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/Montel.lean | UniformlyBoundedOn.equicontinuousOn | [32, 1] | [58, 40] | refine ⟨δ ⊓ ε / M, gt_iff_lt.2 (lt_inf_iff.2 ⟨hδ, div_pos hε hMp⟩), λ w hw i => ?_⟩ | case mk.intro.intro.intro.intro
ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
δ : ℝ
hδ : 0 < δ
h : closedBall z δ ⊆ U
M : ℝ
hMp : M > 0
hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M)
ε : ℝ
hε : ε > 0
⊢ ∃ δ > 0,
∀ (x : ↑K),
dist x { val := z, property := hz } < δ →
∀ (i : ι), dist ((restrict K ∘ F) i { val := z, property := hz }) ((restrict K ∘ F) i x) < ε | case mk.intro.intro.intro.intro
ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
δ : ℝ
hδ : 0 < δ
h : closedBall z δ ⊆ U
M : ℝ
hMp : M > 0
hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M)
ε : ℝ
hε : ε > 0
w : ↑K
hw : dist w { val := z, property := hz } < δ ⊓ ε / M
i : ι
⊢ dist ((restrict K ∘ F) i { val := z, property := hz }) ((restrict K ∘ F) i w) < ε |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.equicontinuousOn | [32, 1] | [58, 40] | simp | case mk.intro.intro.intro.intro
ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
δ : ℝ
hδ : 0 < δ
h : closedBall z δ ⊆ U
M : ℝ
hMp : M > 0
hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M)
ε : ℝ
hε : ε > 0
w : ↑K
hw : dist w { val := z, property := hz } < δ ⊓ ε / M
i : ι
⊢ dist ((restrict K ∘ F) i { val := z, property := hz }) ((restrict K ∘ F) i w) < ε | case mk.intro.intro.intro.intro
ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
δ : ℝ
hδ : 0 < δ
h : closedBall z δ ⊆ U
M : ℝ
hMp : M > 0
hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M)
ε : ℝ
hε : ε > 0
w : ↑K
hw : dist w { val := z, property := hz } < δ ⊓ ε / M
i : ι
⊢ dist (F i z) (F i ↑w) < ε |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.equicontinuousOn | [32, 1] | [58, 40] | have e1 : ∀ x ∈ closedBall z δ, DifferentiableAt ℂ (F i) x :=
λ x hx => (h2 i).differentiableAt (hU.mem_nhds (h hx)) | case mk.intro.intro.intro.intro
ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
δ : ℝ
hδ : 0 < δ
h : closedBall z δ ⊆ U
M : ℝ
hMp : M > 0
hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M)
ε : ℝ
hε : ε > 0
w : ↑K
hw : dist w { val := z, property := hz } < δ ⊓ ε / M
i : ι
⊢ dist (F i z) (F i ↑w) < ε | case mk.intro.intro.intro.intro
ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
δ : ℝ
hδ : 0 < δ
h : closedBall z δ ⊆ U
M : ℝ
hMp : M > 0
hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M)
ε : ℝ
hε : ε > 0
w : ↑K
hw : dist w { val := z, property := hz } < δ ⊓ ε / M
i : ι
e1 : ∀ x ∈ closedBall z δ, DifferentiableAt ℂ (F i) x
⊢ dist (F i z) (F i ↑w) < ε |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.equicontinuousOn | [32, 1] | [58, 40] | have e2 : ∀ x ∈ closedBall z δ, ‖_root_.deriv (F i) x‖ ≤ M := by simpa [MapsTo] using hM i | case mk.intro.intro.intro.intro
ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
δ : ℝ
hδ : 0 < δ
h : closedBall z δ ⊆ U
M : ℝ
hMp : M > 0
hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M)
ε : ℝ
hε : ε > 0
w : ↑K
hw : dist w { val := z, property := hz } < δ ⊓ ε / M
i : ι
e1 : ∀ x ∈ closedBall z δ, DifferentiableAt ℂ (F i) x
⊢ dist (F i z) (F i ↑w) < ε | case mk.intro.intro.intro.intro
ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
δ : ℝ
hδ : 0 < δ
h : closedBall z δ ⊆ U
M : ℝ
hMp : M > 0
hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M)
ε : ℝ
hε : ε > 0
w : ↑K
hw : dist w { val := z, property := hz } < δ ⊓ ε / M
i : ι
e1 : ∀ x ∈ closedBall z δ, DifferentiableAt ℂ (F i) x
e2 : ∀ x ∈ closedBall z δ, ‖_root_.deriv (F i) x‖ ≤ M
⊢ dist (F i z) (F i ↑w) < ε |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.equicontinuousOn | [32, 1] | [58, 40] | have e3 : z ∈ closedBall z δ := mem_closedBall_self hδ.le | case mk.intro.intro.intro.intro
ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
δ : ℝ
hδ : 0 < δ
h : closedBall z δ ⊆ U
M : ℝ
hMp : M > 0
hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M)
ε : ℝ
hε : ε > 0
w : ↑K
hw : dist w { val := z, property := hz } < δ ⊓ ε / M
i : ι
e1 : ∀ x ∈ closedBall z δ, DifferentiableAt ℂ (F i) x
e2 : ∀ x ∈ closedBall z δ, ‖_root_.deriv (F i) x‖ ≤ M
⊢ dist (F i z) (F i ↑w) < ε | case mk.intro.intro.intro.intro
ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
δ : ℝ
hδ : 0 < δ
h : closedBall z δ ⊆ U
M : ℝ
hMp : M > 0
hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M)
ε : ℝ
hε : ε > 0
w : ↑K
hw : dist w { val := z, property := hz } < δ ⊓ ε / M
i : ι
e1 : ∀ x ∈ closedBall z δ, DifferentiableAt ℂ (F i) x
e2 : ∀ x ∈ closedBall z δ, ‖_root_.deriv (F i) x‖ ≤ M
e3 : z ∈ closedBall z δ
⊢ dist (F i z) (F i ↑w) < ε |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.equicontinuousOn | [32, 1] | [58, 40] | have e4 : w.1 ∈ closedBall z δ := by simpa using (lt_inf_iff.1 hw).1.le | case mk.intro.intro.intro.intro
ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
δ : ℝ
hδ : 0 < δ
h : closedBall z δ ⊆ U
M : ℝ
hMp : M > 0
hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M)
ε : ℝ
hε : ε > 0
w : ↑K
hw : dist w { val := z, property := hz } < δ ⊓ ε / M
i : ι
e1 : ∀ x ∈ closedBall z δ, DifferentiableAt ℂ (F i) x
e2 : ∀ x ∈ closedBall z δ, ‖_root_.deriv (F i) x‖ ≤ M
e3 : z ∈ closedBall z δ
⊢ dist (F i z) (F i ↑w) < ε | case mk.intro.intro.intro.intro
ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
δ : ℝ
hδ : 0 < δ
h : closedBall z δ ⊆ U
M : ℝ
hMp : M > 0
hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M)
ε : ℝ
hε : ε > 0
w : ↑K
hw : dist w { val := z, property := hz } < δ ⊓ ε / M
i : ι
e1 : ∀ x ∈ closedBall z δ, DifferentiableAt ℂ (F i) x
e2 : ∀ x ∈ closedBall z δ, ‖_root_.deriv (F i) x‖ ≤ M
e3 : z ∈ closedBall z δ
e4 : ↑w ∈ closedBall z δ
⊢ dist (F i z) (F i ↑w) < ε |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.equicontinuousOn | [32, 1] | [58, 40] | rw [dist_eq_norm] | case mk.intro.intro.intro.intro
ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
δ : ℝ
hδ : 0 < δ
h : closedBall z δ ⊆ U
M : ℝ
hMp : M > 0
hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M)
ε : ℝ
hε : ε > 0
w : ↑K
hw : dist w { val := z, property := hz } < δ ⊓ ε / M
i : ι
e1 : ∀ x ∈ closedBall z δ, DifferentiableAt ℂ (F i) x
e2 : ∀ x ∈ closedBall z δ, ‖_root_.deriv (F i) x‖ ≤ M
e3 : z ∈ closedBall z δ
e4 : ↑w ∈ closedBall z δ
⊢ dist (F i z) (F i ↑w) < ε | case mk.intro.intro.intro.intro
ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
δ : ℝ
hδ : 0 < δ
h : closedBall z δ ⊆ U
M : ℝ
hMp : M > 0
hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M)
ε : ℝ
hε : ε > 0
w : ↑K
hw : dist w { val := z, property := hz } < δ ⊓ ε / M
i : ι
e1 : ∀ x ∈ closedBall z δ, DifferentiableAt ℂ (F i) x
e2 : ∀ x ∈ closedBall z δ, ‖_root_.deriv (F i) x‖ ≤ M
e3 : z ∈ closedBall z δ
e4 : ↑w ∈ closedBall z δ
⊢ ‖F i z - F i ↑w‖ < ε |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.equicontinuousOn | [32, 1] | [58, 40] | refine ((convex_closedBall _ _).norm_image_sub_le_of_norm_deriv_le e1 e2 e4 e3).trans_lt ?_ | case mk.intro.intro.intro.intro
ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
δ : ℝ
hδ : 0 < δ
h : closedBall z δ ⊆ U
M : ℝ
hMp : M > 0
hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M)
ε : ℝ
hε : ε > 0
w : ↑K
hw : dist w { val := z, property := hz } < δ ⊓ ε / M
i : ι
e1 : ∀ x ∈ closedBall z δ, DifferentiableAt ℂ (F i) x
