url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.equicontinuousOn | [32, 1] | [58, 40] | refine β¨Ξ΄ β Ξ΅ / M, gt_iff_lt.2 (lt_inf_iff.2 β¨hΞ΄, div_pos hΞ΅ hMpβ©), Ξ» w hw i => ?_β© | case mk.intro.intro.intro.intro
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Q : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
M : β
hMp : M > 0
hM : β (i : ΞΉ), MapsTo (_root_.deriv (F i)) (... | case mk.intro.intro.intro.intro
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Q : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
M : β
hMp : M > 0
hM : β (i : ΞΉ), MapsTo (_root_.deriv (F i)) (... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.intro.intro.intro.intro
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Q : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
M ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.equicontinuousOn | [32, 1] | [58, 40] | simp | case mk.intro.intro.intro.intro
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Q : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
M : β
hMp : M > 0
hM : β (i : ΞΉ), MapsTo (_root_.deriv (F i)) (... | case mk.intro.intro.intro.intro
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Q : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
M : β
hMp : M > 0
hM : β (i : ΞΉ), MapsTo (_root_.deriv (F i)) (... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.intro.intro.intro.intro
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Q : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
M ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.equicontinuousOn | [32, 1] | [58, 40] | have e1 : β x β closedBall z Ξ΄, DifferentiableAt β (F i) x :=
Ξ» x hx => (h2 i).differentiableAt (hU.mem_nhds (h hx)) | case mk.intro.intro.intro.intro
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Q : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
M : β
hMp : M > 0
hM : β (i : ΞΉ), MapsTo (_root_.deriv (F i)) (... | case mk.intro.intro.intro.intro
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Q : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
M : β
hMp : M > 0
hM : β (i : ΞΉ), MapsTo (_root_.deriv (F i)) (... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.intro.intro.intro.intro
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Q : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
M ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.equicontinuousOn | [32, 1] | [58, 40] | have e2 : β x β closedBall z Ξ΄, β_root_.deriv (F i) xβ β€ M := by simpa [MapsTo] using hM i | case mk.intro.intro.intro.intro
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Q : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
M : β
hMp : M > 0
hM : β (i : ΞΉ), MapsTo (_root_.deriv (F i)) (... | case mk.intro.intro.intro.intro
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Q : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
M : β
hMp : M > 0
hM : β (i : ΞΉ), MapsTo (_root_.deriv (F i)) (... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.intro.intro.intro.intro
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Q : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
M ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.equicontinuousOn | [32, 1] | [58, 40] | have e3 : z β closedBall z Ξ΄ := mem_closedBall_self hΞ΄.le | case mk.intro.intro.intro.intro
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Q : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
M : β
hMp : M > 0
hM : β (i : ΞΉ), MapsTo (_root_.deriv (F i)) (... | case mk.intro.intro.intro.intro
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Q : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
M : β
hMp : M > 0
hM : β (i : ΞΉ), MapsTo (_root_.deriv (F i)) (... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.intro.intro.intro.intro
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Q : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
M ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.equicontinuousOn | [32, 1] | [58, 40] | have e4 : w.1 β closedBall z Ξ΄ := by simpa using (lt_inf_iff.1 hw).1.le | case mk.intro.intro.intro.intro
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Q : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
M : β
hMp : M > 0
hM : β (i : ΞΉ), MapsTo (_root_.deriv (F i)) (... | case mk.intro.intro.intro.intro
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Q : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
M : β
hMp : M > 0
hM : β (i : ΞΉ), MapsTo (_root_.deriv (F i)) (... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.intro.intro.intro.intro
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Q : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
M ... |
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
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Q : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
M : β
hMp : M > 0
hM : β (i : ΞΉ), MapsTo (_root_.deriv (F i)) (... | case mk.intro.intro.intro.intro
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Q : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
M : β
hMp : M > 0
hM : β (i : ΞΉ), MapsTo (_root_.deriv (F i)) (... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.intro.intro.intro.intro
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Q : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
M ... |
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
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Q : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
M : β
hMp : M > 0
hM : β (i : ΞΉ), MapsTo (_root_.deriv (F i)) (... | case mk.intro.intro.intro.intro
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Q : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
M : β
hMp : M > 0
hM : β (i : ΞΉ), MapsTo (_root_.deriv (F i)) (... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.intro.intro.intro.intro
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Q : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
M ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.equicontinuousOn | [32, 1] | [58, 40] | have : βz - w.valβ < Ξ΅ / 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
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Q : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
M : β
hMp : M > 0
hM : β (i : ΞΉ), MapsTo (_root_.deriv (F i)) (... | case mk.intro.intro.intro.intro
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Q : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
M : β
hMp : M > 0
hM : β (i : ΞΉ), MapsTo (_root_.deriv (F i)) (... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.intro.intro.intro.intro
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Q : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
M ... |
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
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Q : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
M : β
hMp : M > 0
hM : β (i : ΞΉ), MapsTo (_root_.deriv (F i)) (... | case h.e'_4
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Q : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
M : β
hMp : M > 0
hM : β (i : ΞΉ), MapsTo (_root_.deriv (F i)) (closedBall z Ξ΄) (clo... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.intro.intro.intro.intro
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Q : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
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
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Q : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
M : β
hMp : M > 0
hM : β (i : ΞΉ), MapsTo (_root_.deriv (F i)) (closedBall z Ξ΄) (clo... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Q : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
M : β
hMp : M > 0
hM :... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.equicontinuousOn | [32, 1] | [58, 40] | obtain β¨Q, hQ1, hQ2β© := h1.deriv hU h2 (closedBall z Ξ΄) β¨h, isCompact_closedBall _ _β© | ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Q : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
β’ β M > 0, β (i : ΞΉ), MapsTo (_root_.deriv (F i)) (closedBall z Ξ΄) (closedBall 0 M) | case intro.intro
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Qβ : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
Q : Set β
hQ1 : IsCompact Q
hQ2 : β (i : ΞΉ), MapsTo ((_root_.deriv β F) i) (c... | Please generate a tactic in lean4 to solve the state.
