
U : Set ℂ
z₀ : ℂ
hU : IsOpen U
hz₀ : z₀ ∈ U
e1 : z₀ ∈ {z₀}
⊢ ContinuousOn (obs z₀) (𝓗 U) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | ContinuousOn_obs | [38, 1] | [43, 29] | have e2 : {z₀} ∈ compacts U := ⟨singleton_subset_iff.2 hz₀, isCompact_singleton⟩ | ι : Type u_1
l : Filter ι
U : Set ℂ
z₀ : ℂ
hU : IsOpen U
hz₀ : z₀ ∈ U
e1 : z₀ ∈ {z₀}
⊢ ContinuousOn (obs z₀) (𝓗 U) | ι : Type u_1
l : Filter ι
U : Set ℂ
z₀ : ℂ
hU : IsOpen U
hz₀ : z₀ ∈ U
e1 : z₀ ∈ {z₀}
e2 : {z₀} ∈ compacts U
⊢ ContinuousOn (obs z₀) (𝓗 U) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | ContinuousOn_obs | [38, 1] | [43, 29] | apply continuous_norm.comp_continuousOn | ι : Type u_1
l : Filter ι
U : Set ℂ
z₀ : ℂ
hU : IsOpen U
hz₀ : z₀ ∈ U
e1 : z₀ ∈ {z₀}
e2 : {z₀} ∈ compacts U
⊢ ContinuousOn (obs z₀) (𝓗 U) | ι : Type u_1
l : Filter ι
U : Set ℂ
z₀ : ℂ
hU : IsOpen U
hz₀ : z₀ ∈ U
e1 : z₀ ∈ {z₀}
e2 : {z₀} ∈ compacts U
⊢ ContinuousOn (fun x => deriv x z₀) (𝓗 U) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | ContinuousOn_obs | [38, 1] | [43, 29] | exact (UniformOnFun.uniformContinuous_eval_of_mem _ _ e1 e2).continuous.comp_continuousOn
(ContinuousOn_uderiv hU) | ι : Type u_1
l : Filter ι
U : Set ℂ
z₀ : ℂ
hU : IsOpen U
hz₀ : z₀ ∈ U
e1 : z₀ ∈ {z₀}
e2 : {z₀} ∈ compacts U
⊢ ContinuousOn (fun x => deriv x z₀) (𝓗 U) | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | main | [45, 1] | [66, 104] | obtain ⟨z₀, hz₀⟩ : U.Nonempty := good_domain.is_nonempty | ι : Type u_1
l : Filter ι
U : Set ℂ
z₀ : ℂ
inst✝ : good_domain U
⊢ ∃ f ∈ 𝓘 U, f '' U = ball 0 1 | case intro
ι : Type u_1
l : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
z₀ : ℂ
hz₀ : z₀ ∈ U
⊢ ∃ f ∈ 𝓘 U, f '' U = ball 0 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | main | [45, 1] | [66, 104] | have hU : IsOpen U := good_domain.is_open | case intro
ι : Type u_1
l : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
z₀ : ℂ
hz₀ : z₀ ∈ U
⊢ ∃ f ∈ 𝓘 U, f '' U = ball 0 1 | case intro
ι : Type u_1
l : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
z₀ : ℂ
hz₀ : z₀ ∈ U
hU : IsOpen U
⊢ ∃ f ∈ 𝓘 U, f '' U = ball 0 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | main | [45, 1] | [66, 104] | have hU' : IsPreconnected U := good_domain.is_preconnected | case intro
ι : Type u_1
l : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
z₀ : ℂ
hz₀ : z₀ ∈ U
hU : IsOpen U
⊢ ∃ f ∈ 𝓘 U, f '' U = ball 0 1 | case intro
ι : Type u_1
l : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
z₀ : ℂ
hz₀ : z₀ ∈ U
hU : IsOpen U
hU' : IsPreconnected U
⊢ ∃ f ∈ 𝓘 U, f '' U = ball 0 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | main | [45, 1] | [66, 104] | have h1 : ContinuousOn (obs z₀) (𝓙 U) := ((ContinuousOn_obs hU hz₀).mono (λ f hf => hf.1.1)) | case intro
ι : Type u_1
l : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
z₀ : ℂ
hz₀ : z₀ ∈ U
hU : IsOpen U
hU' : IsPreconnected U
⊢ ∃ f ∈ 𝓘 U, f '' U = ball 0 1 | case intro
ι : Type u_1
l : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
z₀ : ℂ
hz₀ : z₀ ∈ U
hU : IsOpen U
hU' : IsPreconnected U
h1 : ContinuousOn (obs z₀) (𝓙 U)
⊢ ∃ f ∈ 𝓘 U, f '' U = ball 0 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | main | [45, 1] | [66, 104] | obtain ⟨f, hf, hfg⟩ := IsCompact_𝓙.exists_forall_ge (𝓘_nonempty.mono 𝓘_subset_𝓙) h1 | case intro
ι : Type u_1
l : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
z₀ : ℂ
hz₀ : z₀ ∈ U
hU : IsOpen U
hU' : IsPreconnected U
h1 : ContinuousOn (obs z₀) (𝓙 U)
⊢ ∃ f ∈ 𝓘 U, f '' U = ball 0 1 | case intro.intro.intro
ι : Type u_1
l : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
z₀ : ℂ
hz₀ : z₀ ∈ U
hU : IsOpen U
hU' : IsPreconnected U
h1 : ContinuousOn (obs z₀) (𝓙 U)
f : 𝓒 U
hf : f ∈ 𝓙 U
hfg : ∀ y ∈ 𝓙 U, obs z₀ y ≤ obs z₀ f
⊢ ∃ f ∈ 𝓘 U, f '' U = ball 0 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | main | [45, 1] | [66, 104] | have h7 : ¬ ∃ w, EqOn f (λ _ => w) U := by
obtain ⟨g, hg⟩ : (𝓘 U).Nonempty := 𝓘_nonempty
specialize hfg g (𝓘_subset_𝓙 hg)
have := norm_pos_iff.1 ((norm_pos_iff.2 (deriv_ne_zero_of_inj hU hg.1.1 hg.2 hz₀)).trans_le hfg)
contrapose! this
obtain ⟨w, hw : EqOn f (λ _ => w) U⟩ := this
simpa only [deriv_const'] using (hw.eventuallyEq_of_mem (hU.mem_nhds hz₀)).deriv_eq | case intro.intro.intro
ι : Type u_1
l : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
z₀ : ℂ
hz₀ : z₀ ∈ U
hU : IsOpen U
hU' : IsPreconnected U
h1 : ContinuousOn (obs z₀) (𝓙 U)
f : 𝓒 U
hf : f ∈ 𝓙 U
hfg : ∀ y ∈ 𝓙 U, obs z₀ y ≤ obs z₀ f
⊢ ∃ f ∈ 𝓘 U, f '' U = ball 0 1 | case intro.intro.intro
ι : Type u_1
l : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
z₀ : ℂ
hz₀ : z₀ ∈ U
hU : IsOpen U
hU' : IsPreconnected U
h1 : ContinuousOn (obs z₀) (𝓙 U)
f : 𝓒 U
hf : f ∈ 𝓙 U
hfg : ∀ y ∈ 𝓙 U, obs z₀ y ≤ obs z₀ f
h7 : ¬∃ w, EqOn f (fun x => w) U