e2 : ∀ x ∈ closedBall z δ, ‖_root_.deriv (F i) x‖ ≤ M
e3 : z ∈ closedBall z δ
e4 : ↑w ∈ closedBall z δ
⊢ ‖F i z - F i ↑w‖ < ε | case mk.intro.intro.intro.intro
ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
δ : ℝ
hδ : 0 < δ
h : closedBall z δ ⊆ U
M : ℝ
hMp : M > 0
hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M)
ε : ℝ
hε : ε > 0
w : ↑K
hw : dist w { val := z, property := hz } < δ ⊓ ε / M
i : ι
e1 : ∀ x ∈ closedBall z δ, DifferentiableAt ℂ (F i) x
e2 : ∀ x ∈ closedBall z δ, ‖_root_.deriv (F i) x‖ ≤ M
e3 : z ∈ closedBall z δ
e4 : ↑w ∈ closedBall z δ
⊢ M * ‖z - ↑w‖ < ε |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.equicontinuousOn | [32, 1] | [58, 40] | have : ‖z - w.val‖ < ε / M := by
have := (lt_inf_iff.1 hw).2
rwa [dist_comm, Subtype.dist_eq, dist_eq_norm] at this | case mk.intro.intro.intro.intro
ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
δ : ℝ
hδ : 0 < δ
h : closedBall z δ ⊆ U
M : ℝ
hMp : M > 0
hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M)
ε : ℝ
hε : ε > 0
w : ↑K
hw : dist w { val := z, property := hz } < δ ⊓ ε / M
i : ι
e1 : ∀ x ∈ closedBall z δ, DifferentiableAt ℂ (F i) x
e2 : ∀ x ∈ closedBall z δ, ‖_root_.deriv (F i) x‖ ≤ M
e3 : z ∈ closedBall z δ
e4 : ↑w ∈ closedBall z δ
⊢ M * ‖z - ↑w‖ < ε | case mk.intro.intro.intro.intro
ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
δ : ℝ
hδ : 0 < δ
h : closedBall z δ ⊆ U
M : ℝ
hMp : M > 0
hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M)
ε : ℝ
hε : ε > 0
w : ↑K
hw : dist w { val := z, property := hz } < δ ⊓ ε / M
i : ι
e1 : ∀ x ∈ closedBall z δ, DifferentiableAt ℂ (F i) x
e2 : ∀ x ∈ closedBall z δ, ‖_root_.deriv (F i) x‖ ≤ M
e3 : z ∈ closedBall z δ
e4 : ↑w ∈ closedBall z δ
this : ‖z - ↑w‖ < ε / M
⊢ M * ‖z - ↑w‖ < ε |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.equicontinuousOn | [32, 1] | [58, 40] | convert mul_lt_mul' le_rfl this (norm_nonneg _) hMp | case mk.intro.intro.intro.intro
ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
δ : ℝ
hδ : 0 < δ
h : closedBall z δ ⊆ U
M : ℝ
hMp : M > 0
hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M)
ε : ℝ
hε : ε > 0
w : ↑K
hw : dist w { val := z, property := hz } < δ ⊓ ε / M
i : ι
e1 : ∀ x ∈ closedBall z δ, DifferentiableAt ℂ (F i) x
e2 : ∀ x ∈ closedBall z δ, ‖_root_.deriv (F i) x‖ ≤ M
e3 : z ∈ closedBall z δ
e4 : ↑w ∈ closedBall z δ
this : ‖z - ↑w‖ < ε / M
⊢ M * ‖z - ↑w‖ < ε | case h.e'_4
ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
δ : ℝ
hδ : 0 < δ
h : closedBall z δ ⊆ U
M : ℝ
hMp : M > 0
hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M)
ε : ℝ
hε : ε > 0
w : ↑K
hw : dist w { val := z, property := hz } < δ ⊓ ε / M
i : ι
e1 : ∀ x ∈ closedBall z δ, DifferentiableAt ℂ (F i) x
e2 : ∀ x ∈ closedBall z δ, ‖_root_.deriv (F i) x‖ ≤ M
e3 : z ∈ closedBall z δ
e4 : ↑w ∈ closedBall z δ
this : ‖z - ↑w‖ < ε / M
⊢ ε = M * (ε / M) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.equicontinuousOn | [32, 1] | [58, 40] | field_simp [hMp.lt.ne.symm, mul_comm] | case h.e'_4
ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
δ : ℝ
hδ : 0 < δ
h : closedBall z δ ⊆ U
M : ℝ
hMp : M > 0
hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M)
ε : ℝ
hε : ε > 0
w : ↑K
hw : dist w { val := z, property := hz } < δ ⊓ ε / M
i : ι
e1 : ∀ x ∈ closedBall z δ, DifferentiableAt ℂ (F i) x
e2 : ∀ x ∈ closedBall z δ, ‖_root_.deriv (F i) x‖ ≤ M
e3 : z ∈ closedBall z δ
e4 : ↑w ∈ closedBall z δ
this : ‖z - ↑w‖ < ε / M
⊢ ε = M * (ε / M) | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.equicontinuousOn | [32, 1] | [58, 40] | obtain ⟨Q, hQ1, hQ2⟩ := h1.deriv hU h2 (closedBall z δ) ⟨h, isCompact_closedBall _ _⟩ | ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
δ : ℝ
hδ : 0 < δ
h : closedBall z δ ⊆ U
⊢ ∃ M > 0, ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M) | case intro.intro
ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q✝ : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
δ : ℝ
hδ : 0 < δ
h : closedBall z δ ⊆ U
Q : Set ℂ
hQ1 : IsCompact Q
hQ2 : ∀ (i : ι), MapsTo ((_root_.deriv ∘ F) i) (closedBall z δ) Q
⊢ ∃ M > 0, ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.equicontinuousOn | [32, 1] | [58, 40] | obtain ⟨M, hM⟩ := hQ1.isBounded.subset_closedBall 0 | case intro.intro
ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q✝ : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
δ : ℝ
hδ : 0 < δ
h : closedBall z δ ⊆ U
Q : Set ℂ
hQ1 : IsCompact Q
hQ2 : ∀ (i : ι), MapsTo ((_root_.deriv ∘ F) i) (closedBall z δ) Q
⊢ ∃ M > 0, ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M) | case intro.intro.intro
ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q✝ : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
δ : ℝ
hδ : 0 < δ
h : closedBall z δ ⊆ U
Q : Set ℂ
hQ1 : IsCompact Q
hQ2 : ∀ (i : ι), MapsTo ((_root_.deriv ∘ F) i) (closedBall z δ) Q
M : ℝ
hM : Q ⊆ closedBall 0 M
⊢ ∃ M > 0, ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.equicontinuousOn | [32, 1] | [58, 40] | refine ⟨M ⊔ 1, by simp, fun i => ?_⟩ | case intro.intro.intro
ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q✝ : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
δ : ℝ
hδ : 0 < δ
h : closedBall z δ ⊆ U
Q : Set ℂ
hQ1 : IsCompact Q
hQ2 : ∀ (i : ι), MapsTo ((_root_.deriv ∘ F) i) (closedBall z δ) Q
M : ℝ
hM : Q ⊆ closedBall 0 M
⊢ ∃ M > 0, ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M) | case intro.intro.intro
ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q✝ : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
δ : ℝ
hδ : 0 < δ
h : closedBall z δ ⊆ U
Q : Set ℂ
hQ1 : IsCompact Q
hQ2 : ∀ (i : ι), MapsTo ((_root_.deriv ∘ F) i) (closedBall z δ) Q
M : ℝ
hM : Q ⊆ closedBall 0 M
i : ι
⊢ MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 (M ⊔ 1)) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.equicontinuousOn | [32, 1] | [58, 40] | exact ((hQ2 i).mono_right hM).mono_right <| closedBall_subset_closedBall le_sup_left | case intro.intro.intro
ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q✝ : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
δ : ℝ
hδ : 0 < δ
h : closedBall z δ ⊆ U
Q : Set ℂ
hQ1 : IsCompact Q
hQ2 : ∀ (i : ι), MapsTo ((_root_.deriv ∘ F) i) (closedBall z δ) Q
M : ℝ
hM : Q ⊆ closedBall 0 M
i : ι
⊢ MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 (M ⊔ 1)) | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.equicontinuousOn | [32, 1] | [58, 40] | simp | ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q✝ : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
δ : ℝ
hδ : 0 < δ
h : closedBall z δ ⊆ U
Q : Set ℂ
hQ1 : IsCompact Q
hQ2 : ∀ (i : ι), MapsTo ((_root_.deriv ∘ F) i) (closedBall z δ) Q
M : ℝ
hM : Q ⊆ closedBall 0 M
⊢ M ⊔ 1 > 0 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.equicontinuousOn | [32, 1] | [58, 40] | simpa [MapsTo] using hM i | ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
δ : ℝ
hδ : 0 < δ
h : closedBall z δ ⊆ U
M : ℝ
hMp : M > 0
hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M)
ε : ℝ
hε : ε > 0
w : ↑K
hw : dist w { val := z, property := hz } < δ ⊓ ε / M
i : ι
e1 : ∀ x ∈ closedBall z δ, DifferentiableAt ℂ (F i) x
⊢ ∀ x ∈ closedBall z δ, ‖_root_.deriv (F i) x‖ ≤ M | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.equicontinuousOn | [32, 1] | [58, 40] | simpa using (lt_inf_iff.1 hw).1.le | ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