STATE:
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Q : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
β’ β M > 0, β (i : ΞΉ), MapsTo (_roo... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.equicontinuousOn | [32, 1] | [58, 40] | obtain β¨M, hMβ© := hQ1.isBounded.subset_closedBall 0 | case intro.intro
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Qβ : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
Q : Set β
hQ1 : IsCompact Q
hQ2 : β (i : ΞΉ), MapsTo ((_root_.deriv β F) i) (c... | case intro.intro.intro
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Qβ : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
Q : Set β
hQ1 : IsCompact Q
hQ2 : β (i : ΞΉ), MapsTo ((_root_.deriv β F)... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Qβ : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
Q : Set β
hQ1 : ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.equicontinuousOn | [32, 1] | [58, 40] | refine β¨M β 1, by simp, fun i => ?_β© | case intro.intro.intro
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Qβ : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
Q : Set β
hQ1 : IsCompact Q
hQ2 : β (i : ΞΉ), MapsTo ((_root_.deriv β F)... | case intro.intro.intro
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Qβ : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
Q : Set β
hQ1 : IsCompact Q
hQ2 : β (i : ΞΉ), MapsTo ((_root_.deriv β F)... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Qβ : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
Q : Set β
... |
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
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Qβ : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
Q : Set β
hQ1 : IsCompact Q
hQ2 : β (i : ΞΉ), MapsTo ((_root_.deriv β F)... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Qβ : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
Q : Set β
... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.equicontinuousOn | [32, 1] | [58, 40] | simp | ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Qβ : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
Q : Set β
hQ1 : IsCompact Q
hQ2 : β (i : ΞΉ), MapsTo ((_root_.deriv β F) i) (closedBall z Ξ΄) Q
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Qβ : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
Q : Set β
hQ1 : IsCompact Q
hQ2 :... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.equicontinuousOn | [32, 1] | [58, 40] | simpa [MapsTo] using hM i | ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Q : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
M : β
hMp : M > 0
hM : β (i : ΞΉ), MapsTo (_root_.deriv (F i)) (closedBall z Ξ΄) (closedBall 0 M)... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Q : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
M : β
hMp : M > 0
hM : β (i : ΞΉ), ... |
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 | ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Q : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
M : β
hMp : M > 0
hM : β (i : ΞΉ), MapsTo (_root_.deriv (F i)) (closedBall z Ξ΄) (closedBall 0 M)... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Q : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
M : β
hMp : M > 0
hM : β (i : ΞΉ), ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | UniformlyBoundedOn.equicontinuousOn | [32, 1] | [58, 40] | have := (lt_inf_iff.1 hw).2 | ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Q : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
M : β
hMp : M > 0
hM : β (i : ΞΉ), MapsTo (_root_.deriv (F i)) (closedBall z Ξ΄) (closedBall 0 M)... | ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Q : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
M : β
hMp : M > 0
hM : β (i : ΞΉ), MapsTo (_root_.deriv (F i)) (closedBall z Ξ΄) (closedBall 0 M)... | Please generate a tactic in lean4 to solve the state.
STATE:
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Q : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
M : β
hMp : M > 0
hM : β (i : ΞΉ), ... |
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 | ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Q : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
M : β
hMp : M > 0
hM : β (i : ΞΉ), MapsTo (_root_.deriv (F i)) (closedBall z Ξ΄) (closedBall 0 M)... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ΞΉ : Type u_1
U K : Set β
zβ : β
F : ΞΉ β π U
Q : Set β β Set β
h1 : UniformlyBoundedOn F U
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
hK : K β compacts U
z : β
hz : z β K
Ξ΄ : β
hΞ΄ : 0 < Ξ΄
h : closedBall z Ξ΄ β U
M : β
hMp : M > 0
hM : β (i : ΞΉ), ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | uniformlyBoundedOn_π | [60, 1] | [62, 56] | exact fun K hK => β¨Q K, hQ K hK, fun f => f.2.2 K hKβ© | ΞΉ : Type u_1
U K : Set β
z : β
F : ΞΉ β π U
Q : Set β β Set β
hQ : β K β compacts U, IsCompact (Q K)
β’ UniformlyBoundedOn Subtype.val U | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ΞΉ : Type u_1
U K : Set β
z : β
F : ΞΉ β π U
Q : Set β β Set β
hQ : β K β compacts U, IsCompact (Q K)
β’ UniformlyBoundedOn Subtype.val U
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | isCompact_π | [64, 1] | [73, 59] | have l1 (K) (hK : K β compacts U) : EquicontinuousOn ((β) : π U Q β π U) K :=
(uniformlyBoundedOn_π hQ).equicontinuousOn hU (fun f => f.2.1) hK | ΞΉ : Type u_1
U K : Set β
z : β
F : ΞΉ β π U
Q : Set β β Set β
hU : IsOpen U
hQ : β K β compacts U, IsCompact (Q K)
β’ IsCompact (π U Q) | ΞΉ : Type u_1
U K : Set β
z : β
F : ΞΉ β π U
Q : Set β β Set β
hU : IsOpen U
hQ : β K β compacts U, IsCompact (Q K)
l1 : β K β compacts U, EquicontinuousOn Subtype.val K
β’ IsCompact (π U Q) | Please generate a tactic in lean4 to solve the state.
STATE:
ΞΉ : Type u_1
U K : Set β
z : β
F : ΞΉ β π U
Q : Set β β Set β
hU : IsOpen U
hQ : β K β compacts U, IsCompact (Q K)
β’ IsCompact (π U Q)
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | isCompact_π | [64, 1] | [73, 59] | have l2 (K) (hK : K β compacts U) (x) (hx : x β K) : β L, IsCompact L β§ β i : π U Q, i.1 x β L :=
β¨Q K, hQ K hK, fun f => f.2.2 K hK hxβ© | ΞΉ : Type u_1
U K : Set β
z : β
F : ΞΉ β π U
Q : Set β β Set β
hU : IsOpen U
hQ : β K β compacts U, IsCompact (Q K)
l1 : β K β compacts U, EquicontinuousOn Subtype.val K
β’ IsCompact (π U Q) | ΞΉ : Type u_1
U K : Set β
z : β
F : ΞΉ β π U
Q : Set β β Set β
hU : IsOpen U
hQ : β K β compacts U, IsCompact (Q K)
l1 : β K β compacts U, EquicontinuousOn Subtype.val K
l2 : β K β compacts U, β x β K, β L, IsCompact L β§ β (i : β(π U Q)), βi x β L
β’ IsCompact (π U Q) | Please generate a tactic in lean4 to solve the state.
STATE:
ΞΉ : Type u_1
U K : Set β
z : β
F : ΞΉ β π U
Q : Set β β Set β
hU : IsOpen U
hQ : β K β compacts U, IsCompact (Q K)
l1 : β K β compacts U, EquicontinuousOn Subtype.val K
β’ IsCompact (π U Q)
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | isCompact_π | [64, 1] | [73, 59] | rw [isCompact_iff_compactSpace] | ΞΉ : Type u_1
U K : Set β
z : β
F : ΞΉ β π U
Q : Set β β Set β
hU : IsOpen U
hQ : β K β compacts U, IsCompact (Q K)
l1 : β K β compacts U, EquicontinuousOn Subtype.val K
l2 : β K β compacts U, β x β K, β L, IsCompact L β§ β (i : β(π U Q)), βi x β L
β’ IsCompact (π U Q) | ΞΉ : Type u_1
U K : Set β
z : β
F : ΞΉ β π U
Q : Set β β Set β
hU : IsOpen U
hQ : β K β compacts U, IsCompact (Q K)
l1 : β K β compacts U, EquicontinuousOn Subtype.val K
l2 : β K β compacts U, β x β K, β L, IsCompact L β§ β (i : β(π U Q)), βi x β L
β’ CompactSpace β(π U Q) | Please generate a tactic in lean4 to solve the state.