⊢ ∃ f ∈ 𝓘 U, f '' U = ball 0 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | main | [45, 1] | [66, 104] | have h5 : f ∈ 𝓘 U := ⟨hf.1, hf.2.resolve_right h7⟩ | case intro.intro.intro
ι : Type u_1
l : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
z₀ : ℂ
hz₀ : z₀ ∈ U
hU : IsOpen U
hU' : IsPreconnected U
h1 : ContinuousOn (obs z₀) (𝓙 U)
f : 𝓒 U
hf : f ∈ 𝓙 U
hfg : ∀ y ∈ 𝓙 U, obs z₀ y ≤ obs z₀ f
h7 : ¬∃ w, EqOn f (fun x => w) U
⊢ ∃ f ∈ 𝓘 U, f '' U = ball 0 1 | case intro.intro.intro
ι : Type u_1
l : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
z₀ : ℂ
hz₀ : z₀ ∈ U
hU : IsOpen U
hU' : IsPreconnected U
h1 : ContinuousOn (obs z₀) (𝓙 U)
f : 𝓒 U
hf : f ∈ 𝓙 U
hfg : ∀ y ∈ 𝓙 U, obs z₀ y ≤ obs z₀ f
h7 : ¬∃ w, EqOn f (fun x => w) U
h5 : f ∈ 𝓘 U
⊢ ∃ f ∈ 𝓘 U, f '' U = ball 0 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | main | [45, 1] | [66, 104] | refine ⟨f, h5, ?_⟩ | case intro.intro.intro
ι : Type u_1
l : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
z₀ : ℂ
hz₀ : z₀ ∈ U
hU : IsOpen U
hU' : IsPreconnected U
h1 : ContinuousOn (obs z₀) (𝓙 U)
f : 𝓒 U
hf : f ∈ 𝓙 U
hfg : ∀ y ∈ 𝓙 U, obs z₀ y ≤ obs z₀ f
h7 : ¬∃ w, EqOn f (fun x => w) U
h5 : f ∈ 𝓘 U
⊢ ∃ f ∈ 𝓘 U, f '' U = ball 0 1 | case intro.intro.intro
ι : Type u_1
l : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
z₀ : ℂ
hz₀ : z₀ ∈ U
hU : IsOpen U
hU' : IsPreconnected U
h1 : ContinuousOn (obs z₀) (𝓙 U)
f : 𝓒 U
hf : f ∈ 𝓙 U
hfg : ∀ y ∈ 𝓙 U, obs z₀ y ≤ obs z₀ f
h7 : ¬∃ w, EqOn f (fun x => w) U
h5 : f ∈ 𝓘 U
⊢ f '' U = ball 0 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | main | [45, 1] | [66, 104] | have h10 : f '' U ⊆ ball 0 1 := by
have := ((hf.1.1.analyticOn hU).is_constant_or_isOpen hU').resolve_left h7 U subset_rfl hU
simpa [interior_closedBall] using this.subset_interior_iff.2 (mapsTo'.1 hf.1.2) | case intro.intro.intro
ι : Type u_1
l : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
z₀ : ℂ
hz₀ : z₀ ∈ U
hU : IsOpen U
hU' : IsPreconnected U
h1 : ContinuousOn (obs z₀) (𝓙 U)
f : 𝓒 U
hf : f ∈ 𝓙 U
hfg : ∀ y ∈ 𝓙 U, obs z₀ y ≤ obs z₀ f
h7 : ¬∃ w, EqOn f (fun x => w) U
h5 : f ∈ 𝓘 U
⊢ f '' U = ball 0 1 | case intro.intro.intro
ι : Type u_1
l : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
z₀ : ℂ
hz₀ : z₀ ∈ U
hU : IsOpen U
hU' : IsPreconnected U
h1 : ContinuousOn (obs z₀) (𝓙 U)
f : 𝓒 U
hf : f ∈ 𝓙 U
hfg : ∀ y ∈ 𝓙 U, obs z₀ y ≤ obs z₀ f
h7 : ¬∃ w, EqOn f (fun x => w) U
h5 : f ∈ 𝓘 U
h10 : f '' U ⊆ ball 0 1
⊢ f '' U = ball 0 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | main | [45, 1] | [66, 104] | refine (subset_iff_ssubset_or_eq.1 h10).resolve_left ?_ | case intro.intro.intro
ι : Type u_1
l : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
z₀ : ℂ
hz₀ : z₀ ∈ U
hU : IsOpen U
hU' : IsPreconnected U
h1 : ContinuousOn (obs z₀) (𝓙 U)
f : 𝓒 U
hf : f ∈ 𝓙 U
hfg : ∀ y ∈ 𝓙 U, obs z₀ y ≤ obs z₀ f
h7 : ¬∃ w, EqOn f (fun x => w) U
h5 : f ∈ 𝓘 U
h10 : f '' U ⊆ ball 0 1
⊢ f '' U = ball 0 1 | case intro.intro.intro
ι : Type u_1
l : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
z₀ : ℂ
hz₀ : z₀ ∈ U
hU : IsOpen U
hU' : IsPreconnected U
h1 : ContinuousOn (obs z₀) (𝓙 U)
f : 𝓒 U
hf : f ∈ 𝓙 U
hfg : ∀ y ∈ 𝓙 U, obs z₀ y ≤ obs z₀ f
h7 : ¬∃ w, EqOn f (fun x => w) U
h5 : f ∈ 𝓘 U
h10 : f '' U ⊆ ball 0 1
⊢ ¬f '' U ⊂ ball 0 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | main | [45, 1] | [66, 104] | contrapose! hfg | case intro.intro.intro
ι : Type u_1
l : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
z₀ : ℂ
hz₀ : z₀ ∈ U
hU : IsOpen U
hU' : IsPreconnected U
h1 : ContinuousOn (obs z₀) (𝓙 U)
f : 𝓒 U
hf : f ∈ 𝓙 U
hfg : ∀ y ∈ 𝓙 U, obs z₀ y ≤ obs z₀ f
h7 : ¬∃ w, EqOn f (fun x => w) U
h5 : f ∈ 𝓘 U
h10 : f '' U ⊆ ball 0 1
⊢ ¬f '' U ⊂ ball 0 1 | case intro.intro.intro
ι : Type u_1
l : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
z₀ : ℂ
hz₀ : z₀ ∈ U
hU : IsOpen U
hU' : IsPreconnected U
h1 : ContinuousOn (obs z₀) (𝓙 U)
f : 𝓒 U
hf : f ∈ 𝓙 U
h7 : ¬∃ w, EqOn f (fun x => w) U
h5 : f ∈ 𝓘 U
h10 : f '' U ⊆ ball 0 1
hfg : f '' U ⊂ ball 0 1
⊢ ∃ y ∈ 𝓙 U, obs z₀ f < obs z₀ y |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | main | [45, 1] | [66, 104] | obtain ⟨g, hg⟩ := step_2 U hz₀ ⟨f, hf.1.1, h5.2, mapsTo'.2 h10⟩ hfg | case intro.intro.intro
ι : Type u_1
l : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
z₀ : ℂ
hz₀ : z₀ ∈ U
hU : IsOpen U
hU' : IsPreconnected U
h1 : ContinuousOn (obs z₀) (𝓙 U)
f : 𝓒 U
hf : f ∈ 𝓙 U
h7 : ¬∃ w, EqOn f (fun x => w) U
h5 : f ∈ 𝓘 U
h10 : f '' U ⊆ ball 0 1
hfg : f '' U ⊂ ball 0 1
⊢ ∃ y ∈ 𝓙 U, obs z₀ f < obs z₀ y | case intro.intro.intro.intro
ι : Type u_1
l : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
z₀ : ℂ
hz₀ : z₀ ∈ U
hU : IsOpen U
hU' : IsPreconnected U
h1 : ContinuousOn (obs z₀) (𝓙 U)
f : 𝓒 U
hf : f ∈ 𝓙 U
h7 : ¬∃ w, EqOn f (fun x => w) U
h5 : f ∈ 𝓘 U
h10 : f '' U ⊆ ball 0 1
hfg : f '' U ⊂ ball 0 1