δ : ℝ
hδ : 0 < δ
h : closedBall z δ ⊆ U
M : ℝ
hMp : M > 0
hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M)
ε : ℝ
hε : ε > 0
w : ↑K
hw : dist w { val := z, property := hz } < δ ⊓ ε / M
i : ι
e1 : ∀ x ∈ closedBall z δ, DifferentiableAt ℂ (F i) x
e2 : ∀ x ∈ closedBall z δ, ‖_root_.deriv (F i) x‖ ≤ M
e3 : z ∈ closedBall z δ
⊢ ↑w ∈ closedBall z δ | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.equicontinuousOn | [32, 1] | [58, 40] | have := (lt_inf_iff.1 hw).2 | ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
δ : ℝ
hδ : 0 < δ
h : closedBall z δ ⊆ U
M : ℝ
hMp : M > 0
hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M)
ε : ℝ
hε : ε > 0
w : ↑K
hw : dist w { val := z, property := hz } < δ ⊓ ε / M
i : ι
e1 : ∀ x ∈ closedBall z δ, DifferentiableAt ℂ (F i) x
e2 : ∀ x ∈ closedBall z δ, ‖_root_.deriv (F i) x‖ ≤ M
e3 : z ∈ closedBall z δ
e4 : ↑w ∈ closedBall z δ
⊢ ‖z - ↑w‖ < ε / M | ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
δ : ℝ
hδ : 0 < δ
h : closedBall z δ ⊆ U
M : ℝ
hMp : M > 0
hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M)
ε : ℝ
hε : ε > 0
w : ↑K
hw : dist w { val := z, property := hz } < δ ⊓ ε / M
i : ι
e1 : ∀ x ∈ closedBall z δ, DifferentiableAt ℂ (F i) x
e2 : ∀ x ∈ closedBall z δ, ‖_root_.deriv (F i) x‖ ≤ M
e3 : z ∈ closedBall z δ
e4 : ↑w ∈ closedBall z δ
this : dist w { val := z, property := hz } < ε / M
⊢ ‖z - ↑w‖ < ε / M |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.equicontinuousOn | [32, 1] | [58, 40] | rwa [dist_comm, Subtype.dist_eq, dist_eq_norm] at this | ι : Type u_1
U K : Set ℂ
z✝ : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
hK : K ∈ compacts U
z : ℂ
hz : z ∈ K
δ : ℝ
hδ : 0 < δ
h : closedBall z δ ⊆ U
M : ℝ
hMp : M > 0
hM : ∀ (i : ι), MapsTo (_root_.deriv (F i)) (closedBall z δ) (closedBall 0 M)
ε : ℝ
hε : ε > 0
w : ↑K
hw : dist w { val := z, property := hz } < δ ⊓ ε / M
i : ι
e1 : ∀ x ∈ closedBall z δ, DifferentiableAt ℂ (F i) x
e2 : ∀ x ∈ closedBall z δ, ‖_root_.deriv (F i) x‖ ≤ M
e3 : z ∈ closedBall z δ
e4 : ↑w ∈ closedBall z δ
this : dist w { val := z, property := hz } < ε / M
⊢ ‖z - ↑w‖ < ε / M | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | uniformlyBoundedOn_𝓑 | [60, 1] | [62, 56] | exact fun K hK => ⟨Q K, hQ K hK, fun f => f.2.2 K hK⟩ | ι : Type u_1
U K : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
hQ : ∀ K ∈ compacts U, IsCompact (Q K)
⊢ UniformlyBoundedOn Subtype.val U | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | isCompact_𝓑 | [64, 1] | [73, 59] | have l1 (K) (hK : K ∈ compacts U) : EquicontinuousOn ((↑) : 𝓑 U Q → 𝓒 U) K :=
(uniformlyBoundedOn_𝓑 hQ).equicontinuousOn hU (fun f => f.2.1) hK | ι : Type u_1
U K : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
hU : IsOpen U
hQ : ∀ K ∈ compacts U, IsCompact (Q K)
⊢ IsCompact (𝓑 U Q) | ι : Type u_1
U K : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
hU : IsOpen U
hQ : ∀ K ∈ compacts U, IsCompact (Q K)
l1 : ∀ K ∈ compacts U, EquicontinuousOn Subtype.val K
⊢ IsCompact (𝓑 U Q) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | isCompact_𝓑 | [64, 1] | [73, 59] | have l2 (K) (hK : K ∈ compacts U) (x) (hx : x ∈ K) : ∃ L, IsCompact L ∧ ∀ i : 𝓑 U Q, i.1 x ∈ L :=
⟨Q K, hQ K hK, fun f => f.2.2 K hK hx⟩ | ι : Type u_1
U K : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
hU : IsOpen U
hQ : ∀ K ∈ compacts U, IsCompact (Q K)
l1 : ∀ K ∈ compacts U, EquicontinuousOn Subtype.val K
⊢ IsCompact (𝓑 U Q) | ι : Type u_1
U K : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
hU : IsOpen U
hQ : ∀ K ∈ compacts U, IsCompact (Q K)
l1 : ∀ K ∈ compacts U, EquicontinuousOn Subtype.val K
l2 : ∀ K ∈ compacts U, ∀ x ∈ K, ∃ L, IsCompact L ∧ ∀ (i : ↑(𝓑 U Q)), ↑i x ∈ L
⊢ IsCompact (𝓑 U Q) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | isCompact_𝓑 | [64, 1] | [73, 59] | rw [isCompact_iff_compactSpace] | ι : Type u_1
U K : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
hU : IsOpen U
hQ : ∀ K ∈ compacts U, IsCompact (Q K)
l1 : ∀ K ∈ compacts U, EquicontinuousOn Subtype.val K
l2 : ∀ K ∈ compacts U, ∀ x ∈ K, ∃ L, IsCompact L ∧ ∀ (i : ↑(𝓑 U Q)), ↑i x ∈ L
⊢ IsCompact (𝓑 U Q) | ι : Type u_1
U K : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
hU : IsOpen U
hQ : ∀ K ∈ compacts U, IsCompact (Q K)
l1 : ∀ K ∈ compacts U, EquicontinuousOn Subtype.val K
l2 : ∀ K ∈ compacts U, ∀ x ∈ K, ∃ L, IsCompact L ∧ ∀ (i : ↑(𝓑 U Q)), ↑i x ∈ L
⊢ CompactSpace ↑(𝓑 U Q) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | isCompact_𝓑 | [64, 1] | [73, 59] | refine ArzelaAscoli.compactSpace_of_closedEmbedding (fun K hK => hK.2) ?_ l1 l2 | ι : Type u_1
U K : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
hU : IsOpen U
hQ : ∀ K ∈ compacts U, IsCompact (Q K)
l1 : ∀ K ∈ compacts U, EquicontinuousOn Subtype.val K
l2 : ∀ K ∈ compacts U, ∀ x ∈ K, ∃ L, IsCompact L ∧ ∀ (i : ↑(𝓑 U Q)), ↑i x ∈ L
⊢ CompactSpace ↑(𝓑 U Q) | ι : Type u_1
U K : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
hU : IsOpen U
hQ : ∀ K ∈ compacts U, IsCompact (Q K)
l1 : ∀ K ∈ compacts U, EquicontinuousOn Subtype.val K
l2 : ∀ K ∈ compacts U, ∀ x ∈ K, ∃ L, IsCompact L ∧ ∀ (i : ↑(𝓑 U Q)), ↑i x ∈ L
⊢ ClosedEmbedding (⇑(UniformOnFun.ofFun (compacts U)) ∘ Subtype.val) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | isCompact_𝓑 | [64, 1] | [73, 59] | refine ⟨⟨by tauto, fun f g => Subtype.ext⟩, ?_⟩ | ι : Type u_1
U K : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
hU : IsOpen U
hQ : ∀ K ∈ compacts U, IsCompact (Q K)
l1 : ∀ K ∈ compacts U, EquicontinuousOn Subtype.val K
l2 : ∀ K ∈ compacts U, ∀ x ∈ K, ∃ L, IsCompact L ∧ ∀ (i : ↑(𝓑 U Q)), ↑i x ∈ L
⊢ ClosedEmbedding (⇑(UniformOnFun.ofFun (compacts U)) ∘ Subtype.val) | ι : Type u_1
U K : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
hU : IsOpen U
hQ : ∀ K ∈ compacts U, IsCompact (Q K)
l1 : ∀ K ∈ compacts U, EquicontinuousOn Subtype.val K
l2 : ∀ K ∈ compacts U, ∀ x ∈ K, ∃ L, IsCompact L ∧ ∀ (i : ↑(𝓑 U Q)), ↑i x ∈ L
⊢ IsClosed (range (⇑(UniformOnFun.ofFun (compacts U)) ∘ Subtype.val)) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | isCompact_𝓑 | [64, 1] | [73, 59] | simpa [range, UniformOnFun.ofFun] using isClosed_𝓑 hU hQ | ι : Type u_1
U K : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
hU : IsOpen U
hQ : ∀ K ∈ compacts U, IsCompact (Q K)
l1 : ∀ K ∈ compacts U, EquicontinuousOn Subtype.val K
l2 : ∀ K ∈ compacts U, ∀ x ∈ K, ∃ L, IsCompact L ∧ ∀ (i : ↑(𝓑 U Q)), ↑i x ∈ L
⊢ IsClosed (range (⇑(UniformOnFun.ofFun (compacts U)) ∘ Subtype.val)) | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | isCompact_𝓑 | [64, 1] | [73, 59] | tauto | ι : Type u_1
U K : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
hU : IsOpen U
hQ : ∀ K ∈ compacts U, IsCompact (Q K)
l1 : ∀ K ∈ compacts U, EquicontinuousOn Subtype.val K
l2 : ∀ K ∈ compacts U, ∀ x ∈ K, ∃ L, IsCompact L ∧ ∀ (i : ↑(𝓑 U Q)), ↑i x ∈ L
⊢ Inducing (⇑(UniformOnFun.ofFun (compacts U)) ∘ Subtype.val) | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | montel | [75, 1] | [79, 72] | choose! Q hQ1 hQ2 using h1 | ι : Type u_1
U K : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
hU : IsOpen U
h1 : UniformlyBoundedOn F U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
⊢ TotallyBounded (range F) | ι : Type u_1
U K : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q✝ : Set ℂ → Set ℂ