STATE:
ΞΉ : Type u_1
U K : Set β
z : β
F : ΞΉ β π U
Q : Set β β Set β
hU : IsOpen U
hQ : β K β compacts U, IsCompact (Q K)
l1 : β K β compacts U, EquicontinuousOn Subtype.val K
l2 : β K β compacts U, β x β K, β L, IsCompact L β§ β (i : β(π U Q)), βi x β L
β’ IsCompact... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | isCompact_π | [64, 1] | [73, 59] | refine ArzelaAscoli.compactSpace_of_closedEmbedding (fun K hK => hK.2) ?_ l1 l2 | ΞΉ : Type u_1
U K : Set β
z : β
F : ΞΉ β π U
Q : Set β β Set β
hU : IsOpen U
hQ : β K β compacts U, IsCompact (Q K)
l1 : β K β compacts U, EquicontinuousOn Subtype.val K
l2 : β K β compacts U, β x β K, β L, IsCompact L β§ β (i : β(π U Q)), βi x β L
β’ CompactSpace β(π U Q) | ΞΉ : Type u_1
U K : Set β
z : β
F : ΞΉ β π U
Q : Set β β Set β
hU : IsOpen U
hQ : β K β compacts U, IsCompact (Q K)
l1 : β K β compacts U, EquicontinuousOn Subtype.val K
l2 : β K β compacts U, β x β K, β L, IsCompact L β§ β (i : β(π U Q)), βi x β L
β’ ClosedEmbedding (β(UniformOnFun.ofFun (compacts U)) β Subtype.val) | Please generate a tactic in lean4 to solve the state.
STATE:
ΞΉ : Type u_1
U K : Set β
z : β
F : ΞΉ β π U
Q : Set β β Set β
hU : IsOpen U
hQ : β K β compacts U, IsCompact (Q K)
l1 : β K β compacts U, EquicontinuousOn Subtype.val K
l2 : β K β compacts U, β x β K, β L, IsCompact L β§ β (i : β(π U Q)), βi x β L
β’ CompactSp... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | isCompact_π | [64, 1] | [73, 59] | refine β¨β¨by tauto, fun f g => Subtype.extβ©, ?_β© | ΞΉ : Type u_1
U K : Set β
z : β
F : ΞΉ β π U
Q : Set β β Set β
hU : IsOpen U
hQ : β K β compacts U, IsCompact (Q K)
l1 : β K β compacts U, EquicontinuousOn Subtype.val K
l2 : β K β compacts U, β x β K, β L, IsCompact L β§ β (i : β(π U Q)), βi x β L
β’ ClosedEmbedding (β(UniformOnFun.ofFun (compacts U)) β Subtype.val) | ΞΉ : Type u_1
U K : Set β
z : β
F : ΞΉ β π U
Q : Set β β Set β
hU : IsOpen U
hQ : β K β compacts U, IsCompact (Q K)
l1 : β K β compacts U, EquicontinuousOn Subtype.val K
l2 : β K β compacts U, β x β K, β L, IsCompact L β§ β (i : β(π U Q)), βi x β L
β’ IsClosed (range (β(UniformOnFun.ofFun (compacts U)) β Subtype.val)) | Please generate a tactic in lean4 to solve the state.
STATE:
ΞΉ : Type u_1
U K : Set β
z : β
F : ΞΉ β π U
Q : Set β β Set β
hU : IsOpen U
hQ : β K β compacts U, IsCompact (Q K)
l1 : β K β compacts U, EquicontinuousOn Subtype.val K
l2 : β K β compacts U, β x β K, β L, IsCompact L β§ β (i : β(π U Q)), βi x β L
β’ ClosedEmb... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | isCompact_π | [64, 1] | [73, 59] | simpa [range, UniformOnFun.ofFun] using isClosed_π hU hQ | ΞΉ : Type u_1
U K : Set β
z : β
F : ΞΉ β π U
Q : Set β β Set β
hU : IsOpen U
hQ : β K β compacts U, IsCompact (Q K)
l1 : β K β compacts U, EquicontinuousOn Subtype.val K
l2 : β K β compacts U, β x β K, β L, IsCompact L β§ β (i : β(π U Q)), βi x β L
β’ IsClosed (range (β(UniformOnFun.ofFun (compacts U)) β Subtype.val)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ΞΉ : Type u_1
U K : Set β
z : β
F : ΞΉ β π U
Q : Set β β Set β
hU : IsOpen U
hQ : β K β compacts U, IsCompact (Q K)
l1 : β K β compacts U, EquicontinuousOn Subtype.val K
l2 : β K β compacts U, β x β K, β L, IsCompact L β§ β (i : β(π U Q)), βi x β L
β’ IsClosed ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | isCompact_π | [64, 1] | [73, 59] | tauto | ΞΉ : Type u_1
U K : Set β
z : β
F : ΞΉ β π U
Q : Set β β Set β
hU : IsOpen U
hQ : β K β compacts U, IsCompact (Q K)
l1 : β K β compacts U, EquicontinuousOn Subtype.val K
l2 : β K β compacts U, β x β K, β L, IsCompact L β§ β (i : β(π U Q)), βi x β L
β’ Inducing (β(UniformOnFun.ofFun (compacts U)) β Subtype.val) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ΞΉ : Type u_1
U K : Set β
z : β
F : ΞΉ β π U
Q : Set β β Set β
hU : IsOpen U
hQ : β K β compacts U, IsCompact (Q K)
l1 : β K β compacts U, EquicontinuousOn Subtype.val K
l2 : β K β compacts U, β x β K, β L, IsCompact L β§ β (i : β(π U Q)), βi x β L
β’ Inducing ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | montel | [75, 1] | [79, 72] | choose! Q hQ1 hQ2 using h1 | ΞΉ : Type u_1
U K : Set β
z : β
F : ΞΉ β π U
Q : Set β β Set β
hU : IsOpen U
h1 : UniformlyBoundedOn F U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
β’ TotallyBounded (range F) | ΞΉ : Type u_1
U K : Set β
z : β
F : ΞΉ β π U
Qβ : Set β β Set β
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
Q : Set β β Set β
hQ1 : β K β compacts U, IsCompact (Q K)
hQ2 : β K β compacts U, β (i : ΞΉ), MapsTo (F i) K (Q K)
β’ TotallyBounded (range F) | Please generate a tactic in lean4 to solve the state.