g : embedding U 𝔻
hg : ‖deriv { to_fun := f, is_diff := ⋯, is_inj := ⋯, maps_to := ⋯ }.to_fun z₀‖ < ‖deriv g.to_fun z₀‖
⊢ ∃ y ∈ 𝓙 U, obs z₀ f < obs z₀ y |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | main | [45, 1] | [66, 104] | exact ⟨g.to_fun, 𝓘_subset_𝓙 ⟨⟨g.is_diff, g.maps_to.mono_right ball_subset_closedBall⟩, g.is_inj⟩, hg⟩ | case intro.intro.intro.intro
ι : Type u_1
l : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
z₀ : ℂ
hz₀ : z₀ ∈ U
hU : IsOpen U
hU' : IsPreconnected U
h1 : ContinuousOn (obs z₀) (𝓙 U)
f : 𝓒 U
hf : f ∈ 𝓙 U
h7 : ¬∃ w, EqOn f (fun x => w) U
h5 : f ∈ 𝓘 U
h10 : f '' U ⊆ ball 0 1
hfg : f '' U ⊂ ball 0 1
g : embedding U 𝔻
hg : ‖deriv { to_fun := f, is_diff := ⋯, is_inj := ⋯, maps_to := ⋯ }.to_fun z₀‖ < ‖deriv g.to_fun z₀‖
⊢ ∃ y ∈ 𝓙 U, obs z₀ f < obs z₀ y | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | main | [45, 1] | [66, 104] | obtain ⟨g, hg⟩ : (𝓘 U).Nonempty := 𝓘_nonempty | ι : Type u_1
l : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
z₀ : ℂ
hz₀ : z₀ ∈ U
hU : IsOpen U
hU' : IsPreconnected U
h1 : ContinuousOn (obs z₀) (𝓙 U)
f : 𝓒 U
hf : f ∈ 𝓙 U
hfg : ∀ y ∈ 𝓙 U, obs z₀ y ≤ obs z₀ f
⊢ ¬∃ w, EqOn f (fun x => w) U | case intro
ι : Type u_1
l : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
z₀ : ℂ
hz₀ : z₀ ∈ U
hU : IsOpen U
hU' : IsPreconnected U
h1 : ContinuousOn (obs z₀) (𝓙 U)
f : 𝓒 U
hf : f ∈ 𝓙 U
hfg : ∀ y ∈ 𝓙 U, obs z₀ y ≤ obs z₀ f
g : 𝓒 U
hg : g ∈ 𝓘 U
⊢ ¬∃ w, EqOn f (fun x => w) U |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | main | [45, 1] | [66, 104] | specialize hfg g (𝓘_subset_𝓙 hg) | case intro
ι : Type u_1
l : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
z₀ : ℂ
hz₀ : z₀ ∈ U
hU : IsOpen U
hU' : IsPreconnected U
h1 : ContinuousOn (obs z₀) (𝓙 U)
f : 𝓒 U
hf : f ∈ 𝓙 U
hfg : ∀ y ∈ 𝓙 U, obs z₀ y ≤ obs z₀ f
g : 𝓒 U
hg : g ∈ 𝓘 U
⊢ ¬∃ w, EqOn f (fun x => w) U | case intro
ι : Type u_1
l : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
z₀ : ℂ
hz₀ : z₀ ∈ U
hU : IsOpen U
hU' : IsPreconnected U
h1 : ContinuousOn (obs z₀) (𝓙 U)
f : 𝓒 U
hf : f ∈ 𝓙 U
g : 𝓒 U
hg : g ∈ 𝓘 U
hfg : obs z₀ g ≤ obs z₀ f
⊢ ¬∃ w, EqOn f (fun x => w) U |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | main | [45, 1] | [66, 104] | have := norm_pos_iff.1 ((norm_pos_iff.2 (deriv_ne_zero_of_inj hU hg.1.1 hg.2 hz₀)).trans_le hfg) | case intro
ι : Type u_1
l : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
z₀ : ℂ
hz₀ : z₀ ∈ U
hU : IsOpen U
hU' : IsPreconnected U
h1 : ContinuousOn (obs z₀) (𝓙 U)
f : 𝓒 U
hf : f ∈ 𝓙 U
g : 𝓒 U
hg : g ∈ 𝓘 U
hfg : obs z₀ g ≤ obs z₀ f
⊢ ¬∃ w, EqOn f (fun x => w) U | case intro
ι : Type u_1
l : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
z₀ : ℂ
hz₀ : z₀ ∈ U
hU : IsOpen U
hU' : IsPreconnected U
h1 : ContinuousOn (obs z₀) (𝓙 U)
f : 𝓒 U
hf : f ∈ 𝓙 U
g : 𝓒 U
hg : g ∈ 𝓘 U
hfg : obs z₀ g ≤ obs z₀ f
this : deriv f z₀ ≠ 0
⊢ ¬∃ w, EqOn f (fun x => w) U |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | main | [45, 1] | [66, 104] | contrapose! this | case intro
ι : Type u_1
l : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
z₀ : ℂ
hz₀ : z₀ ∈ U
hU : IsOpen U
hU' : IsPreconnected U
h1 : ContinuousOn (obs z₀) (𝓙 U)
f : 𝓒 U
hf : f ∈ 𝓙 U
g : 𝓒 U
hg : g ∈ 𝓘 U
hfg : obs z₀ g ≤ obs z₀ f
this : deriv f z₀ ≠ 0
⊢ ¬∃ w, EqOn f (fun x => w) U | case intro
ι : Type u_1
l : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
z₀ : ℂ
hz₀ : z₀ ∈ U
hU : IsOpen U
hU' : IsPreconnected U
h1 : ContinuousOn (obs z₀) (𝓙 U)
f : 𝓒 U
hf : f ∈ 𝓙 U
g : 𝓒 U
hg : g ∈ 𝓘 U
hfg : obs z₀ g ≤ obs z₀ f
this : ∃ w, EqOn f (fun x => w) U
⊢ deriv f z₀ = 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | main | [45, 1] | [66, 104] | obtain ⟨w, hw : EqOn f (λ _ => w) U⟩ := this | case intro
ι : Type u_1
l : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
z₀ : ℂ
hz₀ : z₀ ∈ U
hU : IsOpen U
hU' : IsPreconnected U
h1 : ContinuousOn (obs z₀) (𝓙 U)
f : 𝓒 U
hf : f ∈ 𝓙 U
g : 𝓒 U
hg : g ∈ 𝓘 U
hfg : obs z₀ g ≤ obs z₀ f
this : ∃ w, EqOn f (fun x => w) U
⊢ deriv f z₀ = 0 | case intro.intro
ι : Type u_1
l : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
z₀ : ℂ
hz₀ : z₀ ∈ U
hU : IsOpen U
hU' : IsPreconnected U
h1 : ContinuousOn (obs z₀) (𝓙 U)
f : 𝓒 U
hf : f ∈ 𝓙 U
g : 𝓒 U
hg : g ∈ 𝓘 U
hfg : obs z₀ g ≤ obs z₀ f
w : ℂ
hw : EqOn f (fun x => w) U
⊢ deriv f z₀ = 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | main | [45, 1] | [66, 104] | simpa only [deriv_const'] using (hw.eventuallyEq_of_mem (hU.mem_nhds hz₀)).deriv_eq | case intro.intro
ι : Type u_1
l : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
z₀ : ℂ
hz₀ : z₀ ∈ U
hU : IsOpen U
hU' : IsPreconnected U
h1 : ContinuousOn (obs z₀) (𝓙 U)
f : 𝓒 U
hf : f ∈ 𝓙 U
g : 𝓒 U
hg : g ∈ 𝓘 U
hfg : obs z₀ g ≤ obs z₀ f
w : ℂ
hw : EqOn f (fun x => w) U