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
Q : Set ℂ → Set ℂ
hQ1 : ∀ K ∈ compacts U, IsCompact (Q K)
hQ2 : ∀ K ∈ compacts U, ∀ (i : ι), MapsTo (F i) K (Q K)
⊢ TotallyBounded (range F) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | montel | [75, 1] | [79, 72] | have l1 : range F ⊆ 𝓑 U Q := by rintro f ⟨i, rfl⟩ ; exact ⟨h2 i, fun K hK => hQ2 K hK i⟩ | ι : Type u_1
U K : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q✝ : Set ℂ → Set ℂ
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
Q : Set ℂ → Set ℂ
hQ1 : ∀ K ∈ compacts U, IsCompact (Q K)
hQ2 : ∀ K ∈ compacts U, ∀ (i : ι), MapsTo (F i) K (Q K)
⊢ TotallyBounded (range F) | ι : Type u_1
U K : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q✝ : Set ℂ → Set ℂ
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
Q : Set ℂ → Set ℂ
hQ1 : ∀ K ∈ compacts U, IsCompact (Q K)
hQ2 : ∀ K ∈ compacts U, ∀ (i : ι), MapsTo (F i) K (Q K)
l1 : range F ⊆ 𝓑 U Q
⊢ TotallyBounded (range F) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | montel | [75, 1] | [79, 72] | exact totallyBounded_subset l1 <| (isCompact_𝓑 hU hQ1).totallyBounded | ι : Type u_1
U K : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q✝ : Set ℂ → Set ℂ
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
Q : Set ℂ → Set ℂ
hQ1 : ∀ K ∈ compacts U, IsCompact (Q K)
hQ2 : ∀ K ∈ compacts U, ∀ (i : ι), MapsTo (F i) K (Q K)
l1 : range F ⊆ 𝓑 U Q
⊢ TotallyBounded (range F) | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | montel | [75, 1] | [79, 72] | rintro f ⟨i, rfl⟩ | ι : Type u_1
U K : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q✝ : Set ℂ → Set ℂ
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
Q : Set ℂ → Set ℂ
hQ1 : ∀ K ∈ compacts U, IsCompact (Q K)
hQ2 : ∀ K ∈ compacts U, ∀ (i : ι), MapsTo (F i) K (Q K)
⊢ range F ⊆ 𝓑 U Q | case intro
ι : Type u_1
U K : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q✝ : Set ℂ → Set ℂ
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
Q : Set ℂ → Set ℂ
hQ1 : ∀ K ∈ compacts U, IsCompact (Q K)
hQ2 : ∀ K ∈ compacts U, ∀ (i : ι), MapsTo (F i) K (Q K)
i : ι
⊢ F i ∈ 𝓑 U Q |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | montel | [75, 1] | [79, 72] | exact ⟨h2 i, fun K hK => hQ2 K hK i⟩ | case intro
ι : Type u_1
U K : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q✝ : Set ℂ → Set ℂ
hU : IsOpen U
h2 : ∀ (i : ι), DifferentiableOn ℂ (F i) U
Q : Set ℂ → Set ℂ
hQ1 : ∀ K ∈ compacts U, IsCompact (Q K)
hQ2 : ∀ K ∈ compacts U, ∀ (i : ι), MapsTo (F i) K (Q K)
i : ι
⊢ F i ∈ 𝓑 U Q | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | isCompact_𝓜 | [83, 1] | [84, 82] | simpa only [𝓑_const] using isCompact_𝓑 hU (fun _ _ => isCompact_closedBall 0 1) | ι : Type u_1
U K : Set ℂ
z : ℂ
F : ι → 𝓒 U
Q : Set ℂ → Set ℂ
hU : IsOpen U
⊢ IsCompact (𝓜 U) | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/defs.lean | neg_in_𝔻 | [19, 1] | [20, 11] | simp [𝔻] | u : ℂ
U V W : Set ℂ
⊢ u ∈ 𝔻 → -u ∈ 𝔻 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/defs.lean | sqrt_𝔻_eq_𝔻 | [22, 1] | [23, 17] | simp [𝔻, ball] | u : ℂ
U V W : Set ℂ
⊢ {z | z ^ 2 ∈ 𝔻} = 𝔻 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/defs.lean | ne_center_of_not_mem_closed_ball | [69, 1] | [72, 16] | contrapose! hz | u : ℂ
U V W : Set ℂ
w : ℂ
r : ℝ
hr : 0 ≤ r
z : ℂ
hz : z ∈ (closedBall w r)ᶜ
⊢ z ≠ w | u : ℂ
U V W : Set ℂ
w : ℂ
r : ℝ
hr : 0 ≤ r
z : ℂ
hz : z = w
⊢ z ∉ (closedBall w r)ᶜ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/defs.lean | ne_center_of_not_mem_closed_ball | [69, 1] | [72, 16] | simp [hz, hr] | u : ℂ
U V W : Set ℂ
w : ℂ
r : ℝ
hr : 0 ≤ r
z : ℂ
hz : z = w
⊢ z ∉ (closedBall w r)ᶜ | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | one_sub_mul_conj_ne_zero | [9, 1] | [12, 65] | rw [mem_𝔻_iff] at hu hz | z u z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
hu : u ∈ 𝔻
hz : z ∈ 𝔻
⊢ 1 - z * (starRingEnd ℂ) u ≠ 0 | z u z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
hu : ‖u‖ < 1
hz : ‖z‖ < 1
⊢ 1 - z * (starRingEnd ℂ) u ≠ 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | one_sub_mul_conj_ne_zero | [9, 1] | [12, 65] | refine sub_ne_zero.mpr (mt (congr_arg Complex.abs) (ne_comm.mp (ne_of_lt ?_))) | z u z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
hu : ‖u‖ < 1
hz : ‖z‖ < 1
⊢ 1 - z * (starRingEnd ℂ) u ≠ 0 | z u z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
hu : ‖u‖ < 1
hz : ‖z‖ < 1
⊢ Complex.abs (z * (starRingEnd ℂ) u) < Complex.abs 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | one_sub_mul_conj_ne_zero | [9, 1] | [12, 65] | simpa using mul_lt_mul'' hz hu (norm_nonneg z) (norm_nonneg u) | z u z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
hu : ‖u‖ < 1
hz : ‖z‖ < 1
⊢ Complex.abs (z * (starRingEnd ℂ) u) < Complex.abs 1 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | one_sub_mul_conj_add_mul_conj_ne_zero | [14, 1] | [18, 22] | have h1 := one_sub_mul_conj_ne_zero hu hu | z u z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
hu : u ∈ 𝔻
⊢ 1 - z * (starRingEnd ℂ) u + (z - u) * (starRingEnd ℂ) u ≠ 0 | z u z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
hu : u ∈ 𝔻
h1 : 1 - u * (starRingEnd ℂ) u ≠ 0
⊢ 1 - z * (starRingEnd ℂ) u + (z - u) * (starRingEnd ℂ) u ≠ 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | one_sub_mul_conj_add_mul_conj_ne_zero | [14, 1] | [18, 22] | ring_nf | z u z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
hu : u ∈ 𝔻
h1 : 1 - u * (starRingEnd ℂ) u ≠ 0
⊢ 1 - z * (starRingEnd ℂ) u + (z - u) * (starRingEnd ℂ) u ≠ 0 | z u z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
hu : u ∈ 𝔻
h1 : 1 - u * (starRingEnd ℂ) u ≠ 0
⊢ 1 - (starRingEnd ℂ) u * u ≠ 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | one_sub_mul_conj_add_mul_conj_ne_zero | [14, 1] | [18, 22] | simp [h1, mul_comm] | z u z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
hu : u ∈ 𝔻
h1 : 1 - u * (starRingEnd ℂ) u ≠ 0
⊢ 1 - (starRingEnd ℂ) u * u ≠ 0 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | normSq_sub_normSq | [20, 1] | [21, 59] | field_simp [← ofReal_inj, normSq_eq_conj_mul_self] | z u z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
⊢ normSq (z - u) - normSq (1 - z * (starRingEnd ℂ) u) = (normSq z - 1) * (1 - normSq u) | z u z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
⊢ ((starRingEnd ℂ) z - (starRingEnd ℂ) u) * (z - u) - (1 - (starRingEnd ℂ) z * u) * (1 - z * (starRingEnd ℂ) u) =
((starRingEnd ℂ) z * z - 1) * (1 - (starRingEnd ℂ) u * u) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | normSq_sub_normSq | [20, 1] | [21, 59] | ring | z u z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
⊢ ((starRingEnd ℂ) z - (starRingEnd ℂ) u) * (z - u) - (1 - (starRingEnd ℂ) z * u) * (1 - z * (starRingEnd ℂ) u) =
((starRingEnd ℂ) z * z - 1) * (1 - (starRingEnd ℂ) u * u) | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | pre_φ_inv | [25, 1] | [28, 7] | rintro z hz | z u z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
hu : u ∈ 𝔻
⊢ LeftInvOn (pre_φ (-u)) (pre_φ u) 𝔻 | z✝ u z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
hu : u ∈ 𝔻
z : ℂ
hz : z ∈ 𝔻
⊢ pre_φ (-u) (pre_φ u z) = z |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | pre_φ_inv | [25, 1] | [28, 7] | field_simp [pre_φ, one_sub_mul_conj_ne_zero hu hz, one_sub_mul_conj_add_mul_conj_ne_zero hu] | z✝ u z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