STATE:
ΞΉ : Type u_1
U K : Set β
z : β
F : ΞΉ β π U
Q : Set β β Set β
hU : IsOpen U
h1 : UniformlyBoundedOn F U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
β’ TotallyBounded (range F)
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | montel | [75, 1] | [79, 72] | have l1 : range F β π U Q := by rintro f β¨i, rflβ© ; exact β¨h2 i, fun K hK => hQ2 K hK iβ© | ΞΉ : Type u_1
U K : Set β
z : β
F : ΞΉ β π U
Qβ : Set β β Set β
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
Q : Set β β Set β
hQ1 : β K β compacts U, IsCompact (Q K)
hQ2 : β K β compacts U, β (i : ΞΉ), MapsTo (F i) K (Q K)
β’ TotallyBounded (range F) | ΞΉ : Type u_1
U K : Set β
z : β
F : ΞΉ β π U
Qβ : Set β β Set β
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
Q : Set β β Set β
hQ1 : β K β compacts U, IsCompact (Q K)
hQ2 : β K β compacts U, β (i : ΞΉ), MapsTo (F i) K (Q K)
l1 : range F β π U Q
β’ TotallyBounded (range F) | Please generate a tactic in lean4 to solve the state.
STATE:
ΞΉ : Type u_1
U K : Set β
z : β
F : ΞΉ β π U
Qβ : Set β β Set β
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
Q : Set β β Set β
hQ1 : β K β compacts U, IsCompact (Q K)
hQ2 : β K β compacts U, β (i : ΞΉ), MapsTo (F i) K (Q K)
β’ TotallyBounded (range F... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | montel | [75, 1] | [79, 72] | exact totallyBounded_subset l1 <| (isCompact_π hU hQ1).totallyBounded | ΞΉ : Type u_1
U K : Set β
z : β
F : ΞΉ β π U
Qβ : Set β β Set β
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
Q : Set β β Set β
hQ1 : β K β compacts U, IsCompact (Q K)
hQ2 : β K β compacts U, β (i : ΞΉ), MapsTo (F i) K (Q K)
l1 : range F β π U Q
β’ TotallyBounded (range F) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ΞΉ : Type u_1
U K : Set β
z : β
F : ΞΉ β π U
Qβ : Set β β Set β
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
Q : Set β β Set β
hQ1 : β K β compacts U, IsCompact (Q K)
hQ2 : β K β compacts U, β (i : ΞΉ), MapsTo (F i) K (Q K)
l1 : range F β π U Q
β’ T... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | montel | [75, 1] | [79, 72] | rintro f β¨i, rflβ© | ΞΉ : Type u_1
U K : Set β
z : β
F : ΞΉ β π U
Qβ : Set β β Set β
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
Q : Set β β Set β
hQ1 : β K β compacts U, IsCompact (Q K)
hQ2 : β K β compacts U, β (i : ΞΉ), MapsTo (F i) K (Q K)
β’ range F β π U Q | case intro
ΞΉ : Type u_1
U K : Set β
z : β
F : ΞΉ β π U
Qβ : Set β β Set β
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
Q : Set β β Set β
hQ1 : β K β compacts U, IsCompact (Q K)
hQ2 : β K β compacts U, β (i : ΞΉ), MapsTo (F i) K (Q K)
i : ΞΉ
β’ F i β π U Q | Please generate a tactic in lean4 to solve the state.
STATE:
ΞΉ : Type u_1
U K : Set β
z : β
F : ΞΉ β π U
Qβ : Set β β Set β
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
Q : Set β β Set β
hQ1 : β K β compacts U, IsCompact (Q K)
hQ2 : β K β compacts U, β (i : ΞΉ), MapsTo (F i) K (Q K)
β’ range F β π U Q
TACTIC... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | montel | [75, 1] | [79, 72] | exact β¨h2 i, fun K hK => hQ2 K hK iβ© | case intro
ΞΉ : Type u_1
U K : Set β
z : β
F : ΞΉ β π U
Qβ : Set β β Set β
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
Q : Set β β Set β
hQ1 : β K β compacts U, IsCompact (Q K)
hQ2 : β K β compacts U, β (i : ΞΉ), MapsTo (F i) K (Q K)
i : ΞΉ
β’ F i β π U Q | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
ΞΉ : Type u_1
U K : Set β
z : β
F : ΞΉ β π U
Qβ : Set β β Set β
hU : IsOpen U
h2 : β (i : ΞΉ), DifferentiableOn β (F i) U
Q : Set β β Set β
hQ1 : β K β compacts U, IsCompact (Q K)
hQ2 : β K β compacts U, β (i : ΞΉ), MapsTo (F i) K (Q K)
i : ΞΉ
β’ F i β ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/Montel.lean | isCompact_π | [83, 1] | [84, 82] | simpa only [π_const] using isCompact_π hU (fun _ _ => isCompact_closedBall 0 1) | ΞΉ : Type u_1
U K : Set β
z : β
F : ΞΉ β π U
Q : Set β β Set β
hU : IsOpen U
β’ IsCompact (π U) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ΞΉ : Type u_1
U K : Set β
z : β
F : ΞΉ β π U
Q : Set β β Set β
hU : IsOpen U
β’ IsCompact (π U)
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/defs.lean | neg_in_π» | [19, 1] | [20, 11] | simp [π»] | u : β
U V W : Set β
β’ u β π» β -u β π» | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
u : β
U V W : Set β
β’ u β π» β -u β π»
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/defs.lean | sqrt_π»_eq_π» | [22, 1] | [23, 17] | simp [π», ball] | u : β
U V W : Set β
β’ {z | z ^ 2 β π»} = π» | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
u : β
U V W : Set β
β’ {z | z ^ 2 β π»} = π»
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/defs.lean | ne_center_of_not_mem_closed_ball | [69, 1] | [72, 16] | contrapose! hz | u : β
U V W : Set β
w : β
r : β
hr : 0 β€ r
z : β
hz : z β (closedBall w r)αΆ
β’ z β w | u : β
U V W : Set β
w : β
r : β
hr : 0 β€ r
z : β
hz : z = w
β’ z β (closedBall w r)αΆ | Please generate a tactic in lean4 to solve the state.
STATE:
u : β
U V W : Set β
w : β
r : β
hr : 0 β€ r
z : β
hz : z β (closedBall w r)αΆ
β’ z β w
TACTIC:
|
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 : β
U V W : Set β
w : β
r : β
hr : 0 β€ r
z : β
hz : z = w
β’ z β (closedBall w r)αΆ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
u : β
U V W : Set β
w : β
r : β
hr : 0 β€ r
z : β
hz : z = w
β’ z β (closedBall w r)αΆ
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | one_sub_mul_conj_ne_zero | [9, 1] | [12, 65] | rw [mem_π»_iff] at hu hz | z u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
hz : z β π»
β’ 1 - z * (starRingEnd β) u β 0 | z u zβ : β
U : Set β
instβ : good_domain U
hu : βuβ < 1
hz : βzβ < 1
β’ 1 - z * (starRingEnd β) u β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
z u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
hz : z β π»
β’ 1 - z * (starRingEnd β) u β 0
TACTIC:
|
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 zβ : β
U : Set β
instβ : good_domain U
hu : βuβ < 1
hz : βzβ < 1
β’ 1 - z * (starRingEnd β) u β 0 | z u zβ : β
U : Set β
instβ : good_domain U
hu : βuβ < 1
hz : βzβ < 1
β’ Complex.abs (z * (starRingEnd β) u) < Complex.abs 1 | Please generate a tactic in lean4 to solve the state.