⊢ deriv f z₀ = 0 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | main | [45, 1] | [66, 104] | have := ((hf.1.1.analyticOn hU).is_constant_or_isOpen hU').resolve_left h7 U subset_rfl hU | ι : Type u_1
l : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
z₀ : ℂ
hz₀ : z₀ ∈ U
hU : IsOpen U
hU' : IsPreconnected U
h1 : ContinuousOn (obs z₀) (𝓙 U)
f : 𝓒 U
hf : f ∈ 𝓙 U
hfg : ∀ y ∈ 𝓙 U, obs z₀ y ≤ obs z₀ f
h7 : ¬∃ w, EqOn f (fun x => w) U
h5 : f ∈ 𝓘 U
⊢ f '' U ⊆ ball 0 1 | ι : Type u_1
l : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
z₀ : ℂ
hz₀ : z₀ ∈ U
hU : IsOpen U
hU' : IsPreconnected U
h1 : ContinuousOn (obs z₀) (𝓙 U)
f : 𝓒 U
hf : f ∈ 𝓙 U
hfg : ∀ y ∈ 𝓙 U, obs z₀ y ≤ obs z₀ f
h7 : ¬∃ w, EqOn f (fun x => w) U
h5 : f ∈ 𝓘 U
this : IsOpen (f '' U)
⊢ f '' U ⊆ ball 0 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | main | [45, 1] | [66, 104] | simpa [interior_closedBall] using this.subset_interior_iff.2 (mapsTo'.1 hf.1.2) | ι : Type u_1
l : Filter ι
U : Set ℂ
z₀✝ : ℂ
inst✝ : good_domain U
z₀ : ℂ
hz₀ : z₀ ∈ U
hU : IsOpen U
hU' : IsPreconnected U
h1 : ContinuousOn (obs z₀) (𝓙 U)
f : 𝓒 U
hf : f ∈ 𝓙 U
hfg : ∀ y ∈ 𝓙 U, obs z₀ y ≤ obs z₀ f
h7 : ¬∃ w, EqOn f (fun x => w) U
h5 : f ∈ 𝓘 U
this : IsOpen (f '' U)
⊢ f '' U ⊆ ball 0 1 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | RMT | [68, 1] | [72, 31] | have : good_domain U := ⟨h1, h2.1, h2.2, h3, (h4.has_logs h1 h2.isPreconnected).has_sqrt⟩ | ι : Type u_1
l : Filter ι
U : Set ℂ
z₀ : ℂ
h1 : IsOpen U
h2 : IsConnected U
h3 : U ≠ univ
h4 : has_primitives U
⊢ ∃ f, DifferentiableOn ℂ f U ∧ InjOn f U ∧ f '' U = ball 0 1 | ι : Type u_1
l : Filter ι
U : Set ℂ
z₀ : ℂ
h1 : IsOpen U
h2 : IsConnected U
h3 : U ≠ univ
h4 : has_primitives U
this : good_domain U
⊢ ∃ f, DifferentiableOn ℂ f U ∧ InjOn f U ∧ f '' U = ball 0 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | RMT | [68, 1] | [72, 31] | obtain ⟨f, hf : f ∈ 𝓘 U, hfU⟩ := main (U := U) | ι : Type u_1
l : Filter ι
U : Set ℂ
z₀ : ℂ
h1 : IsOpen U
h2 : IsConnected U
h3 : U ≠ univ
h4 : has_primitives U
this : good_domain U
⊢ ∃ f, DifferentiableOn ℂ f U ∧ InjOn f U ∧ f '' U = ball 0 1 | case intro.intro
ι : Type u_1
l : Filter ι
U : Set ℂ
z₀ : ℂ
h1 : IsOpen U
h2 : IsConnected U
h3 : U ≠ univ
h4 : has_primitives U
this : good_domain U
f : 𝓒 U
hf : f ∈ 𝓘 U
hfU : f '' U = ball 0 1
⊢ ∃ f, DifferentiableOn ℂ f U ∧ InjOn f U ∧ f '' U = ball 0 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Main.lean | RMT | [68, 1] | [72, 31] | exact ⟨f, hf.1.1, hf.2, hfU⟩ | case intro.intro
ι : Type u_1
l : Filter ι
U : Set ℂ
z₀ : ℂ
h1 : IsOpen U
h2 : IsConnected U
h3 : U ≠ univ
h4 : has_primitives U
this : good_domain U
f : 𝓒 U
hf : f ∈ 𝓘 U
hfU : f '' U = ball 0 1
⊢ ∃ f, DifferentiableOn ℂ f U ∧ InjOn f U ∧ f '' U = ball 0 1 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/uniform.lean | UniformSpace.mem_thickening | [14, 1] | [15, 70] | simp only [thickening, ball, mem_iUnion, mem_preimage, exists_prop] | ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s t : Set α
x u v : Set (α × α)
⊢ a ∈ thickening u s ↔ ∃ x ∈ s, (x, a) ∈ u | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/uniform.lean | UniformSpace.thickening_singleton | [17, 1] | [18, 67] | simp only [thickening, mem_singleton_iff, iUnion_iUnion_eq_left] | ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s t : Set α
x u v : Set (α × α)
⊢ thickening u {a} = ball a u | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/uniform.lean | UniformSpace.monotone_thickening | [20, 1] | [24, 35] | intro u v huv | ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s t : Set α
x u v : Set (α × α)
⊢ Monotone fun x => thickening x s | ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s t : Set α
x u✝ v✝ u v : Set (α × α)
huv : u ≤ v
⊢ (fun x => thickening x s) u ≤ (fun x => thickening x s) v |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/uniform.lean | UniformSpace.monotone_thickening | [20, 1] | [24, 35] | apply iUnion₂_mono | ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s t : Set α
x u✝ v✝ u v : Set (α × α)
huv : u ≤ v
⊢ (fun x => thickening x s) u ≤ (fun x => thickening x s) v | case h
ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s t : Set α
x u✝ v✝ u v : Set (α × α)
huv : u ≤ v
⊢ ∀ i ∈ s, ball i u ⊆ ball i v |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/uniform.lean | UniformSpace.monotone_thickening | [20, 1] | [24, 35] | simp only [ball, le_eq_subset] at huv ⊢ | case h
ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s t : Set α
x u✝ v✝ u v : Set (α × α)
huv : u ≤ v
⊢ ∀ i ∈ s, ball i u ⊆ ball i v | case h
ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s t : Set α
x u✝ v✝ u v : Set (α × α)
huv : u ⊆ v
⊢ ∀ i ∈ s, Prod.mk i ⁻¹' u ⊆ Prod.mk i ⁻¹' v |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/uniform.lean | UniformSpace.monotone_thickening | [20, 1] | [24, 35] | exact λ _ _ => preimage_mono huv | case h
ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s t : Set α
x u✝ v✝ u v : Set (α × α)
huv : u ⊆ v
⊢ ∀ i ∈ s, Prod.mk i ⁻¹' u ⊆ Prod.mk i ⁻¹' v | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/uniform.lean | UniformSpace.thickening_comp | [29, 1] | [30, 31] | ext | ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s t : Set α
x u v : Set (α × α)
⊢ thickening v (thickening u s) = thickening (u ○ v) s | case h
ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s t : Set α
x u v : Set (α × α)
x✝ : α
⊢ x✝ ∈ thickening v (thickening u s) ↔ x✝ ∈ thickening (u ○ v) s |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/uniform.lean | UniformSpace.thickening_comp | [29, 1] | [30, 31] | simp [thickening, ball] | case h
ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s t : Set α
x u v : Set (α × α)
x✝ : α
⊢ x✝ ∈ thickening v (thickening u s) ↔ x✝ ∈ thickening (u ○ v) s | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/uniform.lean | UniformSpace.disjoint_ball_iff | [32, 1] | [34, 6] | rw [← compl_compl (ball a u), disjoint_compl_left_iff_subset] | ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s t : Set α
x u v : Set (α × α)
⊢ Disjoint (ball a u) t ↔ ∀ b ∈ t, (a, b) ∉ u | ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s t : Set α
x u v : Set (α × α)
⊢ t ⊆ (ball a u)ᶜ ↔ ∀ b ∈ t, (a, b) ∉ u |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/uniform.lean | UniformSpace.disjoint_ball_iff | [32, 1] | [34, 6] | rfl | ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s t : Set α
x u v : Set (α × α)
⊢ t ⊆ (ball a u)ᶜ ↔ ∀ b ∈ t, (a, b) ∉ u | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/uniform.lean | UniformSpace.thickening_inter_eq_empty | [36, 1] | [37, 70] | simp [thickening, ← disjoint_iff_inter_eq_empty, disjoint_ball_iff] | ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s t : Set α
x u v : Set (α × α)
⊢ thickening u s ∩ t = ∅ ↔ ∀ a ∈ s, ∀ b ∈ t, (a, b) ∉ u | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/uniform.lean | UniformSpace.thickening_inter_eq_empty_comm | [39, 1] | [42, 89] | simp [thickening_inter_eq_empty, inter_comm s] | ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s t : Set α
x u v : Set (α × α)
hu : SymmetricRel u
⊢ thickening u s ∩ t = ∅ ↔ s ∩ thickening u t = ∅ | ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s t : Set α
x u v : Set (α × α)
hu : SymmetricRel u
⊢ (∀ a ∈ s, ∀ b ∈ t, (a, b) ∉ u) ↔ ∀ a ∈ t, ∀ b ∈ s, (a, b) ∉ u |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/uniform.lean | UniformSpace.thickening_inter_eq_empty_comm | [39, 1] | [42, 89] | apply Iff.intro | ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s t : Set α
x u v : Set (α × α)
hu : SymmetricRel u
⊢ (∀ a ∈ s, ∀ b ∈ t, (a, b) ∉ u) ↔ ∀ a ∈ t, ∀ b ∈ s, (a, b) ∉ u | case mp
ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s t : Set α
x u v : Set (α × α)
hu : SymmetricRel u
⊢ (∀ a ∈ s, ∀ b ∈ t, (a, b) ∉ u) → ∀ a ∈ t, ∀ b ∈ s, (a, b) ∉ u
case mpr
ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s t : Set α
x u v : Set (α × α)
hu : SymmetricRel u
⊢ (∀ a ∈ t, ∀ b ∈ s, (a, b) ∉ u) → ∀ a ∈ s, ∀ b ∈ t, (a, b) ∉ u |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/uniform.lean | UniformSpace.thickening_inter_eq_empty_comm | [39, 1] | [42, 89] | repeat exact λ h a ha b hb hab => h b hb a ha (hu.mk_mem_comm.mp hab) | case mp
ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s t : Set α
x u v : Set (α × α)
hu : SymmetricRel u
⊢ (∀ a ∈ s, ∀ b ∈ t, (a, b) ∉ u) → ∀ a ∈ t, ∀ b ∈ s, (a, b) ∉ u
case mpr
ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s t : Set α
x u v : Set (α × α)
hu : SymmetricRel u
⊢ (∀ a ∈ t, ∀ b ∈ s, (a, b) ∉ u) → ∀ a ∈ s, ∀ b ∈ t, (a, b) ∉ u | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/uniform.lean | UniformSpace.thickening_inter_eq_empty_comm | [39, 1] | [42, 89] | exact λ h a ha b hb hab => h b hb a ha (hu.mk_mem_comm.mp hab) | case mpr
ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s t : Set α
x u v : Set (α × α)
hu : SymmetricRel u
⊢ (∀ a ∈ t, ∀ b ∈ s, (a, b) ∉ u) → ∀ a ∈ s, ∀ b ∈ t, (a, b) ∉ u | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/uniform.lean | UniformSpace.thickening_inter_thickening_eq_empty_of_comp | [44, 1] | [48, 82] | simp only [←thickening_inter_eq_empty_comm hv, thickening_comp] | ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s t : Set α
x u v : Set (α × α)
hv : SymmetricRel v
hvu : v ○ v ⊆ u
hST : thickening u s ∩ t = ∅
⊢ thickening v s ∩ thickening v t = ∅ | ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s t : Set α
x u v : Set (α × α)
hv : SymmetricRel v
hvu : v ○ v ⊆ u
hST : thickening u s ∩ t = ∅