hu : u ∈ 𝔻
z : ℂ
hz : z ∈ 𝔻
⊢ pre_φ (-u) (pre_φ u z) = z | z✝ u z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
hu : u ∈ 𝔻
z : ℂ
hz : z ∈ 𝔻
⊢ z - u + (1 - z * (starRingEnd ℂ) u) * u = z * (1 - z * (starRingEnd ℂ) u + (z - u) * (starRingEnd ℂ) u) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | pre_φ_inv | [25, 1] | [28, 7] | ring | z✝ u z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
hu : u ∈ 𝔻
z : ℂ
hz : z ∈ 𝔻
⊢ z - u + (1 - z * (starRingEnd ℂ) u) * u = z * (1 - z * (starRingEnd ℂ) u + (z - u) * (starRingEnd ℂ) u) | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | φ_deriv | [50, 1] | [57, 24] | have h1 : DifferentiableAt ℂ (fun z => z - u) z := by simp | z u z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
hu : u ∈ 𝔻
hz : z ∈ 𝔻
⊢ deriv (φ hu).to_fun z = (1 - u * (starRingEnd ℂ) u) / (1 - z * (starRingEnd ℂ) u) ^ 2 | z u z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
hu : u ∈ 𝔻
hz : z ∈ 𝔻
h1 : DifferentiableAt ℂ (fun z => z - u) z
⊢ deriv (φ hu).to_fun z = (1 - u * (starRingEnd ℂ) u) / (1 - z * (starRingEnd ℂ) u) ^ 2 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | φ_deriv | [50, 1] | [57, 24] | have h2 : DifferentiableAt ℂ (fun z => 1 - z * conj u) z := by simp [DifferentiableAt.mul_const] | z u z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
hu : u ∈ 𝔻
hz : z ∈ 𝔻
h1 : DifferentiableAt ℂ (fun z => z - u) z
⊢ deriv (φ hu).to_fun z = (1 - u * (starRingEnd ℂ) u) / (1 - z * (starRingEnd ℂ) u) ^ 2 | z u z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
hu : u ∈ 𝔻
hz : z ∈ 𝔻
h1 : DifferentiableAt ℂ (fun z => z - u) z
h2 : DifferentiableAt ℂ (fun z => 1 - z * (starRingEnd ℂ) u) z
⊢ deriv (φ hu).to_fun z = (1 - u * (starRingEnd ℂ) u) / (1 - z * (starRingEnd ℂ) u) ^ 2 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | φ_deriv | [50, 1] | [57, 24] | have h3 : 1 - z * conj u ≠ 0 := one_sub_mul_conj_ne_zero hu hz | z u z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
hu : u ∈ 𝔻
hz : z ∈ 𝔻
h1 : DifferentiableAt ℂ (fun z => z - u) z
h2 : DifferentiableAt ℂ (fun z => 1 - z * (starRingEnd ℂ) u) z
⊢ deriv (φ hu).to_fun z = (1 - u * (starRingEnd ℂ) u) / (1 - z * (starRingEnd ℂ) u) ^ 2 | z u z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
hu : u ∈ 𝔻
hz : z ∈ 𝔻
h1 : DifferentiableAt ℂ (fun z => z - u) z
h2 : DifferentiableAt ℂ (fun z => 1 - z * (starRingEnd ℂ) u) z
h3 : 1 - z * (starRingEnd ℂ) u ≠ 0
⊢ deriv (φ hu).to_fun z = (1 - u * (starRingEnd ℂ) u) / (1 - z * (starRingEnd ℂ) u) ^ 2 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | φ_deriv | [50, 1] | [57, 24] | have h4 : deriv (fun z => z - u) z = 1 := by simp [deriv_sub_const] | z u z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
hu : u ∈ 𝔻
hz : z ∈ 𝔻
h1 : DifferentiableAt ℂ (fun z => z - u) z
h2 : DifferentiableAt ℂ (fun z => 1 - z * (starRingEnd ℂ) u) z
h3 : 1 - z * (starRingEnd ℂ) u ≠ 0
⊢ deriv (φ hu).to_fun z = (1 - u * (starRingEnd ℂ) u) / (1 - z * (starRingEnd ℂ) u) ^ 2 | z u z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
hu : u ∈ 𝔻
hz : z ∈ 𝔻
h1 : DifferentiableAt ℂ (fun z => z - u) z
h2 : DifferentiableAt ℂ (fun z => 1 - z * (starRingEnd ℂ) u) z
h3 : 1 - z * (starRingEnd ℂ) u ≠ 0
h4 : deriv (fun z => z - u) z = 1
⊢ deriv (φ hu).to_fun z = (1 - u * (starRingEnd ℂ) u) / (1 - z * (starRingEnd ℂ) u) ^ 2 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | φ_deriv | [50, 1] | [57, 24] | have h5 : deriv (fun z => 1 - z * conj u) z = - conj u := by simp [deriv_const_sub] | z u z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
hu : u ∈ 𝔻
hz : z ∈ 𝔻
h1 : DifferentiableAt ℂ (fun z => z - u) z
h2 : DifferentiableAt ℂ (fun z => 1 - z * (starRingEnd ℂ) u) z
h3 : 1 - z * (starRingEnd ℂ) u ≠ 0
h4 : deriv (fun z => z - u) z = 1
⊢ deriv (φ hu).to_fun z = (1 - u * (starRingEnd ℂ) u) / (1 - z * (starRingEnd ℂ) u) ^ 2 | z u z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
hu : u ∈ 𝔻
hz : z ∈ 𝔻
h1 : DifferentiableAt ℂ (fun z => z - u) z
h2 : DifferentiableAt ℂ (fun z => 1 - z * (starRingEnd ℂ) u) z
h3 : 1 - z * (starRingEnd ℂ) u ≠ 0
h4 : deriv (fun z => z - u) z = 1
h5 : deriv (fun z => 1 - z * (starRingEnd ℂ) u) z = -(starRingEnd ℂ) u
⊢ deriv (φ hu).to_fun z = (1 - u * (starRingEnd ℂ) u) / (1 - z * (starRingEnd ℂ) u) ^ 2 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | φ_deriv | [50, 1] | [57, 24] | simp [φ, deriv_div h1 h2 h3, h4, h5] | z u z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
hu : u ∈ 𝔻
hz : z ∈ 𝔻
h1 : DifferentiableAt ℂ (fun z => z - u) z
h2 : DifferentiableAt ℂ (fun z => 1 - z * (starRingEnd ℂ) u) z
h3 : 1 - z * (starRingEnd ℂ) u ≠ 0
h4 : deriv (fun z => z - u) z = 1
h5 : deriv (fun z => 1 - z * (starRingEnd ℂ) u) z = -(starRingEnd ℂ) u
⊢ deriv (φ hu).to_fun z = (1 - u * (starRingEnd ℂ) u) / (1 - z * (starRingEnd ℂ) u) ^ 2 | z u z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
hu : u ∈ 𝔻
hz : z ∈ 𝔻
h1 : DifferentiableAt ℂ (fun z => z - u) z
h2 : DifferentiableAt ℂ (fun z => 1 - z * (starRingEnd ℂ) u) z
h3 : 1 - z * (starRingEnd ℂ) u ≠ 0
h4 : deriv (fun z => z - u) z = 1
h5 : deriv (fun z => 1 - z * (starRingEnd ℂ) u) z = -(starRingEnd ℂ) u
⊢ (1 - z * (starRingEnd ℂ) u + (z - u) * (starRingEnd ℂ) u) / (1 - z * (starRingEnd ℂ) u) ^ 2 =
(1 - u * (starRingEnd ℂ) u) / (1 - z * (starRingEnd ℂ) u) ^ 2 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | φ_deriv | [50, 1] | [57, 24] | field_simp [h3] | z u z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
hu : u ∈ 𝔻
hz : z ∈ 𝔻
h1 : DifferentiableAt ℂ (fun z => z - u) z
h2 : DifferentiableAt ℂ (fun z => 1 - z * (starRingEnd ℂ) u) z
h3 : 1 - z * (starRingEnd ℂ) u ≠ 0
h4 : deriv (fun z => z - u) z = 1
h5 : deriv (fun z => 1 - z * (starRingEnd ℂ) u) z = -(starRingEnd ℂ) u
⊢ (1 - z * (starRingEnd ℂ) u + (z - u) * (starRingEnd ℂ) u) / (1 - z * (starRingEnd ℂ) u) ^ 2 =
(1 - u * (starRingEnd ℂ) u) / (1 - z * (starRingEnd ℂ) u) ^ 2 | z u z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
hu : u ∈ 𝔻
hz : z ∈ 𝔻
h1 : DifferentiableAt ℂ (fun z => z - u) z
h2 : DifferentiableAt ℂ (fun z => 1 - z * (starRingEnd ℂ) u) z
h3 : 1 - z * (starRingEnd ℂ) u ≠ 0
h4 : deriv (fun z => z - u) z = 1
h5 : deriv (fun z => 1 - z * (starRingEnd ℂ) u) z = -(starRingEnd ℂ) u
⊢ 1 - z * (starRingEnd ℂ) u + (z - u) * (starRingEnd ℂ) u = 1 - u * (starRingEnd ℂ) u |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | φ_deriv | [50, 1] | [57, 24] | ring | z u z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
hu : u ∈ 𝔻
hz : z ∈ 𝔻
h1 : DifferentiableAt ℂ (fun z => z - u) z
h2 : DifferentiableAt ℂ (fun z => 1 - z * (starRingEnd ℂ) u) z
h3 : 1 - z * (starRingEnd ℂ) u ≠ 0
h4 : deriv (fun z => z - u) z = 1
h5 : deriv (fun z => 1 - z * (starRingEnd ℂ) u) z = -(starRingEnd ℂ) u
⊢ 1 - z * (starRingEnd ℂ) u + (z - u) * (starRingEnd ℂ) u = 1 - u * (starRingEnd ℂ) u | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | φ_deriv | [50, 1] | [57, 24] | simp | z u z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
hu : u ∈ 𝔻
hz : z ∈ 𝔻
⊢ DifferentiableAt ℂ (fun z => z - u) z | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | φ_deriv | [50, 1] | [57, 24] | simp [DifferentiableAt.mul_const] | z u z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
hu : u ∈ 𝔻
hz : z ∈ 𝔻