STATE:
z u zβ : β
U : Set β
instβ : good_domain U
hu : βuβ < 1
hz : βzβ < 1
β’ 1 - z * (starRingEnd β) u β 0
TACTIC:
|
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 zβ : β
U : Set β
instβ : good_domain U
hu : βuβ < 1
hz : βzβ < 1
β’ Complex.abs (z * (starRingEnd β) u) < Complex.abs 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z u zβ : β
U : Set β
instβ : good_domain U
hu : βuβ < 1
hz : βzβ < 1
β’ Complex.abs (z * (starRingEnd β) u) < Complex.abs 1
TACTIC:
|
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 zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
β’ 1 - z * (starRingEnd β) u + (z - u) * (starRingEnd β) u β 0 | z u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
h1 : 1 - u * (starRingEnd β) u β 0
β’ 1 - z * (starRingEnd β) u + (z - u) * (starRingEnd β) u β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
z u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
β’ 1 - z * (starRingEnd β) u + (z - u) * (starRingEnd β) u β 0
TACTIC:
|
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 zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
h1 : 1 - u * (starRingEnd β) u β 0
β’ 1 - z * (starRingEnd β) u + (z - u) * (starRingEnd β) u β 0 | z u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
h1 : 1 - u * (starRingEnd β) u β 0
β’ 1 - (starRingEnd β) u * u β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
z u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
h1 : 1 - u * (starRingEnd β) u β 0
β’ 1 - z * (starRingEnd β) u + (z - u) * (starRingEnd β) u β 0
TACTIC:
|
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 zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
h1 : 1 - u * (starRingEnd β) u β 0
β’ 1 - (starRingEnd β) u * u β 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
h1 : 1 - u * (starRingEnd β) u β 0
β’ 1 - (starRingEnd β) u * u β 0
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | normSq_sub_normSq | [20, 1] | [21, 59] | field_simp [β ofReal_inj, normSq_eq_conj_mul_self] | z u zβ : β
U : Set β
instβ : good_domain U
β’ normSq (z - u) - normSq (1 - z * (starRingEnd β) u) = (normSq z - 1) * (1 - normSq u) | z u zβ : β
U : Set β
instβ : good_domain U
β’ ((starRingEnd β) z - (starRingEnd β) u) * (z - u) - (1 - (starRingEnd β) z * u) * (1 - z * (starRingEnd β) u) =
((starRingEnd β) z * z - 1) * (1 - (starRingEnd β) u * u) | Please generate a tactic in lean4 to solve the state.
STATE:
z u zβ : β
U : Set β
instβ : good_domain U
β’ normSq (z - u) - normSq (1 - z * (starRingEnd β) u) = (normSq z - 1) * (1 - normSq u)
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | normSq_sub_normSq | [20, 1] | [21, 59] | ring | z u zβ : β
U : Set β
instβ : good_domain U
β’ ((starRingEnd β) z - (starRingEnd β) u) * (z - u) - (1 - (starRingEnd β) z * u) * (1 - z * (starRingEnd β) u) =
((starRingEnd β) z * z - 1) * (1 - (starRingEnd β) u * u) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z u zβ : β
U : Set β
instβ : good_domain U
β’ ((starRingEnd β) z - (starRingEnd β) u) * (z - u) - (1 - (starRingEnd β) z * u) * (1 - z * (starRingEnd β) u) =
((starRingEnd β) z * z - 1) * (1 - (starRingEnd β) u * u)
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | pre_Ο_inv | [25, 1] | [28, 7] | rintro z hz | z u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
β’ LeftInvOn (pre_Ο (-u)) (pre_Ο u) π» | zβ u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
z : β
hz : z β π»
β’ pre_Ο (-u) (pre_Ο u z) = z | Please generate a tactic in lean4 to solve the state.
STATE:
z u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
β’ LeftInvOn (pre_Ο (-u)) (pre_Ο u) π»
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | pre_Ο_inv | [25, 1] | [28, 7] | field_simp [pre_Ο, one_sub_mul_conj_ne_zero hu hz, one_sub_mul_conj_add_mul_conj_ne_zero hu] | zβ u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
z : β
hz : z β π»
β’ pre_Ο (-u) (pre_Ο u z) = z | zβ u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
z : β
hz : z β π»
β’ z - u + (1 - z * (starRingEnd β) u) * u = z * (1 - z * (starRingEnd β) u + (z - u) * (starRingEnd β) u) | Please generate a tactic in lean4 to solve the state.
STATE:
zβ u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
z : β
hz : z β π»
β’ pre_Ο (-u) (pre_Ο u z) = z
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | pre_Ο_inv | [25, 1] | [28, 7] | ring | zβ u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
z : β
hz : z β π»
β’ z - u + (1 - z * (starRingEnd β) u) * u = z * (1 - z * (starRingEnd β) u + (z - u) * (starRingEnd β) u) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
zβ u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
z : β
hz : z β π»
β’ z - u + (1 - z * (starRingEnd β) u) * u = z * (1 - z * (starRingEnd β) u + (z - u) * (starRingEnd β) u)
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | Ο_deriv | [50, 1] | [57, 24] | have h1 : DifferentiableAt β (fun z => z - u) z := by simp | z u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
hz : z β π»
β’ deriv (Ο hu).to_fun z = (1 - u * (starRingEnd β) u) / (1 - z * (starRingEnd β) u) ^ 2 | z u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
hz : z β π»
h1 : DifferentiableAt β (fun z => z - u) z
β’ deriv (Ο hu).to_fun z = (1 - u * (starRingEnd β) u) / (1 - z * (starRingEnd β) u) ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
z u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
hz : z β π»
β’ deriv (Ο hu).to_fun z = (1 - u * (starRingEnd β) u) / (1 - z * (starRingEnd β) u) ^ 2
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | Ο_deriv | [50, 1] | [57, 24] | have h2 : DifferentiableAt β (fun z => 1 - z * conj u) z := by simp [DifferentiableAt.mul_const] | z u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
hz : z β π»
h1 : DifferentiableAt β (fun z => z - u) z