⊢ thickening (v ○ v) s ∩ t = ∅ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/uniform.lean | UniformSpace.thickening_inter_thickening_eq_empty_of_comp | [44, 1] | [48, 82] | exact subset_eq_empty (inter_subset_inter_left _ (monotone_thickening hvu)) hST | ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s t : Set α
x u v : Set (α × α)
hv : SymmetricRel v
hvu : v ○ v ⊆ u
hST : thickening u s ∩ t = ∅
⊢ thickening (v ○ v) s ∩ t = ∅ | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/uniform.lean | UniformSpace.uniform_nhds_set_singleton | [69, 1] | [70, 73] | simp only [uniform_nhds_set, thickening_singleton, nhds_eq_uniformity] | ι : Type u_1
α : Type u_2
β : Type u_3
a✝ : α
s t : Set α
x u v : Set (α × α)
inst✝ : UniformSpace α
a : α
⊢ 𝓝ᵘ {a} = 𝓝 a | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/uniform.lean | UniformSpace.mem_uniform_nhds_set_iff | [72, 1] | [73, 42] | simp [uniform_nhds_set, mem_lift'_sets] | ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s t : Set α
x u v : Set (α × α)
inst✝ : UniformSpace α
⊢ s ∈ 𝓝ᵘ t ↔ ∃ u ∈ 𝓤 α, thickening u t ⊆ s | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/uniform.lean | UniformSpace.nhds_le_uniform_nhds_set | [75, 1] | [76, 97] | simpa [← uniform_nhds_set_singleton] using uniform_nhds_set_mono (singleton_subset_iff.mpr ha) | ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s✝ t : Set α
x u v : Set (α × α)
inst✝ : UniformSpace α
s : Set α
ha : a ∈ s
⊢ 𝓝 a ≤ 𝓝ᵘ s | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/uniform.lean | UniformSpace.nhds_set_le_uniform_nhds_set | [78, 1] | [79, 56] | simpa [nhdsSet] using λ _ => nhds_le_uniform_nhds_set | ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s✝ t : Set α
x u v : Set (α × α)
inst✝ : UniformSpace α
s : Set α
⊢ 𝓝ˢ s ≤ 𝓝ᵘ s | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/uniform.lean | UniformSpace.uniform_nhds_inf_uniform_nhds_eq_bot | [81, 1] | [87, 58] | simp_rw [inf_principal_eq_bot, inf_eq_bot_iff, mem_uniform_nhds_set_iff] at h ⊢ | ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s✝ t✝ : Set α
x u v : Set (α × α)
inst✝ : UniformSpace α
s t : Set α
h : 𝓝ᵘ s ⊓ 𝓟 t = ⊥
⊢ 𝓝ᵘ s ⊓ 𝓝ᵘ t = ⊥ | ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s✝ t✝ : Set α
x u v : Set (α × α)
inst✝ : UniformSpace α
s t : Set α
h : ∃ u ∈ 𝓤 α, thickening u s ⊆ tᶜ
⊢ ∃ U, (∃ u ∈ 𝓤 α, thickening u s ⊆ U) ∧ ∃ V, (∃ u ∈ 𝓤 α, thickening u t ⊆ V) ∧ U ∩ V = ∅ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/uniform.lean | UniformSpace.uniform_nhds_inf_uniform_nhds_eq_bot | [81, 1] | [87, 58] | obtain ⟨u, hu, hsu⟩ := h | ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s✝ t✝ : Set α
x u v : Set (α × α)
inst✝ : UniformSpace α
s t : Set α
h : ∃ u ∈ 𝓤 α, thickening u s ⊆ tᶜ
⊢ ∃ U, (∃ u ∈ 𝓤 α, thickening u s ⊆ U) ∧ ∃ V, (∃ u ∈ 𝓤 α, thickening u t ⊆ V) ∧ U ∩ V = ∅ | case intro.intro
ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s✝ t✝ : Set α
x u✝ v : Set (α × α)
inst✝ : UniformSpace α
s t : Set α
u : Set (α × α)
hu : u ∈ 𝓤 α
hsu : thickening u s ⊆ tᶜ
⊢ ∃ U, (∃ u ∈ 𝓤 α, thickening u s ⊆ U) ∧ ∃ V, (∃ u ∈ 𝓤 α, thickening u t ⊆ V) ∧ U ∩ V = ∅ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/uniform.lean | UniformSpace.uniform_nhds_inf_uniform_nhds_eq_bot | [81, 1] | [87, 58] | obtain ⟨v, hv, hvs, hvu⟩ := comp_symm_of_uniformity hu | case intro.intro
ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s✝ t✝ : Set α
x u✝ v : Set (α × α)
inst✝ : UniformSpace α
s t : Set α
u : Set (α × α)
hu : u ∈ 𝓤 α
hsu : thickening u s ⊆ tᶜ
⊢ ∃ U, (∃ u ∈ 𝓤 α, thickening u s ⊆ U) ∧ ∃ V, (∃ u ∈ 𝓤 α, thickening u t ⊆ V) ∧ U ∩ V = ∅ | case intro.intro.intro.intro.intro
ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s✝ t✝ : Set α
x u✝ v✝ : Set (α × α)
inst✝ : UniformSpace α
s t : Set α
u : Set (α × α)
hu : u ∈ 𝓤 α
hsu : thickening u s ⊆ tᶜ
v : Set (α × α)
hv : v ∈ 𝓤 α
hvs : ∀ {a b : α}, (a, b) ∈ v → (b, a) ∈ v
hvu : v ○ v ⊆ u
⊢ ∃ U, (∃ u ∈ 𝓤 α, thickening u s ⊆ U) ∧ ∃ V, (∃ u ∈ 𝓤 α, thickening u t ⊆ V) ∧ U ∩ V = ∅ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/uniform.lean | UniformSpace.uniform_nhds_inf_uniform_nhds_eq_bot | [81, 1] | [87, 58] | refine ⟨_, ⟨v, hv, subset_rfl⟩, _, ⟨v, hv, subset_rfl⟩, ?h⟩ | case intro.intro.intro.intro.intro
ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s✝ t✝ : Set α
x u✝ v✝ : Set (α × α)
inst✝ : UniformSpace α
s t : Set α
u : Set (α × α)
hu : u ∈ 𝓤 α
hsu : thickening u s ⊆ tᶜ
v : Set (α × α)
hv : v ∈ 𝓤 α
hvs : ∀ {a b : α}, (a, b) ∈ v → (b, a) ∈ v
hvu : v ○ v ⊆ u
⊢ ∃ U, (∃ u ∈ 𝓤 α, thickening u s ⊆ U) ∧ ∃ V, (∃ u ∈ 𝓤 α, thickening u t ⊆ V) ∧ U ∩ V = ∅ | case h
ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s✝ t✝ : Set α
x u✝ v✝ : Set (α × α)
inst✝ : UniformSpace α
s t : Set α
u : Set (α × α)
hu : u ∈ 𝓤 α
hsu : thickening u s ⊆ tᶜ
v : Set (α × α)
hv : v ∈ 𝓤 α