h1 : DifferentiableAt ℂ (fun z => z - u) z
⊢ DifferentiableAt ℂ (fun z => 1 - z * (starRingEnd ℂ) u) z | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | φ_deriv | [50, 1] | [57, 24] | simp [deriv_sub_const] | z u z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
hu : u ∈ 𝔻
hz : z ∈ 𝔻
h1 : DifferentiableAt ℂ (fun z => z - u) z
h2 : DifferentiableAt ℂ (fun z => 1 - z * (starRingEnd ℂ) u) z
h3 : 1 - z * (starRingEnd ℂ) u ≠ 0
⊢ deriv (fun z => z - u) z = 1 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | φ_deriv | [50, 1] | [57, 24] | simp [deriv_const_sub] | z u z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
hu : u ∈ 𝔻
hz : z ∈ 𝔻
h1 : DifferentiableAt ℂ (fun z => z - u) z
h2 : DifferentiableAt ℂ (fun z => 1 - z * (starRingEnd ℂ) u) z
h3 : 1 - z * (starRingEnd ℂ) u ≠ 0
h4 : deriv (fun z => z - u) z = 1
⊢ deriv (fun z => 1 - z * (starRingEnd ℂ) u) z = -(starRingEnd ℂ) u | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | set u := f 0 | z u z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
⊢ ‖deriv f 0‖ < 1 | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
⊢ ‖deriv f 0‖ < 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | have u_in_𝔻 : u ∈ 𝔻 := f_img (mem_ball_self zero_lt_one) | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
⊢ ‖deriv f 0‖ < 1 | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
⊢ ‖deriv f 0‖ < 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | let g := φ u_in_𝔻 ∘ f | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
⊢ ‖deriv f 0‖ < 1 | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
⊢ ‖deriv f 0‖ < 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | have g_diff : DifferentiableOn ℂ g 𝔻 := (φ u_in_𝔻).is_diff.comp f_diff f_img | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
⊢ ‖deriv f 0‖ < 1 | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
⊢ ‖deriv f 0‖ < 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | have g_maps : MapsTo g 𝔻 𝔻 := (φ u_in_𝔻).maps_to.comp f_img | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
⊢ ‖deriv f 0‖ < 1 | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
⊢ ‖deriv f 0‖ < 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | have g_0_eq_0 : g 0 = 0 := by simp [g, φ] | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
⊢ ‖deriv f 0‖ < 1 | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
⊢ ‖deriv f 0‖ < 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | by_cases h : ‖deriv g 0‖ = 1 | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
⊢ ‖deriv f 0‖ < 1 | case pos
z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ‖deriv g 0‖ = 1
⊢ ‖deriv f 0‖ < 1
case neg
z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ¬‖deriv g 0‖ = 1
⊢ ‖deriv f 0‖ < 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | case pos =>
have g_lin : EqOn g (λ (z : ℂ) => z • deriv g 0) (ball 0 1) := by
have h2 : MapsTo g (ball 0 1) (ball (g 0) 1) := by rwa [g_0_eq_0]
have h1 : Set.EqOn g (fun z => g 0 + (z - 0) • dslope g 0 0) (Metric.ball 0 1) := by
apply affine_of_mapsTo_ball_of_exists_norm_dslope_eq_div g_diff h2 (mem_ball_self zero_lt_one)
rwa [dslope_same, div_one]
convert h1 using 1
ext1 z
rw [g_0_eq_0, zero_add, sub_zero, dslope_same]
have g'0_ne_0 : deriv g 0 ≠ 0 := λ h' => by simp [h'] at h
have g_inj : InjOn g 𝔻 := λ x hx y hy => by
rw [g_lin hx, g_lin hy]
simp [g'0_ne_0]
cases f_noninj (injOn_of_injOn_comp g_inj) | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ‖deriv g 0‖ = 1
⊢ ‖deriv f 0‖ < 1 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | case neg =>
rw [norm_eq_abs] at h
have g'0_le_1 := abs_deriv_le_one_of_mapsTo_ball g_diff g_maps g_0_eq_0 zero_lt_one
have g'0_lt_1 : abs (deriv g 0) < 1 := Ne.lt_of_le h g'0_le_1
have g'0_eq_mul : deriv g 0 = deriv (φ u_in_𝔻) u * deriv f 0 :=
deriv.comp 0 ((φ u_in_𝔻).is_diff.differentiableAt (isOpen_ball.mem_nhds u_in_𝔻))
(f_diff.differentiableAt (ball_mem_nhds _ zero_lt_one))
have e1 : 1 - (normSq u : ℂ) ≠ 0 := by
simpa [normSq_eq_conj_mul_self, mul_comm] using one_sub_mul_conj_ne_zero u_in_𝔻 u_in_𝔻
have φ'u_u : deriv (φ u_in_𝔻) u = 1 / (1 - normSq u) := by
set w := 1 - conj u * u with hw
have : w ≠ 0 := by simpa [normSq_eq_conj_mul_self, mul_comm u] using e1
rw [φ_deriv u_in_𝔻 u_in_𝔻, normSq_eq_conj_mul_self, mul_comm u, ← hw]
field_simp; ring
have e2 : 0 ≤ normSq u := normSq_nonneg _
have e3 : normSq u < 1 := by
rw [normSq_eq_abs]
have : abs u < 1 := mem_𝔻_iff.mp u_in_𝔻
simp only [sq_lt_one_iff_abs_lt_one, Complex.abs_abs, this]
simp [normSq_eq_abs, ← mem_𝔻_iff]
simp only [φ'u_u, one_div] at g'0_eq_mul
rw [eq_comm, inv_mul_eq_iff_eq_mul₀ e1] at g'0_eq_mul
rw [← norm_eq_abs, g'0_eq_mul, norm_mul, mul_comm, ← one_mul (1 : ℝ)]
refine mul_lt_mul g'0_lt_1 ?_ (norm_pos_iff.mpr e1) zero_le_one
simp at e2 e3 ⊢
norm_cast
rw [abs_sub_le_iff]
refine ⟨?_, ?_⟩; repeat linarith | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ¬‖deriv g 0‖ = 1
⊢ ‖deriv f 0‖ < 1 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | simp [g, φ] | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
⊢ g 0 = 0 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | have g_lin : EqOn g (λ (z : ℂ) => z • deriv g 0) (ball 0 1) := by
have h2 : MapsTo g (ball 0 1) (ball (g 0) 1) := by rwa [g_0_eq_0]
have h1 : Set.EqOn g (fun z => g 0 + (z - 0) • dslope g 0 0) (Metric.ball 0 1) := by
apply affine_of_mapsTo_ball_of_exists_norm_dslope_eq_div g_diff h2 (mem_ball_self zero_lt_one)
rwa [dslope_same, div_one]
convert h1 using 1
ext1 z
rw [g_0_eq_0, zero_add, sub_zero, dslope_same] | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ‖deriv g 0‖ = 1
⊢ ‖deriv f 0‖ < 1 | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ‖deriv g 0‖ = 1
g_lin : EqOn g (fun z => z • deriv g 0) (ball 0 1)
⊢ ‖deriv f 0‖ < 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | have g'0_ne_0 : deriv g 0 ≠ 0 := λ h' => by simp [h'] at h | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ‖deriv g 0‖ = 1
g_lin : EqOn g (fun z => z • deriv g 0) (ball 0 1)
⊢ ‖deriv f 0‖ < 1 | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ‖deriv g 0‖ = 1
g_lin : EqOn g (fun z => z • deriv g 0) (ball 0 1)
g'0_ne_0 : deriv g 0 ≠ 0
⊢ ‖deriv f 0‖ < 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | have g_inj : InjOn g 𝔻 := λ x hx y hy => by
rw [g_lin hx, g_lin hy]
simp [g'0_ne_0] | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ‖deriv g 0‖ = 1
g_lin : EqOn g (fun z => z • deriv g 0) (ball 0 1)
g'0_ne_0 : deriv g 0 ≠ 0
⊢ ‖deriv f 0‖ < 1 | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ‖deriv g 0‖ = 1
g_lin : EqOn g (fun z => z • deriv g 0) (ball 0 1)
g'0_ne_0 : deriv g 0 ≠ 0
g_inj : InjOn g 𝔻
⊢ ‖deriv f 0‖ < 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | cases f_noninj (injOn_of_injOn_comp g_inj) | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ‖deriv g 0‖ = 1
g_lin : EqOn g (fun z => z • deriv g 0) (ball 0 1)
g'0_ne_0 : deriv g 0 ≠ 0
g_inj : InjOn g 𝔻
⊢ ‖deriv f 0‖ < 1 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | have h2 : MapsTo g (ball 0 1) (ball (g 0) 1) := by rwa [g_0_eq_0] | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ‖deriv g 0‖ = 1
⊢ EqOn g (fun z => z • deriv g 0) (ball 0 1) | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ‖deriv g 0‖ = 1
h2 : MapsTo g (ball 0 1) (ball (g 0) 1)