β’ deriv (Ο hu).to_fun z = (1 - u * (starRingEnd β) u) / (1 - z * (starRingEnd β) u) ^ 2 | z u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
hz : z β π»
h1 : DifferentiableAt β (fun z => z - u) z
h2 : DifferentiableAt β (fun z => 1 - z * (starRingEnd β) u) z
β’ deriv (Ο hu).to_fun z = (1 - u * (starRingEnd β) u) / (1 - z * (starRingEnd β) u) ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
z u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
hz : z β π»
h1 : DifferentiableAt β (fun z => z - u) z
β’ deriv (Ο hu).to_fun z = (1 - u * (starRingEnd β) u) / (1 - z * (starRingEnd β) u) ^ 2
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | Ο_deriv | [50, 1] | [57, 24] | have h3 : 1 - z * conj u β 0 := one_sub_mul_conj_ne_zero hu hz | z u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
hz : z β π»
h1 : DifferentiableAt β (fun z => z - u) z
h2 : DifferentiableAt β (fun z => 1 - z * (starRingEnd β) u) z
β’ deriv (Ο hu).to_fun z = (1 - u * (starRingEnd β) u) / (1 - z * (starRingEnd β) u) ^ 2 | z u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
hz : z β π»
h1 : DifferentiableAt β (fun z => z - u) z
h2 : DifferentiableAt β (fun z => 1 - z * (starRingEnd β) u) z
h3 : 1 - z * (starRingEnd β) u β 0
β’ deriv (Ο hu).to_fun z = (1 - u * (starRingEnd β) u) / (1 - z * (starRingEnd β) u) ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
z u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
hz : z β π»
h1 : DifferentiableAt β (fun z => z - u) z
h2 : DifferentiableAt β (fun z => 1 - z * (starRingEnd β) u) z
β’ deriv (Ο hu).to_fun z = (1 - u * (starRingEnd β) u) / (1 - z * (starRingEnd β) u) ^ ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | Ο_deriv | [50, 1] | [57, 24] | have h4 : deriv (fun z => z - u) z = 1 := by simp [deriv_sub_const] | z u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
hz : z β π»
h1 : DifferentiableAt β (fun z => z - u) z
h2 : DifferentiableAt β (fun z => 1 - z * (starRingEnd β) u) z
h3 : 1 - z * (starRingEnd β) u β 0
β’ deriv (Ο hu).to_fun z = (1 - u * (starRingEnd β) u) / (1 - z * (starRingEnd β) u) ^ 2 | z u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
hz : z β π»
h1 : DifferentiableAt β (fun z => z - u) z
h2 : DifferentiableAt β (fun z => 1 - z * (starRingEnd β) u) z
h3 : 1 - z * (starRingEnd β) u β 0
h4 : deriv (fun z => z - u) z = 1
β’ deriv (Ο hu).to_fun z = (1 - u * (starRingEnd β) u) / (1 - z * (starRingEnd ... | Please generate a tactic in lean4 to solve the state.
STATE:
z u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
hz : z β π»
h1 : DifferentiableAt β (fun z => z - u) z
h2 : DifferentiableAt β (fun z => 1 - z * (starRingEnd β) u) z
h3 : 1 - z * (starRingEnd β) u β 0
β’ deriv (Ο hu).to_fun z = (1 - u * (starRingEnd β) ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | Ο_deriv | [50, 1] | [57, 24] | have h5 : deriv (fun z => 1 - z * conj u) z = - conj u := by simp [deriv_const_sub] | z u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
hz : z β π»
h1 : DifferentiableAt β (fun z => z - u) z
h2 : DifferentiableAt β (fun z => 1 - z * (starRingEnd β) u) z
h3 : 1 - z * (starRingEnd β) u β 0
h4 : deriv (fun z => z - u) z = 1
β’ deriv (Ο hu).to_fun z = (1 - u * (starRingEnd β) u) / (1 - z * (starRingEnd ... | z u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
hz : z β π»
h1 : DifferentiableAt β (fun z => z - u) z
h2 : DifferentiableAt β (fun z => 1 - z * (starRingEnd β) u) z
h3 : 1 - z * (starRingEnd β) u β 0
h4 : deriv (fun z => z - u) z = 1
h5 : deriv (fun z => 1 - z * (starRingEnd β) u) z = -(starRingEnd β) u
β’ deriv... | Please generate a tactic in lean4 to solve the state.
STATE:
z u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
hz : z β π»
h1 : DifferentiableAt β (fun z => z - u) z
h2 : DifferentiableAt β (fun z => 1 - z * (starRingEnd β) u) z
h3 : 1 - z * (starRingEnd β) u β 0
h4 : deriv (fun z => z - u) z = 1
β’ deriv (Ο hu).to... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | Ο_deriv | [50, 1] | [57, 24] | simp [Ο, deriv_div h1 h2 h3, h4, h5] | z u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
hz : z β π»
h1 : DifferentiableAt β (fun z => z - u) z
h2 : DifferentiableAt β (fun z => 1 - z * (starRingEnd β) u) z
h3 : 1 - z * (starRingEnd β) u β 0
h4 : deriv (fun z => z - u) z = 1
h5 : deriv (fun z => 1 - z * (starRingEnd β) u) z = -(starRingEnd β) u
β’ deriv... | z u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
hz : z β π»
h1 : DifferentiableAt β (fun z => z - u) z
h2 : DifferentiableAt β (fun z => 1 - z * (starRingEnd β) u) z
h3 : 1 - z * (starRingEnd β) u β 0
h4 : deriv (fun z => z - u) z = 1
h5 : deriv (fun z => 1 - z * (starRingEnd β) u) z = -(starRingEnd β) u
β’ (1 - ... | Please generate a tactic in lean4 to solve the state.
STATE:
z u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
hz : z β π»
h1 : DifferentiableAt β (fun z => z - u) z
h2 : DifferentiableAt β (fun z => 1 - z * (starRingEnd β) u) z
h3 : 1 - z * (starRingEnd β) u β 0
h4 : deriv (fun z => z - u) z = 1
h5 : deriv (fun z... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | Ο_deriv | [50, 1] | [57, 24] | field_simp [h3] | z u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
hz : z β π»
h1 : DifferentiableAt β (fun z => z - u) z
h2 : DifferentiableAt β (fun z => 1 - z * (starRingEnd β) u) z
h3 : 1 - z * (starRingEnd β) u β 0
h4 : deriv (fun z => z - u) z = 1
h5 : deriv (fun z => 1 - z * (starRingEnd β) u) z = -(starRingEnd β) u
β’ (1 - ... | z u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
hz : z β π»
h1 : DifferentiableAt β (fun z => z - u) z
h2 : DifferentiableAt β (fun z => 1 - z * (starRingEnd β) u) z
h3 : 1 - z * (starRingEnd β) u β 0
h4 : deriv (fun z => z - u) z = 1
h5 : deriv (fun z => 1 - z * (starRingEnd β) u) z = -(starRingEnd β) u
β’ 1 - z... | Please generate a tactic in lean4 to solve the state.