hvs : ∀ {a b : α}, (a, b) ∈ v → (b, a) ∈ v
hvu : v ○ v ⊆ u
⊢ thickening v s ∩ thickening v t = ∅ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/uniform.lean | UniformSpace.uniform_nhds_inf_uniform_nhds_eq_bot | [81, 1] | [87, 58] | apply thickening_inter_thickening_eq_empty_of_comp (symmetricRel_of hvs) hvu | case h
ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s✝ t✝ : Set α
x u✝ v✝ : Set (α × α)
inst✝ : UniformSpace α
s t : Set α
u : Set (α × α)
hu : u ∈ 𝓤 α
hsu : thickening u s ⊆ tᶜ
v : Set (α × α)
hv : v ∈ 𝓤 α
hvs : ∀ {a b : α}, (a, b) ∈ v → (b, a) ∈ v
hvu : v ○ v ⊆ u
⊢ thickening v s ∩ thickening v t = ∅ | case h
ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s✝ t✝ : Set α
x u✝ v✝ : Set (α × α)
inst✝ : UniformSpace α
s t : Set α
u : Set (α × α)
hu : u ∈ 𝓤 α
hsu : thickening u s ⊆ tᶜ
v : Set (α × α)
hv : v ∈ 𝓤 α
hvs : ∀ {a b : α}, (a, b) ∈ v → (b, a) ∈ v
hvu : v ○ v ⊆ u
⊢ thickening u s ∩ t = ∅ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/uniform.lean | UniformSpace.uniform_nhds_inf_uniform_nhds_eq_bot | [81, 1] | [87, 58] | exact (subset_compl_iff_disjoint_right.mp hsu).inter_eq | case h
ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s✝ t✝ : Set α
x u✝ v✝ : Set (α × α)
inst✝ : UniformSpace α
s t : Set α
u : Set (α × α)
hu : u ∈ 𝓤 α
hsu : thickening u s ⊆ tᶜ
v : Set (α × α)
hv : v ∈ 𝓤 α
hvs : ∀ {a b : α}, (a, b) ∈ v → (b, a) ∈ v
hvu : v ○ v ⊆ u
⊢ thickening u s ∩ t = ∅ | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/uniform.lean | UniformSpace.nhds_inf_uniform_nhds_eq_bot | [89, 1] | [91, 48] | rw [← uniform_nhds_set_singleton] at hf ⊢ | ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s✝ t : Set α
x u v : Set (α × α)
inst✝ : UniformSpace α
s : Set α
hf : 𝓝 a ⊓ 𝓟 s = ⊥
⊢ 𝓝 a ⊓ 𝓝ᵘ s = ⊥ | ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s✝ t : Set α
x u v : Set (α × α)
inst✝ : UniformSpace α
s : Set α
hf : 𝓝ᵘ {a} ⊓ 𝓟 s = ⊥
⊢ 𝓝ᵘ {a} ⊓ 𝓝ᵘ s = ⊥ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/uniform.lean | UniformSpace.nhds_inf_uniform_nhds_eq_bot | [89, 1] | [91, 48] | exact uniform_nhds_inf_uniform_nhds_eq_bot hf | ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s✝ t : Set α
x u v : Set (α × α)
inst✝ : UniformSpace α
s : Set α
hf : 𝓝ᵘ {a} ⊓ 𝓟 s = ⊥
⊢ 𝓝ᵘ {a} ⊓ 𝓝ᵘ s = ⊥ | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/uniform.lean | lemma0 | [104, 1] | [106, 77] | simp_rw [comap_principal, uniform_nhds_set, tendsto_lift', eventually_inf_principal] | ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s t : Set α
x u v : Set (α × α)
p : Filter ι
inst✝ : UniformSpace α
⊢ Tendsto Prod.snd (𝓤 α ⊓ Filter.comap Prod.fst (𝓟 s)) (𝓝ᵘ s) | ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s t : Set α
x u v : Set (α × α)
p : Filter ι
inst✝ : UniformSpace α
⊢ ∀ s_1 ∈ 𝓤 α, ∀ᶠ (x : α × α) in 𝓤 α, x ∈ Prod.fst ⁻¹' s → x.2 ∈ thickening s_1 s |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/uniform.lean | lemma0 | [104, 1] | [106, 77] | exact λ U hU => mem_of_superset hU (λ ⟨x, y⟩ hxy hx => mem_biUnion hx hxy) | ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s t : Set α
x u v : Set (α × α)
p : Filter ι
inst✝ : UniformSpace α
⊢ ∀ s_1 ∈ 𝓤 α, ∀ᶠ (x : α × α) in 𝓤 α, x ∈ Prod.fst ⁻¹' s → x.2 ∈ thickening s_1 s | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/uniform.lean | lemma1 | [111, 1] | [115, 73] | rw [tendstoUniformlyOn_iff_tendsto] at hF | ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s t : Set α
x u v : Set (α × α)
p : Filter ι
F : ι → α → β
f : α → β
inst✝ : UniformSpace β
hF : TendstoUniformlyOn F f p s
⊢ Tendsto (fun q => (f q.2, F q.1 q.2)) (p ×ˢ 𝓟 s) (𝓟 (f '' s) ×ˢ 𝓝ᵘ (f '' s)) | ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s t : Set α
x u v : Set (α × α)
p : Filter ι
F : ι → α → β
f : α → β
inst✝ : UniformSpace β
hF : Tendsto (fun q => (f q.2, F q.1 q.2)) (p ×ˢ 𝓟 s) (𝓤 β)
⊢ Tendsto (fun q => (f q.2, F q.1 q.2)) (p ×ˢ 𝓟 s) (𝓟 (f '' s) ×ˢ 𝓝ᵘ (f '' s)) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/uniform.lean | lemma1 | [111, 1] | [115, 73] | refine tendsto_prod_iff'.mpr ⟨lemma2, ?h⟩ | ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s t : Set α
x u v : Set (α × α)
p : Filter ι
F : ι → α → β
f : α → β
inst✝ : UniformSpace β
hF : Tendsto (fun q => (f q.2, F q.1 q.2)) (p ×ˢ 𝓟 s) (𝓤 β)
⊢ Tendsto (fun q => (f q.2, F q.1 q.2)) (p ×ˢ 𝓟 s) (𝓟 (f '' s) ×ˢ 𝓝ᵘ (f '' s)) | case h
ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s t : Set α
x u v : Set (α × α)
p : Filter ι
F : ι → α → β
f : α → β
inst✝ : UniformSpace β
hF : Tendsto (fun q => (f q.2, F q.1 q.2)) (p ×ˢ 𝓟 s) (𝓤 β)
⊢ Tendsto (fun n => (f n.2, F n.1 n.2).2) (p ×ˢ 𝓟 s) (𝓝ᵘ (f '' s)) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/uniform.lean | lemma1 | [111, 1] | [115, 73] | exact lemma0.comp (tendsto_inf.mpr ⟨hF, tendsto_comap_iff.mpr lemma2⟩) | case h