⊢ EqOn g (fun z => z • deriv g 0) (ball 0 1) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | have h1 : Set.EqOn g (fun z => g 0 + (z - 0) • dslope g 0 0) (Metric.ball 0 1) := by
apply affine_of_mapsTo_ball_of_exists_norm_dslope_eq_div g_diff h2 (mem_ball_self zero_lt_one)
rwa [dslope_same, div_one] | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ‖deriv g 0‖ = 1
h2 : MapsTo g (ball 0 1) (ball (g 0) 1)
⊢ EqOn g (fun z => z • deriv g 0) (ball 0 1) | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ‖deriv g 0‖ = 1
h2 : MapsTo g (ball 0 1) (ball (g 0) 1)
h1 : EqOn g (fun z => g 0 + (z - 0) • dslope g 0 0) (ball 0 1)
⊢ EqOn g (fun z => z • deriv g 0) (ball 0 1) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | convert h1 using 1 | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ‖deriv g 0‖ = 1
h2 : MapsTo g (ball 0 1) (ball (g 0) 1)
h1 : EqOn g (fun z => g 0 + (z - 0) • dslope g 0 0) (ball 0 1)
⊢ EqOn g (fun z => z • deriv g 0) (ball 0 1) | case h.e'_4
z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ‖deriv g 0‖ = 1
h2 : MapsTo g (ball 0 1) (ball (g 0) 1)
h1 : EqOn g (fun z => g 0 + (z - 0) • dslope g 0 0) (ball 0 1)
⊢ (fun z => z • deriv g 0) = fun z => g 0 + (z - 0) • dslope g 0 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | ext1 z | case h.e'_4
z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ‖deriv g 0‖ = 1
h2 : MapsTo g (ball 0 1) (ball (g 0) 1)
h1 : EqOn g (fun z => g 0 + (z - 0) • dslope g 0 0) (ball 0 1)
⊢ (fun z => z • deriv g 0) = fun z => g 0 + (z - 0) • dslope g 0 0 | case h.e'_4.h
z✝ u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ‖deriv g 0‖ = 1
h2 : MapsTo g (ball 0 1) (ball (g 0) 1)
h1 : EqOn g (fun z => g 0 + (z - 0) • dslope g 0 0) (ball 0 1)
z : ℂ
⊢ z • deriv g 0 = g 0 + (z - 0) • dslope g 0 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | rw [g_0_eq_0, zero_add, sub_zero, dslope_same] | case h.e'_4.h
z✝ u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ‖deriv g 0‖ = 1
h2 : MapsTo g (ball 0 1) (ball (g 0) 1)
h1 : EqOn g (fun z => g 0 + (z - 0) • dslope g 0 0) (ball 0 1)
z : ℂ
⊢ z • deriv g 0 = g 0 + (z - 0) • dslope g 0 0 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | rwa [g_0_eq_0] | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ‖deriv g 0‖ = 1
⊢ MapsTo g (ball 0 1) (ball (g 0) 1) | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | apply affine_of_mapsTo_ball_of_exists_norm_dslope_eq_div g_diff h2 (mem_ball_self zero_lt_one) | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ‖deriv g 0‖ = 1
h2 : MapsTo g (ball 0 1) (ball (g 0) 1)
⊢ EqOn g (fun z => g 0 + (z - 0) • dslope g 0 0) (ball 0 1) | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ‖deriv g 0‖ = 1
h2 : MapsTo g (ball 0 1) (ball (g 0) 1)
⊢ ‖dslope g 0 0‖ = 1 / 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | rwa [dslope_same, div_one] | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ‖deriv g 0‖ = 1
h2 : MapsTo g (ball 0 1) (ball (g 0) 1)
⊢ ‖dslope g 0 0‖ = 1 / 1 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | simp [h'] at h | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ‖deriv g 0‖ = 1
g_lin : EqOn g (fun z => z • deriv g 0) (ball 0 1)
h' : deriv g 0 = 0
⊢ False | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | rw [g_lin hx, g_lin hy] | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ‖deriv g 0‖ = 1
g_lin : EqOn g (fun z => z • deriv g 0) (ball 0 1)
g'0_ne_0 : deriv g 0 ≠ 0
x : ℂ
hx : x ∈ 𝔻
y : ℂ
hy : y ∈ 𝔻
⊢ g x = g y → x = y | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ‖deriv g 0‖ = 1
g_lin : EqOn g (fun z => z • deriv g 0) (ball 0 1)
g'0_ne_0 : deriv g 0 ≠ 0
x : ℂ
hx : x ∈ 𝔻
y : ℂ
hy : y ∈ 𝔻
⊢ (fun z => z • deriv g 0) x = (fun z => z • deriv g 0) y → x = y |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | simp [g'0_ne_0] | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ‖deriv g 0‖ = 1
g_lin : EqOn g (fun z => z • deriv g 0) (ball 0 1)
g'0_ne_0 : deriv g 0 ≠ 0
x : ℂ
hx : x ∈ 𝔻
y : ℂ
hy : y ∈ 𝔻
⊢ (fun z => z • deriv g 0) x = (fun z => z • deriv g 0) y → x = y | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | rw [norm_eq_abs] at h | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ¬‖deriv g 0‖ = 1
⊢ ‖deriv f 0‖ < 1 | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ¬Complex.abs (deriv g 0) = 1
⊢ ‖deriv f 0‖ < 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | have g'0_le_1 := abs_deriv_le_one_of_mapsTo_ball g_diff g_maps g_0_eq_0 zero_lt_one | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ¬Complex.abs (deriv g 0) = 1
⊢ ‖deriv f 0‖ < 1 | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ¬Complex.abs (deriv g 0) = 1
g'0_le_1 : Complex.abs (deriv g 0) ≤ 1
⊢ ‖deriv f 0‖ < 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | have g'0_lt_1 : abs (deriv g 0) < 1 := Ne.lt_of_le h g'0_le_1 | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ¬Complex.abs (deriv g 0) = 1
g'0_le_1 : Complex.abs (deriv g 0) ≤ 1
⊢ ‖deriv f 0‖ < 1 | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ¬Complex.abs (deriv g 0) = 1
g'0_le_1 : Complex.abs (deriv g 0) ≤ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
⊢ ‖deriv f 0‖ < 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | have g'0_eq_mul : deriv g 0 = deriv (φ u_in_𝔻) u * deriv f 0 :=
deriv.comp 0 ((φ u_in_𝔻).is_diff.differentiableAt (isOpen_ball.mem_nhds u_in_𝔻))
(f_diff.differentiableAt (ball_mem_nhds _ zero_lt_one)) | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ¬Complex.abs (deriv g 0) = 1
g'0_le_1 : Complex.abs (deriv g 0) ≤ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
⊢ ‖deriv f 0‖ < 1 | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ¬Complex.abs (deriv g 0) = 1
g'0_le_1 : Complex.abs (deriv g 0) ≤ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
g'0_eq_mul : deriv g 0 = deriv (φ u_in_𝔻).to_fun u * deriv f 0
⊢ ‖deriv f 0‖ < 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | have e1 : 1 - (normSq u : ℂ) ≠ 0 := by
simpa [normSq_eq_conj_mul_self, mul_comm] using one_sub_mul_conj_ne_zero u_in_𝔻 u_in_𝔻 | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ¬Complex.abs (deriv g 0) = 1
g'0_le_1 : Complex.abs (deriv g 0) ≤ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
g'0_eq_mul : deriv g 0 = deriv (φ u_in_𝔻).to_fun u * deriv f 0
⊢ ‖deriv f 0‖ < 1 | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ¬Complex.abs (deriv g 0) = 1
g'0_le_1 : Complex.abs (deriv g 0) ≤ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
g'0_eq_mul : deriv g 0 = deriv (φ u_in_𝔻).to_fun u * deriv f 0
e1 : 1 - ↑(normSq u) ≠ 0
⊢ ‖deriv f 0‖ < 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | have φ'u_u : deriv (φ u_in_𝔻) u = 1 / (1 - normSq u) := by
set w := 1 - conj u * u with hw
have : w ≠ 0 := by simpa [normSq_eq_conj_mul_self, mul_comm u] using e1
rw [φ_deriv u_in_𝔻 u_in_𝔻, normSq_eq_conj_mul_self, mul_comm u, ← hw]
field_simp; ring | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ¬Complex.abs (deriv g 0) = 1
g'0_le_1 : Complex.abs (deriv g 0) ≤ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
g'0_eq_mul : deriv g 0 = deriv (φ u_in_𝔻).to_fun u * deriv f 0
e1 : 1 - ↑(normSq u) ≠ 0
⊢ ‖deriv f 0‖ < 1 | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ¬Complex.abs (deriv g 0) = 1
g'0_le_1 : Complex.abs (deriv g 0) ≤ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