STATE:
z u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
hz : z β π»
h1 : DifferentiableAt β (fun z => z - u) z
h2 : DifferentiableAt β (fun z => 1 - z * (starRingEnd β) u) z
h3 : 1 - z * (starRingEnd β) u β 0
h4 : deriv (fun z => z - u) z = 1
h5 : deriv (fun z... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | Ο_deriv | [50, 1] | [57, 24] | ring | z u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
hz : z β π»
h1 : DifferentiableAt β (fun z => z - u) z
h2 : DifferentiableAt β (fun z => 1 - z * (starRingEnd β) u) z
h3 : 1 - z * (starRingEnd β) u β 0
h4 : deriv (fun z => z - u) z = 1
h5 : deriv (fun z => 1 - z * (starRingEnd β) u) z = -(starRingEnd β) u
β’ 1 - z... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
hz : z β π»
h1 : DifferentiableAt β (fun z => z - u) z
h2 : DifferentiableAt β (fun z => 1 - z * (starRingEnd β) u) z
h3 : 1 - z * (starRingEnd β) u β 0
h4 : deriv (fun z => z - u) z = 1
h5 : deriv (fun z... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | Ο_deriv | [50, 1] | [57, 24] | simp | z u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
hz : z β π»
β’ DifferentiableAt β (fun z => z - u) z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
hz : z β π»
β’ DifferentiableAt β (fun z => z - u) z
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | Ο_deriv | [50, 1] | [57, 24] | simp [DifferentiableAt.mul_const] | z u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
hz : z β π»
h1 : DifferentiableAt β (fun z => z - u) z
β’ DifferentiableAt β (fun z => 1 - z * (starRingEnd β) u) z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
hz : z β π»
h1 : DifferentiableAt β (fun z => z - u) z
β’ DifferentiableAt β (fun z => 1 - z * (starRingEnd β) u) z
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | Ο_deriv | [50, 1] | [57, 24] | simp [deriv_sub_const] | z u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
hz : z β π»
h1 : DifferentiableAt β (fun z => z - u) z
h2 : DifferentiableAt β (fun z => 1 - z * (starRingEnd β) u) z
h3 : 1 - z * (starRingEnd β) u β 0
β’ deriv (fun z => z - u) z = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
hz : z β π»
h1 : DifferentiableAt β (fun z => z - u) z
h2 : DifferentiableAt β (fun z => 1 - z * (starRingEnd β) u) z
h3 : 1 - z * (starRingEnd β) u β 0
β’ deriv (fun z => z - u) z = 1
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | Ο_deriv | [50, 1] | [57, 24] | simp [deriv_const_sub] | z u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
hz : z β π»
h1 : DifferentiableAt β (fun z => z - u) z
h2 : DifferentiableAt β (fun z => 1 - z * (starRingEnd β) u) z
h3 : 1 - z * (starRingEnd β) u β 0
h4 : deriv (fun z => z - u) z = 1
β’ deriv (fun z => 1 - z * (starRingEnd β) u) z = -(starRingEnd β) u | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z u zβ : β
U : Set β
instβ : good_domain U
hu : u β π»
hz : z β π»
h1 : DifferentiableAt β (fun z => z - u) z
h2 : DifferentiableAt β (fun z => 1 - z * (starRingEnd β) u) z
h3 : 1 - z * (starRingEnd β) u β 0
h4 : deriv (fun z => z - u) z = 1
β’ deriv (fun z =>... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | set u := f 0 | z u zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
β’ βderiv f 0β < 1 | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
β’ βderiv f 0β < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
z u zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
β’ βderiv f 0β < 1
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | have u_in_π» : u β π» := f_img (mem_ball_self zero_lt_one) | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
β’ βderiv f 0β < 1 | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
β’ βderiv f 0β < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
β’ βderiv f 0β < 1
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | let g := Ο u_in_π» β f | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
β’ βderiv f 0β < 1 | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
β’ βderiv f 0β < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
β’ βderiv f 0β < 1
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | have g_diff : DifferentiableOn β g π» := (Ο u_in_π»).is_diff.comp f_diff f_img | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
β’ βderiv f 0β < 1 | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
β’ βderiv f 0β < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
β’ βderiv f 0β < 1
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | have g_maps : MapsTo g π» π» := (Ο u_in_π»).maps_to.comp f_img | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
β’ βderiv f 0β < 1 | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
β’ βderiv f 0β < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
β’ βderiv f 0β < 1
TACTIC:
|
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, Ο] | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
β’ βderiv f 0β < 1 | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
β’ βderiv f 0β < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
β’ β... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | by_cases h : βderiv g 0β = 1 | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
β’ βderiv f 0β < 1 | case pos
z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : βderiv g 0β = 1
β’ βderiv f 0β < ... | Please generate a tactic in lean4 to solve the state.
STATE:
z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | case pos =>
have g_lin : EqOn g (Ξ» (z : β) => z β’ 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) β’ dslope g 0 0) (Metric.ball 0 1) := by
apply affine_of_mapsTo_ball_of_exists_norm_dslope_eq_div g_diff h2 (mem_bal... | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : βderiv g 0β = 1
β’ βderiv f 0β < 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0... |
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 (Ο u_in_π») u * deriv f 0 :=
deriv.comp 0 ((Ο u_in_π»).is_diff.differentiableAt (isOpen_ball... | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : Β¬βderiv g 0β = 1
β’ βderiv f 0β < 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | simp [g, Ο] | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
β’ g 0 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
β’ g... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | have g_lin : EqOn g (Ξ» (z : β) => z β’ 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) β’ 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)
... | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : βderiv g 0β = 1
β’ βderiv f 0β < 1 | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : βderiv g 0β = 1
g_lin : EqOn g (fun z => ... | Please generate a tactic in lean4 to solve the state.
STATE:
z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0... |
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 β 0 := Ξ» h' => by simp [h'] at h | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : βderiv g 0β = 1
g_lin : EqOn g (fun z => ... | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : βderiv g 0β = 1
g_lin : EqOn g (fun z => ... | Please generate a tactic in lean4 to solve the state.
STATE:
z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | have g_inj : InjOn g π» := Ξ» x hx y hy => by
rw [g_lin hx, g_lin hy]
simp [g'0_ne_0] | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : βderiv g 0β = 1
g_lin : EqOn g (fun z => ... | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : βderiv g 0β = 1
g_lin : EqOn g (fun z => ... | Please generate a tactic in lean4 to solve the state.
STATE:
z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0... |
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 uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : βderiv g 0β = 1
g_lin : EqOn g (fun z => ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0... |
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 uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : βderiv g 0β = 1
β’ EqOn g (fun z => z β’ de... | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : βderiv g 0β = 1
h2 : MapsTo g (ball 0 1) ... | Please generate a tactic in lean4 to solve the state.
STATE:
z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0... |
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) β’ 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 uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : βderiv g 0β = 1
h2 : MapsTo g (ball 0 1) ... | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : βderiv g 0β = 1
h2 : MapsTo g (ball 0 1) ... | Please generate a tactic in lean4 to solve the state.
STATE:
z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | convert h1 using 1 | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : βderiv g 0β = 1
h2 : MapsTo g (ball 0 1) ... | case h.e'_4
z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : βderiv g 0β = 1
h2 : MapsTo g... | Please generate a tactic in lean4 to solve the state.