ι : Type u_1
α : Type u_2
β : Type u_3
a : α
s t : Set α
x u v : Set (α × α)
p : Filter ι
F : ι → α → β
f : α → β
inst✝ : UniformSpace β
hF : Tendsto (fun q => (f q.2, F q.1 q.2)) (p ×ˢ 𝓟 s) (𝓤 β)
⊢ Tendsto (fun n => (f n.2, F n.1 n.2).2) (p ×ˢ 𝓟 s) (𝓝ᵘ (f '' s)) | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | HasFPowerSeriesAt.eventually_differentiable_at | [19, 1] | [22, 91] | obtain ⟨r, hp⟩ := hp | E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
p : FormalMultilinearSeries ℂ ℂ E
U : Set ℂ
f : ℂ → E
z₀ : ℂ
hp : HasFPowerSeriesAt f p z₀
⊢ ∀ᶠ (z : ℂ) in 𝓝 z₀, DifferentiableAt ℂ f z | case intro
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
p : FormalMultilinearSeries ℂ ℂ E
U : Set ℂ
f : ℂ → E
z₀ : ℂ
r : ENNReal
hp : HasFPowerSeriesOnBall f p z₀ r
⊢ ∀ᶠ (z : ℂ) in 𝓝 z₀, DifferentiableAt ℂ f z |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | HasFPowerSeriesAt.eventually_differentiable_at | [19, 1] | [22, 91] | exact hp.differentiableOn.eventually_differentiableAt (EMetric.ball_mem_nhds _ hp.r_pos) | case intro
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
p : FormalMultilinearSeries ℂ ℂ E
U : Set ℂ
f : ℂ → E
z₀ : ℂ
r : ENNReal
hp : HasFPowerSeriesOnBall f p z₀ r
⊢ ∀ᶠ (z : ℂ) in 𝓝 z₀, DifferentiableAt ℂ f z | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | circleIntegral.integral_add | [31, 1] | [33, 84] | simp only [circleIntegral, smul_add, intervalIntegral.integral_add hf.out hg.out] | E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f g : ℂ → E
c : ℂ
R : ℝ
hf : CircleIntegrable f c R
hg : CircleIntegrable g c R
⊢ (∮ (z : ℂ) in C(c, R), f z + g z) = (∮ (z : ℂ) in C(c, R), f z) + ∮ (z : ℂ) in C(c, R), g z | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | circle_integral_sub_center_inv_smul | [49, 1] | [51, 72] | simp [circleIntegral.integral_sub_inv_of_mem_ball (mem_ball_self hr)] | E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f g : ℂ → E
r : ℝ
U : Set ℂ
c : ℂ
v : E
hr : 0 < r
⊢ (∮ (z : ℂ) in C(c, r), (z - c)⁻¹ • v) = (2 * ↑π * I) • v | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | DifferentiableOn.iterate_dslope | [60, 1] | [64, 83] | induction n generalizing f | E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
p : FormalMultilinearSeries ℂ ℂ E
U : Set ℂ
z₀ c : ℂ
n : ℕ
hf : DifferentiableOn ℂ f U
hU : IsOpen U
hc : c ∈ U
⊢ DifferentiableOn ℂ ((swap dslope c)^[n] f) U | case zero
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
p : FormalMultilinearSeries ℂ ℂ E
U : Set ℂ
z₀ c : ℂ
hU : IsOpen U
hc : c ∈ U
f : ℂ → E
hf : DifferentiableOn ℂ f U
⊢ DifferentiableOn ℂ ((swap dslope c)^[zero] f) U
case succ
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
p : FormalMultilinearSeries ℂ ℂ E
U : Set ℂ
z₀ c : ℂ
hU : IsOpen U
hc : c ∈ U
n✝ : ℕ
n_ih✝ : ∀ {f : ℂ → E}, DifferentiableOn ℂ f U → DifferentiableOn ℂ ((swap dslope c)^[n✝] f) U
f : ℂ → E
hf : DifferentiableOn ℂ f U
⊢ DifferentiableOn ℂ ((swap dslope c)^[succ n✝] f) U |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | DifferentiableOn.iterate_dslope | [60, 1] | [64, 83] | case zero => exact hf | E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
p : FormalMultilinearSeries ℂ ℂ E
U : Set ℂ
z₀ c : ℂ
hU : IsOpen U
hc : c ∈ U
f : ℂ → E
hf : DifferentiableOn ℂ f U
⊢ DifferentiableOn ℂ ((swap dslope c)^[zero] f) U | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | DifferentiableOn.iterate_dslope | [60, 1] | [64, 83] | case succ n_ih => exact n_ih ((differentiableOn_dslope (hU.mem_nhds hc)).mpr hf) | E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
p : FormalMultilinearSeries ℂ ℂ E
U : Set ℂ
z₀ c : ℂ
hU : IsOpen U
hc : c ∈ U
n✝ : ℕ
n_ih : ∀ {f : ℂ → E}, DifferentiableOn ℂ f U → DifferentiableOn ℂ ((swap dslope c)^[n✝] f) U
f : ℂ → E
hf : DifferentiableOn ℂ f U
⊢ DifferentiableOn ℂ ((swap dslope c)^[succ n✝] f) U | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | DifferentiableOn.iterate_dslope | [60, 1] | [64, 83] | exact hf | E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
p : FormalMultilinearSeries ℂ ℂ E
U : Set ℂ
z₀ c : ℂ
hU : IsOpen U
hc : c ∈ U
f : ℂ → E
hf : DifferentiableOn ℂ f U
⊢ DifferentiableOn ℂ ((swap dslope c)^[zero] f) U | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/cindex.lean | DifferentiableOn.iterate_dslope | [60, 1] | [64, 83] | exact n_ih ((differentiableOn_dslope (hU.mem_nhds hc)).mpr hf) | E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
p : FormalMultilinearSeries ℂ ℂ E
U : Set ℂ
z₀ c : ℂ
hU : IsOpen U
hc : c ∈ U
n✝ : ℕ
n_ih : ∀ {f : ℂ → E}, DifferentiableOn ℂ f U → DifferentiableOn ℂ ((swap dslope c)^[n✝] f) U
f : ℂ → E
hf : DifferentiableOn ℂ f U
⊢ DifferentiableOn ℂ ((swap dslope c)^[succ n✝] f) U | no goals |
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