g'0_eq_mul : deriv g 0 = deriv (φ u_in_𝔻).to_fun u * deriv f 0
e1 : 1 - ↑(normSq u) ≠ 0
φ'u_u : deriv (φ u_in_𝔻).to_fun u = 1 / (1 - ↑(normSq u))
⊢ ‖deriv f 0‖ < 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | have e2 : 0 ≤ normSq u := normSq_nonneg _ | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ¬Complex.abs (deriv g 0) = 1
g'0_le_1 : Complex.abs (deriv g 0) ≤ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
g'0_eq_mul : deriv g 0 = deriv (φ u_in_𝔻).to_fun u * deriv f 0
e1 : 1 - ↑(normSq u) ≠ 0
φ'u_u : deriv (φ u_in_𝔻).to_fun u = 1 / (1 - ↑(normSq u))
⊢ ‖deriv f 0‖ < 1 | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ¬Complex.abs (deriv g 0) = 1
g'0_le_1 : Complex.abs (deriv g 0) ≤ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
g'0_eq_mul : deriv g 0 = deriv (φ u_in_𝔻).to_fun u * deriv f 0
e1 : 1 - ↑(normSq u) ≠ 0
φ'u_u : deriv (φ u_in_𝔻).to_fun u = 1 / (1 - ↑(normSq u))
e2 : 0 ≤ normSq u
⊢ ‖deriv f 0‖ < 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | have e3 : normSq u < 1 := by
rw [normSq_eq_abs]
have : abs u < 1 := mem_𝔻_iff.mp u_in_𝔻
simp only [sq_lt_one_iff_abs_lt_one, Complex.abs_abs, this] | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ¬Complex.abs (deriv g 0) = 1
g'0_le_1 : Complex.abs (deriv g 0) ≤ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
g'0_eq_mul : deriv g 0 = deriv (φ u_in_𝔻).to_fun u * deriv f 0
e1 : 1 - ↑(normSq u) ≠ 0
φ'u_u : deriv (φ u_in_𝔻).to_fun u = 1 / (1 - ↑(normSq u))
e2 : 0 ≤ normSq u
⊢ ‖deriv f 0‖ < 1 | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ¬Complex.abs (deriv g 0) = 1
g'0_le_1 : Complex.abs (deriv g 0) ≤ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
g'0_eq_mul : deriv g 0 = deriv (φ u_in_𝔻).to_fun u * deriv f 0
e1 : 1 - ↑(normSq u) ≠ 0
φ'u_u : deriv (φ u_in_𝔻).to_fun u = 1 / (1 - ↑(normSq u))
e2 : 0 ≤ normSq u
e3 : normSq u < 1
⊢ ‖deriv f 0‖ < 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | simp [normSq_eq_abs, ← mem_𝔻_iff] | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ¬Complex.abs (deriv g 0) = 1
g'0_le_1 : Complex.abs (deriv g 0) ≤ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
g'0_eq_mul : deriv g 0 = deriv (φ u_in_𝔻).to_fun u * deriv f 0
e1 : 1 - ↑(normSq u) ≠ 0
φ'u_u : deriv (φ u_in_𝔻).to_fun u = 1 / (1 - ↑(normSq u))
e2 : 0 ≤ normSq u
e3 : normSq u < 1
⊢ ‖deriv f 0‖ < 1 | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ¬Complex.abs (deriv g 0) = 1
g'0_le_1 : Complex.abs (deriv g 0) ≤ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
g'0_eq_mul : deriv g 0 = deriv (φ u_in_𝔻).to_fun u * deriv f 0
e1 : 1 - ↑(normSq u) ≠ 0
φ'u_u : deriv (φ u_in_𝔻).to_fun u = 1 / (1 - ↑(normSq u))
e2 : 0 ≤ normSq u
e3 : normSq u < 1
⊢ Complex.abs (deriv f 0) < 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | simp only [φ'u_u, one_div] at g'0_eq_mul | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ¬Complex.abs (deriv g 0) = 1
g'0_le_1 : Complex.abs (deriv g 0) ≤ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
g'0_eq_mul : deriv g 0 = deriv (φ u_in_𝔻).to_fun u * deriv f 0
e1 : 1 - ↑(normSq u) ≠ 0
φ'u_u : deriv (φ u_in_𝔻).to_fun u = 1 / (1 - ↑(normSq u))
e2 : 0 ≤ normSq u
e3 : normSq u < 1
⊢ Complex.abs (deriv f 0) < 1 | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ¬Complex.abs (deriv g 0) = 1
g'0_le_1 : Complex.abs (deriv g 0) ≤ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
e1 : 1 - ↑(normSq u) ≠ 0
φ'u_u : deriv (φ u_in_𝔻).to_fun u = 1 / (1 - ↑(normSq u))
e2 : 0 ≤ normSq u
e3 : normSq u < 1
g'0_eq_mul : deriv g 0 = (1 - ↑(normSq u))⁻¹ * deriv f 0
⊢ Complex.abs (deriv f 0) < 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | rw [eq_comm, inv_mul_eq_iff_eq_mul₀ e1] at g'0_eq_mul | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ¬Complex.abs (deriv g 0) = 1
g'0_le_1 : Complex.abs (deriv g 0) ≤ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
e1 : 1 - ↑(normSq u) ≠ 0
φ'u_u : deriv (φ u_in_𝔻).to_fun u = 1 / (1 - ↑(normSq u))
e2 : 0 ≤ normSq u
e3 : normSq u < 1
g'0_eq_mul : deriv g 0 = (1 - ↑(normSq u))⁻¹ * deriv f 0
⊢ Complex.abs (deriv f 0) < 1 | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ¬Complex.abs (deriv g 0) = 1
g'0_le_1 : Complex.abs (deriv g 0) ≤ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
e1 : 1 - ↑(normSq u) ≠ 0
φ'u_u : deriv (φ u_in_𝔻).to_fun u = 1 / (1 - ↑(normSq u))
e2 : 0 ≤ normSq u
e3 : normSq u < 1
g'0_eq_mul : deriv f 0 = (1 - ↑(normSq u)) * deriv g 0
⊢ Complex.abs (deriv f 0) < 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | rw [← norm_eq_abs, g'0_eq_mul, norm_mul, mul_comm, ← one_mul (1 : ℝ)] | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ¬Complex.abs (deriv g 0) = 1
g'0_le_1 : Complex.abs (deriv g 0) ≤ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
e1 : 1 - ↑(normSq u) ≠ 0
φ'u_u : deriv (φ u_in_𝔻).to_fun u = 1 / (1 - ↑(normSq u))
e2 : 0 ≤ normSq u
e3 : normSq u < 1
g'0_eq_mul : deriv f 0 = (1 - ↑(normSq u)) * deriv g 0
⊢ Complex.abs (deriv f 0) < 1 | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ¬Complex.abs (deriv g 0) = 1
g'0_le_1 : Complex.abs (deriv g 0) ≤ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
e1 : 1 - ↑(normSq u) ≠ 0
φ'u_u : deriv (φ u_in_𝔻).to_fun u = 1 / (1 - ↑(normSq u))
e2 : 0 ≤ normSq u
e3 : normSq u < 1
g'0_eq_mul : deriv f 0 = (1 - ↑(normSq u)) * deriv g 0
⊢ ‖deriv g 0‖ * ‖1 - ↑(normSq u)‖ < 1 * 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | refine mul_lt_mul g'0_lt_1 ?_ (norm_pos_iff.mpr e1) zero_le_one | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ¬Complex.abs (deriv g 0) = 1
g'0_le_1 : Complex.abs (deriv g 0) ≤ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
e1 : 1 - ↑(normSq u) ≠ 0
φ'u_u : deriv (φ u_in_𝔻).to_fun u = 1 / (1 - ↑(normSq u))
e2 : 0 ≤ normSq u
e3 : normSq u < 1
g'0_eq_mul : deriv f 0 = (1 - ↑(normSq u)) * deriv g 0
⊢ ‖deriv g 0‖ * ‖1 - ↑(normSq u)‖ < 1 * 1 | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ¬Complex.abs (deriv g 0) = 1
g'0_le_1 : Complex.abs (deriv g 0) ≤ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
e1 : 1 - ↑(normSq u) ≠ 0
φ'u_u : deriv (φ u_in_𝔻).to_fun u = 1 / (1 - ↑(normSq u))
e2 : 0 ≤ normSq u
e3 : normSq u < 1
g'0_eq_mul : deriv f 0 = (1 - ↑(normSq u)) * deriv g 0
⊢ ‖1 - ↑(normSq u)‖ ≤ 1 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | simp at e2 e3 ⊢ | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ¬Complex.abs (deriv g 0) = 1
g'0_le_1 : Complex.abs (deriv g 0) ≤ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
e1 : 1 - ↑(normSq u) ≠ 0
φ'u_u : deriv (φ u_in_𝔻).to_fun u = 1 / (1 - ↑(normSq u))
e2 : 0 ≤ normSq u
e3 : normSq u < 1
g'0_eq_mul : deriv f 0 = (1 - ↑(normSq u)) * deriv g 0
⊢ ‖1 - ↑(normSq u)‖ ≤ 1 | z u✝ z₀ : ℂ
U : Set ℂ
inst✝ : good_domain U
f : ℂ → ℂ
f_diff : DifferentiableOn ℂ f 𝔻
f_img : MapsTo f 𝔻 𝔻
f_noninj : ¬InjOn f 𝔻
u : ℂ := f 0
u_in_𝔻 : u ∈ 𝔻
g : ℂ → ℂ := (φ u_in_𝔻).to_fun ∘ f
g_diff : DifferentiableOn ℂ g 𝔻
g_maps : MapsTo g 𝔻 𝔻
g_0_eq_0 : g 0 = 0
h : ¬Complex.abs (deriv g 0) = 1
g'0_le_1 : Complex.abs (deriv g 0) ≤ 1
g'0_lt_1 : Complex.abs (deriv g 0) < 1
e1 : 1 - ↑(normSq u) ≠ 0
φ'u_u : deriv (φ u_in_𝔻).to_fun u = 1 / (1 - ↑(normSq u))
e2 : 0 ≤ normSq u
e3 : normSq u < 1
g'0_eq_mul : deriv f 0 = (1 - ↑(normSq u)) * deriv g 0
⊢ Complex.abs (1 - ↑(normSq u)) ≤ 1 |
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