STATE:
z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_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 uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : βderiv g 0β = 1
h2 : MapsTo g... | case h.e'_4.h
zβ uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : βderiv g 0β = 1
h2 : MapsT... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo... |
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
zβ uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : βderiv g 0β = 1
h2 : MapsT... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4.h
zβ uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : Map... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | rwa [g_0_eq_0] | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : βderiv g 0β = 1
β’ MapsTo g (ball 0 1) (ba... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0... |
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 uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : βderiv g 0β = 1
h2 : MapsTo g (ball 0 1) ... | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : βderiv g 0β = 1
h2 : MapsTo g (ball 0 1) ... | Please generate a tactic in lean4 to solve the state.
STATE:
z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | rwa [dslope_same, div_one] | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : βderiv g 0β = 1
h2 : MapsTo g (ball 0 1) ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | simp [h'] at h | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : βderiv g 0β = 1
g_lin : EqOn g (fun z => ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0... |
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 uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : βderiv g 0β = 1
g_lin : EqOn g (fun z => ... | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : βderiv g 0β = 1
g_lin : EqOn g (fun z => ... | Please generate a tactic in lean4 to solve the state.
STATE:
z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | simp [g'0_ne_0] | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : βderiv g 0β = 1
g_lin : EqOn g (fun z => ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | rw [norm_eq_abs] at h | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : Β¬βderiv g 0β = 1
β’ βderiv f 0β < 1 | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : Β¬Complex.abs (deriv g 0) = 1
β’ βderiv f 0... | Please generate a tactic in lean4 to solve the state.
STATE:
z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0... |
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 uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : Β¬Complex.abs (deriv g 0) = 1
β’ βderiv f 0... | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : Β¬Complex.abs (deriv g 0) = 1
g'0_le_1 : C... | Please generate a tactic in lean4 to solve the state.
STATE:
z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0... |
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 uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : Β¬Complex.abs (deriv g 0) = 1
g'0_le_1 : C... | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : Β¬Complex.abs (deriv g 0) = 1
g'0_le_1 : C... | Please generate a tactic in lean4 to solve the state.
STATE:
z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0... |
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 (Ο u_in_π») u * deriv f 0 :=
deriv.comp 0 ((Ο u_in_π»).is_diff.differentiableAt (isOpen_ball.mem_nhds u_in_π»))
(f_diff.differentiableAt (ball_mem_nhds _ zero_lt_one)) | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : Β¬Complex.abs (deriv g 0) = 1
g'0_le_1 : C... | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : Β¬Complex.abs (deriv g 0) = 1
g'0_le_1 : C... | Please generate a tactic in lean4 to solve the state.
STATE:
z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | have e1 : 1 - (normSq u : β) β 0 := by
simpa [normSq_eq_conj_mul_self, mul_comm] using one_sub_mul_conj_ne_zero u_in_π» u_in_π» | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : Β¬Complex.abs (deriv g 0) = 1
g'0_le_1 : C... | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : Β¬Complex.abs (deriv g 0) = 1
g'0_le_1 : C... | Please generate a tactic in lean4 to solve the state.
STATE:
z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | have Ο'u_u : deriv (Ο u_in_π») u = 1 / (1 - normSq u) := by
set w := 1 - conj u * u with hw
have : w β 0 := by simpa [normSq_eq_conj_mul_self, mul_comm u] using e1
rw [Ο_deriv u_in_π» u_in_π», normSq_eq_conj_mul_self, mul_comm u, β hw]
field_simp; ring | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : Β¬Complex.abs (deriv g 0) = 1
g'0_le_1 : C... | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : Β¬Complex.abs (deriv g 0) = 1
g'0_le_1 : C... | Please generate a tactic in lean4 to solve the state.
STATE:
z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | have e2 : 0 β€ normSq u := normSq_nonneg _ | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : Β¬Complex.abs (deriv g 0) = 1
g'0_le_1 : C... | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : Β¬Complex.abs (deriv g 0) = 1
g'0_le_1 : C... | Please generate a tactic in lean4 to solve the state.
STATE:
z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0... |
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_π»_iff.mp u_in_π»
simp only [sq_lt_one_iff_abs_lt_one, Complex.abs_abs, this] | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : Β¬Complex.abs (deriv g 0) = 1
g'0_le_1 : C... | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : Β¬Complex.abs (deriv g 0) = 1
g'0_le_1 : C... | Please generate a tactic in lean4 to solve the state.
STATE:
z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | simp [normSq_eq_abs, β mem_π»_iff] | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : Β¬Complex.abs (deriv g 0) = 1
g'0_le_1 : C... | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : Β¬Complex.abs (deriv g 0) = 1
g'0_le_1 : C... | Please generate a tactic in lean4 to solve the state.
STATE:
z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | simp only [Ο'u_u, one_div] at g'0_eq_mul | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : Β¬Complex.abs (deriv g 0) = 1
g'0_le_1 : C... | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : Β¬Complex.abs (deriv g 0) = 1
g'0_le_1 : C... | Please generate a tactic in lean4 to solve the state.
STATE:
z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0... |
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_mulβ e1] at g'0_eq_mul | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : Β¬Complex.abs (deriv g 0) = 1
g'0_le_1 : C... | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : Β¬Complex.abs (deriv g 0) = 1
g'0_le_1 : C... | Please generate a tactic in lean4 to solve the state.
STATE:
z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | rw [β norm_eq_abs, g'0_eq_mul, norm_mul, mul_comm, β one_mul (1 : β)] | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : Β¬Complex.abs (deriv g 0) = 1
g'0_le_1 : C... | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : Β¬Complex.abs (deriv g 0) = 1
g'0_le_1 : C... | Please generate a tactic in lean4 to solve the state.
STATE:
z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0... |
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 uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : Β¬Complex.abs (deriv g 0) = 1
g'0_le_1 : C... | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : Β¬Complex.abs (deriv g 0) = 1
g'0_le_1 : C... | Please generate a tactic in lean4 to solve the state.
STATE:
z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/etape2.lean | non_injective_schwarz | [62, 1] | [112, 37] | simp at e2 e3 β’ | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : Β¬Complex.abs (deriv g 0) = 1
g'0_le_1 : C... | z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0_eq_0 : g 0 = 0
h : Β¬Complex.abs (deriv g 0) = 1
g'0_le_1 : C... | Please generate a tactic in lean4 to solve the state.
STATE:
z uβ zβ : β
U : Set β
instβ : good_domain U
f : β β β
f_diff : DifferentiableOn β f π»
f_img : MapsTo f π» π»
f_noninj : Β¬InjOn f π»
u : β := f 0
u_in_π» : u β π»
g : β β β := (Ο u_in_π»).to_fun β f
g_diff : DifferentiableOn β g π»
g_maps : MapsTo g π» π»
g_0... |
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